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/416497 | 6 | I was recently trying to find a numerical solution to a thermodynamics problem and the expression $x\ln x$ appeared in one of the computations. I did not have to find its value very near $0$, so the computer managed fine, but it got me thinking - can one make a stable numerical algorithm to compute $x\ln x$ for values ... | https://mathoverflow.net/users/114143 | How to numerically compute $x \ln x$ and related functions near $0$? | Modern mathematics libraries should be able to find $\log x$ precisely for all floating-point numbers, as the algorithms for doing that have long been known and adopted. My experiments on a fairly recent Intel chip with gnu mathematics library and gcc 10 compiler confirm that.
Multiplication is even more definite: co... | 8 | https://mathoverflow.net/users/9025 | 416518 | 169,729 |
https://mathoverflow.net/questions/416464 | 2 | In all introductory texts I'm aware of, convex polytopes are dealt with strictly within $\mathbb R^n$. Being used to quite larger levels of generality elsewhere, I'm wondering if the same can be done here. Specifically, I'm looking for a generalization of convex polytopes in other spaces such that their face lattices s... | https://mathoverflow.net/users/147705 | Convex polytopes in other spaces | Some time ago I was having the same question and I found these two:
* Robert Morelli *"[A theory of polyhedra](https://doi.org/10.1006/aima.1993.1001)"* (Adv. Math. 97, No. 1, 1-73 (1993)). He works in modules over ordered rings (and vector spaces over ordered fields). This was very helpful personally to understand v... | 1 | https://mathoverflow.net/users/109085 | 416519 | 169,730 |
https://mathoverflow.net/questions/416517 | 2 | Let $M\_n$ denote the set of all $n\times n$ complex matrices. Let $1\leq p<\infty.$ For $A\in M\_n$ define $\|A\|\_p:=(Tr(A^\*A)^{p/2})^{1/p}$ where $Tr$ denotes the usual trace of a matrix. Then $\|.\|\_p$ is indeed a norm. We write $S\_p^n$ to be the Banach space whose vector space is $M\_n$ and norm is given by $\|... | https://mathoverflow.net/users/136860 | Banach-Mazur distance between Schatten-$p$ classes | Same order as $d(\ell\_p^n, \ell\_q^n)$. It is $n^{1/p-1/q}$ when they are on the same side of 2, and $n^{(1/2-1/q)\vee (1/p-1/2)}$ for $1\le p<2<q\le\infty$.
See Section 45 of [Tomczak-Jaegermann's book Banach Mazur distances and Finite dimensional Operator ideals](https://sites.ualberta.ca/%7Entj/bm_book/index.html... | 4 | https://mathoverflow.net/users/3675 | 416525 | 169,731 |
https://mathoverflow.net/questions/416540 | 1 | Let $V = (v\_1,...,v\_n)$ be a set of vertices for a directed graph. We denote an edge between the two vertices $v\_i$ and $v\_j$ by $(i,j)$. The edge set $E$ for a graph $(V,E)$ is then a set of tuples $E \in P(\{(i,j) \mid i,j \in \{1,...,n\}\})$.
My Question is if for fixed $n \in \mathbb{N}$ we can efficiently de... | https://mathoverflow.net/users/409412 | Is there a permutation invariant for graphs? | You are essentially asking about the graph isomorphism problem. Or, more precisely, you are asking about canonical representation of graphs. There is a huge literature about this. See <https://en.wikipedia.org/wiki/Graph_canonization>.
| 4 | https://mathoverflow.net/users/20598 | 416541 | 169,734 |
https://mathoverflow.net/questions/416547 | 2 | Does there exist any C\*-algebra $(A,\|\cdot\|)$ enjoying the following property?
$\bullet$ There exists a norm $|\cdot|$ on $A$ with $\|\cdot\|\leq|\cdot|$ such that $(A,|\cdot|)$ is a pre C\*-algebra (necessarily non-complete).
| https://mathoverflow.net/users/84390 | A C*-algebra enjoying some different C*-norms | No that's not possible (except the trivial case). Any $\*$-homomorphism between $C^\*$-algebras is automatically contractive, and if it is injective then it is isometric. You can apply this to the identity map of $A$ seens as a map from $(A,\Vert \cdot \Vert)$ to the completion of $(A,\vert \cdot \vert)$ and conclude t... | 10 | https://mathoverflow.net/users/22131 | 416550 | 169,736 |
https://mathoverflow.net/questions/416548 | 3 | Let $ f \colon E \to B $ be a morphism between (weak) homotopy types such that each (homotopy) fiber has a homotopy type of some finite CW complex.
Then does $ \Sigma f \colon \Sigma E \to \Sigma B $ have the same property?
| https://mathoverflow.net/users/119076 | Is the suspension of a finite fibration again finite? | Assuming you work with unpointed spaces (but the example can easily be adapted to the pointed case) the map $1 \to 2$ gives a counterexample : its fiber are $1$ and $\varnothing$ so they are both finite, but on applying the suspension you get $1 \to S^1$ whose homotopy fibers identitifes with $\mathbb{Z}$.
| 8 | https://mathoverflow.net/users/22131 | 416551 | 169,737 |
https://mathoverflow.net/questions/414575 | 1 | Intro
-----
I'm referring to the original paper [Fast unfolding of communities in large
networks](https://perso.uclouvain.be/vincent.blondel/publications/08BG.pdf) by Blondel et al. in this question and adopt their notation. Explanation for the used symbols is on page 4 below equation (2).
I'm wondering why Blondel... | https://mathoverflow.net/users/475789 | Louvain method: Why do they drop coefficient 1/m in the official implementation? | The function is used when they decide whether the node should stay in the community or not. Therefore, they only need to compare quality value differences and have no need for absolute numbers. The division factor is constant during the computation and therefore can be omitted since it does not affect the result of the... | 2 | https://mathoverflow.net/users/477406 | 416566 | 169,744 |
https://mathoverflow.net/questions/416565 | 3 | ### Situation
Suppose that we have
* a commutative ring (or an $E\_{\infty}$-ring) $R$ and
* a homotopy type $X$.
Then we get a canonical morphism
$$
f \colon K(R ^ {\Sigma ^ \infty X\_+}) \to K(R) ^ {\Sigma ^ \infty X\_+}
$$
where $K$ denotes the **nonconnective** algebraic $K$-theory.
### Question
Is $f$ an... | https://mathoverflow.net/users/119076 | (Nonconnective algebraic) K-theory cohomology = K-theory of cohomology? | The answer is typically going to be no.
Suppose for example that $R$ is discrete. Then $R^{\Sigma^\infty\_+X}= C^\*(X;R)$ is the usual cochain algebra, and perfect modules over that algebra are equivalent (via the homotopy fixed points functor) to the thick full stable subcategory of $Fun(X, Perf(R))$ generated by $R... | 4 | https://mathoverflow.net/users/102343 | 416573 | 169,746 |
https://mathoverflow.net/questions/416561 | 3 | This question was originally at Math Stackexchange, but had no answers:
<https://math.stackexchange.com/questions/4348707/is-projection-of-locally-connected-compact-subset-locally-connected>
### Problem
Let $(X, \mathcal{T}\_X)$ and $(Y, \mathcal{T}\_Y)$ be topological spaces, $Z = X \times Y$, $\mathcal{T}\_Z$ b... | https://mathoverflow.net/users/32487 | Is projection of locally connected compact subset locally connected? | A counterexample to this question can be constructed as follows.
Let $X=[0,1]$ be the closed interval with the standard Euclidean topology.
Let $Y=\omega$ and $\mathcal T\_Y$ be the topology on $Y$ consisting of the sets $W\subseteq Y$ satisfying two conditions:
$\bullet$ if $0\in W$, then $W=\omega$;
$\bullet$... | 3 | https://mathoverflow.net/users/61536 | 416575 | 169,747 |
https://mathoverflow.net/questions/416574 | 0 | I am looking for the best way to approximate $\exp(x)$ on a finite domain $[0,M]$ with a piecewise-linear function. My initial approach is to take $K$ evenly-spaced segments from $0$ to $M$. For each segment $[s\_k, s\_{k+1}]$, I add the line from point $(s\_k, \exp(s\_k))$ to point $(s\_{k+1}, \exp(s\_{k+1}))$.
But ... | https://mathoverflow.net/users/476491 | Approximating exp(x) with a piecewise-linear function accurately | For $b$ not much larger than $a$, the maximum difference between $e^x$ and the line from $(a,e^a)$ to $(b,e^b)$, for $x\in[a,b]$, is approximately $\frac 18 (b-a)^2 e^a$.
So you should choose your points so that $(b-a)^2 e^a$ is approximately constant for each interval. After selecting the first interval, use this ru... | 2 | https://mathoverflow.net/users/9025 | 416582 | 169,750 |
https://mathoverflow.net/questions/354899 | 7 | For this question a manifold is a locally upper-Euclidean Hausdorff space; paracompactness or second-countability is not assumed, and boundary may be present.
Let $M$ be a manifold. What are sufficient conditions for there to exist open $U \subset M$ such that $\partial M \subset U$ and $U$ (strongly?) deformation re... | https://mathoverflow.net/users/32487 | When is a manifold boundary a deformation retract of its open neighborhood? | The following theorem and example give (partial) answers. By an *$n$-manifold* we understand a Hausdorff space whose any point has a neighborhood, homeomorphic to an open subspace of $\mathbb R^n\_+=\{(x\_1,\dots,x\_n)\in\mathbb R^n:x\_n\ge 0\}$.
**Theorem.** For each 1-manifold $M$ the boundary $\partial M$ is a def... | 3 | https://mathoverflow.net/users/61536 | 416586 | 169,752 |
https://mathoverflow.net/questions/416474 | 0 | In the case of a square lattice, the exact number of points within a circle of radius r centered in the center is (see: [http://mathworld.wolfram.com/GausssCircleProblem.html](https://mathworld.wolfram.com/GausssCircleProblem.html):
$$N(r)=1+4Floor(r)+4 \sum\_{x=1}^{Floor(r)}{Floor(\sqrt{r^2−x^2)}}$$
And in the case of... | https://mathoverflow.net/users/97558 | The exact number of points within a circle of radius r centered on a lattice point in a hexagonal lattice? Review expression Gauss circle problem | By identifying the lattice points with numbers of the form $x - y\omega$, $\omega = e^{2\pi i / 3}$, $x, y \in \mathbb{Z}$, we find that we want to count Diophantine solutions to $x^2 + xy + y^2 \le r^2$. From $$\sum\_{m, n \in \mathbb{Z}} q^{m^2 + mn + n^2} = 1 + 6 \sum\_{n \ge 0} \left(\frac{q^{3n+1}}{1 - q^{3n+1}} -... | 5 | https://mathoverflow.net/users/46140 | 416587 | 169,753 |
https://mathoverflow.net/questions/416585 | 2 | Consider the circle $\mathbb{T}^1= \frac{\mathbb{R}}{\mathbb{Z}}$. We represent it as a union of disjoint subsegments $M\_j=[t\_j,t\_{j+1})$, $j = 0, \cdots, n$, $t\_0=t\_n$ and define the map $S$ by the formula
\begin{align}
S(t)=t+c\_j \quad \text{mod 1}, \quad t\in M\_j
\end{align}
Here $c\_j$ are real values. Such ... | https://mathoverflow.net/users/172647 | A lemma in approximating sequences | Let me answer your question.
The cxdimension of the space $L$ is finite (this is a subspace of ${\mathbb R}^N$ for some $N$.
Let us assume for simplicity that $codim L=1$ i.e. there is only one equation (\*). Then one of coordinates (let it be $c\_{n}$) can be expressed
as
$$c\_n=a\_1 t\_1+\ldots+ a\_n t\_n+b\_1 c\... | 2 | https://mathoverflow.net/users/477424 | 416591 | 169,756 |
https://mathoverflow.net/questions/416042 | 2 | Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as
* $\operatorname{Rep}(G)$ i.e. the category of representations for a finite group $G$; I've also seen references to things like $\operatorname{... | https://mathoverflow.net/users/135817 | Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request] | In general you should not expect to have explicit formulas for the decomposition of tensor products. The situation is a bit better for Lie algebras than for finite groups. As Ehud Meir pointed out, even calculating the Kronecker coefficients for symmetric groups
<https://en.wikipedia.org/wiki/Kronecker_coefficient>
... | 3 | https://mathoverflow.net/users/129060 | 416600 | 169,758 |
https://mathoverflow.net/questions/416596 | 0 | *[Repost from MSE.](https://math.stackexchange.com/questions/4374009/a-series-expansion-of-a-function-at-a-non-differentiable-point)*
The function $x\mapsto (\frac{x}{1+x})^{x+x^2}$ is not differentiable in $0$, but nevertheless I'm interested in a series expansion at $0$. [Wolfram Alpha](https://www.wolframalpha.com... | https://mathoverflow.net/users/78650 | A series expansion of a function at a non-differentiable point | Let $$f(x)=\left(\frac{x}{1+x}\right)^{x+x^2}.$$
Expand first
$$g(x)=\log f(x)=(x+x^2)(\log x-\log(1+x))=x\log x+x^2\log x-x^2-\frac{1}{2}x^3-\ldots.$$
This has only two terms with $\log$, the rest are powers.
So powers of $g$ have the same property: they contain powers of $\log$ and ordinary powers. And then use
$$f=e... | 5 | https://mathoverflow.net/users/25510 | 416609 | 169,759 |
https://mathoverflow.net/questions/416608 | 8 | $\newcommand{\KL}{\operatorname{KL}}$Let $X$ be a Polish metric space and $P(X)$ the space of probability measures on $X$. Given $\mu, \nu\in P(X)$, recall that
$$\KL(\mu\parallel\nu) = \begin{cases}\mathbb E\_\mu[\log\tfrac{d\mu}{d\nu}]&\text{if $\mu\ll\nu$;}\\+\infty&\text{otherwise.}\end{cases}$$
I know that both $\... | https://mathoverflow.net/users/475450 | Is the square root of the Kullback-Leibler divergence a convex map? | $\newcommand\de\delta\newcommand{\KL}{\operatorname{KL}}\newcommand{\p}{\,\|\,}$The maps
$$\mu\mapsto\sqrt{\KL(\mu\p\nu)}$$
and
$$\nu\mapsto\sqrt{\KL(\mu\p\nu)}$$
are not convex in general.
Indeed, let $\mu\_p:=p\de\_0+(1-p)\de\_1$, where $p\in(0,1)$ and $\de\_a$ is the Dirac measure supported on $\{a\}$.
Then the ... | 10 | https://mathoverflow.net/users/36721 | 416612 | 169,760 |
https://mathoverflow.net/questions/416611 | 6 | I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.
Let
$$
T^n \to M \to T^m
$$
be a principal torus bundle over a torus. Then $ M $ is a solvmanifold, even a nilmanifold (in fact $ M $ is the ... | https://mathoverflow.net/users/387190 | Torus bundles and compact solvmanifolds | A group $G$ is isomorphic to the fundamental groups of a compact solvmanifold if and only if it fits into the short exact sequence $1\to N\to G\to\mathbb Z^n\to 1$ where $N$ is a finitely generated torsion-free nilpotent group.
This is stated on p.253 and explained is chapter III of Auslander's [An exposition of the st... | 7 | https://mathoverflow.net/users/1573 | 416616 | 169,763 |
https://mathoverflow.net/questions/416598 | 25 | What is known about the existence of other pairs of spheres (such as $S^2$ and $S^3$) whose homotopy groups coincide starting from some index.
A sufficient condition for this is the existence of a fiber bundle $S^m \to S^n$ with fiber having a finite number of nonzero homotopy groups (as in the case of the Hopf fibra... | https://mathoverflow.net/users/148161 | Spheres with the same homotopy groups | It is a result from Serre's thesis that for $n\geq 3$ and a prime $p$, the first $p$-torsion in $\pi\_\*S^n$ occurs precisely for $\* = n+2p-3$. This shows that $(m,n) = (2,3)$ is the only pair of (edit: positive integers) $m<n$ with $\pi\_\*S^m \cong \pi\_\*S^n$ for $\*$ large enough.
| 51 | https://mathoverflow.net/users/2039 | 416618 | 169,764 |
https://mathoverflow.net/questions/416624 | 5 | Let $(M, g\_M)$ where $M= B \times\_f F$ and $g\_M=g\_B + f^2g\_F$, an Einstein warped product manifold (i.e., $Ric\_M= \lambda g\_M$), with Ricci flat fiber-manifold $F$, i.e., $Ric\_F=0$.
Then $M$ can admit only constant negative Ricci curvature or zero Ricci curvature (i e., $\lambda \le 0$) or $M$ could also have p... | https://mathoverflow.net/users/111304 | Possible sign of scalar curvature for Einstein warped product manifold with Ricci-flat | It can not have constant positive Ricci curvature. By Bonnet-Myers constant positive Ricci curvature implies that $M$ is compact.
If $V$ is a vertical vector then by the formula for Ricci curvature of warped product (page 266 in Besse's book)
$$
Ric(V,V)=Ric\_F(V,V) -|V|^2(\frac{\Delta f}{f}+(p-1)\frac{|\nabla f|^2... | 4 | https://mathoverflow.net/users/18050 | 416629 | 169,767 |
https://mathoverflow.net/questions/416627 | 9 | For $S \subset \mathbb{N}$ define $S-S=\{x-y:x \in S, y \in S\}$.
As noted below there is a simple example showing that a set $S \subset \mathbb{N}$ with positive upper density has a sumset $S+S=\{x+y:x \in S, y \in S\}$ with $S+S$ containing only finite length arithmetic progressions. However the case for the differ... | https://mathoverflow.net/users/7113 | A set with positive upper density whose difference set does not contain an infinite arithmetic progression | Let $\langle x\rangle$ denote the fractional part of a real number $x$ (i.e. $\langle x \rangle := x- \lfloor x\rfloor $, where $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$).
Let $\alpha \in \mathbb R$ be irrational and let $S:=\{n\in \mathbb Z: \langle n\alpha \rangle \in (0,1/4)\}$. The upp... | 13 | https://mathoverflow.net/users/10457 | 416632 | 169,769 |
https://mathoverflow.net/questions/416630 | 1 | Can I calculate the volume of a frustum if all I know is the volume of the pyramid the height of the pyramid and the height of the frustum?
| https://mathoverflow.net/users/477453 | Volume of a frustum knowing the volume and height of the pyramid and the height of the frustum | Let the pyramid $P$ have volume $V\_P$ and height $H$. Let the frustum $F$ that is a sliced part of this pyramid have height $h$, so $0 \leq h \leq H$. You want to know the volume of the frustum. Call it $V\_F$.
By calculus, $V\_P = (1/3)AH$, where $A$ is the area of the base of the pyramid. The part of the pyramid a... | 3 | https://mathoverflow.net/users/3272 | 416633 | 169,770 |
https://mathoverflow.net/questions/416628 | 3 | Let $X$ be a scheme and $\mathcal{F}$ a presheaf on $X\_{ét}$.
For each $x\_{i}\in X$, pick a geometric point $\bar{x}\_{i}$ over $x$ and denote by $i\_{\bar{x}\_{i}}:\text{Spec}(k\_{i})\_{\text{ét}}\rightarrow X\_{ét}$ the morphism of sites induced by the geometric point $\bar{x}\_{i}$ where $k\_{i}$ is algebraicall... | https://mathoverflow.net/users/211978 | Sheafifcation for the étale site | It's not diagonal. A section of $\mathcal F$ on $U$ has a germ at each element of $\text{Hom}\_{X}(\bar{x},U)$, and you take the product of all those germs, which may be different.
It's not so hard to see that if you tried to do it diagonally, it wouldn't be functorial for automorphisms of $U$, say in the case when $... | 4 | https://mathoverflow.net/users/18060 | 416637 | 169,773 |
https://mathoverflow.net/questions/415718 | 3 | Is there any standard name for semigroups $S$ in which $xS=Sx$ for all $x\in S$?
Examples of such semigroups are commutative semigroups and Clifford inverse semigroups.
| https://mathoverflow.net/users/61536 | A name for semigroups in which left and right principal ideals coincide | People in factorization theory call a monoid $H$ *normalizing* if $aH = Ha$ for every $a \in H$; see, e.g.,
>
> A. Geroldinger, *Non-commutative Krull monoids: A divisor-theoretic approach and their arithmetic*, Osaka J. Math. 50 (2013), 503-539.
>
>
>
However, it makes a lot of sense to refer to the same obje... | 2 | https://mathoverflow.net/users/16537 | 416640 | 169,774 |
https://mathoverflow.net/questions/416473 | 5 | Let $X$ be a smooth variety over a characteristic zero field $k$. It seems to be well-known that
"A coherent $\mathcal{D}\_X$-module is holonomic if and only if 'every element is annihilated by a nonzero differential operator'."
I know that, when $X=\mathbb{A}^1$, this is true in the sense that "a finitely generate... | https://mathoverflow.net/users/131975 | Holonomic = annihilated by some differential operator | "A coherent $D\_X$-module is holonomic if and only if 'every element is annihilated by a nonzero differential operator'."
This is very much not true in dim>1.
For example, let $D=k[x\_1,x\_2]\langle \partial\_1,\partial\_2\rangle$ be the ring of differential operators on $\mathbb A^2$, and consider the left $D$-mod... | 8 | https://mathoverflow.net/users/7762 | 416643 | 169,777 |
https://mathoverflow.net/questions/416644 | 4 | Let $X \subseteq \mathbb{C}^n$ be an irreducible algebraic set that forms a cone, and let $I=I(X) \subseteq \mathbb{C}[x\_1,...,x\_n]$. Let $m < n$ and $k\leq m$ be positive integers. Is it true that for a generic collection of elements $v\_1,..., v\_k \in X$ and $v\_{k+1},..., v\_m \in \mathbb{C}^n$, it holds that $I ... | https://mathoverflow.net/users/150898 | Is the sum of a radical ideal and the ideal of a generic linear space intersecting that ideal radical? | In general no. I'm only familiar with this question with $k=0$. Here are two basic counter-examples (both with $k=0$)
(1) Take $X$ singular and dimension $n-m$, so we are intersecting $X$ with a generic codimension $n-m$-plane. Then the intersection is a point of multiplicity equal to the multiplicity of $0$ as a sin... | 5 | https://mathoverflow.net/users/297 | 416660 | 169,779 |
https://mathoverflow.net/questions/416668 | -3 | We have two particles $A$ and $B$ in a maximally entangled state $|\Psi\rangle \in \cal{H}\_A \times \cal{H}\_B$
$$
\left|\Psi\right\rangle = \frac{1}{\sqrt{2}} ( \left| 0
\right\rangle\_A\otimes \left| 0 \right\rangle\_B + \left| 1
\right\rangle\_A\otimes \left| 1 \right\rangle\_B ),
$$
where $\_A\left\langle i | j ... | https://mathoverflow.net/users/477478 | SU(2) and entangled particles | This is presumably simply an issue of notation -
\begin{eqnarray}
U\otimes I
\left|\Psi\right\rangle = \frac{1}{\sqrt{2}} \left( (U\_{11} \left| 0 \right\rangle\_A + U\_{21} \left| 1 \right\rangle\_A ) \otimes \left| 0 \right\rangle\_B \\
\hspace{3cm} + ( U\_{12} \left| 0 \right\rangle\_A + U\_{22} \left| 1
\right\ran... | 0 | https://mathoverflow.net/users/134299 | 416670 | 169,783 |
https://mathoverflow.net/questions/416653 | 1 | For a bounded smooth domain $\Omega$, let $H\_2^{s}(\Omega)$ be the usual Sobolev space on $\Omega$.
Define $A:=\{f\in H\_2^{s}(\Omega)| \lVert f\rVert\_{L^p(\Omega)}=1\}$ where $2<p<2\_{s}^\*$.
Can we show $A$ is a $C^2$-Hilbert manifold since I need to use the Morse lemma?
| https://mathoverflow.net/users/166368 | Is an $L^p$-sphere in Sobolev space $H_2^{s}(\Omega)$ a Hilbert manifold? | The norm map
$$
f\mapsto \int\_{\Omega} f^p dvol
$$
is a $C^k$-map for $k<p$ and in fact smooth if $p$ is an even integer as is shown in [this paper](http://www.jstor.org/stable/24901438.) of Bonic and Frampton. The implicit function theorem for maps of Banach spaces then applies to give your desired result.
| 3 | https://mathoverflow.net/users/12605 | 416676 | 169,785 |
https://mathoverflow.net/questions/416659 | 16 | Is it possible to find the determinant of an $n\times n$- matrix, only given the determinant of all $p\times p$ sub-matrices in it? Here $p\leq n$ is fixed. This is obviously true if $p=1,n$. But what happens in other cases?
| https://mathoverflow.net/users/44215 | Find the determinant of a matrix given the determinant of all $p\times p$ sub-matrices? | As [mentioned](https://mathoverflow.net/questions/416659/find-the-determinant-of-a-matrix-given-the-determinant-of-all-p-times-p-sub-ma#comment1069136_416659) by Will Sawin, a necessary condition is that $p$ divides $n$. Thus let us assume that $n=pk$. Denoting $e\_1,\dotsc,e\_n$ the canonical basis, the knowledge of t... | 20 | https://mathoverflow.net/users/8799 | 416677 | 169,786 |
https://mathoverflow.net/questions/408166 | 7 | Federer and Fleming's notion of "currents" is well established so far, and starting from the seminal work of Ambrosio and Kirchheim, the notion of metric currents is well studied also. The main underlying idea of Federer and Flaming's notion of Euclidean current generalizes to metric space thanks to the fact that one f... | https://mathoverflow.net/users/170982 | Currents in sub-Riemannian geometry | I want to try to give you an answer, although it will be very sketchy. Take into account that the topic is still unclear to the experts themselves.
The key word is “Rumin currents”: they are defined on all Heisenberg groups (and contact manifolds) starting with a pre-selection of differential forms. So, instead of ta... | 5 | https://mathoverflow.net/users/97130 | 416680 | 169,787 |
https://mathoverflow.net/questions/416678 | 1 | This question is motivated by [Linear Feedback Shift Registers](https://en.wikipedia.org/wiki/Linear-feedback_shift_register), which cycle through $\{0,1\}^n \setminus \{(0,\ldots,0)\}$ by shifting and applying a small set of XOR operations.
Let $n>1$ be an integer. To any map $f:\{0,1\}^{n} \to \{0,1\}$ we associate... | https://mathoverflow.net/users/8628 | Cycling through $\{0,1\}^n$ by shifting and applying a $n$-ary function | Such $f$ exist for all $n.$
The resulting sequences are de Bruijn sequences. Let $a=(a₀, a₁,\ldots )$ be a periodic sequence of period
$T$ with symbols taken from a finite alphabet $A.$
**Definition** The sequence $a$ is a de Bruijn sequence of span $n$ if every block of length $n$
occurs exactly once in (each peri... | 4 | https://mathoverflow.net/users/17773 | 416683 | 169,788 |
https://mathoverflow.net/questions/416684 | 7 | Let $(M^n,g)$ be a smooth Riemannian manifold. It is well known that if $sec(M)\geq \kappa$ then $Ric(M)\geq (n-1)\kappa$.
If I understand correctly in dimensions $n\geq 4$ a lower bound on $Ric(M)$ does not imply a lower bound on $sec(M)$. However in $n=2$ this implication is true for trivial reason.
**Is it true ... | https://mathoverflow.net/users/16183 | Relation between Ricci curvature and sectional curvature for 3-manifolds | This is definitely false. In dimension 3 if $\lambda\_1,\lambda\_2,\lambda\_3$ are eigenvalues of the curvature operator then Ricci curvatures of eigenvectors are $\lambda\_1+\lambda\_2, \lambda\_1+\lambda\_3, \lambda\_2+\lambda\_3$. If one of $\lambda\_i$'s is very negative but the other two are very positive then all... | 11 | https://mathoverflow.net/users/18050 | 416687 | 169,789 |
https://mathoverflow.net/questions/156257 | 10 | Let $K$ be a $p$-adic field and $\Omega^{(n)}\_K$ the $n$-dimensional Drinfeld upper half space over $K$ (which is a rigid analytic space over $K$).
>
> Is the Picard group of $\Omega^{(n)}\_K$ known? More generally, I would like to know
> the Picard group of $\Omega^{(n)}\_K$ base changed to any finite extension ... | https://mathoverflow.net/users/519 | Picard group of Drinfeld upper half space | In [this](https://arxiv.org/abs/2012.12729) paper of Junger, all these Picard groups are shown to be zero.
I think this was known for a long time for $n=1$, see for example the book *Rigid Analytic Geometry and its Applications* by Fresnel and van der Put.
| 2 | https://mathoverflow.net/users/144285 | 416688 | 169,790 |
https://mathoverflow.net/questions/416686 | 0 | Update: Thanks to GJC20's answer on the existence and uniqueness. Let me reformulate my questions 3/4 as follows: There exists a unique non-increasing and continuously differentiable function $f:\mathbb R \to [0,1]$ s.t.
$$X\_t=X\_0+W\_t-1+f(t) \mbox{ and } f(t)= \mathbb E \left[ \exp\left(-\frac 1 \epsilon \int\_0^t... | https://mathoverflow.net/users/nan | Has this "stochastic differential equation" been studied? | Updates : This is an answer to the simplest question above, existence and uniqueness.
If the existence holds, then any solution must be a strong solution. Let $X, Y$ be two arbitrary solution, it appears that $|X\_t-Y\_t|$ is deterministic and satisfies further
$$\mathbb E|X\_t-Y\_t| = |X\_t-Y\_t| \le \mathbb E\lef... | 1 | https://mathoverflow.net/users/261243 | 416698 | 169,794 |
https://mathoverflow.net/questions/416691 | 7 | Bases of a topological space in point set topology will in general form a coverage on its category of inclusion on open subsets and on its category of inclusion on basic opens, but it takes a bit more work to check whether either forms a Grothendieck pretopology. Is there a useful or natural criterion for when a (point... | https://mathoverflow.net/users/174368 | When is a basis of a topological space a Grothendieck pretopology? | This is a matter of expanding the definition, in this case Definition II.1.3 in SGA 4, which defines pretopologies.
By a “base” in this answer I mean what appears to be the most common definition: a collection of subsets of a fixed set $A$ such that any finite intersection of elements in the base is a union of elemen... | 11 | https://mathoverflow.net/users/402 | 416706 | 169,796 |
https://mathoverflow.net/questions/416709 | 1 | This is a [cross-post from MSE.](https://math.stackexchange.com/questions/4376145/affine-semigroup-generating-a-lattice)
Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that $S$ generates $N$ as a grou... | https://mathoverflow.net/users/143549 | Affine semigroup generating a lattice | (Write $\mathbf{N}=\{0,1,2,\dots\}$.)
What about the submonoid of $\mathbf{Z}^2$ generated by $\{(2,0),(3,0),(0,1)\}$? It equals $\mathbf{N}^2\smallsetminus (\{1\}\times\mathbf{N}$). So here the complement is infinite.
| 3 | https://mathoverflow.net/users/14094 | 416712 | 169,797 |
https://mathoverflow.net/questions/416704 | 3 | $\newcommand{\R}{\mathbb{R}}$I have a map $f\in C^0(X,\mathbb{R})$, where $X$ is a compact and Hausdorff topological space, which is a manifold outside of a compact subset $K\subset X$.
I would like to modify $f$, to obtain $\tilde f\in C^0(X,\mathbb{R})$ such that
1. $\tilde f\rvert\_K = f\rvert\_K$ and
2. $\tilde... | https://mathoverflow.net/users/99042 | Smoothing a map $f:X\to \mathbb{R}$ while fixing it over a closed $C\subset X$ | I claim that it is always possible (as long as one does not require the derivatives of $\tilde{f}\_{X\setminus K}$ to be bounded). This is actually an easy consequence of the approximation theorem of manifolds.
To state the approximation theorem of manifolds, suppose that
$0\leq s<\infty$, and $M,N$ are manifolds. Le... | 5 | https://mathoverflow.net/users/22277 | 416718 | 169,799 |
https://mathoverflow.net/questions/416707 | 4 | Let $n>1$ be an integer. For $a,b\in \{0,1\}^n$ let $d\_h(a, b)$ denote the [Hamming distance](https://en.wikipedia.org/wiki/Hamming_distance) of $a$ and $b$. For $k\in \{1,\ldots,n-1\}$ let $H(n,k)$ be the graph on $\{0,1\}^n$ given by the edge set $$E(n,k) = \{(a, b)\in (\{0,1\}^n)^2 : d\_h(a, b) = k\}.$$
**Questio... | https://mathoverflow.net/users/8628 | Hamilton cycles in $\{0,1\}^n$ with fixed Hamming distance | I believe these are exactly odd $k$.
Indeed, it can be easily seen that if $k$ is even, then a cycle starting at $0^n$ can visit only those vectors that have even number of $1$'s, and so it cannot be Hamiltonian.
On the other hand, there exist [long-run Gray codes](https://www.combinatorics.org/ojs/index.php/eljc/a... | 4 | https://mathoverflow.net/users/7076 | 416720 | 169,800 |
https://mathoverflow.net/questions/416681 | 3 | Let $X$ be a smooth projective variety of dimension $2n+1$, let $i\colon Y\subset X$ be an ample hypersurface, by Lefschetz hyperplane theorem, the pullback $i^\*\colon H^{2n}(X,\mathbb{Z})\to H^{2n}(Y,\mathbb{Z})$ is an injection.
When $X$ is a product of two projective spaces or more general (e.g., projective bundl... | https://mathoverflow.net/users/nan | Primitivity of the Lefschetz embedding | This is actually a general consequence of the Lefschetz hyperplane theorem and the universal coefficient theorem.
For simplicity, write $A\_{\operatorname{tf}} = A/A\_{\operatorname{tors}}$. Note that this association is functorial, additive, and preserves injections and surjections, but is not exact in the middle! I... | 1 | https://mathoverflow.net/users/82179 | 416721 | 169,801 |
https://mathoverflow.net/questions/416708 | 7 | I am cross-posting this question from math stack exchange (link below) as it has not received any comments or answers over the past month. A *regular epimorphism* is a morphism $f: X \to Y$ that is the coequalizer of some parallel pair $a,b: Z \to X$. I believe regular epimorphisms in the category of topological groups... | https://mathoverflow.net/users/46484 | Regular epimorphisms in the category of topological rings | It is true and hods more generlaly for models of any algebraic theory with a Mal'tsev operation, i.e. a trinary operation $f$ such that $f(x,x,y)=y=f(y,z,z)$. For groups, you can take $f(x,y,z)=xy^{-1}z$. This is Theorem 10 in A.I. Mal'tsev, On the general theory of algebraic systems, Mat. Sb. (N.S.), 1954, Volume (35)... | 9 | https://mathoverflow.net/users/75650 | 416730 | 169,803 |
https://mathoverflow.net/questions/416699 | 1 | Here's the problem: start with $2^n$, then take away $\frac{1}{2}a^2+\frac{3}{2}a$ starting with $a=1$, and running up to $a = 2^{n+1}-2$, evaluating modulo $2^n$. Does the resulting sequence contain representatives for all the congruence classes module $2^n$?
Some examples of these sequences:
\begin{align\*}
n=2\qua... | https://mathoverflow.net/users/265714 | Modular arithmetic problem | This is correct. Denote $f(x)=x(x+3)/2 \pmod {2^n}$. Then $f(x)=f(y)$ if and only if $x(x+3)-y(y+3)=(x-y)(x+y+3)$ is divisible by $2^{n+1}$. Since $x-y$ and $x+y+3$ have different parity, this in turn happens if and only if one of them is divisible by $2^{n+1}$. So, if $x=1,2,\ldots,2^{n+1}$, then $f(x)$ takes each val... | 4 | https://mathoverflow.net/users/4312 | 416758 | 169,812 |
https://mathoverflow.net/questions/416757 | 13 | There are a number of criteria for determining whether a polynomial $\in \mathbb{Z} [X]$ is irreducible over $\mathbb{Q}$ (the traditional ones being Eisenstein criterion and irreducibility over a prime finite field).
I was wondering if the decision problem of "Given an arbitrary polynomial $\in \mathbb{Z} [X]$, is i... | https://mathoverflow.net/users/111528 | Is irreducibility of polynomials $\in \mathbb{Z} [X]$ over $\mathbb{Q}$ an undecidable problem? | There is a polynomial-time algorithm that decomposes any non-zero polynomial in $\mathbb{Q}[X]$ into irreducible factors. The algorithm is due to Lenstra–Lenstra–Lovász ([Factoring Polynomials with Rational Coefficients](http://www.digizeitschriften.de/dms/img/?PID=PPN235181684_0261%7Clog91)).
| 22 | https://mathoverflow.net/users/11919 | 416759 | 169,813 |
https://mathoverflow.net/questions/416765 | 1 | Suppose $X\sim \mathcal{N}(0,\sigma^2)$, find the expectation $\mathbb{E}\left[\frac{1}{(1+X^2)^a}\right]$ where $a$ is a fixed positive real number.
Is there an explicit formula for the above expectation?
| https://mathoverflow.net/users/111204 | Expected value of a function of normal random variable | According to Mathematica, the expectation is
$$\frac{\Gamma \left(a-\frac{1}{2}\right) \, \_1F\_1\left(\frac{1}{2};\frac{3}{2}-a;\frac{1}{2 \sigma^2}\right)}{\sqrt{2} \sigma
\Gamma (a)}+\frac{2^{-a} \sigma^{-2 a} \Gamma \left(\frac{1}{2}-a\right) \, \_1F\_1\left(a;a+\frac{1}{2};\frac{1}{2
\sigma^2}\right)}{\sqrt{\pi ... | 1 | https://mathoverflow.net/users/36721 | 416772 | 169,817 |
https://mathoverflow.net/questions/416754 | 24 | Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. A standard fact is that $G$ is generated by $\exp(\mathfrak{g})$, i.e. every $g \in G$ can be written as $g=\exp(x\_1)\cdots\exp(x\_n)$ for some $x\_i\in\mathfrak{g}$. A natural follow-up question is whether there is a bound on the number $n$ of Lie alge... | https://mathoverflow.net/users/385475 | Lie groups generated by finitely many Lie algebra elements | $\def\fg{\mathfrak{g}}$The following answer is a paraphrase of material from
*Philips, Christopher N.*, [**How many exponentials?**](http://dx.doi.org/10.2307/2375057), Am. J. Math. 116, No. 6, 1513-1543 (1994). [ZBL0839.46054](https://zbmath.org/?q=an:0839.46054).
We'll say that a Lie group $G$ has **exponential r... | 26 | https://mathoverflow.net/users/297 | 416776 | 169,819 |
https://mathoverflow.net/questions/396064 | 17 | In the 1970s Ol'shanskii constructed a non-cyclic finitely generated group $G$ with the following properties:
1. Every proper, non-trivial subgroup of $G$ is infinite cyclic.
2. If $X^m=Y^n$ for $X, Y\in G$ with $m,n\neq0$, then $\langle X, Y\rangle$ is cyclic i.e., any two maximal subgroups of $G$ have trivial inter... | https://mathoverflow.net/users/6503 | Are groups with every proper, non-trivial subgroup infinite cyclic simple? | The answer is no, there exist non-simple torsion-free Tarski monsters. Theorem 31.4 of Ol’shanskii's book "Geometry of defining relations in groups" [ZBL0676.20014](https://zbmath.org/?q=an:0676.20014) [MR1191619](https://mathscinet.ams.org/mathscinet-getitem?mr=1191619) is:
**Theorem.** There is a non-abelian group ... | 9 | https://mathoverflow.net/users/24447 | 416786 | 169,822 |
https://mathoverflow.net/questions/416762 | 5 | For which $(a,b,n) \in \mathbb{Z}^3$ satisfying $a+b=n$ does $\frac{\Gamma(z+1)}{\Gamma(x+1)\Gamma(y+1)}$ approach a limit as $(x,y,z) \rightarrow (a,b,n)$ in $\mathbb{C}^3$, and what is that limit? (The case where $a$, $b$, and $n$ are all negative is the problematic one.) What if we restrict to $\{(x,y,z) \in \mathbb... | https://mathoverflow.net/users/3621 | Extended binomial coefficients and the gamma function | There's nothing special about the gamma function; the failure of the limit to exist when $a$, $b$, and $n$ are negative is exactly the same as the failure of
$$\lim\_{(x,y,z)\to(0,0,0)} \frac{xy}{z}$$ to exist.
I will renormalize in a way that I think makes clearer what's going on. First let's look at
$$\lim\_{(x,y,z... | 6 | https://mathoverflow.net/users/10744 | 416792 | 169,824 |
https://mathoverflow.net/questions/416793 | 3 | A [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H =(V, E)$ consists of a set $V$ and a set $E \subseteq {\cal P}(V)$ of subsets of $V$. A *hypergraph coloring* is a map $c: V\to \kappa$, where $\kappa \neq \emptyset$ is a cardinal and the restriction $c\restriction\_e: e \to \kappa$ is non-constant whenever $... | https://mathoverflow.net/users/8628 | Conflict-free coloring of $\mathbb{R}$ with the Euclidean topology | By colouring every element of $\Bbb Q$ with a unique colour and every element of $\Bbb R\setminus\Bbb Q$ with the same colour (different from all the colours used so far), we see that $\chi\_\mathrm{cf}(\Bbb R)\leq\aleph\_0$.
But in fact $\aleph\_0$ is a lower bound in any reasonable space.
**Lemma 1:** Let $X$ be ... | 6 | https://mathoverflow.net/users/49381 | 416798 | 169,825 |
https://mathoverflow.net/questions/361757 | 1 | Let $ (\Lambda\_n) $ be a family of lattices, $ \Lambda\_n \subset \mathbb{Z}^n $, with $ \det\Lambda\_n \sim n $ as $ n \to \infty $ (meaning $ \lim\_{n\to\infty} n^{-1} \det\Lambda\_n = 1$). I am interested in the asymptotics of the number of points of $ \Lambda\_n $ in the hypercube $ [0,2)^n $. In particular, is it... | https://mathoverflow.net/users/83189 | Lattice points in hypercubes | The following answer is from [A Reverse Minkowski Theorem](https://arxiv.org/pdf/1611.05979.pdf). It deals with a sphere, rather than a hypercube. I am unaware of extensions to hypercubes, but the lower bound I will quote from it at least seems non-trivial, and of interest.
Minkowski's first theorem has a "point-coun... | 1 | https://mathoverflow.net/users/101207 | 416810 | 169,830 |
https://mathoverflow.net/questions/416818 | 4 | Let $X$ denote the sequence [A200246](https://oeis.org/A200246): an $i$-th element of $X$ is equal to $w(p\_i) \bmod 2$, where $w(p\_i)$ is the number of ones in the base-$2$ representation of an $i$-th prime.
The first $564163$ bits of $X$ (corresponding to all primes less than $2^{23}$) contain $274615$ zeros and $... | https://mathoverflow.net/users/122796 | Parities of binary weights of primes | [Mauduit and Rivat](https://annals.math.princeton.edu/2010/171-3/p04) showed that the sum of the binary digits of primes are equally distributed between the possibilities $0\bmod 2$ and $1\bmod 2$. So the limit you want is $1$. More generally they consider the distribution $\bmod q$ of the sums of digits of primes writ... | 8 | https://mathoverflow.net/users/38624 | 416824 | 169,834 |
https://mathoverflow.net/questions/416807 | 10 | In modern treatments of statistical mechanics, the natural base is conventionally used for the Gibbs and Boltzmann entropy without careful justification. While I am aware that the properties of the Shannon entropy are invariant to the choice of base of the logarithm, I suspect that physicists might have careful theoret... | https://mathoverflow.net/users/56328 | The origin of the natural base in statistical mechanics | As Matt F. points out, we could just absorb a change of base of the logarithm into the coefficient. The reason that is not convenient in physics is that we would like the same coefficient $k$ to appear in the [ideal gas law](https://en.wikipedia.org/wiki/Boltzmann_constant#Roles_of_the_Boltzmann_constant) $pV=NkT$.
N... | 13 | https://mathoverflow.net/users/11260 | 416840 | 169,838 |
https://mathoverflow.net/questions/416764 | 1 | Let $X\_k$ be a sequence of *iid* Bernoulli random variables of parameter $p$ and let $\hat{X}\_n=\frac1n\sum\_{k=1}^nX\_k$. Hoeffding's inequality states that for any $n$:
$$\mathbb{P}(\hat{X}\_n - p \ge \epsilon) \le e^{-2n\epsilon^2},$$
or said otherwise:
$$\mathbb{P}(\hat{X}\_n - p \ge \frac{\epsilon}{\sqrt{n}}) \l... | https://mathoverflow.net/users/90045 | Hoeffing inequality is not true for stopping time | In fact, the answer is indeed contained in the question:
Let $N$ be the stopping time:
$$ N = \inf\{n : \hat{X}\_n - p \ge \frac{\epsilon}{\sqrt{n}}\}.$$
$N$ is finite almost surely (for instance, this is a consequence of the law of iterated logarithm). This shows that $$P( \hat{X}\_N - p \ge \frac{\epsilon}{\sqrt{N}... | 1 | https://mathoverflow.net/users/90045 | 416851 | 169,841 |
https://mathoverflow.net/questions/416855 | 2 | I am looking for an upper bound $R=R\_{n,\varepsilon}$ such that for given $\varepsilon>0$ and real numbers $\alpha\_1, \dotsc, \alpha\_n$ in, say, $[1,2]$, there is $x\in [1,R]$ such that
$$
\frac 1n\sum\_{k-1}^n\cos(\alpha\_k x)\geq 1-\varepsilon
$$
Dirichlet's principle states that $x$ can be (an integer) smaller ... | https://mathoverflow.net/users/16934 | When does a random trigonometric sum approximate $1$? | $\newcommand\ep\varepsilon\newcommand\al\alpha$Let $X\_k:=\cos(\alpha\_kx)$. Then $X\_1,X\_2,\dots$ are iid random variables such that $P(X\_1<1)=1$ and hence $\mu:=EX\_1<1$, so that $\mu<1-\varepsilon$ if $\varepsilon\in(0,1-\mu)$. So, by the law of large numbers, with high probability we get the inequality
$$
\frac 1... | 3 | https://mathoverflow.net/users/36721 | 416856 | 169,843 |
https://mathoverflow.net/questions/416823 | 2 | Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to
$$-\Delta u=f,\quad u\_{|\partial\Omega}=0$$
Now consider the elliptic operator $\Delta + b\cdot\nabla(-\Delta)^{-1}$ associated with, s... | https://mathoverflow.net/users/166785 | Spectral analysis for nonlocal elliptic operator | So that's
$$
-\Delta u -b\cdot\nabla v = \lambda u
$$
$$
-\Delta v + u =0
$$
which you can write, with $U=(u,v)$
$$
-\Delta U + B \nabla U + C U = \lambda Q U
$$
where $\Delta$ is applied line wise, and so is the gradient, whereas
$$
B=\begin{pmatrix} 0& b\cdot \\ 0 &0\end{pmatrix}, C=\begin{pmatrix} 0& 0 \\ 1 &0\end{p... | 2 | https://mathoverflow.net/users/40120 | 416861 | 169,845 |
https://mathoverflow.net/questions/416868 | 2 | Let $X$ be a compact metric space and $Y\subset X$ be a compact set. Assume that $f\_1, f\_2: Y \to \mathbb{P}\mathbb{R}^2$ are continuous functions. Let $N \subset \mathbb{P}\mathbb{R}^2$ be a countable set.
### Question
Is there a Borel measurable function $T:f\_1(Y) \to f\_2(Y)$ such that if $x \in Y$ and $f\_1(... | https://mathoverflow.net/users/168351 | Existence of a Borel measurable function | Let $Y=[-1,1]$. We can also view $Y$ by homeomorphic embedding and abuse of notation as a subset of $\mathbb{P}\mathbb{R}^2$. For all $x\in Y$, let $f\_1(x)=|x|$ and $f\_2(x)=x$. For all but the countably many $x$ such that $f\_1(x)=|x|\in N$, one would need $$x=f\_2(x)=T(f\_1(x))=T(|x|)=T(|-x|)=T(f\_1(-x))=f\_2(-x)=-x... | 2 | https://mathoverflow.net/users/35357 | 416871 | 169,847 |
https://mathoverflow.net/questions/416490 | 3 | I define a subset $M$ of $\mathbb R^n$ to be a "homogeneous Euclidean manifold" if:
* it is a closed connected smooth submanifold of $\mathbb R^n$,
* for every $p, q$ in $M$, there is a Euclidean isometry $f$ of $\mathbb R^n$ sending $p$ to $q$ fixing $M$ (i.e., $f(M)=M$ and $f(p)=q$).
The problem is to classify th... | https://mathoverflow.net/users/477343 | Classification of "homogeneous" submanifolds of ℝⁿ | Such $M$ are called extrinsically homogeneous submanifolds of euclidean space.
Reference: for example the book by Berndt-Console-Olmos, Submanifolds and holonomy.
The set of all euclidean isometries that fix $M$ is a closed subgroup $G$ of the group of euclidean motions $E(n)$, and $M$ is an orbit of $G$. So the qu... | 1 | https://mathoverflow.net/users/127309 | 416872 | 169,848 |
https://mathoverflow.net/questions/416556 | 4 | For an elliptic curve $E/\mathbb{Q}$, let us denote by $\Delta\_{\min}(E)$ the minimal discriminant of $E$ and $N(E)$ the conductor of $E$. Then it is well-known that $N(E) | \Delta\_\min(E)$.
The *Szpiro ratio* of $E$ is defined as the ratio
$$\displaystyle \beta\_E = \frac{\log |\Delta\_\min(E)|}{\log N(E)}.$$
Th... | https://mathoverflow.net/users/10898 | Szpiro ratios of elliptic curves over $\mathbb{Q}$ | The answer is "yes". Use one of the various families with Szpiro ratio exceeding $6$ and large conductor $N$, and then twist by a prime $p$ of appropriate size, coprime to $N$ (which increases the conductor by $p^2$ and the minimal discriminant by $p^6$). With a certain amount of care, this gives density in the interva... | 4 | https://mathoverflow.net/users/7302 | 416876 | 169,850 |
https://mathoverflow.net/questions/416883 | 5 | The Kodaira($-$Iitaka) dimension of a line bundle $L$ on a complex manifold $X$ can be defined either in three ways:
1. The maximal dimension of the image of the rational maps $φ\_{|mL|} : X \dashrightarrow \mathbb{P}(H^0(X, m L)^\*)$
2. The unique integer $k$ such that $h^0(m L) = O(m^k)$.
3. $\operatorname{trdeg} \... | https://mathoverflow.net/users/123448 | Equivalent definitions of Kodaira dimension | For $(1) \iff (3)$, we have the following chain of identities:
1. $\operatorname{trdeg} \operatorname{Frac} \left( \bigoplus\_{m ≥ 0} H^0(X, mL) \right)$ is equal to
2. $\max \operatorname{trdeg}(F)$ where $F$ is a finitely generated subfield of $ \operatorname{Frac} \left( \bigoplus\_{m ≥ 0} H^0(X, mL) \right)$, whi... | 10 | https://mathoverflow.net/users/18060 | 416885 | 169,851 |
https://mathoverflow.net/questions/416884 | 1 | Suppose a first order set theory $S$ has a countable model. Does it follow that there is a countable ordinal $\alpha$ so that $L\_\alpha$ in the constructible hierarchy is a model of $S$?
| https://mathoverflow.net/users/37385 | Are countable models constructible? | For the question in the text, the answer is no. Let $S$ consist of (1) a sufficient finite number of ZF axioms to provide an absolute (i.e., $\Delta\_1$ in the Lévy hierarchy) definition of the constructible hierarchy and (2) the sentence formalizing $V\neq L$.
For the question in the title, asking only for a constru... | 8 | https://mathoverflow.net/users/6794 | 416887 | 169,852 |
https://mathoverflow.net/questions/416888 | 1 | $\DeclareMathOperator\SO{SO}$I posted this on [MSE](https://math.stackexchange.com/questions/4381194/s-4-subgroups-and-so-3-mathbbr) 10 days ago and it got 3 upvotes but no answers or comments, so I'm cross-posting to MO.
Background: The group of rotations $ \SO\_3(\mathbb{R}) $ has a 24 element subgroup of integer p... | https://mathoverflow.net/users/387190 | $ S_4 $ subgroups and $ \operatorname{SO}_3(\mathbb{R}) $ | Consider the group of matrices of the form $w \oplus \det(w)^{-1}$, with $w$ a $4\times4$ permutation matrix. This is an $\operatorname S\_4$ inside $\operatorname{SL}\_5(\mathbb R) \subseteq \operatorname{SL}\_5(\mathbb C)$, and $\mathbb C^5$ decomposes as a sum of the trivial, the sign, and the reflection representat... | 4 | https://mathoverflow.net/users/2383 | 416889 | 169,853 |
https://mathoverflow.net/questions/416836 | 9 | Everyone knows the Heine-Borel theorem characterizing compact subsets of Euclidean space. For any $n \in \mathbb N$ a set $A \subseteq \mathbb R^n$ is compact just in case it is closed and bounded (in the sense that it is contained in some large enough finite diameter ball).
There is a similar theorem for the irratio... | https://mathoverflow.net/users/114946 | Structure theorems for compact sets of rationals | It is easy to describe these up to homeomorphism:
* Every countable compact Hausdorff space is homeomorphic to a countable successor ordinal, see [Milliet - A remark on Cantor derivative](https://arxiv.org/abs/1104.0287v1).
* Conversely, it's easy to embed all countable limit ordinals in the rationals by transfinite ... | 16 | https://mathoverflow.net/users/123634 | 416900 | 169,857 |
https://mathoverflow.net/questions/416902 | 2 | Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ such that there are $c,C,d,D>0$ satisfying: for every $x \in \mathbb{R}^n$ and every $r>0$
$$
c r^d \leq \mu(B(x,r)) \leq Cr^D.
$$
Let's call such a measure **almost Ahlors Regular.**
***Note:** when $d=D$ then we say that the measure is [*Ahlfors regular*](h... | https://mathoverflow.net/users/36886 | Examples of "almost" Ahlfors regular measures | $\newcommand\R{\mathbb R}\newcommand\la{\lambda}$A measure $\mu$ is Ahlfors regular (according to your definition) iff it has a density (with respect to the Lebesgue measure $\la$) bounded away from $0$ and $\infty$ (if $B(x,r)$ denotes the ball centered at $x$ and of radius $r$).
Indeed, by inscribing and circumscri... | 1 | https://mathoverflow.net/users/36721 | 416911 | 169,861 |
https://mathoverflow.net/questions/406120 | 10 | A monohedron is a convex polyhedron with all faces mutually congruent but with no other symmetry necessarily needed. So obviously, this is a wide class of polyhedrons that includes the Platonic solids and isohedra (<https://mathworld.wolfram.com/Isohedron.html>). An earlier post is <https://math.stackexchange.com/quest... | https://mathoverflow.net/users/142600 | Are there Monohedra with odd number of faces? | The answer to the question in the title is negative in dimension 3: it was shown by Grunbaum that every 3-polytope with congruent facets has an even number of facets. See p. 414 of his book "Convex Polytopes", 2nd edition. The original reference is:
Grunbaum, B. On polyhedra in $\mathbb{E}^3$ having all faces congrue... | 5 | https://mathoverflow.net/users/110306 | 416912 | 169,862 |
https://mathoverflow.net/questions/416703 | 4 | I'm currently reading the article ["A Generalization of a Poincaré–Bendixson Theorem to Closed Two-Dimensional Manifolds" by Arthur Shwartz](https://www.jstor.org/stable/2373135). The paper first establishes a result for minimal sets, which are closed, non-empty subsets of $M$ which are invariant under the flow of some... | https://mathoverflow.net/users/111199 | Poincaré–Bendixson Theorem on a compact, connected, orientable, two-dimensional manifold | To address your two questions:
1. Why is $q \in N$?
If I understand correctly, they could have (and should have) just started with $q \in \Sigma$, since for any $q \in \Sigma$ the orbit $\phi\_t(x)$ approaches $q$ arbitrarily closely.
To be precise, because $\Sigma \subset \omega(x)$, there's a $t\_1 > 0$ such th... | 1 | https://mathoverflow.net/users/1227 | 416917 | 169,865 |
https://mathoverflow.net/questions/416901 | 5 | This is a dublicate from stackexchange:
Consider two families of hyperplanes $F\_1$ and $F\_2$ in $\mathbb{R}^d$ both containing $n$ hyperplanes. We have that for all $f \in F\_1$ and $g \in F\_2$ that $f$ and $g$ intersect. Further we know that for each point $p \in \mathbb{R}^d$ we have that at most $c \cdot n$ hyp... | https://mathoverflow.net/users/216305 | Counting intersections of hyperplanes | As by my comment above, it is enough to solve the problem for $d=2$.
Using point-line duality, we get the following:
Given $n$ red and $n$ blue points in the plane, such that no line contains more than $cn$ points, prove that through at least $\varepsilon n$ of the red points there are at least $\tau n$ lines tha... | 5 | https://mathoverflow.net/users/955 | 416920 | 169,867 |
https://mathoverflow.net/questions/416921 | 3 | Denote by $ \binom{n}{k}\_q = \prod\_{i=0}^{k-1} \frac{ q^{n-i} - 1 }{ q^{k-i} - 1 } $, $ k = 0, 1, \ldots, n $, the $ q $-binomial (Gaussian) coefficients. These numbers are symmetric, in the sense that $ \binom{n}{k}\_q = \binom{n}{n-k}\_q $, unimodal, and hence maximized for $ k = \lfloor n/2 \rfloor $ (or $ k = \lc... | https://mathoverflow.net/users/83189 | Sum of $q$-binomial coefficients | Let's look at the ratio of two adjacent $q$-binomials as we move away from the center, for simplicity I'll do the even case.
$\binom{2n}{n-a}\_q / \binom{2n}{n-a-1}\_q = \frac{[n+a+1]\_q}{[n-a]\_q} > q^{2a+1}$
In particular these are decaying faster than geometrically. With some possible exceptions for say $q =2$ o... | 4 | https://mathoverflow.net/users/39120 | 416926 | 169,870 |
https://mathoverflow.net/questions/416895 | 2 | I have been studying hyperplane arrangements from R. Stanley's notes and so far, I have read until lecture 5. But, Lecture 6 and end of Lecture 5 seems very combinatorial. I'm more interested in the "algebraic" direction. More specifically, in representation-theoretic perspective (if there exists any).
So, I was wond... | https://mathoverflow.net/users/338456 | Possible "algebraic" direction in hyperplane arrangements | If I recall correctly, Stanley does not discuss the so-called "Tits monoid" associated to a hyperplane arrangement. The relationship between the Tits monoid and the lattice of flats of a hyperplane arrangement is the main focus of the book "Topics in hyperplane arrangements" by Aguiar and Mahajan (it's free online at [... | 2 | https://mathoverflow.net/users/25028 | 416928 | 169,871 |
https://mathoverflow.net/questions/416905 | 1 | This question was previously posted [on MSE.](https://math.stackexchange.com/questions/4389798/eigenvectors-of-the-dual-of-positive-irreducible-operators)
Let $E$ be a Banach lattice such that $E$ is an $M$-space. Assume that $T\colon E\to E$ is a positive bounded non-compact irreducible linear operator with positive... | https://mathoverflow.net/users/125982 | Eigenvectors of the dual of positive irreducible operators | No, the dual operator $T'$ does have a positive eigenvector, in general.
As a counterexample, consider the space $E = c\_0(\mathbb{Z})$ if scalar-valued sequences indexed over the integers, endowed with the sup norm and the pointwise order. This is a Banach lattice and an M-space. Its dual space $E'$ can be identified ... | 1 | https://mathoverflow.net/users/102946 | 416932 | 169,873 |
https://mathoverflow.net/questions/414577 | 2 | I'm interested in Sobolev space estimates for commutators involving a pseudodifferential operator and a Fourier multiplier. More specifically, suppose $p = p(x,\xi) \in S\_{1,0}^{m\_1}$ and let $q = q(\xi) \in S\_{1,0}^{m\_2}$, where $S\_{\rho,\delta}^m$ denotes the standard Hörmander symbol class. Does anyone know of ... | https://mathoverflow.net/users/114164 | Reference for commutator estimate | Given your assumptions on $p$ and $q$, $[Op(p),Op(q)]\in S^{m\_1+m\_2-1}\_{1,0}.$ Given any $A\in OPS^m\_{\rho,\delta}$ with $0\leq\delta<\rho\leq 1$, one has that $A:H^s\rightarrow H^{s-m}$ boundedly (follows from a combination of Calderon-Valliancourt and the Fourier multiplier $\Lambda\_s$ with symbol $\langle \xi\r... | 2 | https://mathoverflow.net/users/477714 | 416946 | 169,879 |
https://mathoverflow.net/questions/416892 | 5 | We define a norm on $\mathbb C^2$ as $\|(\alpha,\beta)\|:=\max\left\{|\alpha|,|\beta|,\big|\frac{\alpha+\beta}{\sqrt{2}}\big|\right\}.$ Can the dual norm be calculated explicitly?
| https://mathoverflow.net/users/136860 | Dual norm of a subspace of $\ell_\infty^3$ | $\newcommand{\C}{\mathbb C}\newcommand{\R}{\mathbb R}\newcommand{\si}{\sigma}\newcommand{\al}{\alpha}\newcommand{\be}{\beta}$This is to detail and correct the answer by Onur Oktay, which is based on a nice idea, leading to simplified and speedier calculations of the dual norm. (That Onur Oktay's answer contains at leas... | 4 | https://mathoverflow.net/users/36721 | 416948 | 169,881 |
https://mathoverflow.net/questions/416844 | 5 | Let $S$ and $\rm ku$ denote the sphere spectrum and the connective K-theory spectrum, respectively.
Then is the morphism ${\rm ku} \to F(F({\rm ku},S),S)$ an equivalence?
Here $F$ denotes the mapping spectrum.
| https://mathoverflow.net/users/119076 | Is ku reflexive as a spectrum? | Writing $D(-) = F(-, S)$, I believe that there is an equivalence $$D ku \simeq \prod\_{n=0}^\infty \Sigma^{-2n-1} H(\widehat{\mathbb Z} / \mathbb Z),$$ where $\widehat{\mathbb Z}$ is the profinite completion of the integers. This would split as a similar product after dualizing a second time, so can't be equivalent to ... | 8 | https://mathoverflow.net/users/1094 | 416949 | 169,882 |
https://mathoverflow.net/questions/416939 | 5 | Let $\mu$ be a finite non negative singular measure on $\mathbf{R}^d$. I would like to know if there exists some result on the infimum of the absolute value of its Fourier Transform $$\hat{\mu}(t)=\displaystyle\int \mathrm{e}^{2i\pi t\cdot x} ~\mu(dx).$$
It is known that we don't necessarily have $\hat{\mu}(t)\righta... | https://mathoverflow.net/users/477699 | Infimum of Fourier transform of singular measure | Yes, that's true. By Wiener's lemma we have $\sum\_{x\in \mathbb{R}^d} |\mu(\{x\})|^2 = \lim\_{R\to \infty} \frac{1}{(2R)^d} \int\_{[-R,R]^{d}} |\widehat{\mu}(t)|^2 dt$. If $\mu$ is continuous, then the left-hand side is $0$, so the infimum of $|\widehat{\mu}|$ has to be zero as well, since the right-hand side is bound... | 4 | https://mathoverflow.net/users/24953 | 416958 | 169,885 |
https://mathoverflow.net/questions/416837 | 2 | It is a classical result of Wakimoto that the sheaf of chiral differential operators $D\_{ch}$ on $\mathbf{P}^1$ has global sections
$$D\_{ch}(\mathbf{P}^1)\ \simeq\ L\_{-2}(\mathfrak{sl}\_2)$$
the simple quotient affine vertex algebra at critical level $-2$. Apparently the map
$$L\_{-2}(\mathfrak{sl}\_2)\ \to\ D\_{ch}... | https://mathoverflow.net/users/119012 | Confusion about Wakimoto's chiral differential operators on $\mathbf{P}^1$ | There are a couple of errors in the statement of the question. First in the transformation formula that I suppose you took from MSV "Chiral de Rham complex". Let us denote $\tilde{x} = x^{-1}$ and $\tilde{\partial} = x^2 \partial + 2 x'$. We assume the standard OPE $\partial(z) \cdot x(w) \sim \frac{1}{z-w}$ as the onl... | 3 | https://mathoverflow.net/users/17980 | 416972 | 169,888 |
https://mathoverflow.net/questions/416987 | 4 | Let $\zeta$ be the Riemann zeta-function, and let $t> 0$. I'm interested in the growth rate of
$$
\left|\frac{\zeta'}{\zeta}\left(-\frac{1}{2}+it\right)\right|
$$
as $t\to\infty$. It is easy to find references in the literature to estimations like $O(\log t)$. What is the actual growth rate?
| https://mathoverflow.net/users/3199 | Correct growth rate of logarithmic derivative of zeta, outside critical strip | It is $\log t + O(1)$. Without the absolute value, $-(\zeta'/\zeta)(-1/2 + it) = \log |t| + O(1)$ as $|t| \to \infty$.
First, by taking the logarithmic derivative of the functional equation
$$
\zeta(1-s) = 2(2\pi)^{-s}\Gamma(s)\cos\left(\frac{\pi}{2}s\right)\zeta(s)
$$
we have
$$
-\frac{\zeta'(1-s)}{\zeta(1-s)} = -\l... | 10 | https://mathoverflow.net/users/3272 | 416990 | 169,891 |
https://mathoverflow.net/questions/416985 | 1 | I am wondering if there is a way to formulate or generate a matrix $X \in R^{n\times n}$ whose column vectors $\{x\_1,\dotsc,x\_i,\dotsc,x\_n\}$ are such that $x\_i$ and $x\_j$ are orthogonal iff $\lvert i-j\rvert>m$. The matrix $X^T X$ would be a band matrix with zeros off-band. I do not have a particular requirement ... | https://mathoverflow.net/users/173974 | Generate an ordered set of mostly orthogonal vectors $\{x_i\}$ where $x_i \cdot x_j =0$ iff $\lvert i-j\rvert >m$ | It seems that the band matrix example is an essentially sufficient characterization, up to a choice of basis.
The orthogonality constraint forces zeroes in the QR decomposition $X=QR$: $R$ is an upper triangular $m$-band matrix (and $Q$ is our usual orthogonal matrix.)
Conversely, any such pair $Q$, $R$ generates a... | 2 | https://mathoverflow.net/users/69359 | 416992 | 169,892 |
https://mathoverflow.net/questions/315115 | 2 | For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function away from the diagonal $\{x=y\}$. If I calculated correctly, we have estimates of the form
$$|R(\lambda;x, y)| \leq C ... | https://mathoverflow.net/users/16702 | Exponential decay of resolvent kernel | The local statement is correct for any positive self-adjoint extension of the Laplace operator on a Riemannian manifold (not necessarily complete). It is an old result by Hoermander. See the proof of Prop. 4.8 in
"L. HOERMANDER, On the Riesz means of spectral functions and eigenfunction expansions
for elliptic differen... | 1 | https://mathoverflow.net/users/471159 | 417008 | 169,897 |
https://mathoverflow.net/questions/416989 | 2 | Let $M=(E,I)$ be a paving matroid with rank $n$. Let $A\subset E$ be an $n-1$ subset. How many bases of $M$ containing $A$ exist? (Note that every $n-1$ subset of $E$ is independent.)
| https://mathoverflow.net/users/165074 | Counting certain bases of a paving matroids | In an $n$-dimensionsl space, take a hyperplane $H$. Take a set of vectors in $H$ in a general position, and add one vector $x$ outside $H$. This is a paving matroid of rank $n$.
Now, if $|A|=n-1$ and $x\notin A$, then the only vector which can be added without breaking independency is $x$, so the minimal number of so... | 2 | https://mathoverflow.net/users/17581 | 417016 | 169,899 |
https://mathoverflow.net/questions/416965 | 9 | This question is inspired by [xkcd #2585 (*Rounding*)](https://xkcd.com/2585/):
Let $u\_0,\ldots,u\_n$ be positive real numbers (we can assume w.l.o.g. that $u\_0=1$) or “units”.
Consider the following directed graph: its vertices are pairs $(i,k)$ where $0\leq i\leq n$ designates a unit, and $k\in\mathbb{Z}$ is a ... | https://mathoverflow.net/users/17064 | How far away can we get by multiple rounding and unit change? | For a vertex $(i,k)$, it is reasonable to define its *value* as $ku\_i$. We show that the value cannot grow too large, which establishes a negative answer to the question.
Let $v$ be the current value, and let $V>v$ be a real number such that $\{V/u\_i\}<1/2$ for all $i$. Such a $V$ exists: we can start by choosing a... | 7 | https://mathoverflow.net/users/17581 | 417018 | 169,900 |
https://mathoverflow.net/questions/416976 | 4 | Let $\tau\_X$ denote the collection of open subsets of a topological space $X$ and let $\mathsf{RO}(X)$ be the subset of $\tau\_X$ made up of regular open subsets. With this terminology, the inequality $|\mathsf{RO}(X)|\leq |\tau\_X|$ is obvious. Furthermore, if $X$ is a semiregular space, it is easy to see that $|\tau... | https://mathoverflow.net/users/146942 | $|\mathsf{RO}(X)|$ vs. $|\tau_X|$ for Tychonoff spaces | The compact Hausdorff space $X = \beta\mathbb{N}$ is another example. Every regular open subset is the closure of a subset of $\mathbb{N}$ and there are only $\frak{c}$ such subsets but $X$ has $2^{\frak{c}}$ points, and for each such point $p$, the set $X \setminus \{p\}$ is an open set.
For the second question, a s... | 4 | https://mathoverflow.net/users/89233 | 417024 | 169,902 |
https://mathoverflow.net/questions/417001 | 0 | Let $G$ be a finitely generated group and $\varphi:G\to \operatorname{Aut}(\mathbb C)$ a homomorphism, where $\operatorname{Aut}(\mathbb C)$ is the group of complex affine transfromations $a z+b$.
Can we find a torsion free-subgroup $H$ of $G$ with finite index? And can we find a normal subgroup $H$ which is torsion-... | https://mathoverflow.net/users/356774 | Torsion-free subgroup of affine group | Yes. More generally, for any field $K$ we have an embedding of $\operatorname{Aff}(K^n)$ in $\operatorname{GL}\_{n+1}(K)$, and so if $K$ has characteristic zero we can apply Selberg's lemma to conclude that a finitely generated group of affine transformations of $K^n$ is virtually torsion-free. The normal core of any f... | 6 | https://mathoverflow.net/users/24447 | 417027 | 169,903 |
https://mathoverflow.net/questions/416998 | 0 | The game allows you to bet only \$1 at a time, and if you win, you end up \$1 richer, otherwise you end up \$1 poorer. Probability of winning is $p=0.45$. If you start with \$1 - what is expected number of games before bust (i.e., \$0 dollars)?
My initial approach was essentially just:
$\sum\_{n=1}^{\infty} (2n-1)(0.... | https://mathoverflow.net/users/477770 | Expected number of games until bust | Let $x$ be the expected time to get to \$0, starting from \$1.
What's the expected time to get to \$0, starting from \$2? It's $2x$. That's because to get from \$2 to \$0, we first need to get from \$2 to \$1, and then we need to get from \$1 to \$0. And getting from \$2 to \$1 behaves exactly the same as getting fro... | 12 | https://mathoverflow.net/users/5784 | 417035 | 169,906 |
https://mathoverflow.net/questions/417012 | 1 | Let $u\leq v$ be two locally bounded subharmonic functions in a domain in $\mathbb{R}^n$. Assume that $u=v$ on a dense subset.
**Is it true that $u=v$ everywhere?**
| https://mathoverflow.net/users/16183 | Equality of two subharmonic functions | This is not the case. Here is a counterexample in the case where $n=2$.
Suppose that $A$ is a countable dense subset of $\mathbb{R}^{2}=\mathbb{C}$. Let $r\_{a}>0$ for each $a\in A$, and suppose that $\sum\_{a\in A}r\_{a}\cdot(1+\max(0,\log(a)))<\infty$.
Let $p(z)=\sum\_{a\in A}r\_{a}\cdot\log(|z-a|)$. Let $s:\math... | 2 | https://mathoverflow.net/users/22277 | 417044 | 169,909 |
https://mathoverflow.net/questions/416993 | 1 | Is there a standard phrase for the largest odd factor of a positive integer $n$, or more generally for $n$ divided by the largest power of $p$ that divides it (with $p$ some fixed prime)? Five minutes of online search failed to yield an answer.
(I’m hoping for an answer to my terminology question, but general advice ... | https://mathoverflow.net/users/3621 | Nomenclature for largest odd factor | Call it the $2$-free part of $n$ or the odd part of $n$. For a general prime $p$, the $p$-free part of $n$ is what you have after you divide $n$ by the highest power of $p$ dividing it.
| 5 | https://mathoverflow.net/users/3272 | 417047 | 169,912 |
https://mathoverflow.net/questions/416638 | 0 | Is it possible to get a good upper bound for $$\sum\_{1\leq |h|\leq q}\frac{c\_{q}(a-h)}{h}$$ with $(a,q)=1$ and $1\leq a\leq q$.
| https://mathoverflow.net/users/155294 | Ramanujan's type sum | If I'm not mistaken, the $h>0$ part of the sum is
\[ \sum \_{d|q}d\mu (q/d)\sum \_{h\leq q\atop {h\equiv a(d)}}\frac {1}{h}\]
and here the inner sum is
\[ \sum \_{0\leq h\leq (q-a)/d}\frac {1}{hd+a}\leq \frac {\log q}{d}+\frac {1}{a}\]
so that the whole sum is
\[ \leq \sum \_{d|q}d\left (\frac {\log q}{d}+\frac {1}{a}\... | 2 | https://mathoverflow.net/users/110603 | 417059 | 169,916 |
https://mathoverflow.net/questions/417077 | 1 | I've been trying to read through the following paper:
<https://arxiv.org/pdf/0704.1751.pdf>
And I've been stuck on the proposition in the middle of page 8 which says that if a r.v. $X$ has independent entries, then:
$$\textbf{J}(X) = \text{diag}(J(X\_i))\_i$$
Where $\textbf{J}(X) := \text{Cov}(S(X))$ and $J(X):... | https://mathoverflow.net/users/477840 | Covariance matrix of score of random vector with independent entries is diagonal | The equality $ES(X\_i)S(X\_j)=0$ for $i\ne j$ follows because $X\_i$ and $X\_j$ are independent and $ES(X\_i)=0$, so that
$$ES(X\_i)S(X\_j)=ES(X\_i)\,ES(X\_j)=0.$$
(The equality $ES(X\_i)=0$ holds because for the density $p\_i$ of $X\_i$ we have
$$ES(X\_i)=E\frac{p\_i'(X)}{p\_i(X)}=\int\_{-\infty}^\infty\frac{p\_i'(x... | 0 | https://mathoverflow.net/users/36721 | 417078 | 169,918 |
https://mathoverflow.net/questions/417068 | 2 | A simple, undirected graph is vertex-transitive if for any pair of vertices $x,y$, there exists an automorphism (adjacency-preserving self-bijection) $\phi$ such that $\phi(x)=y$.
What if, instead of **taking** $x$ to $y$ as above, we require the automorphism $\phi$ to **exchange** $x$ and $y$, i.e. $\phi(x)=y$ and $... | https://mathoverflow.net/users/477827 | Imposing reciprocity in the definition of vertex-transitivity | A permutation group $G$ acting on a set $X$ is called *generously transitive* if for any two elements $x$, $y \in X$ there is a permutation $g \in G$ such that $x^g = y$ and $y^g = x$.
It is fairly easy to find examples of vertex-transitive graphs whose automorphism group is not generously transitive, by searching Ma... | 2 | https://mathoverflow.net/users/1492 | 417095 | 169,927 |
https://mathoverflow.net/questions/415789 | 0 | By the [Tennenbaum's theorem](https://en.wikipedia.org/wiki/Tennenbaum%27s_theorem),
there are no non-standard countable models of
[Peano Arithmetic](https://en.wikipedia.org/wiki/Peano_axioms#Peano_arithmetic_as_first-order_theory) that are computable using Turing machines. What about models of infinitary computation ... | https://mathoverflow.net/users/116605 | Is there a non-standard model of PA computable with infinitary computation? | To move this off the unanswered queue, the answer is **yes** in a very strong way.
The relevant notion is that of a **PA degree**. There are many equivalent definitions of PA-ness, as well as many interesting results about them, but for the current purposes the relevant ones are the following I think:
* **Definitio... | 4 | https://mathoverflow.net/users/8133 | 417098 | 169,930 |
https://mathoverflow.net/questions/417022 | 3 | I believe I found a complicated proof by bounding the spectral norm $||uu^T-vv^T||^2\_2:=\max\_{||x||=1}|(u^Tx)^2-(v^Tx)^2|$.
Using the fact that $dist(x,y):=\sin|x-y|$ is a distance function over unit vectors, we may prove
$$(v^Tx)^2-(u^Tx)^2=\sin^2(u,x)-\sin^2(v,x)=dist^2(u,x)-dist^2(v,x)
=(dist(u,x)-dist(v,x))(dist(... | https://mathoverflow.net/users/476110 | Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$ | This is a variation of Fedor Petrov's proof (found independently). First, it suffices to prove the statement when $x$ lies in $V:=\mathbb{R}u+\mathbb{R}v$. Indeed, we see this reduction readily by decomposing $x$ as $cy+z$, where $c\in[0,1]$ is a scalar, $y\in V$ is a unit vector, and $z\in V^\perp$. By this reduction,... | 7 | https://mathoverflow.net/users/11919 | 417107 | 169,933 |
https://mathoverflow.net/questions/417102 | 9 | Ellis Lemma on idempotent elements asserts that:
>
> **Lemma (Ellis).** Every compact semigroup has an idempotent.
>
>
>
The proof below is excerpted from Todorcevic's *Introduction to Ramsey Spaces*, [Lemma 2.1](https://books.google.com/books?id=Yhyi8vRfPNAC&pg=PA27).
>
> Let $S$ be a compact semigroup. P... | https://mathoverflow.net/users/146831 | Why is choice needed in Ellis' Lemma? | The issue is that $R$ could be empty. When you apply Zorn's lemma, a nested intersection of non-empty compact sets is non-empty, guaranteeing minimal non-empty elements.
| 9 | https://mathoverflow.net/users/11552 | 417109 | 169,934 |
https://mathoverflow.net/questions/417085 | 1 | Let $X$ be a topological space and consider a continuous function $f:X\to [0,+\infty)$. For $c\geq 0$ set $X\_c := f^{-1} ([0,c])$.
Furthermore, suppose that $X\_0 \neq \emptyset$ and $f$ is proper.
**Motivating example**
If $X$ is a closed manifold and $f$ is smooth and $0$ is a regular value, then $X\_0$ is a smoot... | https://mathoverflow.net/users/99042 | Vanishing of $H^*(f^{-1}[0,c], f^{-1}(0))$ for small $c$, and $f\in C^0(X, [0,+\infty))$ | With this degree of generality I don't think you can say anything. Here are two pathological examples:
* If X is the [Hawaiian earring](https://en.wikipedia.org/wiki/Hawaiian_earring) and $f(x,y) = x$, then for any $c>0$, $f^{-1}([0,c])$ retracts on a copy of the Hawaiian earring (hence has very large $H^1$).
* If X ... | 1 | https://mathoverflow.net/users/173096 | 417115 | 169,935 |
https://mathoverflow.net/questions/417108 | 6 | [Solèr’s theorem](https://arxiv.org/abs/math/9504224) says that for every star division ring $R$ and every $R$-module $H$ with an orthomodular Hermitian form $\langle (-),(-) \rangle:H \times H \to R$ such that there exists an infinite orthonormal sequence $e:\mathbb{N} \to H$, $R$ is either the real numbers $\mathbb{R... | https://mathoverflow.net/users/nan | Is Solèr’s theorem true in constructive mathematics? | Suppose you have a classical classification theorem saying
>
> Each structure (of a certain kind) is either an $A$ or a $B$.
>
>
>
Then you cannot exhibit constructively a $C$ which is neither $A$ nor $B$ because every constructive proof is also classical, and so that would contradict the classical classificat... | 8 | https://mathoverflow.net/users/1176 | 417128 | 169,941 |
https://mathoverflow.net/questions/417123 | 2 | Two $n\times n$ positive definite symetric matrices $A,B$ define two normed spaces $E\_A=({\mathbb R}^n;\|\cdot\|\_A)$ and $E\_B=({\mathbb R}^n;\|\cdot\|\_B)$, where
$$\|x\|\_A=\sqrt{x^TAx},\qquad \|x\|\_B=\sqrt{x^TBx}.$$
>
> Is it true that the interpolation space $[E\_A,E\_B]\_{1/2}$ equals $E\_{A\sharp B}$, wher... | https://mathoverflow.net/users/8799 | Interpolation of normed spaces *vs* geometrical mean of positive matrices | Yes. The proof of Theorem 1.1 from John E. McCarthy's "Geometric interpolation between Hilbert spaces," Ark. Mat. 30, 321-330 (1991) works for this case. Let $A\_i$, $B\_i$, be SPD matrices, $i = 1, 2$, and let $$X\_i = A\_i^{1/2}(A\_i^{-1/2} B\_i A\_i^{-1/2})^{1/2}A\_i^{1/2}$$ be the geometric means of each pair $(A\_... | 3 | https://mathoverflow.net/users/70005 | 417152 | 169,945 |
https://mathoverflow.net/questions/416969 | 3 | Let $(M, \bar g)$, where $M=\mathbb{R}^n$, with coordinates $(x\_1, x\_2, .. x\_n)$, $f:\mathbb{R}^n \rightarrow \mathbb{R}$ and $\phi:\mathbb{R}^n \rightarrow \mathbb{R}$ with $n \geq 3$ and $f$ a positive function.
Let $g=\epsilon\_i \delta\_{ij}$ and $\bar g=\frac{1}{\phi^2}g$.
I denote $\phi\_{, x\_i x\_j}$, $\... | https://mathoverflow.net/users/111304 | PDE system solution on manifold with conformal metric | I don't know what the OP means by **canonical answer**. I assume that what's desired is to understand the solutions of the system of *three* second order ordinary differeential equations for *two* unknown positive functions $\phi(\xi)$ and $f(\xi)$.
At first glance, there's no reason to believe that any nontrivial so... | 5 | https://mathoverflow.net/users/13972 | 417154 | 169,946 |
https://mathoverflow.net/questions/417100 | 5 | For ordinals $\alpha<\beta$, we say $\alpha<\_{el}\beta$, if there is an elementary embedding with domain $L\_\beta$ and critical point $\alpha$.
Is $<\_{el}$ transitive?
| https://mathoverflow.net/users/170286 | Is this relation about elementary embedding transitive? | EDIT: If the codomain $N$ is allowed to be illfounded, then the answer is yes.
(Here if $\kappa=\mathrm{crit}(j)$ then this will mean that $\kappa\subseteq N$, but it might be that $N$ is illfounded and $\kappa$ is exactly the wellfounded part of the codomain, in which case $\kappa\notin N$.)
For let $\kappa\_0<\beta... | 11 | https://mathoverflow.net/users/160347 | 417157 | 169,948 |
https://mathoverflow.net/questions/416774 | 5 | $\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}$Let $S\_\infty$ be the group of all permutations of a countable infinite set (enriched with the product Polish topology).
Does there exist a (continuous) group embedding $\phi: S\_\infty\to S\_\infty$ such that $\phi(G)$ is conjugate to $\phi(H)$ for every pai... | https://mathoverflow.net/users/128723 | Are isomorphic subgroups of the symmetric group conjugate in some larger group? | No, there's no such endomorphism. More generally:
>
> **Proposition.** for every infinite set $X$, there exists no set $Y$ and nontrivial continuous homomorphism $f:S\_X\to S\_Y$ for which the following assertion holds: for any two isomorphic discrete infinite cyclic subgroups of $S\_X$, the images $f(S\_X)$ and $f... | 5 | https://mathoverflow.net/users/14094 | 417163 | 169,950 |
https://mathoverflow.net/questions/417105 | 4 | Given a finite field $\mathbb{F}$ with $|\mathbb{F}|=q=p^m\geq4$ where $p=\text{char}(\mathbb{F})$, I'm wondering is there a characterization of the kernel of the map $f:H\_3(\text{GL}\_3(\mathbb{F}))\to H\_3(\text{GL}\_4(\mathbb{F}))$?
Is it an isomorphism?(Here $H\_n(G)$ means the $n$-th integral homology of the grou... | https://mathoverflow.net/users/471160 | The third homology stability of general linear groups over finite fields | The original paper by Suslin is available [here](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=1914&option_lang=eng) and with some effort you should be able to read it.
If $|\mathbb{F}| = p^r$ then the homology groups $H\_\*(GL\_n(\mathbb{F});\mathbb{F}\_\ell)$ with $\mathbb{F}\_\ell$-coefficie... | 7 | https://mathoverflow.net/users/798 | 417165 | 169,951 |
https://mathoverflow.net/questions/417184 | 2 | Could any one give a proof for this inequility here? I just know its some kind of Gagliardo-Nirenberg inequility, but where does the second term come from? Thx~
$$
\int\_{B\_r}|u|^q\le C\left(\int\_{B\_r}|\nabla u|^2\right)^a \left(\int\_{B\_r}|u|^2\right)^{\frac{q}{2}-a}+\frac{C}{r^{2a}} \left(\int\_{B\_r}|u|^2\right)... | https://mathoverflow.net/users/295572 | A kind of Gagliardo-Nirenberg inequality proof | By context, I infer that you are working in 3 dimensions.
**Step 1: reduce to case $r = 1$**
Consider the mappings $S\_r$ that map
$$ S\_r u(x) := r^{-3/q} u(r^{-1} x) $$
We have that $S\_r: L^q(B\_1) \to L^q(B\_r)$ is an isometric bijection.
Note that
$$ \|\nabla S\_r u \|\_{L^2(B\_r)} = r^{-3/q - 1 + 3/2} \|\na... | 2 | https://mathoverflow.net/users/3948 | 417190 | 169,958 |
https://mathoverflow.net/questions/417191 | 5 | "A Primer on Mapping Class Groups" wrote
>
> Let $\mathrm{Homeo}\_+(S, \partial S)$ denote the group of orientation-preserving
> homeomorphisms of $S$ that restrict to the identity on $\partial S$. $\mathrm{Mod}(S)$ is the
> group of isotopy classes of elements of $\mathrm{Homeo}\_+(S, \partial S)$, where isotopies... | https://mathoverflow.net/users/nan | Mapping class group and pure mapping class group | I do not recall the conventions adopted in the Primer, but there is a wide difference between boundary components (which will be embedded circles or lines) and punctures (which are “missing points” and are typically given by specifying a “punctured disk”). Under most conventions punctures are not “part” of the boundary... | 10 | https://mathoverflow.net/users/1650 | 417196 | 169,961 |
https://mathoverflow.net/questions/417167 | 1 | Before I begin, I apologize for the bad wording. Consider the following "sorting algorithm":
Suppose there are $n$ books on the bookshelf labeled $1$-$n$, and ordered from left to right in a random way. Your goal is to end up with a sorted bookshelf from left to right with $1$ at the left and $n$ at the right. The al... | https://mathoverflow.net/users/142201 | Runtime for Terrible "Sorting Algorithm"? | Here is a quote of [my 2009 proof @ AoPS](https://artofproblemsolving.com/community/c6h256829p1419072) that the algorithm will eventually terminate:
For a permutation $ p$, consider a function $ f(p) := \sum\_{j=1}^n |p\_j - j|$, i.e., the sum of absolute differences of elements and their positions in $ p$.
In the so... | 4 | https://mathoverflow.net/users/7076 | 417201 | 169,962 |
https://mathoverflow.net/questions/417199 | 0 | Consider the equation
$$ -\Delta f+mf+\lambda f^p=0$$
on $\mathbb{R}^d$, where $d>2$,$m>0$, $p>1$ is integer, and $\lambda \in \mathbb{R}$. Are there any known results regarding the non-existence of non-zero solutions $f$ of this equation in $\mathcal{S}(\mathbb{R}^d)$ (the Schwartz space of rapidly decreasing smooth f... | https://mathoverflow.net/users/17971 | Non-existence of rapidly decaying solutions of certain elliptic semilinear equations | (To substantiate my comment above)
If $\lambda \geq 0$ and $p$ is odd, assume $f\in \mathcal{S}(\mathbb{R}^d)$ is a solution, then you can multiply the equation by $f$ and integrate by parts (everything converges appropriately) to find
$$ \int |\nabla f|^2 + m f^2 + \lambda f^{p+1} = 0 $$
Noting that $p+1$ is eve... | 5 | https://mathoverflow.net/users/3948 | 417210 | 169,965 |
https://mathoverflow.net/questions/417204 | -1 | I have the following question related to multivariable moment generating functions. Given two vectors $u,X\in\mathbb{R}^d$, I want to characterize the quantity $e^{u^T X}$ in terms of "powers" of the vector $X$. To see what I mean, note that by the power series expansion of the exponential:
$$e^{u^T X} = \sum\_{n=0}^\i... | https://mathoverflow.net/users/477962 | Multi-variable expansion of $e^{u^T X}$ as a power series in terms of tensors | Define the tensor $v^{\otimes k}$ with elements $[v^{\otimes k}]\_{i\_1,i\_2,\ldots i\_k}=v\_{i\_1}v\_{i\_2}\cdots v\_{i\_k}$ and denote the full contraction of two tensors by $A\odot B=\sum\_{i\_1,i\_2,\ldots i\_k}A\_{i\_1,i\_2,\ldots i\_k}B\_{i\_k,\ldots i\_2,i\_1}$, then you can write
$$e^{u^T X} = \sum\_{n=0}^\inft... | 1 | https://mathoverflow.net/users/11260 | 417213 | 169,967 |
https://mathoverflow.net/questions/414896 | 6 | I have been reading the famous [paper of Alves, Bonatti, and Viana](https://link.springer.com/article/10.1007/s002220000057) where they proved that there is an SRB measure for partially hyperbolic systems. Since I am new to this field, I have some basic questions.
1)All results were proved for $C^2$ diff, but I saw a... | https://mathoverflow.net/users/127839 | SRB measure and Gibbs u-state | The results are true for $C^{1+}$. [This paper](https://arxiv.org/pdf/1403.2937.pdf) that you mentioned uses $C^{1+}$ regularity. Actually, in this business the main thing that usually comes from $C^{1+}$ condition is control of distortion along an unstable disc.
For you second question, consider the following. Let $... | 1 | https://mathoverflow.net/users/117630 | 417216 | 169,968 |
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