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https://mathoverflow.net/questions/327119 | 4 | Let $X$ be a connected, locally finite graph with vertex set $V(X)$ and $G$ a group acting freely on $X$ such that $X/G$ is a finite graph. Fix a vertex $x$ and for $k\in\mathbb N$ set
$$
N(k)=\#\{ g\in G: d(gx,x)\le k\},
$$
where $d$ is the vertex distance in the graph $X$.
Further set
$$
A(k)=\#\{y\in V(X):d(x,y)\le ... | https://mathoverflow.net/users/nan | Asymptotics of orbits on graphs | It is possible that the limit does not exist at all: Consider the free group on two generators acting on the $(4,2)$-biregular tree in the obvious way. This action is free and has 3 orbits (one containing all vertices of degree 4, and the other two containing "half" of the vertices of degree 2).
Let $x$ be a vertex o... | 4 | https://mathoverflow.net/users/97426 | 327131 | 140,766 |
https://mathoverflow.net/questions/327124 | 8 | It is well-known that a compact Hausdorff $X$ space is scattered if and only if admits no continuous maps onto the unit interval $[0,1]$.
Surprisingly, but I cannot find a good reference to this well-known fact (desirably some textbook).
In the survey paper "[Scattered spaces](https://books.google.com/books?id=JWy... | https://mathoverflow.net/users/61536 | A reference to a well-known characterization of scattered compact spaces | The proof in the direction that there is no continuous surjection from a compact scattered $X$ onto $[0,1]$ may be found here (Theorem 1):
W. Rudin, *Continuous functions on compact spaces without perfect subsets*, Proc. Amer. Math. Soc. 8 (1957), 39-42.
And the full characterisation may be found in (Theorem 8.5.4,... | 13 | https://mathoverflow.net/users/15860 | 327135 | 140,768 |
https://mathoverflow.net/questions/326113 | 6 | Related to [Why symplectic geometry gives Poisson geometry](https://mathoverflow.net/questions/255034/why-symplectic-geometry-gives-poisson-geometry) by coming at it from the other side.
This isn't as fully formalized as it probably should be, but I think enough of the idea is there to ask the question.
Given an al... | https://mathoverflow.net/users/44191 | Bracket systems (generalization of Poisson brackets) | I'm no expert on operads but it seems that
* a "bracket system" can be formalized as an [operad](https://en.wikipedia.org/wiki/Operad); see e.g. [the Poisson operad here](https://mathoverflow.net/a/261448/61197),
* the "product-complete" condition could be related to having a Hopf operad (see previous link),
* the "L... | 4 | https://mathoverflow.net/users/61197 | 327143 | 140,769 |
https://mathoverflow.net/questions/327152 | 3 | Given is a connected graph with undirected edges such that deleting any vertex decomposes the graph into $\le k$ components, for some $k\ge 2$. Is it true that the graph has a spanning tree with degree at most $k$?
The claim is clearly true if the graph itself is a tree. I checked some articles on vertex connectivit... | https://mathoverflow.net/users/137930 | Deleting vertex decomposes graph | No, this is false. Take $K\_{n, n+2}$. Removing any vertex gives a connected graph, so in particular the graph satisfies the condition with $k=2$. However, it does not contain a spanning tree with maximum degree $2$, since $K\_{n, n+2}$ does not contain a Hamiltonian path.
Here is a counterexample that works for all ... | 3 | https://mathoverflow.net/users/2233 | 327154 | 140,771 |
https://mathoverflow.net/questions/325362 | 3 | $\textbf{Background:}$ Consider a particle in $\mathbb{N}^2$ starting at a point $(x,y)$ that can only move one step at a time along the lattice. The particle moves each up or down in continuous time with rate $y$, and each left or right with rate $x$. It will therefore be moving faster, the further away from the origi... | https://mathoverflow.net/users/20343 | A 2d recurrence equation representing a step-free continuous-time Markov process on N^2 with frequency-dependent rates | Some experimenting reveals that the hitting time $t\_1(x)$ of the 1D process started at $x\in\mathbb{N}$ (in the notation of OP's comment) has a simple cdf,
$$\mathbb{P}(t\_1(x) < t ) = \left(\frac{t}{t+1}\right)^x.$$
I don't know an easy combinatorial proof, but one may check explicitly that it solves
$$\mathbb{P}(t\_... | 4 | https://mathoverflow.net/users/47484 | 327161 | 140,773 |
https://mathoverflow.net/questions/327146 | 5 | Let $\mathcal{O}$ be a topological operad and $X$ an algebra over $\mathcal{O}$. Then $H\_\*(X)$ is an algebra (in the category of $\mathbb{Z}$-graded $R$-modules) over $H\_\*(\mathcal{O})$. Each $e\in H\_n(\mathcal{O}(2))$ gives us a graded product
$$H\_p(X)\otimes H\_q(X)\to H\_{p+q+n}(X),a\otimes b\mapsto e(a\otimes... | https://mathoverflow.net/users/124042 | Graded commutativity of the $n$th Browder bracket | Both papers choose a normalization that is different from yours: they define
$$
[a,b] = (-1)^{na+1} s(a \otimes b).
$$
This is definition 5.7 in Cohen's paper that you mentioned.
The reason for this convention is that it is more consistent with writing the Browder bracket as a binary operation, rather than a function... | 3 | https://mathoverflow.net/users/360 | 327173 | 140,774 |
https://mathoverflow.net/questions/327068 | 3 | If $A=K(H)$, where $H$ is an infinite dimensional separable Hilbert space, then $A$ is simple and nuclear, and the multiplier algebra $M(A)$ of $A$ is not nuclear.
My question is: can we find a non-unital simple nuclear $\mathbb C$ $\*$- algebra $B$ such that the multiplier algebra of $B$ is also nuclear?
| https://mathoverflow.net/users/63864 | multiplier algebra of a simple $C^*$ algebra | If $A$ is a $\sigma$-unital, simple, non-unital $C^\ast$-algebra, then $M(A)$ is non-exact (in particular, it is non-nuclear). The following argument is modelled after Yemon's idea in the comments above. Also, it uses the very deep theorem of Kirchberg, that quotients of exact $C^\ast$-algebras are exact (this can prob... | 7 | https://mathoverflow.net/users/126109 | 327180 | 140,776 |
https://mathoverflow.net/questions/327149 | 10 | Is there any reference for calculation of the rational homology of the free loop space $H\_\*(\mathcal{L}Gr(k,n),\mathbb{Q})$ of a complex Grassmanian? More precisely, I am interested in computing ranks $$r\_i=rk(H\_i(\mathcal{L}Gr(2,4),\mathbb{Q}))$$ (Here $Gr(k,n)$ stands for k-planes in n-space).
| https://mathoverflow.net/users/114985 | Homology of the free loop space of a Grassmanian | The complex Grassmannian $Gr(2,4)$ can be realized up to homotopy as the homotopy fiber of the map $BU(2) \times BU(2) \rightarrow BU(4)$ which corresponds to the Whitney sum of two complex rank 2 bundles. To obtain the rational cohomology of it (and then of its free loop space), we'll calculate its minimal model as th... | 17 | https://mathoverflow.net/users/104342 | 327184 | 140,779 |
https://mathoverflow.net/questions/327193 | 0 | <https://www.ams.org/journals/proc/1976-060-01/S0002-9939-1976-0423137-6/S0002-9939-1976-0423137-6.pdf>
This paper gives a sufficient condition for the uniqueness of SCHAUDER fix point. I wonder if there are any sufficient and necessary results, whether published or not, for the uniqueness of Brouwer or Schauder fix ... | https://mathoverflow.net/users/109527 | Sufficient and necessary condition for the global uniqueness of fix-points | Well, here's a necessary and sufficient condition for uniqueness of the fixed point of a continuous function from $[0,1]$ to itself:
For every $x \in [0,1]$, if $f(x) \ge x$ then $f(t) > t$ for all $t < x$, and if $f(x) \le x$ then $f(t) < t$ for all $t > x$.
Of course it's not likely to be a **useful** necessary a... | 1 | https://mathoverflow.net/users/13650 | 327195 | 140,784 |
https://mathoverflow.net/questions/327024 | 6 | I am now reading Higher Topos Theory. I have recently met the following questions that I am not able to figure out and I am here to look for some answer or help.
First, in Lemma 2.2.3.6, while proving $(a)\implies (c)$, Lurie constructed a class $\scr U$ of simplicial sets which contains all simplicial sets $A$ satis... | https://mathoverflow.net/users/123746 | A few questions while reading Higher Topos Theory | $\newcommand{\SSet}{\mathsf{SSet}}\DeclareMathOperator{\Map}{Map}$First, let's record the fact that for any $A$ in $\SSet\_{/S}$ and any right fibration $p : X \to S$, the simplicial set $\Map\_{\SSet\_{/S}}(A, X)$ is a Kan complex. This follows from Lemma 2.2.3.4 by applying it to the inclusion $\emptyset \subset S$. ... | 6 | https://mathoverflow.net/users/126667 | 327196 | 140,785 |
https://mathoverflow.net/questions/326997 | 4 | I am puzzled over something I read in Quillen's *[On the Cohomology and K-Theory of the General Linear Groups Over a Finite Field](https://www.jstor.org/stable/1970825)*.
On page 557, when computing the $E\_2$ page of a case of the Eilenberg-Moore spectral sequence, he first shows that
$$E\_2 \cong \text{Tor}^{A\_... | https://mathoverflow.net/users/135687 | Grading in Eilenberg-Moore spectral sequence | Different people use different notation on gradings, for example I would have called the bigrdading of $e\_i$ $E\_2^{1,2i}$. Supposing that this is not a typo, Quillen meant
by $k$ in $E\_s^{j,k}$ the total degree, not the internal degree of the element.
In other words, he has a spectral sequence $E\_2^{s,t}\Rightar... | 3 | https://mathoverflow.net/users/43326 | 327208 | 140,786 |
https://mathoverflow.net/questions/327202 | 3 | All references below are from McCleary's book, second edition.
Suppose that we have a filtered complex where the filtration is unbounded. Suppose that the associated spectral sequence is weakly convergent, as per Theorem 3.2. Then the spectral sequence converges to the filtered quotients of the cohomology, but this ... | https://mathoverflow.net/users/31631 | Question about spectral sequences associated to filtered complexes with unbounded filtrations | Suppose we have $\bigoplus\_\mathbb{N}\mathbb{Z}$ and $\prod\_\mathbb{N}\mathbb{Z}$. Both modules hava a canonical decreasing filtration where the $k$-th filtration step consists of all elements whose first k coordinates vanish. Both filtrations have the same subquotients. From this information you cannot distinguish $... | 7 | https://mathoverflow.net/users/3969 | 327209 | 140,787 |
https://mathoverflow.net/questions/327207 | 6 | Consider the following property of an undirected graph: For any four distinct vertices $a,b,c,d$, there is a path from $a$ to $b$ and a path from $c$ to $d$ such that the two paths do not share any vertex.
Is there a name for this property, or has it been studied? It looks related to vertex connectivity but not quite... | https://mathoverflow.net/users/137930 | Disjoint paths between four vertices | The property that you are describing is called *$2$-linked*. More generally, we say that a graph is *$k$-linked* if it has at least $2k$ vertices and for all distinct vertices $s\_1, \dots s\_k, t\_1, \dots, t\_k$ there are $k$-vertex disjoint paths connecting $s\_i$ to $t\_i$ for all $i \in [k]$. Note that every $k$-l... | 8 | https://mathoverflow.net/users/2233 | 327213 | 140,789 |
https://mathoverflow.net/questions/327175 | 3 | I apologize in advance if this is a naive question.
**Def:** A topological chain complex is a chain complex of topological $\mathbb{R}$-vector spaces such that the boundary maps are continuous.
Let $C$ be a topological chain complex.
one can in a natural way consider the dual $C$\* of the chain complex $C$ (igno... | https://mathoverflow.net/users/32135 | Is the continuous dual of a topological chain complex chain equivalent to the algebraic dual? | In general they are not chain equivalent, even if $H\_\ast(C)$ is finite dimensional. (I need the axiom of choice)
For a counterexample let $p \in (0,1)$ and let $f: L^p([0,1]) \to \mathbb{R}$ be a non-zero linear functional. $W := \ker(f)$ is a topological space with the subspace topology and the inclusion $W \hook... | 4 | https://mathoverflow.net/users/18571 | 327216 | 140,790 |
https://mathoverflow.net/questions/327171 | 2 | By "backtrack" I mean a subword of a relator in a group presentation of the form $x x^{-1}$.
Let $X = \langle a \rangle$ as a presentation complex.
Let $Y = \langle a$ | $aa^{-1} \rangle$ as a presentation complex.
Now we see that $X$ is a circle and $Y$ is a pinched torus, and these two spaces clearly do not ... | https://mathoverflow.net/users/119181 | A Backtrack as a Single Word in a Group Presentation yields a Complex that isn't of the Same Homotopy Type? | Let me elaborate on my comment a little. It's a standard exercise that attaching a cell by homotopic maps leads to homotopy-equivalent spaces. (See p.13 of Hatcher, for instance.) In particular, it certainly is true that deleting backtracks leads to a homotopy equivalence of presentation complexes.
From this we concl... | 7 | https://mathoverflow.net/users/1463 | 327232 | 140,797 |
https://mathoverflow.net/questions/327134 | 1 | I was reading the proof of Riesz-Thorin interpolation theorem in <http://www.math.kit.edu/iana3/lehre/fourierana2014w/media/rieszthorinproof.pdf>
and get stuck at the last step. We construct the complex function
\begin{equation\*}
F(z)=\int g\_zTf\_z=\sum\_{i,j}|a\_i|^{\frac{p}{p(z)}}\frac{a\_i}{|a\_i|}|b\_j|^{\frac{q... | https://mathoverflow.net/users/78326 | A question about the proof of Riesz-Thorin interpolation theorem | Concerning your Question (3): A nice proof of the Riesz--Thorin theorem is given in Section 3.5 in [Lax's book](https://bookstore.ams.org/ulect-58/). A chapter devoted to interpolation of linear operators, including different versions of the Riesz--Thorin theorem, with further references therein, is contained in [Mashr... | 2 | https://mathoverflow.net/users/36721 | 327233 | 140,798 |
https://mathoverflow.net/questions/327234 | 6 | I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a potential function (which was addressed in a separate [question](https://mathoverflow.net/questions/122308/when-a-riemannian-ma... | https://mathoverflow.net/users/125275 | Can the number of solutions to a system of PDEs be bounded using the characteristic variety? | The wave equation in the plane is $\partial^2\_x-\partial^2\_y=(\partial\_x+\partial\_y)(\partial\_x-\partial\_y)$, so two points in the characteristic variety, but infinite dimensional family of solutions.
Each isolated point in the characteristic variety represents a hypersurface foliation inside every sufficiently... | 6 | https://mathoverflow.net/users/13268 | 327235 | 140,799 |
https://mathoverflow.net/questions/327192 | 3 | Let $(M,g)$ be a compact orientable hyperbolic manifold with dimension at least $4$. Are there any topological conditions on $M$ which guarantee the existence of a hyperbolic surface in $M$? i.e. a $2$-dimensional submanifold $\Sigma$ such that the restriction of $g$ to $\Sigma$ is hyperbolic. (apologies if this is ver... | https://mathoverflow.net/users/99732 | submanifold of a hyperbolic manifold | The thing is that every compact Riemannian surface admits a $C^\infty$ isometric embedding in ${\mathbb E}^5$, see Michael Albanese's answer [here](https://mathoverflow.net/questions/325842/isometric-embedding-of-a-genus-g-surface); the result is a version of the Nash isometric embedding theorem due to Gromov.
Next, ... | 10 | https://mathoverflow.net/users/21684 | 327239 | 140,803 |
https://mathoverflow.net/questions/327187 | 5 | Assume GCH. Suppose $j: V\to M$ is an elementary embedding such that $\mathrm{crit}(j)=\kappa$, ${}^\kappa M \subset M$, $M\supset V\_{\kappa+2}$. We can assume $j$ is defined from some extender of length $\kappa^{++}$. Hence, in particular, we know $\kappa^{++}<j(\kappa) < \kappa^{+++}$.
For each $\delta<j(\kappa)$,... | https://mathoverflow.net/users/119731 | Number of ultrafilters in an extender | TLDR: Yes, all the ultrafilters are different.
Suppose $M$ is a transitive class and $j : V\to M$ is an elementary embedding.
*Some notation.*
* For any $x\in M$, let $H\_x = \{j(f)(x) : f\text{ is a function}\}\prec M.$
* Let $j\_x : V\to M\_x$ be the ultrapower by the derived ultrafilter $U\_x = \{A\subseteq ... | 5 | https://mathoverflow.net/users/102684 | 327248 | 140,806 |
https://mathoverflow.net/questions/327219 | 2 | Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already:
1. $S$ is set of measurable functions $f : \Omega \rightarrow X$ on fixed probability space $\langle \Omega, \mathfrak{I}, \mathbb{P} \rangle$ w... | https://mathoverflow.net/users/48726 | Proving that family of sets has non-empty intersection | One approach is to choose a weaker topology, where you might get compactness.
For instance, let's identify your finite set $X$ with $\{1,2,\dots,N\} \subset \mathbb{R}$. Then we can view your functions $f$ as elements of $L^2(\Omega, \mathbb{P})$ and equip this space with the weak topology. Since all your functions $... | 2 | https://mathoverflow.net/users/4832 | 327257 | 140,812 |
https://mathoverflow.net/questions/327249 | 3 | Let $X$, $Y$ be integral separated schemes of finite type over $\mathbb{C}$, $Y$ be normal, $f:X\rightarrow Y$ be a surjective morphism of schemes. Can the non-flat locus of $f$ be non-empty and have codimension $\geq 2$ in $X$?
| https://mathoverflow.net/users/nan | Codimension of non-flat locus | Let $Y$ be a cone over a smooth projective variety which is a product and projectively normal (so $Y$ is normal). For example let $Y$ be the cone over $\mathbb P^1\times \mathbb P^1\subseteq \mathbb P^3$. In general, say $Y$ is a cone over $V\times W$.
Next let $H\subseteq W$ be an effective Cartier divisor (In the $... | 4 | https://mathoverflow.net/users/10076 | 327262 | 140,817 |
https://mathoverflow.net/questions/327204 | 9 | Hyperbolic PDE (like the wave equation) are roughly speaking, PDE that satisfy the “finite propagation speed of information” property. They are ubiquitous in mathematical physics (essentially, most fundamental laws of nature are hyperbolic).
However, do hyperbolic PDE occur in any other areas of mathematics that do ... | https://mathoverflow.net/users/3709 | Hyperbolic PDE in mathematics | Hyperbolic PDEs arise unexpectedly in some differential geometric questions involving prescribed data. What's weird in these cases is that there is no natural time coordinate in the PDEs. Here are some examples:
1. Bryant; Griffiths; Yang.
Characteristics and existence of isometric embeddings.
Duke Math. J. 50 (1983... | 6 | https://mathoverflow.net/users/613 | 327273 | 140,821 |
https://mathoverflow.net/questions/327281 | 1 | Let $S(x)$ be continuous, differentiable, and such that $S(x)=O(x/\log x)$. Let $J(x)=\int\_x^{\infty} \frac{S(y)(1+\log y)}{y^2\log^2 y}dy$ and let $K(x)=\int\_x^{\infty}\frac{S(y)}{y^2}dy$. Let $K(2)>J(2)>0$, and let $J(x)>0$ for $x>2$. Does it follow that $K(x)>0$ for $x>2$? If this is not the case, could someone su... | https://mathoverflow.net/users/130113 | Comparison of two integrals | An integration by parts shows that
$$
K(x) = \frac{\log^2 x}{1+\log x} J(x) + \int\_x^{\infty} J(y) \frac{\log^2 y+2\log y}{y(1+\log y)^2}\, dy >0 .
$$
(I'm also assuming here that $S(x)/x^2$ is integrable, but we need this anyway to make $K$ well defined.)
| 3 | https://mathoverflow.net/users/48839 | 327286 | 140,826 |
https://mathoverflow.net/questions/278797 | 1 | Given a finite dimensional local commutative algebra over a finite field $K$ and a finite dimensional module $M$. What is the fastest/best way to obtain all submodule from $M$ using a Computer algebra system
like GAP/QPA? It might be also interesting to hear how you might do that by hand.
| https://mathoverflow.net/users/61949 | Finding all submodules | Here is how you can do it with QPA:
```
gap> A := NakayamaAlgebra(GF(3), [4]);
<GF(3)[<quiver with 1 vertices and 1 arrows>]/<two-sided ideal in <GF(3)[<quiver with 1 vertices and 1 arrows>]>, (1 generators)>>
gap> M := IndecProjectiveModules(A)[1];
<[ 4 ]>
gap> subs := AllSubmodulesOfModule(M);
[ [ <<[ 0 ]> ---> <[... | 2 | https://mathoverflow.net/users/130741 | 327301 | 140,831 |
https://mathoverflow.net/questions/327267 | 5 | Let $u: \Omega\subset \mathbb{R}^N \to \mathbb{R}^M$ be a $BV$ function.
>
> Is the Hausdorff dimension of the graph of $u$ equal to $N$? How can we prove it?
>
>
>
---
**Update.**
1. In an [answer to this post](http://tps://mathoverflow.net/a/327310/122620), it has been showed that there exist a repres... | https://mathoverflow.net/users/122620 | Hausdorff dimension of the graph of a BV function | A partial answer here.
Let us recall the following Lusin type result (see e.g. Theorem 5.34 in *Functions of bounded variation* by L. Ambrosio, N. Fusco and D. Pallara):
>
> **Theorem 1.** There exists a constant $\kappa>0$ such that for every $u\in BV(\mathbb R^N)$ and $\lambda>0$ there exists a Lipschitz functi... | 4 | https://mathoverflow.net/users/44463 | 327310 | 140,833 |
https://mathoverflow.net/questions/327228 | 6 | Consider the interval $[0,1]$ and let $\mu\_k(t)$ with $k=1,\ldots,n$ be continuous functions such that they are all strictly increasing on the interval $[0,1]$ and such that $\mu\_1(t)<\mu\_2(t)<\ldots<\mu\_n(t)$ for all $t \in [0,1]$.
Consider for each $k \geq 0$, the functions $$f\_k(t)=\sum\_{j=1}^n (\mu\_j(t))^k... | https://mathoverflow.net/users/50438 | uniform approximation by a particular set of functions | Assume that the linear combinations of $\{f\_i\}$ are not dense in $C[0,1]$. Then by Hahn - Banach theorem and Riesz - Markov - Kakutani theorem there exists a non-trivial Borel finite sign measure $\eta$ on $[0,1]$ such that $0=\int f\_k(t)d\eta=\sum\_{i=1}^n \int (\mu\_i(t))^kd\eta(t)=\sum\_{i=1}^n \int\_{\mathbb{R}}... | 3 | https://mathoverflow.net/users/4312 | 327312 | 140,834 |
https://mathoverflow.net/questions/327321 | 5 | How could we find the large-$n$ asymptotic of $$\int\_{1}^{e^n}\left(1-\frac{\ln x}{n}\right)^n\,dx.$$
I have a suspicion that this is $\sqrt{n}$.
| https://mathoverflow.net/users/118090 | Asymptotic of integral $\int_{1}^{e^n}(1-\frac{\ln(x)}{n})^n\,dx$ | Denote $t=\ln(x)/n$, then $t$ varies from 0 to 1 and the the integral reads as $$n\int\_0^1 ((1-t) e^{t})^ndt.$$
We have $(1-t)e^t=(1-t)(1+t+t^2/2+\ldots)=1-t^2/2+O(t^3)=\exp(-t^2/2+O(t^3))$ for small $t$ and $(1-t)e^t<1$ for $0<t\leqslant 1$. Thus by Laplace method the asymptotics is the same as for the integral $$n\i... | 11 | https://mathoverflow.net/users/4312 | 327322 | 140,835 |
https://mathoverflow.net/questions/327268 | 17 | Let $X$ be a CW complex and let $\Sigma^\infty X$ denote its suspension spectrum. By definition, the $n$th singular homology group of $\Sigma^\infty X$ with coefficients in $\mathbb{Z}$ is $\pi\_n(\Sigma^\infty X \wedge H\mathbb{Z})$.
Now, connective spectra are in bijection with infinite loop spaces. The infinite lo... | https://mathoverflow.net/users/nan | Homology of spectra vs homology of infinite loop spaces | $\newcommand{\H}{\mathrm{H}} \newcommand{\Z}{\mathbf{Z}}$Let $X$ be a space. Then the $E$-(co)homology of $X$ is the same as the $E$-(co)homology of its suspension spectrum, i.e., $E\_\ast(X) \cong E\_\ast(\Sigma^\infty\_+ X)$ (and you remove the basepoint in $\Sigma^\infty\_+ X$ to get reduced $E$-homology). In the ca... | 17 | https://mathoverflow.net/users/102390 | 327326 | 140,837 |
https://mathoverflow.net/questions/323083 | 2 | Given a connected quiver algebra $A$ over a finite field $K$.
>
> Question : Is there an effective/quick method to obtain all $d$-dimensional indecomposable representations for a fixed $d$ with a computer algebra system such as the GAP package QPA?
>
>
>
| https://mathoverflow.net/users/61949 | Finding all $d$-dimensional indecomposable representations | Now this is implemented in QPA version 1. There is an error above, one needs to consider both `ExtOverAlgebra(S[i],iso[j])` and `ExtOverAlgebra(iso[j],S[i])` in order to get all modules of the given length. It looks like this in QPA:
```
gap> A := KroneckerAlgebra(GF(3),2);;
gap> AllModulesOfLengthAtMost( A, 4 ); ... | 1 | https://mathoverflow.net/users/130741 | 327328 | 140,838 |
https://mathoverflow.net/questions/327330 | 11 | Assume that $f:\mathbb R\to\mathbb R$ is continuous.
Given a real symmetric matrix $A\in\text{Sym}(n)$, we can define $f(A)$ by applying $f$ to its spectrum. More explicitly,
$$ f(A):=\sum f(\lambda)P\_\lambda,\qquad A=\sum\lambda P\_\lambda. $$
Here both sums are finite, and the second one is the decomposition of $A$ ... | https://mathoverflow.net/users/36952 | Smoothness of finite-dimensional functional calculus | Yes. The can be derived from the resolvent formalism.
I'll just do the $C^1$ case and leave higher derivatives as an exercise - ask if it's not clear how to generalize. I am basically using formula (2.7) of "On differentiability of symmetric matrix valued functions", Alexander Shapiro, <http://www.optimization-online... | 10 | https://mathoverflow.net/users/112284 | 327335 | 140,840 |
https://mathoverflow.net/questions/327293 | 4 | What is a method to compute the asymptotics of a sum resulting from the inclusion-exclusion principle? Each term of the sum can be approximated perhaps by Stirling's formula or the Gaussian distribution. However the alternating sign should effect some cancellation. As an example, the [answer to this combinatorial probl... | https://mathoverflow.net/users/32660 | Asymptotics for the sums from the inclusion-exclusion principle | Another way to approach this from the generating function perspective. Notice that
$$\sum\_{k=l}^u (-1)^k [x^k]\, f(x) = [x^u]\,\frac{f(-x)}{1-x} - [x^{l-1}]\,\frac{f(-x)}{1-x},$$
and so the question reduces to studying the asymptotic of the coefficients of $\frac{f(-x)}{1-x}$.
In your example, we have
$$(j-k+1)\bin... | 2 | https://mathoverflow.net/users/7076 | 327353 | 140,844 |
https://mathoverflow.net/questions/327340 | 9 | Consider the complex projective variety given by $X^n = 0$, where $X\in \mathrm{M}\_n(\mathbb{C})$ and, say, $n\geq 3$. Some basic properties of it are already mentioned in this question:
<https://math.stackexchange.com/questions/405291/variety-of-nilpotent-matrices>
>
> I would like to know if its geometry has ... | https://mathoverflow.net/users/1849 | Has the geometry of the variety of nilpotent matrices over $\mathbb{C}$ been studied? | Consider $PGL\_n$ acting by conjugation on the space of $n\times n$ matrices $M\_n$, and let the GIT quotient map be $\pi:M\_n\to M\_n//PGL\_n$. I think you are asking about the geometry of the fibre of zero.
Regardless, I believe this is a special case of the nullcone for a reductive group action on a vector space $... | 9 | https://mathoverflow.net/users/12218 | 327359 | 140,846 |
https://mathoverflow.net/questions/327346 | 4 | It is well known that the disk algebra (viewed as an algebra on the circle) is uncomplemented in $C(\mathbb T)$. What can be said about the pair
$(A(\mathbb D), H^\infty(\mathbb D))$?
| https://mathoverflow.net/users/61993 | Is the disk algebra a complemented subspace of the algebra of bounded analytic functions? | Call the smaller algebra $A$ and the larger algebra $B$ for convenience. Here is a ludicrously over-the-top way to prove that $A$ is not complemented in $B$: invoke Bourgain's result that $B$ is a [*Grothendieck space*](https://en.wikipedia.org/wiki/Grothendieck_space), which means that every bounded linear map from $B... | 4 | https://mathoverflow.net/users/763 | 327368 | 140,848 |
https://mathoverflow.net/questions/327382 | 6 | Let $A$ be a $n \times n$ matrix so that the Frobenius norm squared $\|A\|\_F^2$ is $\Theta(n)$, the spectral norm squared $\|A\|\_2^2=1$. Is it true that $\sum\_{i=1}^n\max\_{1\leq j\leq n} |A\_{ij}|^2$ is $\Omega(n)$? Assume that $n$ is sufficiently large.
I cannot find a relation between matrix norms that can show... | https://mathoverflow.net/users/138033 | Relation between Frobenius, spectral norm and sum of maxima | This is false in general, but true for matrices with non-negative entries.
For a counterexample, suppose that $n=p$ is prime, and consider the matrix
$$ A=\left\|p^{-1/2}\left(\frac{i-j}p\right)\right\|\_{i,j=0,\dotsc,p-1} $$
where $(\cdot/p)$ is the Legendre symbol. This is a circulant matrix; its non-zero eigenval... | 6 | https://mathoverflow.net/users/9924 | 327387 | 140,852 |
https://mathoverflow.net/questions/325666 | 1 | I want to justify why the Chaos game works to produce Sierpinski triangle. I use a theorem taken from Massopust *Interpolation and Approximation with Splines and Fractals*.
>
> Suppose that $(X,d)$ is a compact metric space and $(X,F,P)$ is an IFS with probabilities. Futher assume that $m \in P(X)$ is the invariant... | https://mathoverflow.net/users/82839 | Formal justification of the Chaos game in the Sierpinski triangle | The following is a small correction to Massopust *Interpolation and Approximation with Splines and Fractals.*
**Relation between the fractal generated by the IFS $A$ and the invariant measure $m$**
If the involved probabilities are strictly positive $A = supp \; m$.
**Support of a Borel probability measure on a c... | 0 | https://mathoverflow.net/users/82839 | 327390 | 140,853 |
https://mathoverflow.net/questions/327384 | -2 | How one could prove, that q pochhammer symbol $(1,1/n) = \prod\_{k = 1}^{\infty}(1-\frac{1}{n^k}) \geq 1 - \frac{1}{n-1}$
| https://mathoverflow.net/users/118090 | Lower bound of q pochhammer symbol | $$\prod (1-x\_i)=1-x\_1-x\_2(1-x\_1)-x\_3(1-x\_1)(1-x\_2)-\ldots \geqslant 1-\sum x\_i, \forall x\_i\in [0,1],$$
use this for $x\_i=1/n^i$.
| 3 | https://mathoverflow.net/users/4312 | 327392 | 140,854 |
https://mathoverflow.net/questions/327318 | 8 | It is [known](https://en.wikipedia.org/wiki/Stallings_theorem_about_ends_of_groups) that the number of ends of a finitely generated group is 0,1, 2 or $\infty$.
>
> **Problem 1.** What is known about the number of ends of infinite finitely generated torsion groups?
>
>
>
In particular,
>
> **Problem 2.** I... | https://mathoverflow.net/users/61536 | Ends of finitely generated torsion groups | Stallings' Theorem, given in his book [Group theory and three-dimensinal manifolds](https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=stallings&s5=&s6=&s7=three%20dimensional&s8=All&sort=Newest&vfpref=html&... | 5 | https://mathoverflow.net/users/38698 | 327393 | 140,855 |
https://mathoverflow.net/questions/327061 | 5 | To expand upon the title, I am wondering if there is a specific name for square matrices of the form: $$M = \begin{bmatrix} a\_{11} & 0 & \cdots & 0 & \cdots & & 0 & b\_{1n} \\ 0 & a\_{22} & 0 & \cdots & & b\_{2 \ i} & & 0 \\ 0 & \ddots & \ddots & & \cdot^{\textstyle \cdot^{\textstyle \cdot}} & 0 & & 0 \\ & & & a\_{} &... | https://mathoverflow.net/users/133954 | Do matrices with only elements along the main and anti-diagonals have a name? | I am continuing in the answer box, to get this out of the "unanswered" queue. The OP asks "for a more standard terminology that is perhaps present in the literature."
The name "X-matrices" or "X-form matrices" has also been used in the published literature, for example, [Properties of Central Symmetric X-Form Matric... | 6 | https://mathoverflow.net/users/11260 | 327406 | 140,860 |
https://mathoverflow.net/questions/327404 | 2 | Let $\theta$ be a real number. We define $A\_{\theta}$, the algebra of continuous functions on a noncommutative $2$-torus, to be the universal $C^\*$-algebra generated by two generators $U$ and $V$ which satisfy
$$
UV=e^{2\pi i\theta}VU, UU\*=U^\*U=VV^\*=V^\*V=1.
$$
In literature people define smooth functions on a n... | https://mathoverflow.net/users/24965 | Is the algebra of Schwarz functions on a noncommutative torus the maximal algebra of smooth functions? | The two basic derivations are infinitesimal generators of an action $\phi$ of $\mathbb{T}^2$ on $A\_\theta$, and the Fourier coefficients of $A \in A\_\theta$ are recovered by $a\_{rs}U^rV^s = \int\_{\mathbb{T}^2} \phi\_{xy}(A) e^{rx + sy} dxdy$. This shows that the $a\_{rs}$ are bounded by $\|A\|$. The two generators ... | 2 | https://mathoverflow.net/users/23141 | 327417 | 140,865 |
https://mathoverflow.net/questions/327418 | 8 | Let $X$ be a Banach space and $B(X)$ be its space of all (bounded) operators. A *nuclear functional* on $B(X)$ is a linear functional $u:B(X)\to{\mathbb C}$ that can be represented in the form
$$
u(A)=\sum\_{n=1}^\infty \lambda\_n\cdot f\_n(Ax\_n),\qquad A\in B(X),
$$
where $\lambda\_n\in{\mathbb C}$, $x\_n\in X$, $f\_... | https://mathoverflow.net/users/18943 | When does the dual to the space $K(X)$ of compact operators consist of nuclear functionals? | This is the question of duality of injective and projective tensor products of Banach spaces, and the natural question would be whether the dual of $K(X)$ can be represented by the functionals in $N(X^\*)$. A quick answer is: If $X^\*$ or $X^{\*\*}$ has the approximation property and if $X^\*$ or $X^{\*\*}$ has the Rad... | 7 | https://mathoverflow.net/users/127871 | 327430 | 140,867 |
https://mathoverflow.net/questions/327426 | 8 | What makes Graph invariants so useful/important? If I were trying to create a useful graph invariant, what principles should I follow?
My understanding is that they allow one to isolate and study specific properties of graphs algebraically or to classify graphs up to isomorphism (although, it seems to me that canoni... | https://mathoverflow.net/users/136371 | What makes Graph invariants so useful/important? | We probably wouldn’t ask what makes graph **properties** useful. In many ways we consider isomorphic graphs as “the same.” Invariants are just properties that respect this sameness. The specific vertex set is not an invariant. The number of vertices is. You can certainly make up unmotivated invariants like “the number ... | 8 | https://mathoverflow.net/users/8008 | 327437 | 140,873 |
https://mathoverflow.net/questions/327441 | 1 | A semifinite trace $\tau$ on $M\_{+}$ (for a von Neumann algebra $M$) is said to be normal if $\tau(\sup x\_i ) = \sup \tau(x\_i)$ for an bounded increasing net of positive operators $(x\_i)\_{i \in I}$.
Is it true that $\tau(\inf x\_i) = \inf \tau(x\_i)$ for a bounded decreasing net of positive operators $(x\_i)\_{i... | https://mathoverflow.net/users/127523 | infimum and supremum for normal semifinite trace | No, let $\tau$ be integration against counting measure on $l^\infty$ and let $x\_n$ be the characteristic function of $\{i: i \geq n\}$.
| 3 | https://mathoverflow.net/users/23141 | 327442 | 140,876 |
https://mathoverflow.net/questions/327306 | 3 | Let $X,Y$ be two (dependent) random variables with $\mathbb{P}(X\ge 0)=\mathbb{P}(Y\ge 0)=1$.
>
> I want to find a tight lower bound of $\mathbb{E}(XY)$ when $X,Y$ are non-negative, almost surely.
>
>
>
Note that one trivial lower bound is $0$. But I want to know if there is an elegant method in literature f... | https://mathoverflow.net/users/64194 | Is there a tight lower bound for the expectation of the product of two positive valued random variables? | $\newcommand{\de}{\delta}
\newcommand{\si}{\sigma}
\newcommand{\ep}{\varepsilon}$
Let us present the exact lower bound on $EXY$ in terms of $\mu\_1:=\mu\_X$, $\mu\_2:=\mu\_Y$, $\si\_1:=\si\_X$, $\si\_2:=\si\_Y$, as follows:
>
> The minimum of $EXY$ over all nonnegative random variables (r.v.'s) $X$ and $Y$ with pre... | 4 | https://mathoverflow.net/users/36721 | 327450 | 140,878 |
https://mathoverflow.net/questions/327447 | 13 | Let $m$ be a positive integer and $\chi$ a primitive character mod $m$. Let $x$ be such that $\chi(p)\ne 1$ for all primes $p<x$. Assume GRH. How can one bound $x$ in terms of $m$ ? I do not need the best possible bound, but I need a good quality bound which is **totally explicit** in all parameters.
A related quest... | https://mathoverflow.net/users/138069 | Least quadratic residue under GRH: an explicit bound | See the work of [Lamzouri, Li, and Soundararajan](https://arxiv.org/abs/1309.3595) (I link the arXiv version; the paper appeared in Math. Comp.). Assuming that $\chi$ is a primitive quadratic character (as the title suggests) then Theorem 1.4 of that paper gives an explicit bound on the least prime quadratic residue on... | 22 | https://mathoverflow.net/users/38624 | 327451 | 140,879 |
https://mathoverflow.net/questions/327456 | 7 | Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences
$$S(n,m,t)=\sum\_{k\_1+\cdots+k\_n=m}\binom{m}{k\_1,\dots,k\_n}^t$$
where the sum runs over non-negative integers $k\_1,\dots,k\_n$. These numbers are related to average distances (from the origin) of uniform unit-step random walks on the pla... | https://mathoverflow.net/users/66131 | Divisibility of sum of multinomials | We count the number of $t$-tuples $(\xi\_1,\ldots,\xi\_t)$ of the colorings of $\{1,\ldots,m\}$ with $n$ given colors, for which any two colorings use the same multisets of colors. If $M$ is the number of such tuples in which 1 is colored red in the coloring $\xi\_1$ (red is one of our $n$ colors), the total number of ... | 9 | https://mathoverflow.net/users/4312 | 327458 | 140,881 |
https://mathoverflow.net/questions/327394 | 3 | Let $A$ be a sequence of $n \times n$ matrices so that the Frobenius norm squared satisfies $\|A\|\_F^2 \simeq n$ and the infinity norm squared is $\|A\|\_{\infty}^2 = 1$. Is the following true?
$$\sum\_{i=1}^n\max\_{1\leq j\leq n} |A\_{ij}|^2\gtrsim n$$
I cannot find a relation between matrix norms that can show t... | https://mathoverflow.net/users/138033 | Relation between Frobenius norm, infinity norm and sum of maxima | The answer is **Yes**. This is not really a problem about matrices. The best way to analyse it is to rewrite it in terms of the row vectors $u\_i\in{\mathbb C}^n$. Let me denote $\|\cdot\|\_p$ the $\ell^p$-norm over ${\mathbb C}^n$ ; when $p=2$, this is the standard Euclidian norm. The Frobenius norm is $\sum\_i\|u\_i\... | 4 | https://mathoverflow.net/users/8799 | 327459 | 140,882 |
https://mathoverflow.net/questions/327402 | 3 | Assume that $F(e^{it})=e^{if(t)}$ is a diffeomorphism of the unit circle onto itself and let $A=\left|\int\_0^{2\pi}(1-F^2)\,dt\right|$ and $B=\left|\int\_0^{2\pi} F^2(1-F^2) \,dt\right|$. It seems that $A>B$ but so far I cannot prove.
Thanks
| https://mathoverflow.net/users/124426 | An integral inequality for diffeomorphisms | It looks false. Note that $1-\cos x<1+\cos x-2\cos^2 x=\cos x-\cos 2x$ for $|x|\in (0,\pi/2)$. Therefore if say $|2f(t)|\in (\pi/10,\pi/3)$ for almost all $t$, the real part of $\int F^2(t)(1-F^2(t))$ is greater than that of $\int 1-F^2(t)$. If also $2f(t+\pi)=-2f(t)$ for $t\in [0,\pi]$, the imaginary part of $\int 1-F... | 1 | https://mathoverflow.net/users/4312 | 327467 | 140,883 |
https://mathoverflow.net/questions/327460 | 3 | Just trying to locate a copy of Hecke’s 1938 lecture notes. I’m aware of the notes by Berndt that are based on Hecke’s lectures, but I really would like to find (pdf or for purchase) a copy of the original.
Worldcat entry, for reference: <https://www.worldcat.org/title/lectures-by-erich-hecke-on-dirichlet-series-modu... | https://mathoverflow.net/users/41582 | Hecke’s 1938 IAS lectures - pdf or print copy? | To answer the second question of the OP, here are the table of contents of the two books, without any significant overlap.
| 2 | https://mathoverflow.net/users/11260 | 327471 | 140,886 |
https://mathoverflow.net/questions/327341 | 4 | Let $(W,S)$ be an arbitrary Coxeter system. We consider the following **scenario**:
>
> Let $\mathcal{O}$ be a conjugacy class of an element $w$ in $W$ which is finite and which generates the whole group $W$.
>
>
>
The only **example** of the above scenario I know arises in simply laced irreducible finite Coxe... | https://mathoverflow.net/users/66288 | Coxeter groups generated by one finite conjugacy class | Besides the deeper comment of YCor above which points out further solutions/directions, I add a very simple consideration as answer to "close the case".
Let $(W,S)$ be infinite and $\mathcal{O}\_w$ a finite conjugacy class of some $w\in W$. Let $G$ be the group generated by $\mathcal{O}\_w$.
First note that $\mathc... | 0 | https://mathoverflow.net/users/66288 | 327486 | 140,888 |
https://mathoverflow.net/questions/316503 | 25 | The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic projective space $\mathbb{HP}^n$ is a quaternionic Kähler manifold, and so its de Rham complex carries a local represen... | https://mathoverflow.net/users/125941 | The de Rham complex of the octonionic projective spaces | A description of the octonionic projective plane in terms of the octonionic algebra is described by John Baez in his notes on [Octonionic projective geometry:](http://math.ucr.edu/home/baez/octonions/node9.html)
The [Jordan algebra](https://en.wikipedia.org/wiki/Jordan_algebra) of 3×3 Hermitian octonionic matrices, w... | 4 | https://mathoverflow.net/users/11260 | 327487 | 140,889 |
https://mathoverflow.net/questions/327498 | 6 | Let's view $\mathbb{R}$ as a vector space over $\mathbb{Q}$, and pick some basis $(v\_\alpha)\_{0 \leq \alpha < \mathfrak{c}}$ of it. We can then consider the subspace $L$ spanned by $(v\_\alpha)\_{0 < \alpha < \mathfrak{c}}$ (ie leaving out one vector from the basis). Given the horrible way we have built $L$, I don't ... | https://mathoverflow.net/users/15002 | Measure of rational hyperplanes of $\mathbb{R}$ | This is the (a version of the) Vitali set, which is not Lebesgue measurable. A quick reason is:
$\mathbb{R}= (v\_0\mathbb{Q})\oplus\_\mathbb{Q} V$ shows that $\mathbb{R}$ is a countable union of translates of $V$, so $V$ cannot be a Lebesgue set of measure zero, but must have positive outer measure. To show that the... | 8 | https://mathoverflow.net/users/6101 | 327501 | 140,893 |
https://mathoverflow.net/questions/327020 | 1 | Consider some finite string $x=(x\_1,x\_2,...,x\_{n-1},x\_n)$ that is drawn from a *non-stationary* process. Would it be possible to use the algorithmic probability formula, defined by Solomonoff as,
$$
P\_M(x)=\sum\_{i=1}^{\infty}2^{-|s\_{i}(x)|}
$$
to predict $x\_n$, by computing the probability $$P\_M(x\_n | x\_1,..... | https://mathoverflow.net/users/102097 | Find probability of non-stationary inputs into Turing machine? | The equation you have for algorithmic probability states how to compute (or, more specifically, lower semicompute) $P\_M(x)$ for any finite string $x$, so it can also be used to predict $x\_n$ by evaluating $P\_M(x\_n \vert x\_1, \dots , x\_{n-1})$. It holds for *any* finite string $x$ -- it does not matter what distri... | 1 | https://mathoverflow.net/users/76565 | 327509 | 140,895 |
https://mathoverflow.net/questions/327510 | 5 | Let $\mathcal{C}$ be a semiadditive $\infty$-category, complete and cocomplete, and let $G$ be a finite group. Then for any $X \in Fun(BG,\mathcal{C})$, there is a norm map $N\_X: X\_G \to X^G$. For each $X \in \mathcal{C}$, we have the constant $X^{triv} \in Fun(BG,\mathcal{C})$. Suppose that $N\_{X^{triv}}$ is an equ... | https://mathoverflow.net/users/2362 | If the Tate construction vanishes for all trivial $G$-actions, then does it vanish for all $G$-actions? | There is a reference in my comment above (Lemma 2.1.5 in arxiv.org/pdf/1811.02057.pdf). Let me repeat the argument from there in the answer here.
Let $X$ be a $\pi$-finite space and $q:X\to pt$ be the projection to the point. We assume that the norm map $Nm\_q : q\_! \to q\_\*$ is an isomorphism when evaluated at "t... | 6 | https://mathoverflow.net/users/115052 | 327513 | 140,897 |
https://mathoverflow.net/questions/327515 | 7 | It's pretty clear that for a simplicial complex $\Delta$, shellability of the complex implies that it is homotopy equivalent to a wedge of spheres. However, does the converse hold? That is, does $\Delta$ being homotopy equivalent to a wedge of spheres imply shellability? Thanks!
| https://mathoverflow.net/users/138105 | Does homotopy equivalence to a wedge of spheres imply shellability? | No. There are unshellable n-spheres for all n≥3:
*Lickorish, W. B. R.*, [**Unshellable triangulations of spheres**](http://dx.doi.org/10.1016/S0195-6698(13)80103-5), Eur. J. Comb. 12, No. 6, 527-530 (1991). [ZBL0746.57007](https://zbmath.org/?q=an:0746.57007).
| 13 | https://mathoverflow.net/users/353 | 327517 | 140,898 |
https://mathoverflow.net/questions/326810 | 13 | Let $k$ be a field (of possibly positive characteristic), let $U\_n$ denote the space of all $n \times n$ unipotent upper triangular matrices over $k$, and let $G$ be an algebraic subgroup of $U\_n$ (hence a unipotent algebraic group itself). Then each $X \in \text{Lie}(G)$ (thought of as a member of $\text{Lie}(U\_n)$... | https://mathoverflow.net/users/19048 | exponential/logarithm for unipotent algebraic groups | This is false in characteristic $p$, no matter how large $p$ is. The counterexample is the group parameterized by
$\begin{pmatrix} 1 & t & t^p \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
Its Lie algebra is generated by the matrix
$\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$
whose exponential... | 13 | https://mathoverflow.net/users/18060 | 327519 | 140,899 |
https://mathoverflow.net/questions/327551 | 2 | This question is a development of [my previous question](https://mathoverflow.net/questions/327318/ends-of-finitely-generated-torsion-groups).
Let $G$ be a finitely generated group acting transitively on an infinite set $X$ so that for every $g\in G$ and $x\in X$ the $g$-orbit $\{g^nx:n\in\mathbb Z\}$ of $x$ is finit... | https://mathoverflow.net/users/61536 | Ends of G-spaces with action of a finitely generated group | No, the Grigorchuk group, which is torsion, admits a 2-ended connected Schreier graph. See for instance <http://www.math.tamu.edu/~yvorobet/Research/Schreier.pdf>
---
*Edit:* Here's a more elementary example, just assuming the bare existence of an infinite finitely generated torsion group, and also produces examp... | 6 | https://mathoverflow.net/users/14094 | 327553 | 140,905 |
https://mathoverflow.net/questions/327566 | 2 | Denote by $\mathfrak{g}$ a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$.
Let $\Phi$
be the root system of $(\mathfrak{g},\mathfrak{h})$, write $W$ for the corresponding Weyl group of $\Phi$, and denote by $\mathfrak{g}\_\alpha$ the root subspace of $\mathfrak{g}$ cor... | https://mathoverflow.net/users/110229 | Equivalence of definition of category $\mathcal{O}^\mathfrak{p}$ | By the PBW theorem we can write $U(\mathfrak{p}) = U(\mathfrak{l})U(\mathfrak{u}).$ By our assumption $U(\mathfrak{u})m$ is finite dimensional for any $m\in M$ and by the second point of the definition $U(\mathfrak{l})n$ is finite-dimensional for any $n \in U(\mathfrak{u})m.$
| 3 | https://mathoverflow.net/users/6818 | 327569 | 140,910 |
https://mathoverflow.net/questions/327567 | 2 | In quantum mechanics, given an $N$-qubit ($2^N$-dimensional) Hamiltonian $\hat{H}$, I'm fairly sure that the variance in energies of randomly drawn pure states (i.e. norm-$1$ vectors) may be calculated to be $\|\hat{H}\|\_F^2/4^N$, where $\|\hat{H}\|\_F$ is the Frobenius norm.
Does anyone know a good reference to cit... | https://mathoverflow.net/users/138132 | Reference request: Energy variance of randomly drawn state | I may find a reference, but the derivation is simple once you know that for large ${\cal N}=2^N$ a random vector $X\in\mathbb{C}^{\cal N}$ of unit length has independently distributed complex Gaussian elements of zero mean and variance $1/{\cal N}$. Then you just perform the Gaussian averages to obtain the first and se... | 1 | https://mathoverflow.net/users/11260 | 327573 | 140,912 |
https://mathoverflow.net/questions/327571 | 3 | **The question**
What is the order of magnitude for the function $n\mapsto |{\rm cd}(S\_n)|$?
**The motivation**
In my research on character degrees of finite groups, I have in recent years been focusing on symmetric groups as a test bed for general conjectures. Among my more recent investigations, I have come up... | https://mathoverflow.net/users/128140 | Growth rate of $|{\rm cd}(S_n)|$ | If $cd(S\_n)$ is the set of degrees of irreducible, complex-valued characters on the symmetric group $S\_n$, then by the basic representation theory of the symmetric group this is the same as the set $\{f^{\lambda}\colon \lambda\vdash n\}$, where for a partition $\lambda\vdash n$, $f^{\lambda}$ is the number of Standar... | 3 | https://mathoverflow.net/users/25028 | 327580 | 140,915 |
https://mathoverflow.net/questions/327575 | 1 |
>
> Given an hyperbolic IFS $(X,\{f\_i:i=1,\ldots,N\})$ and denoting its code space by $\Sigma\_N = \{1,\ldots,N\}^{\mathbb{N}}$ and the generated fractal set by $\mathcal{A}$.
>
>
> There is a continuous and surjective mapping $\gamma: \Sigma\_N \to \mathcal{A}$ given by $\gamma(\sigma) = \lim\limits\_{n \to \inft... | https://mathoverflow.net/users/82839 | Pointless characterization relating between a fractal and its code space | $\newcommand{\de}{\delta}
\newcommand{\ga}{\gamma}
\newcommand{\si}{\sigma}
\newcommand{\ep}{\varepsilon}$
First here, $\bigcap\limits\_{n \in \mathbb{N}} A\_{\sigma(n)}$ is a subset of $X$, whereas $\gamma(\sigma)$ is an element of $X$. So, in your second highlighted statement $\gamma(\sigma)$ should be replaced by $\... | 2 | https://mathoverflow.net/users/36721 | 327582 | 140,916 |
https://mathoverflow.net/questions/327585 | 3 | Let $C\_1,\dots, C\_m$ be a family of ordered set partitions of $[n]$ with exactly $k$ blocks.
Write $C\_i = \{B\_{i1}, \dots, B\_{ik}\}$ for $i=1,\dots, m$ where $B\_{ij}$ are the blocks of the ordered set partition $C\_i$.
Suppose this family also has the property that for each $j=1,\dots, k$
$$B\_{1j} \cup \c... | https://mathoverflow.net/users/94267 | Families of ordered set partitions with disjoint blocks | We have $$mn=\sum\_i\sum\_j |B\_{ij}|=\sum\_j\sum\_i |B\_{ij}|=kn,$$
thus $m=k$.
| 4 | https://mathoverflow.net/users/4312 | 327588 | 140,920 |
https://mathoverflow.net/questions/327591 | 20 | The metric in sub-Riemannian geometry is often called the Carnot-Carathéodory metric.
>
> **Question 1.** What is the origin of this name? Who was the first to introduce it?
>
>
>
I believe that the "Carathéodory" part of the name could be related to his work in theoretical thermodynamics [1], but I do not rea... | https://mathoverflow.net/users/121665 | Carnot-Carathéodory metric | [Pierre Pansu](http://www.math.polytechnique.fr/xups/textes-provisoires18/pansu.pdf) tells us that the terminology of the Carnot-Carathéodory metric is due to Mikhail Gromov [1].
Gromov himself [explains](https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/carnot_caratheodory.pdf) the choice of the name:
>
> T... | 22 | https://mathoverflow.net/users/11260 | 327594 | 140,923 |
https://mathoverflow.net/questions/327604 | 2 | For a fixed $u \in BV(\mathbb{R}^N)$, consider the function
$h:(0,+\infty) \to BV(\mathbb{R}^N)$, given by
$h(t) = u (tx)$.
Is $h$ continuous?
| https://mathoverflow.net/users/138146 | About the continuity of a function on BV | Not in the strong topology: for instance take $u:=1\_{(-1,1)}$, i.e. the characteristic function of the interval $(-1,1)$ on $\mathbb R$ (in this example $N=1$). Then $\|h(t)-h(s)\|\_{BV}\ge 2$ whenever $s\neq t$, since the distributional derivative of the element $h(s)\in BV(\mathbb R)$ is $\delta\_{-1/s}-\delta\_{1/s... | 4 | https://mathoverflow.net/users/36952 | 327606 | 140,927 |
https://mathoverflow.net/questions/327562 | 8 | Is it true that for every sufficiently large positive integer $n$, one can always find at most $k=\lfloor\pi(n)/2\rfloor$ integers, $a\_1,a\_2,a\_3,a\_3,\dots a\_k$, between $1$ and $n$, such that each of the $\pi(n)$ primes not greater that $n$ divides at least one of the integers $a\_1,a\_1+1,a\_2,a\_2+1,a\_3,a\_3+1,... | https://mathoverflow.net/users/60732 | Covering the primes with pairs of consecutive integers | For large enough $n$ no such set of integers exist.
First of all, let us say that $a$ *covers* prime $p$ if $a$ or $a+1$ is divisible by $p$.
Now, if $p$ is a prime with $\frac{n+1}{2}<p\leq n$, $a\leq n$ can cover $p$ only if $a=p$ or $a=p-1$, because otherwise $a\geq 2p-1>n$. Let $p\_1,p\_2,\ldots,p\_m$ be the set of... | 9 | https://mathoverflow.net/users/101078 | 327612 | 140,929 |
https://mathoverflow.net/questions/327558 | 1 | Let $R$ be a PID with infinitely many prime ideals. Suppose we have two integral locally closed subschemes of $\mathrm{Spec}\,R[x]$ such that
* both have non-empty intersection with the affine open $\mathrm{Spec}\, R[x, \frac{1}{x}]$; moreover, both intersections have the same set of points;
* both have non-empty int... | https://mathoverflow.net/users/nan | Subschemes of the affine line over PID | The answer is "yes" (and I believe the argument works with minor modifications if you do not assume that there are infinitely many prime ideals). By definition, either of your two subschemes has at least 2 points. This means that their closures can not be zero-dimensional (a closed subscheme of a Noetherian scheme is N... | 0 | https://mathoverflow.net/users/nan | 327616 | 140,931 |
https://mathoverflow.net/questions/327595 | 2 | Can the derivative of a BV function $f:\mathbb{R}^n\to\mathbb{R}^n$ be controlled by the symmetric part of the derivative $\frac{1}{2}(Df+(Df)^T)$?
| https://mathoverflow.net/users/122620 | Control the derivative of a BV function by its symmetric part | This is not a full answer, but some comments that might put you on the right track.
If a function $f=(f\_1,\ldots,f\_n):\mathbb{R}^n\to\mathbb{R}^n$ is sufficiently smooth and has compact support, then
$$
f\_k=\frac{2}{n\omega\_n}\sum\_{1\leq i\leq j\leq n}\left(\epsilon\_{jk}\*\frac{\partial K\_{ij}}{\partial x\_i}... | 3 | https://mathoverflow.net/users/121665 | 327617 | 140,932 |
https://mathoverflow.net/questions/327615 | 4 | Suppose $f,g:X \rightarrow Y$ are finite morphisms between connected smooth curves over $\mathbb{C}$, with $Y$ of genus at least $2$.
If $f$ and $g$ induce the same morphism $H^\*(Y,\mathbb{C}) \rightarrow H^\*(X,\mathbb{C})$, does $f=g$?
| https://mathoverflow.net/users/77909 | Can distinct morphisms between curves induce the same morphism on singular cohomology? | Yes. Since $Y$ embeds into its Jacobian $B$, it is enough to prove the statement for pairs of maps to an abelian variety $f, g\colon X\to B$ sending a base point $x\in X$ to $0\in B$. Every such map factors uniquely through the Albanese variety $A$ of $X$, so we reduce further to the case of pairs of maps $f, g\colon A... | 10 | https://mathoverflow.net/users/3847 | 327618 | 140,933 |
https://mathoverflow.net/questions/327636 | 3 | I'm wondering if there are any known result for the maximum large cardinal strength
which can be preserved by Radin forcing? For instance, with any large cardinal hypothesis in the ground model, can one show that in $V[G]$ which is a generic for a Radin forcing of some length $\kappa$ is $V\_{\kappa + 2}$-strong?
| https://mathoverflow.net/users/38228 | Radin forcing preserving large cardinals | If you start with a strong (or a supercompact cardinal) and if you force with Radin forcing
$\mathbb{R}\_u$, for some suitable $u$, then you can preserve the full strength (or supercompactness) of $\kappa$.
To preserve partial strength, (weak) repeat points are sufficient (see Radin's paper [Adding closed cofinal seq... | 4 | https://mathoverflow.net/users/11115 | 327643 | 140,941 |
https://mathoverflow.net/questions/327316 | 6 | Let $\mathcal M\_m$ be the set of $2$-by-$2$ **primitive** (relatively prime entries) matrices with determinant $m$. Let $\alpha \in \mathcal M\_m$ and let $\Gamma\subset \operatorname{SL}\_2(\mathbb Z)$. Define
$$R\_{\alpha,\Gamma}=\lbrace M\in \Gamma\backslash\mathcal M\_m : M \equiv \alpha \text{ in } \Gamma\backsla... | https://mathoverflow.net/users/122104 | Galois theory of modular functions | **EDIT**. The answer is yes as soon as $f$ and $f \circ \alpha$ both have rational Fourier coefficients.
To see this, we recall Shimura's theorem: for any modular form $f$ of level $N$, any $g \in \mathrm{SL}\_2(\mathbb{Z})$ and any $\sigma \in \mathrm{Aut}(\mathbb{C})$, we have $(f | g)^\sigma = f^\sigma | g\_\lambd... | 2 | https://mathoverflow.net/users/6506 | 327655 | 140,947 |
https://mathoverflow.net/questions/327650 | 5 | Probably this question can be phrased in a much greater generality, but I will just state it in the generality I require. I work over $\mathbb{C}$.
>
> Let $C\_1, C\_2 \subset \mathbb{P}^1$ be non-empty open subsets and $f: X \to C\_1 \times C\_2$ a non-trivial finite etale cover. Does there exist $i\in \{1,2\}$ su... | https://mathoverflow.net/users/5101 | Finite etale covers of products of curves | The answer is *no*, at least in general, as shown by the following counterexample.
Take a double cover $\bar{f} \colon \bar{X} \to \mathbb{P}^1 \times \mathbb{P}^1$, branched over a reducible
curve of the form $B=L\_1 + L\_2 + M\_1 + M\_2$ (here $|L|$ and $|M|$ are the two pencil of lines on the quadric).
Such a ... | 6 | https://mathoverflow.net/users/7460 | 327663 | 140,949 |
https://mathoverflow.net/questions/327671 | 2 |
>
> For a prime $p$ and $a\_1,\dotsc,a\_n\in\mathbb F\_p^\times$, consider the system of equations
> $$ \begin{cases}
> \begin{align}
> a\_1 + \dotsb + a\_n &= 0 \\
> a\_1x\_1 + \dotsb + a\_nx\_n &= 0 \\
> \qquad &\ \vdots \\
> a\_1x\_1^K + \dotsb + a\_nx\_n^K &= 0
> \end{align} \end{cases}
> $$
> How large ... | https://mathoverflow.net/users/9924 | A Vandermonde-type system | If $P(x)$ is divisible by $(x-1)^K$ and $\deg P<p$, then $P$ has at least $K+1$ non-zero coefficients. Proof: divide $P$ by maximal possible power of $x$, take the derivative and use induction in $K$. Or what I get wrong?
| 3 | https://mathoverflow.net/users/4312 | 327676 | 140,951 |
https://mathoverflow.net/questions/327680 | 0 | It is well known that for any smooth bounded (connected) domain $\Omega\subset\mathbb R^d$ with $d\ge2$, we can define a Green's function $G:\Omega\times\Omega\to\mathbb R$ in $\Omega$ which is smooth on $\mathring\Omega\times\mathring\Omega\setminus\Delta$, such that
$$ G\*\phi(x) = \int\_\Omega\! G(x,y)\phi(y)\,\mat... | https://mathoverflow.net/users/94022 | Boundary behavior of Greens functions on smooth bounded (planar) domains | This follows from the so-called (Eberhard) Hopf Minimum Principle. If you have a positive (super-) harmonic function $u$ in a ball, and $u(z\_0)=0$ for some boundary point $z\_0$,
then the normal derivative at $z\_0$ is non-zero. This is a simple exercise:
If the ball ix $|x|<R$ then
$$m(r)=\min\{u(x):|x|=r\}$$
is con... | 1 | https://mathoverflow.net/users/25510 | 327684 | 140,953 |
https://mathoverflow.net/questions/327557 | 2 | The semigroup of all order-preserving and decreasing transformations in full transformations semigroup $T\_n$ is denoted $C\_n$.
I consider the idempotent set $A=\{\begin{bmatrix}2\\1 \end{bmatrix},\begin{bmatrix}3\\2 \end{bmatrix},\cdots, \begin{bmatrix}n\\n-1 \end{bmatrix}\}$.
It is well known that $\langle A \ran... | https://mathoverflow.net/users/132399 | A question about semigroup union | $\DeclareMathOperator\im{im}$For ease of notation, let me denote $a\_i=\genfrac[]{0pt}{}{i+1}i$, so that $A=\{a\_1,\dots,a\_{n-1}\}$. I will write multiplication so that $(fg)(x)=f(g(x))$; reverse everything if you want the opposite convention.
>
> **Proposition:** For any $f\in C\_n$, the least $k$ such that $f$ c... | 1 | https://mathoverflow.net/users/12705 | 327685 | 140,954 |
https://mathoverflow.net/questions/327581 | 3 | Let $u \in C^0([-1,1])$ such that $u(0)=0$. Suppose that $u$ satisfies the following property:
For every $\{t\_k\}\subset \mathbb{R}$ such that $t\_k \to 0$, there exist a real number $\alpha$ (depending on the sequence $\{t\_k\}$), and a subsequence $\{t\_{k\_n}\}$, such that, if we set $u\_k(x)={u(t\_k x) \over t\_... | https://mathoverflow.net/users/131957 | Differentiability of the blow-up of a function | Set $f(x)=x\cos(\log(\log(1/|x|)))$ (and 0 at 0).
Let $\epsilon>0$. Now if $0<t<\epsilon^{1/\epsilon}$, consider
$\Delta\_t(x):=|f(tx)/t - x\cos(\log\log(1/t)|$.
On $[-\epsilon,\epsilon]$, this is bounded above by $2\epsilon$.
If $\epsilon<|x|\le 1$, we have
\begin{align\*}
\Delta\_t(x)&=|x[\cos(\log\log(1/(t|... | 1 | https://mathoverflow.net/users/11054 | 327688 | 140,955 |
https://mathoverflow.net/questions/327651 | 5 | Let $G$ be a semisimple Lie group with finite center, $K$ a maximal compact
subgroup, and $d=\dim(G/K)$. Let $\Gamma$ be a non-cocompact discrete subgroup of $G$. [Edit: assume that $\Gamma$ is virtually torsion-free (which is automatic if $G$ is linear and $\Gamma$ is finitely generated).]
>
> Is it true that the ... | https://mathoverflow.net/users/14094 | "Dimension" of discrete subgroups of infinite covolume in Lie groups | First of all, since you are not assuming finite generation, you should at least assume that $\Gamma$ is virtually torsion-free. (Otherwise, you need to work rationally and reprove Whitehead's lemma, in the setting of orbifolds/orbicomplexes; see below.)
With this extra assumption, here is a proof:
WLOG, $\Gamma$ i... | 4 | https://mathoverflow.net/users/21684 | 327693 | 140,957 |
https://mathoverflow.net/questions/327285 | 6 | How can one prove (or where can I find a proof) that if $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$, then $u$ cannot have a $(N-1)$-manifold of discontinuity points?
| https://mathoverflow.net/users/122620 | Sobolev functions on $\mathbb{R}^N$ cannot be discontinuous on a $(N-1)$-dimensional submanifold | Any function in $W^{1,p}$, $p>N$, has a continuous representative by the Sobolev embedding theorem so there is no issue here. However:
>
> **Proposition.** There is a function $f\in W^{1,p}$, $p\leq N$, that is essentially discontinuous everywhere. In fact you can find a function such that the essential supremum o... | 8 | https://mathoverflow.net/users/121665 | 327703 | 140,959 |
https://mathoverflow.net/questions/327697 | 10 | Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the books of Dickson and Hardy.
Motivation for this question. Some person came up to me before class and asked, are you the... | https://mathoverflow.net/users/126532 | Reference request: Oldest number theory books with (unsolved) exercises? | I wonder if you are already aware of [R. D. Carmichael](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=4918)'s "The theory of numbers" (John Wiley & Sons, Inc., NY, 1914).
Apropos of the exercises in this monograph, one can read the following in the preface:
>
> Numerous problems are supplied throughout the ... | 19 | https://mathoverflow.net/users/1593 | 327719 | 140,963 |
https://mathoverflow.net/questions/327742 | 0 | In [this MO question](https://mathoverflow.net/questions/149011/order-of-magnitude-of-sum-frac1-logp), it says that we have
$$ \sum\_{p<n} \frac{1}{\log{p}} =\frac{n}{\log^2 n}+O\left(\frac{n\log\log n}{\log^3 n}\right).$$
where the sum is on all primes $p$, up to some max prime $n$. This is derived from the prime ... | https://mathoverflow.net/users/24611 | Order of magnitude of $\sum \frac{1}{\log^2{p}}$, or $\sum \frac{1}{\log^a{p}}$ for arbitrary power $a$ | For $a > 0$, partial summation yields $$\sum\_{p \leq x} \frac{1}{\log^a{p}} = \frac{\pi(x)}{\log^a x} + a \int\_2^x \frac{\pi(t)}{t \log^{a+1} t} \,dt.$$ The first term is $\frac{\text{li(x)}}{\log^a x} + O\left(\frac{x}{\log^a x}e^{-c \sqrt{\log x}}\right)$ by the prime number theorem.
Cut the integral in half : $$... | 3 | https://mathoverflow.net/users/133679 | 327745 | 140,972 |
https://mathoverflow.net/questions/327733 | 52 | The first ever black hole was "pictured" recently, per an announcement made on 10th April, 2019. See for example: <https://www.bbc.com/news/science-environment-47873592> .
It has been claimed that state-of-the-art imaging algorithms were an enabler for this historic success. Does anybody care to describe the difficul... | https://mathoverflow.net/users/30684 | Mathematics of imaging the black hole | Essential elements$^\*$ of the reconstruction algorithm were developed at MIT under the name CHIRP = *Continuous High-resolution Image Reconstruction using Patch priors*, as described in [Computational Imaging for VLBI Image Reconstruction](https://arxiv.org/abs/1512.01413) (2015).
The difficulty of VLBI (*Very Long ... | 45 | https://mathoverflow.net/users/11260 | 327746 | 140,973 |
https://mathoverflow.net/questions/327729 | -1 | Thompson's Group has two well known presentations:
$\langle x\_0,x\_1, ... $ | $ x\_k^{-1} x\_n x\_k = x\_{n+1}\forall k < n \rangle$
$\langle A,B $ | $ [AB^{-1}, A^{-1}BA], [AB^{-1}, A^{-2}BA^2] \rangle$
where $x\_0=A$ and $x\_n = A^{1-n}BA^{n-1}$
It is also known that every finite group exists as a subgroup o... | https://mathoverflow.net/users/119181 | Groups With Arbitrarily Large Torsion | There exist many variations of Thompson's group $V$ which are finitely presented (or even of type $F\_\infty$) and which contain all the finite groups. For instance:
* Higman-Thompson groups $V\_{n,r}$ (same definition as $V$ but with $n$-adic subdivisions of $r$ disjoint copies of the Cantor set).
* Higher dimension... | 3 | https://mathoverflow.net/users/122026 | 327747 | 140,974 |
https://mathoverflow.net/questions/327252 | 10 | Let us say that a topological space $X$ is a **Kreisel-Putnam** space when it satisfies the following property:
>
> For all open sets $V\_1, V\_2$ and regular open set $W$ of $X$, if a point $x\in X$ has a neighborhood $N$ such that $N \cap W \subseteq V\_1 \cup V\_2$ then in fact it has a neighborhood $N'$ such th... | https://mathoverflow.net/users/17064 | Examples of Kreisel-Putnam topological spaces | Some observations (hopefully correct):
1. It is enough to consider $x ∈ ∂W$ and $V\_1, V\_2 ⊆ W$. So for fixed $x, W$ we want that $\mathcal{N}\_{x, W} := \{N ∩ W: N$ open neighborhood of $x\}$ is a prime filter (in the distributive lattice of open subsets of $W$).
- Extremally disconnected spaces are exactly those w... | 6 | https://mathoverflow.net/users/112373 | 327750 | 140,975 |
https://mathoverflow.net/questions/327660 | 6 | Let $R$ be a commutative ring with identity. If $P$ is a prime ideal of $R$ that is minimal over some zerodivisor of $R$, then must $P$ consist only of zerodivisors? I suspect not but I can't figure out how to construct a counterexample.
(The full context of the situation I am in is this. Suppose that $P$ is a prime ... | https://mathoverflow.net/users/17218 | prime ideals minimal over a zerodivisor | For the first question, consider $R = \mathbb{Z}[X]/(X^3 - 1)$, $P = (x+ 1)$ and $a = (x + 1)(1 + x + x^2)$ where $x$ denotes the image of $X$ in $R$.
In order to see that it provides us with a reduced one-dimensional Noetherian **counter-example**, note that $R/P \simeq \mathbb{Z}/2\mathbb{Z}$ and that $(x - 1)a = 0... | 4 | https://mathoverflow.net/users/84349 | 327752 | 140,977 |
https://mathoverflow.net/questions/327753 | 0 | I can't find much on the internet about this, but apparently vectors of naturals are called hyperscalars. It's not hard to bijectively map naturals to 2D hyperscalars and with that to prove that any-dimensional hyperscalars are countable and thus bijectively mappable to naturals.
More specifically, I'm interested in ... | https://mathoverflow.net/users/138214 | Mapping naturals to pairs of naturals and viceversa | $$a\_n=\{Y\_n,X\_n\}$$
where $X\_n$ is sequence [A319572](https://oeis.org/A319572) and $Y\_n$ is sequence [A319573](https://oeis.org/A319573) in the OEIS database. These are the coordinates of the stripe enumeration of $N \times N$ where $N = \{0, 1, 2, \ldots\}$. A *"Stripe Enumeration"* function to produce these s... | 0 | https://mathoverflow.net/users/11260 | 327756 | 140,979 |
https://mathoverflow.net/questions/327737 | 8 | Suppose we have two Gibbs measures with densities
$$
p\_f(x) \propto \exp(f(x)),\quad q\_g(x)\propto \exp(g(x)).
$$
Consider the KL-divergence between $p\_f$ and $q\_g$, as a functional of $f$ and $g$, that is,
$$
D(f, g) := \text{KL}(p\_f \| q\_g).
$$
Question: Do we have the following lower bound:
$$
D(f, g) \geq ... | https://mathoverflow.net/users/82358 | Lower Bound of KL-Divergence Between Two Gibbs Measures | The answer is no. Indeed, your question can be restated as follows: Is it true that for some constant $c>0$ we have
$$\int P\ln\frac PQ\,d\mu\ge c\int(\ln P-\ln Q)^2\,d\mu, \tag{1}
$$
where $P$ and $Q$ are probability densities with respect to a measure $\mu$?
Let $\mu$ be the counting measure on the set $\{0,1\}$,... | 5 | https://mathoverflow.net/users/36721 | 327760 | 140,981 |
https://mathoverflow.net/questions/327761 | 1 | Let $f:\Omega \subset \mathbb{R}^N \to \mathbb{R}^N$ be a vector field such that $f \in BV(\Omega)$.
Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure and that its density is in $L^\infty$.
What does this imply about the derivative of $f$? For example, about its Cantor pa... | https://mathoverflow.net/users/122620 | BV function with absolutely continuous divergence | According to G. Alberti (*Proc. Roy. Soc. Edinburgh* Sect. A **123** (1993), no. 2, 239–274), the singular part of $Df$ is a rank-one measure $D\_Sf$. This is true for every BV vector field. When ${\rm div}\,f$ is a.c. with respect to the Lebesgue measure, then the trace of $D\_Sf$ vanishes. However, I don't see what k... | 3 | https://mathoverflow.net/users/8799 | 327763 | 140,982 |
https://mathoverflow.net/questions/327714 | 2 | Let $\Phi$ be the unique solution of
$$\begin{cases}
\frac{d}{dt}\Phi(x,t) = f(\Phi(x,t),t) \quad t >0 \\
\Phi(x,0) = x \quad x \in \mathbb{R}^N
\end{cases}$$
where we have assumed $f$ smooth.
How do you prove that $\Phi(\cdot, t)$ is smooth and in particular that the following holds?
$$\begin{cases}
\frac{d}{dt} ... | https://mathoverflow.net/users/nan | Derivative and Jacobian determinant of solution of ODE | The smoothness of $\Phi$ usually is addressed in the textbooks on ODEs (some of them are discussed on [MO](https://mathoverflow.net/questions/81221/graduate-ode-textbook) and on [MSE](https://math.stackexchange.com/questions/175908/ode-book-recommendation) as well).
An application of the chain rule yields the equatio... | 2 | https://mathoverflow.net/users/44463 | 327773 | 140,985 |
https://mathoverflow.net/questions/292298 | 2 | For the minimal counter-example to union closed sets conjecture, we have the lower bound $\mid$$\mathcal{A}$$\mid$ $\geq$ $4q-1$ ($\mathcal{A}$ denotes the minimal counter-example family, $q$ denotes the number of elements in $\cup$$\mathcal{A}$). Is there any better lower bound? Is there any research/development happe... | https://mathoverflow.net/users/120490 | Inequality in Frankl's conjecture | The 2018 paper [A lower bound for the minimal counter-example to Frankl’s conjecture by Ankush Hore](https://ajc.maths.uq.edu.au/pdf/72/ajc_v72_p350.pdf) improved the bound to:
$$\mid\mathcal{A}\mid \geq 4q+1$$
| 4 | https://mathoverflow.net/users/136218 | 327782 | 140,987 |
https://mathoverflow.net/questions/326863 | 1 | In Mac Lane's Categories for the Working Mathematician, page 110, second edition, he states that, in the category of groups $Grp$, $F\_n$ being groups and $x\_n, y\_n$ being two cones on these groups (thus a family of group morphisms from X (resp. Y) to $F\_n$), these cones form a group by multiplication $x\_n y\_n$.
... | https://mathoverflow.net/users/29853 | Product of two group morphisms not a group morphism | From the comments:
The *cones* are in $\mathrm{Set}$, but the set of *cones*, under pointwise multiplication is the limit in $\mathrm{Grp}$. In fancier language, MacLane's saying that limits in $\mathrm{Grp}$ are created in $\mathrm{Set}$.
| 0 | https://mathoverflow.net/users/14094 | 327787 | 140,989 |
https://mathoverflow.net/questions/327776 | 1 | In our group we are working with a probability distribution $X$ defined on a non-negative domain, satisfying the following property
$$
P\left[X>a\right]\ge P\left[X>a+t \mid X>t\right],
$$
where $a,t\ge 0$. We noticed some interesting properties that follow, and wonder if the above property is known in the community. N... | https://mathoverflow.net/users/37757 | The weak version of the memoriless property | By googling
>
> "decreasing conditional survival" function
>
>
>
we find this term used e.g. in [this paper](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=2ahUKEwiMwLPCrcjhAhVLs6wKHVpiAiAQFjAAegQIABAB&url=http%3A%2F%2Fshodhganga.inflibnet.ac.in%2Fbitstream%2F10603%2F41218%... | 1 | https://mathoverflow.net/users/36721 | 327788 | 140,990 |
https://mathoverflow.net/questions/327791 | 3 | The usual definition of complete positivity (e.g. [Stinespring (1955)](https://www.ams.org/journals/proc/1955-006-02/S0002-9939-1955-0069403-4/home.html), or Holevo's Statistical Structure of Quantum Theory) is that a linear map between (sub $C^\*$ algebras of) the bounded operators on some Hilbert spaces $\phi:\mathca... | https://mathoverflow.net/users/130032 | Complete positivity with infinite dimensional auxillary spaces | You have to be a little careful because there may be different ways of tensoring with $B(K)$ when $K$ is infinite dimensional. But if you work with the spatial tensor product, say, then I think "extra complete positivity" is equivalent to ordinary complete positivity. This follows from Stinespring's theorem: if $\phi: ... | 3 | https://mathoverflow.net/users/23141 | 327796 | 140,992 |
https://mathoverflow.net/questions/327711 | 0 | Suppose I have an allergy to non-Hausdorff spaces but I really want to do, say, arithmetic geometry. I wonder if there is some perverse way I could develop scheme theory that would accomodate for my needs.
I am not sure what is the best way to formalize this question but here is an attempt. Does there exist a functo... | https://mathoverflow.net/users/nan | Doing scheme theory with Hausdorff spaces | It's not clear to me that any of these properties are even attained by a functor from schemes to *all* topological spaces. For instance, the functor sending a scheme to its underlying space isn't full or faithful, nor does it preserve limits (in particular, it doesn't preserve the terminal object or binary products) --... | 4 | https://mathoverflow.net/users/2362 | 327803 | 140,994 |
https://mathoverflow.net/questions/327781 | 4 | On page 3 of [this preprint](https://arxiv.org/pdf/1611.05928.pdf), after recalling the definition of flow generated by a vector field, the authors remark that "a necessary condition for a flow $\varphi\_t(\cdot)$ generated by $a(t, \cdot)$ to be measure preserving is: $\mathrm{div}\, a = 0$ in a suitable sense".
Ass... | https://mathoverflow.net/users/122620 | Prove that the flow of a divergence-free vector field is measure preserving | Let $\mu\_t = (\varphi\_t)\_{\sharp} \mu$ denote the image of the measure $\mu$ by the flow of $a$ (where $\mu$ can be Lebesgue measure).
It is well-known that the family of measures $\{\mu\_t\}\_{t\in \mathbb R}$ satisfies Liouville equation (aka continuity equation)
$$
\partial\_t \mu\_t + \operatorname{div\,} (a \mu... | 5 | https://mathoverflow.net/users/44463 | 327805 | 140,995 |
https://mathoverflow.net/questions/327802 | 6 | Fix a compact, symplectic four-manifold ($X$, $\omega$).
Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; \mathbb{Z})$ defined by weighted counts of pseudoholomorphic curves in $X$. In particular, this count combines curves of any genus.
On the other hand, the Gromov-Witten invariant... | https://mathoverflow.net/users/43158 | Relationship between Gromov-Witten and Taubes' Gromov invariant | Ionel and Parker worked out the relationship between Taubes' Gromov invariant and the usual Gromov--Witten invariants (which they refer to as "Ruan--Tian invariants") in this paper:
<https://arxiv.org/abs/alg-geom/9702008>
| 6 | https://mathoverflow.net/users/35353 | 327806 | 140,996 |
https://mathoverflow.net/questions/327659 | 7 | If $K$ is a p-adic field, with maximal unramified subfield $K\_0$, and $X$ is a proper semi-stable $O\_K$-scheme, then there's a canonical way to make the special fibre $X\_k$ into a log-scheme; and there's a theory of log-rigid cohomology adapted to such things, which gives $K\_0$-vector spaces $H^i\_{log-rig}(X\_k / ... | https://mathoverflow.net/users/2481 | Rigid versus log-rigid cohomology for semistable varieties | $\require{AMScd}$I'll expand a little on my comment to give an answer to David's follow up question:
Firstly, the general relationship is described in Chiarellotto's Duke 1999 paper *"Rigid cohomology and invariant cycles for a semistable log scheme"*. He shows that the weight-monodromy conjecture implies that there ... | 6 | https://mathoverflow.net/users/76409 | 327818 | 141,001 |
https://mathoverflow.net/questions/327716 | 9 | Let $X$ be a topological space (assume additional assumptions if needed) and denote by $\mathcal O \_X$ its sheaf of $\Bbbk$-valued continuous functions where $\Bbbk$ is $\mathbb{R}$ or $\mathbb{C}$ with standard topology.
Then, as it is done in the differentiable setting or in algebraic geometry, one can define the ... | https://mathoverflow.net/users/4721 | The (co)tangent sheaf of a topological space | Your $$ is always $0$. If $$ is a derivation and $$ is a function, then for every point $$ $$ vanishes at $$; it suffices to prove this when $()=0$, and in that case $=ℎ$ where $()=0=ℎ()$, so $=ℎ+ℎ$ vanishes at $$.
| 11 | https://mathoverflow.net/users/6666 | 327819 | 141,002 |
https://mathoverflow.net/questions/327698 | 2 | Let $u: \Omega\subset \mathbb{R} \to \mathbb{R}$ be a function of bounded variation.
**Question 1.**
How can we prove that the Hausdorff dimension of the [essential graph](https://mathoverflow.net/questions/327698/hausdorff-dimension-of-the-graph-of-a-bv-function-in-1-dimensional-setting?noredirect=1#comment818424... | https://mathoverflow.net/users/122620 | Hausdorff dimension of the graph of a BV function (in 1 dimensional setting) | I will assume $\Omega$ is an interval $[a,b]$, say. I assume this was intended as part of your question.
Let $\text{Var}f=M$. Let $r>0$. The function has at most countably many discontinuities, which are necessarily of jump type. The magnitude of the discontinuities sums to at most $M$. In particular, there are at m... | 2 | https://mathoverflow.net/users/11054 | 327820 | 141,003 |
https://mathoverflow.net/questions/327735 | 6 | I would appreciate if someone could point me to some introductory literature/resources where I can learn about **Poincaré's uniformization theorem** at a basic level.
Any good powerpoint notes, short papers or video lectures would be nice. I want to learn about the result in general, the proof, how it relates to othe... | https://mathoverflow.net/users/138205 | Reference request: uniformization theorem | On a basic level:
W. Abikoff, The uniformization theorem, Amer. Math. Monthly 88 (1981), no. 8, 574–592.
L. Ahlfors, Conformal invariants, last chapter.
S. Donaldson, Riemann surfaces, Oxford, 2011. Very nice. Modern.
R. Courant, Function theory (if you read German or Russian, this is the second part of the fam... | 9 | https://mathoverflow.net/users/25510 | 327824 | 141,005 |
https://mathoverflow.net/questions/327828 | 2 | Is there a ''nice'' classification of **even** nilpotent elements in $\mathfrak{sl}\_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $e$, such that for any (= every) $\mathfrak{sl}\_2$-triple $(e,f,h)$ in $\mathfrak{sl}\_n$, $ad(h)$ acts only with even ... | https://mathoverflow.net/users/114985 | Characterisation of even nilpotent elements in $\mathfrak{sl}_n$ | Per Collingwood-McGovern ([1993](https://mathscinet.ams.org/mathscinet-getitem?mr=1251060), Corollary 3.8.8) an element is even iff all labels on its weighted Dynkin diagram are 0 or 2. For partition $[d\_1,\dots,d\_k,0,\dots,0]$ these labels are the $h\_i-h\_{i+1}$, where $h\_1\geqslant h\_2\geqslant\dots\geqslant h\_... | 4 | https://mathoverflow.net/users/19276 | 327831 | 141,009 |
https://mathoverflow.net/questions/327774 | 10 | We have $N$ students and $n$ exams. We need to select $n$ out of the students using the grade of those exams. The procedure is as follows:
1- We set some ordering on the exams.
2- Going through this order, In every exam, the best student will be selected and we forget about his/her grade in the next exams.
Now we... | https://mathoverflow.net/users/51778 | Lower bound for a combinatorial problem ($N$ students taking $n$ exams) | **My initial argument was erroneous. I'm rewriting everything completely.**
Denote by $f(n)$ the maximal value of $N$ for that value of $n$. I claim that $$
f(k+\ell)\geq f(k)+f(\ell)+k \qquad\text{for all $k\leq \ell$.}
\qquad\qquad(\*)
$$
In a standard way, this shows that $f(n)\geq 1+\frac12n\log\_2 n$.
To pro... | 3 | https://mathoverflow.net/users/17581 | 327834 | 141,010 |
https://mathoverflow.net/questions/327654 | 3 | Let $G$ be a simple compact connected Lie group and let $K$ be a connected closed subgroup of $G$.
Let $\widehat G$ and $\widehat K$ denote the corresponding unitary duals, that is, the (equivalence classes of) irreducible representations.
By the Highest Weight Theorem, elements in $\widehat G$ are in correspondence w... | https://mathoverflow.net/users/20052 | Distinguished dominant integral weight related to a branching problem | It is true that there exists such a $\omega$. Indeed, $every$ $\omega \neq 0$ works. Fix a representation $\pi \_{\omega}$ which has a nonzero $K$ fixed vector, with $\omega \neq 0$. One can replace $G,K$ by their complexifications. Fix a Borel subgroup $B$ of $G(\mathbb C)$ and view the vectors in the representations ... | 1 | https://mathoverflow.net/users/23291 | 327850 | 141,015 |
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