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https://mathoverflow.net/questions/439855 | 2 | According to the standard definition, $\mathcal{S}\_0^{\beta}(\mathbb{R})$ is a subspace of smooth functions on $\mathbb{R}$ with the property that
\begin{equation}
\lvert x^k f^{(q)}(x) \rvert \leq CA^kB^q q^{\beta q}
\end{equation}
for some constants A, B, C that depend on each $f$. I am aware that $f$ must be compac... | https://mathoverflow.net/users/56524 | Specific estimation of the norm for a linearly transformed function in $\mathcal{S}_0^{\beta}(\mathbb{R}^n)$ | $\newcommand\be\beta\newcommand\A{\mathfrak A}\newcommand\R{\mathbb R}\newcommand\N{\mathbb N}$Suppose that
\begin{equation}
|x\_j^k F^{(q)}(x)| \le CA^kB^{|q|} q^{\beta q}
\end{equation}
for some positive real $A,B,C,\be$, all $k\in\N\_0:=\{0,1,\dots\}$, all $j\in[n]:=\{1,\dots,n\}$, all $x=[x\_1,\dots,x\_n]^\top\in\R... | 2 | https://mathoverflow.net/users/36721 | 439877 | 177,627 |
https://mathoverflow.net/questions/421682 | 2 | Let $G$ be a compact matrix Lie group under the Killing form metric $\langle \xi, \eta \rangle\_g = -\frac{1}{2}\text{tr}((g^{-1}\xi)^T(g^{-1}\eta))$ for $g \in G$ and $\xi,\eta \in T\_gG$. Let $C \subset G$ be a geodesically convex set. Pick finitely many $g\_1,...,g\_N \in C$ and define $\Omega$ to be the smallest cl... | https://mathoverflow.net/users/141449 | What does the boundary of convex hulls look like in matrix Lie groups? | I guess you wanted to say *smallest geodesic polytope* (not *polygon*).
It is unclear what is polytope in a the matrix Lie group, but it seems to require geodesic hypersurfaces.
They do not exist in most Riemannian manifold starting from dimension 3 and matrix Lie groups are not exceptional.
BTW, if you are interes... | 1 | https://mathoverflow.net/users/1441 | 439889 | 177,635 |
https://mathoverflow.net/questions/3347 | 38 | The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be better for him to try adjacent and less technical questions. So it's not of importance for my own work per se, but I'd be ... | https://mathoverflow.net/users/763 | Is the set of primes "translation-finite"? | As pointed out to me by the OP, Theorem 1.5 of [my recent paper with Tamar Ziegler](https://arxiv.org/abs/2301.10303) answers this question in the negative. (The situation has changed since the 2009 and 2013 comments due to the breakthrough [work of Zhang on prime gaps](https://mathscinet.ams.org/mathscinet-getitem?mr=... | 18 | https://mathoverflow.net/users/766 | 439890 | 177,636 |
https://mathoverflow.net/questions/439893 | 2 | In "Number: The Language of Science" (1930), Tobias Dantzig refers to what we call the base case of mathematical induction as "the induction step" (and refers to what we call the induction step as "the recurrence step" or "the proof of the hereditary property"). Was this standard terminology a century ago, or was Dantz... | https://mathoverflow.net/users/3621 | Terminology associated with mathematical induction | Not really an answer (too long for a comment) but I hope that this will be helpful:
In Jeff Miller's "Earliest Uses of Some Words of Mathematics" (<https://mathshistory.st-andrews.ac.uk/Miller/mathword/>) we find the following:
The term MATHEMATICAL INDUCTION was introduced by Augustus de Morgan (1806-1871) in 1838... | 2 | https://mathoverflow.net/users/10744 | 439896 | 177,638 |
https://mathoverflow.net/questions/439917 | 3 | Let $X$ be a smooth projective curve over a field $k$ and $K\_X$ be its canonical line bundle. By the Serre duality, $\text{H}^1(X,K\_X)$ is a one-dimensional $k$-vector space. On the other hand, $\text{H}^1(X,K\_X)=\text{Ext}^1(O\_X,K\_X)$ so this vector space corresponds to the set of extensions of $O\_X$ by $K\_X$. ... | https://mathoverflow.net/users/95601 | Extension of the trivial bundle by the canonical bundle on a curve | If $X$ is a smooth curve, the canonical bundle $\omega\_X$ is nothing but the bundle of differentials $\Omega\_X^1$, and the corresponding extension defines the **jet bundle**
$$
0 \to \Omega\_X^1 \to J\_X \to \mathcal{O}\_X \to 0.
$$
| 4 | https://mathoverflow.net/users/4428 | 439921 | 177,648 |
https://mathoverflow.net/questions/304119 | 9 |
>
> Given a bi-Lipschitz homeomorphism
> $\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), can one find a bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ such that
> $$
> \Psi|\_{\mathbb{B}^n(0,\frac{1}{2})}=\Phi|\_{\mathbb{B}^n(0,\frac{1}{2})}\ \ ?
> $$
>
>
>
It... | https://mathoverflow.net/users/121665 | Bi-Lipschitz extension | The following result answers the question by providing a locally bi-Lipschitz extension.
>
> **Theorem.** Given a bi-Lipschitz homeomorphism
> $\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), one can find a locally bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ su... | 1 | https://mathoverflow.net/users/121665 | 439924 | 177,650 |
https://mathoverflow.net/questions/439951 | 3 | I'm studing C. Petit's work "Faster algorithms for isogeny problems using torsion point images" ([link](https://eprint.iacr.org/2017/571.pdf)) and he talks about meet-in-the-middle approach/strategy for solve some isogenies problems.
Well, what does Petit mean with meet-in-the-middle approach/strategy?
I've read th... | https://mathoverflow.net/users/497497 | What is meant by a meet-in-the-middle approach? | The "meet in the middle approach", also known as "bidirectional search", is a method to find shortest paths in graphs. It was proposed by Pohl [1] and first used by Galbraith [2] to construct isogenies between elliptic curves $E$ and $E'$. One builds two trees of isogenies from both sides of $E$ and $E'$, and finds a c... | 3 | https://mathoverflow.net/users/11260 | 439953 | 177,659 |
https://mathoverflow.net/questions/439864 | 1 | I recently encountered a particular delay PDE in my work, the solution of which corresponds to the Laplace transform of some probability distribution. I'm having trouble to solve this equation. The solution $f(q,s)$ defined on $q\geq 0, s>1$ should verify $f(0,s) = 1 \forall s$ and $f(q, s\to \infty) = \delta\_{q,0}$ (... | https://mathoverflow.net/users/498690 | Solving a particular delay PDE $\partial_q f(q,s-1) = -\sqrt{s(2+s)}f(q,s)$ | Inspired by your comparison with the Hurwitz zeta function, which has the integral representation
$$ \zeta (s,a)={\frac {1}{\Gamma (s)}}\int \_{0}^{\infty }{\frac {x^{s-1}e^{-ax}}{1-e^{-x}}}\mathrm{d}x \:, $$
I tried to come up with an integral representation for your function $f$. The factor $s$ that comes out when ta... | 2 | https://mathoverflow.net/users/498819 | 439964 | 177,665 |
https://mathoverflow.net/questions/439918 | 7 | We know about volume: The $L\_{\infty}$ ball of radius one-half, i.e. the hypercube, has volume $1$ in all dimensions. On the other hand, I believe that for every $1 \leq p < \infty$, the volume of the inscribed $L\_p$ ball $\{x : \|x\|\_p \leq \tfrac{1}{2}\}$ goes to zero as the dimension $d \to \infty$. (For example,... | https://mathoverflow.net/users/29697 | Does the surface area of the unit Lp ball go to zero for all $p < \infty$? | The surface area of the unit $L^p$ ball (of radius $1$) goes to $0$ as $n$ goes to infinity, if $p<\infty$. Below I show it for $p>2$, but this is enough because if we have convex sets $A\subseteq B$, then the surface area of $B$ is bigger than that of $A$, which follows from the proof in [this MSE answer](https://math... | 5 | https://mathoverflow.net/users/172802 | 439968 | 177,666 |
https://mathoverflow.net/questions/439279 | 9 | From discussions [1](https://mathoverflow.net/questions/377032/more-identities-for-the-lambert-w-function), [2](https://mathoverflow.net/questions/377012/an-identity-for-the-lambert-w-function), @HenriCohen wrote a paper on [Lambert $W$-Function Branch Identities](https://arxiv.org/pdf/2012.11698.pdf) which includes id... | https://mathoverflow.net/users/113397 | Is $(m,n)=(2,3)$ the only solution to $\sum_\limits{k \in\Bbb Z}\frac1{(W_k(x)+1)^m}=\sum_\limits{k \in\Bbb Z}\frac1{(W_k(x)+1)^n}$? | The answer is "yes":
Define the generating function
$$
G(x,t) = \sum\_{k \in \Bbb Z}\log\left[1- \frac{t}{W\_k(x)}\right].\tag{1}
$$
Then,
$$
\sum\_{k \in \Bbb Z} \frac 1{(W\_k(x)+1)^m}
= -\frac{1}{\Gamma(m)} \, G^{(0,m)}(x,-1),\tag{2}
$$
where $G^{(0,m)}(x,-1)$ denotes the $m$-th derivative w.r.t. $t$ at $t=-1$.
Fro... | 5 | https://mathoverflow.net/users/90413 | 439969 | 177,667 |
https://mathoverflow.net/questions/439973 | 5 | For (hopefully) simplicity, let a premouse be defined coarsely as in Martin and Steel's 1994 paper, *Iteration Trees*.
Is (or is it consistent that) there is a premouse that is $\omega\_1$-iterable but not $(\omega\_1+1)$-iterable?
Note: if adding some extra conditions will get partial answers, that would also be v... | https://mathoverflow.net/users/9324 | Are there premice that are $\omega_1$-iterable but not $(\omega_1+1)$-iterable? | It is consistent (relative to large cardinals). There is an example given in Example 3.6 [here](https://arxiv.org/abs/2012.07185). For a brief summary: the model is the minimal proper class mouse $S$ such that $\mathbb{R}^S$ is closed under the $M\_1^\#$-operator. In particular, $M\_1^\#$ is a proper segment of $S$. An... | 8 | https://mathoverflow.net/users/160347 | 439979 | 177,670 |
https://mathoverflow.net/questions/439976 | 11 | For a finite graph $X$, let $A\_X$ denote the associated right-angled Artin group. Thus $A\_X$ is generated by the vertices of $X$ subject to the relations $[v,w]=1$ whenever vertices $v$ and $w$ are connected by an edge.
I have seen references to the following theorem in several places, but I can neither figure out ... | https://mathoverflow.net/users/498832 | Right-angled Artin groups that split as direct products | The place I've seen this is in Koberda's RAAG notes [here](https://users.math.yale.edu/users/koberda/raagcourse.pdf), see Corollary 2.15. This relies on the description of centralizers in Proposition 2.14, which is also proved in Behrstock and Charney's paper [here](https://link.springer.com/article/10.1007/s00208-011-... | 8 | https://mathoverflow.net/users/164670 | 439992 | 177,673 |
https://mathoverflow.net/questions/439966 | 2 | In Beauville's "Variétés de Prym et jacobiennes intermédiaires", Proposition 3.5, it is claimed that $\textrm{Corr}(T)$ is torsion-free for a smooth projective variety $T$. Here $$\textrm{Corr}(T) := \textrm{Pic}(T \times T)/ (\pi\_1^\*\textrm{Pic}(T) \oplus\pi\_2^\*\textrm{Pic}(T))$$ Why is this true? A reference woul... | https://mathoverflow.net/users/160814 | Reference for torsion-freeness of the group of correspondences on a smooth projective variety | It's not just torsion-free, we can actually compute it in terms of the Picard and Albanese variety of $T$, by the following classical result:
**Lemma.** *If $X$ and $Y$ are smooth projective varieties over a field $k$, then there is a canonical short exact sequence of group schemes*
$$0 \to \mathbf{Pic}\_X \times \ma... | 2 | https://mathoverflow.net/users/82179 | 439994 | 177,674 |
https://mathoverflow.net/questions/439972 | 11 | I am interested in seeing examples of research problems which fall into one of the two following categories:
1. A problem which is solved in the case of primes (or prime powers), but which remains open in the case of composite integers.
2. A problem which historically was first solved for primes, and then significant... | https://mathoverflow.net/users/138628 | What are examples of problems we know how to solve for primes (or prime powers), but not for composites? | There is a projective plane of order $N$ for every prime power $N$. The existence of projective planes of other orders is an open question; in particular, it is not known whether there is a projective plane of order $12$. See, e.g., <https://en.wikipedia.org/wiki/Projective_plane#Finite_projective_planes>
| 14 | https://mathoverflow.net/users/3684 | 439999 | 177,676 |
https://mathoverflow.net/questions/439983 | 1 | After the GNS representation for $C^{\*}$-algebras is presented in Thirring's book [Quantum mathematical physics](https://doi.org/10.1007/978-3-662-05008-8), the author states the following theorem.
>
> **The Spectral Theorem:** For any given Hermitian (self-adjoint) element $a$ of a $C^{\*}$-algebra $A$, every rep... | https://mathoverflow.net/users/152094 | GNS Representation — A theorem from Thirring’s book | The C${}^\*$-algebra $A$ is a red herring here. All the result is really saying is that if $T$ is a self-adjoint operator on a Hilbert space $H$ then we can find a family of measures $\mu\_i$ on $\sigma(T)$ and an isomorphism $H \cong \bigoplus L^2(\sigma(T), d\mu\_i)$ which takes $T$ to the operator of multiplication ... | 2 | https://mathoverflow.net/users/23141 | 440000 | 177,677 |
https://mathoverflow.net/questions/440010 | 3 | All manifolds will be assumed to be closed, oriented, and connected.
Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi\_1(f)$ is surjective.
>
> What is an example of a *non* $\pi\_1$-injective, degree one, self-map of a three-manifold?
>
>
>
If $\dim(M) = 2$, then $\pi\_1(f)$... | https://mathoverflow.net/users/363264 | Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold | In the book [Three-manifold groups](https://arxiv.org/abs/1205.0202) we find a giant flow chart showing what is known to follow from the assumption that $N$ is an irreducible, compact, orientable three-manifold with empty or toroidal boundary (such that $\pi\_1(N)$ is neither finite nor solvable). As one particular con... | 6 | https://mathoverflow.net/users/1650 | 440015 | 177,680 |
https://mathoverflow.net/questions/440012 | 3 | In Squier's short, yet influential, paper about the Burau representation, he made two conjectures that might have provided a proof for the faithfulness of the Burau representation (which we now know to be false by work of Bigelow, for instance).
**Intro.** The (reduced) Burau representation $\beta$ sends the Artin ge... | https://mathoverflow.net/users/151664 | Squier's conjecture on Burau at roots of unity | Presumably, this conjecture appears from the observation that the suitable powers of the standard generator $\sigma\_1$ lies in the kernel, although it looks that Squier missed other obvious elements of kernels as you mentioned.
I mention the paper by Funar and Kohno
On Burau’s representations at roots of unity.
<h... | 5 | https://mathoverflow.net/users/193957 | 440017 | 177,682 |
https://mathoverflow.net/questions/439932 | 4 | The question in the title seems a natural one to ask, and I suspect that it is already considered, even completely solved, somewhere in the literature. Although I prefer some explanation of the idea(s), relevant references are also appreciated.
| https://mathoverflow.net/users/40789 | Is every Riemannian metric conformally equivalent to one that is geodesically complete? | This is not hard to prove. The idea of a proof is as follows. Take any metric $g$ on a manifold $M$. Define $d:M\to \mathbb{R}\_+$ by saying that $d(x)$ is the infimum of all lengths of curves $\gamma:[0,b)\to M$ with $\gamma(0)=x$, such that $\gamma$ is proper (In particular: For every compact subset K\subset M, there... | 6 | https://mathoverflow.net/users/110127 | 440031 | 177,688 |
https://mathoverflow.net/questions/440035 | 6 | Let $A$ be an $n \times n$ invertible complex matrix. Let $Gr(k)=Gr(k,\mathbb{C}^n)$ be the complex $k$-Grassmannian, $1\leq k \leq n$. Since $A$ is invertible, it maps a $k$-dimensional subspace to a $k$-dimensional subspace, so it gives a function (which I'll call $A\_k$) on $Gr(k)$. The fact that matrices have eigen... | https://mathoverflow.net/users/322473 | Invariant subspaces for matrices via fixed points on Grassmannians | I think the Lefschetz fixed point theorem still applies. If a self-map M→M of a compact orientable manifold M has no fixed points than the Euler characteristic of M is zero. But if M is a complex Grassmannian then its odd Betti numbers vanish so the Euler characteristic is positive (since b\_0=1).
| 7 | https://mathoverflow.net/users/14830 | 440036 | 177,689 |
https://mathoverflow.net/questions/439996 | 7 | This is a curiosity question that came out of teaching abstract algebra.
Let $F$ be a field, and $n>1$ an integer.
Let $F^{n \leq n}$ be the $F$-algebra of all upper-triangular $n\times n$-matrices $\begin{pmatrix} a\_{1,1} & a\_{1,2} & \cdots & a\_{1,n} \\ 0 & a\_{2,2} & \cdots & a\_{2,n} \\ \vdots & \vdots & \ddo... | https://mathoverflow.net/users/2530 | Smallest faithful representation of an upper-triangular matrix quotient | Here's an elementary proof that $2n-2$ is a lower bound.
Suppose that
$$V\_1\xrightarrow{\alpha\_1}V\_2\xrightarrow{\alpha\_2}\dots\xrightarrow{\alpha\_{n-2}}V\_{n-1}\xrightarrow{\alpha\_{n-1}}V\_n$$
is a representation of the linearly ordered $A\_n$ quiver $Q$ that is a representation of $FQ/I$, where $I$ is the one... | 4 | https://mathoverflow.net/users/22989 | 440046 | 177,695 |
https://mathoverflow.net/questions/439995 | 1 | Let $k$ be a field, $m$ be a positive integer and $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x,y]$. Let $B$ be the quotient ring $R/xR$. Then $B$ is the finitely generated $k$-algebra $k[xy,xy^2,…,xy^m]$.
How do I compute the associate reduced ring for the $k$-algebra $B$? Is this reduced ring... | https://mathoverflow.net/users/477848 | How to compute the associated reduced ring for this finitely generated algebra? | In $R/xR$, all $xy^l, l<m$ are nilpotent. To see this, notice that $(xy^l)^m= x^{m-l}(xy^m)^l$. Thus, modulo nilpotents, $R/xR$ is generated by the single element $xy^m$ over $k$. The rest is clear.
| 1 | https://mathoverflow.net/users/9502 | 440053 | 177,698 |
https://mathoverflow.net/questions/440048 | 0 | Consider complex smooth hypersurfaces $X\subset\mathbb{P}^n$ and $Y\subset\mathbb{P}^m$ for $m,n\geq 4$ and a morphism $f\colon X\rightarrow Y$ which satisfies one of the properties
1. $f\_\*\mathcal{O}\_X\cong\mathcal{O}\_Y$ and $f\_\*\mathcal{O}\_X(1)\cong\mathcal{O}\_Y(1)$
2. $f\_\*\mathcal{O}\_X(i)\cong\mathcal{O... | https://mathoverflow.net/users/nan | morphisms between smooth hypersurfaces that preserve many line bundles | Your assumption about $f$ is very strong, it implies that $X \cong Y$ and $f$ is an isomorphism.
Indeed, let $Z \subset X$ be a general fiber of $f$. Then the assumption that $f\_\*\mathcal{O}\_X(1)$ is a line bundle implies that
$$
\dim H^0(Z, \mathcal{O}\_X(1)\vert\_Z) = 1.
$$
But $\mathcal{O}\_X(1)\vert\_Z$ is a v... | 4 | https://mathoverflow.net/users/4428 | 440054 | 177,699 |
https://mathoverflow.net/questions/440033 | 3 | Given a partition of the edges of $K\_{n,n}$ into $n$ colours, where each colour appears exactly $n$ times, prove that there exists a vertex incident to at least $\sqrt{n}$ colours.
| https://mathoverflow.net/users/497926 | For every partition of $E(K_{n,n})$ into $n$ colour classes of size $n$, there is a vertex incident to at least $\sqrt{n}$ colours | Here is a short proof. Thanks to David Speyer for simplifying an earlier proof of mine (see the comments below).
For each colour $i \in [n]$, let $a\_i$ be the number of vertices incident to an edge of colour $i$. Observe that $a\_i \geq 2\sqrt{n}$ for all $i$, where equality is obtained if and only if the edges of c... | 4 | https://mathoverflow.net/users/2233 | 440058 | 177,702 |
https://mathoverflow.net/questions/440069 | 1 | Let $X$ be a closed Riemannian manifold and consider the function $f\_n : X \times \cdots \times X \to \mathbb{R}$ where the domain of $f\_n$ is the $n$-fold cartesian product of $X$ and where $f\_n(p\_1,...,p\_n) = \sum\_{i \neq j} d(p\_i, p\_j)$ where $d$ is the distance function on $X$. The function $f\_n$ is invari... | https://mathoverflow.net/users/419791 | Isolated maxima for sum of distances of points on a manifold | Not necessarily. Consider the sphere $(\mathbb{S}^2,g)$ with its usual metric and give it a new metric $hg$, where $h\leq1$, $h$ has three local minima $h(p\_1)=0.7,h(p\_2)=0.8,h(p\_3)=0.9$ (where $p,q,r$ are close to one another) and $h=1$ outside very small neighborhoods of $p\_1,p\_2,p\_3$. Then Isom$(\mathbb{S}^2,h... | 1 | https://mathoverflow.net/users/172802 | 440070 | 177,705 |
https://mathoverflow.net/questions/440081 | 3 | Let $S$ be a separable irreducible Noetherian scheme and let $X$ be a projective smooth curve over $S$. Let $\mathcal F$ be a coherent sheaf on $X$ which is flat over $S$. Suppose the restriction $\mathcal F\mid\_{X\_s}$ of $\mathcal F$ on the fiber $X\_s$ is locally free for some point $s\in S$.
**Question:** is $\m... | https://mathoverflow.net/users/11750 | Is a coherent and flat sheaf locally free? | As pointed out in the comments, being a vector bundle at a point is (by definition) an open property and for coherent sheaves flat over a base, being a vector bundle at a point can be checked on fibers. This means,
**Prop** Let $f : X \to Y$ be a flat map of schemes and $\mathcal{F}$ a coherent $\mathcal{O}\_{X}$-mod... | 2 | https://mathoverflow.net/users/154157 | 440094 | 177,715 |
https://mathoverflow.net/questions/440102 | 4 | Fix a group $G$ and fix a presentation of $G$ as $\langle X\mid R\rangle$. A *presentationally finite* extension of $G$ is any group that can be presented as $H=\langle X\cup X'\mid R\cup R'\rangle$, where $X',R'$ are finite. (Mind, the natural homomorphism $G\to H$ may not be injective. If necessary, we can restrict o... | https://mathoverflow.net/users/54415 | Presentationally finite group "extensions" | The kind of presentation you are giving is called a relative presentation and has been studied extensively by Steve Pride and his coauthors. So relatively finitely presented over $G$ is probably the best name (I didn't search his papers to see if he uses this term.)
| 6 | https://mathoverflow.net/users/15934 | 440105 | 177,721 |
https://mathoverflow.net/questions/440040 | 8 | The categories of modules over commutative rings are especially notable Abelian categories. Wanting to extend this class a bit, I thought of this question:
Let $R$ be a commutative ring with $1$. Does there exist a Grothendieck topos $E$ such that $\mathrm{Ab}(E) \simeq R\text{-}\mathrm{Mod}$ (equivalence of categori... | https://mathoverflow.net/users/148161 | Is the category of modules over a commutative ring the category of abelian objects in a topos? | **No — in particular, not if $R$ has any non-integer rationals.**
Briefly: From $\newcommand{\Z}{\mathbb{Z}}\newcommand{\Ab}{\mathrm{Ab}}\newcommand{\RMod}{{R\text{-}\mathrm{Mod}}}\RMod$, we can recover $R$ as the ring of endomorphisms of the identity functor, $\newcommand{\End}{\mathrm{End}}\newcommand{\id}{\mathrm{... | 7 | https://mathoverflow.net/users/2273 | 440110 | 177,724 |
https://mathoverflow.net/questions/440099 | 7 | Let us define a pure vector state of a quantum system as a vector $\psi$ in a Hilbert space $\mathscr{H}$ with norm $\|\psi\| = 1$. Let $\mathscr{B}(\mathscr{H})$ be the Banach space of bounded linear operators on $\mathscr{H}$.
In the $C^{\*}$-algebra formulation of quantum mechanics, one defines observables as the ... | https://mathoverflow.net/users/152094 | Interpretation of spectral measures in quantum mechanics | (Small correction: We can take the observables to be the *self-adjoint* elements of $B(H)$, or any C${}^\*$-algebra, and in your whole discussion $A$ should be assumed self-adjoint.)
This can be reduced to the following more fundamental principle. Let $v$ be a unit vector in some Hilbert space $H$ and let $E$ be a cl... | 7 | https://mathoverflow.net/users/23141 | 440113 | 177,726 |
https://mathoverflow.net/questions/440131 | 0 | Let $\mu$ be a Radon measure on $[0, 1]$, and $f: [0, 1] \to \mathbb R$ a Borel measurable function.
**Question:** Is it true that for $\mu$ almost every $x \in [0, 1]$, we have
$$f(x) \leq \mu\text{-esssup}\_{[0, x]} \, f?$$
Here the esssup is taken with respect to $\mu$.
| https://mathoverflow.net/users/173490 | An inequality involving the essential supremum | Yes. It suffices to prove that for every rationals $p<q$ the set $A$ of those $x$ for which simultaneosly $\mu\text{-esssup}\_{[0,x]} f<p$ and $q<f(x)$ satisfies $\mu(A)=0$. Note that if $x\in A$, then $\mu(A\cap [0,x])=0$, otherwise we would get $\mu\text{-esssup}\_{[0,x]} f\geqslant q$. It remains to note that $A$ is... | 3 | https://mathoverflow.net/users/4312 | 440132 | 177,731 |
https://mathoverflow.net/questions/440120 | 2 | I am looking for a proof for the following fact: for $U>0,\beta>0,$ there exists $C>0,\epsilon>0$ such that $$\forall u\in [0,U],\alpha\in\left]0,1\right],\int\_0^u\int\_{[-1,1]^2}\int\_{[-1,1]^2} \frac{1}{r}e^{-\alpha^2|x-y|^2/r} \, dx\,dy\,dr\leq Cu^{\epsilon}\alpha^{-2\beta},$$ where $|\cdot|$ is the euclidean norm.... | https://mathoverflow.net/users/138491 | $\int_0^u\int_{[-1,1]^2}\int_{[-1,1]^2}\frac{1}{r}e^{-\alpha^2|x-y|^2/r} \, dx\,dy\,dr\leq Cu^{\epsilon}\alpha^{-2\beta}$ | $\newcommand{\al}{\alpha}\newcommand{\be}{\beta}\newcommand\ep\epsilon$The integral in question is
\begin{equation\*}
I(u,\al):=\int\_0^u \frac{dr}r \int\_{[-1,1]^2} dy \int\_{[-1,1]^2} dx\,e^{-\al^2|x-y|^2/r}.
\end{equation\*}
Using the substitution $r=\al^2 s$ (suggested by Giorgio Metafune), we get
\begin{equation... | 2 | https://mathoverflow.net/users/36721 | 440145 | 177,737 |
https://mathoverflow.net/questions/440152 | 5 | I am an undergrad senior math major taking a gap year looking to become an actuary. However, I still want to continue learning pure math. I've been looking for a relatively high level text to self study for the next few months. I'm already a good chunk through the first chapter of Lang's "Algebra" and everything is flo... | https://mathoverflow.net/users/498786 | Lang's "Algebra" as a self-study book | In my opinion, Lang's *Algebra* has an excellent choice of topics for someone who wants to do further work in algebraic number theory or algebraic geometry. I'm not sure whether I'd recommend it for self-study, however. My feeling is that the exposition, and the exercises, are rather uneven. If you're studying from it ... | 13 | https://mathoverflow.net/users/3106 | 440154 | 177,741 |
https://mathoverflow.net/questions/440088 | 5 | Background
----------
For a finite graph $G$, let $\tilde{G}$ denote the [universal cover](https://en.wikipedia.org/wiki/Covering_graph#Universal_cover) of $G$. For a vertex $v$, let $p\_{2n}(v)$ denote the number of paths of length $2n$ that start and end at $v$. The spectral radius of $\tilde{G}$, denoted $\rho(\ti... | https://mathoverflow.net/users/12176 | Can we calculate the spectral radius of the universal cover for specific graphs? | For the complete graph minus an edge $K\_n-e$, the spectral radius is the largest zero of
\begin{align\*}&x^{14}+(30-10 n) x^{12}+(2 n^{3}+21 n^{2}-202 n +357) x^{10}\\
&+(-10 n^{4}+26 n^{3}+456 n^{2}-2288 n +2888) x^{8} \\
&+(n^{6}-4 n^{5}+76 n^{4}-1520 n^{3}+9320 n^{2}-23056 n +20360) x^{6}\\
&+(-4 n^{7}+48 n^{6}-272... | 5 | https://mathoverflow.net/users/9025 | 440155 | 177,742 |
https://mathoverflow.net/questions/439306 | 3 | We assume ZFC+U.
A *category* is an ordered pair $(\operatorname{Ob} \mathcal{C},\operatorname{Mor} \mathcal{C},\operatorname{dom},\operatorname{codom},e,∘)$ of sets (not classes) and maps satifying some conditions.
Let $\mathbb{U}$ be a Grothendieck universe.
An element of $\mathbb{U}$ is called a *$\mathbb{U}$-set*... | https://mathoverflow.net/users/137654 | On the definition of small categories in SGA4 | You’re correct: read literally, those definitions are mismatched, for the reasons you give. The solution is to fix the definition of “$U$-small category” to say that “$\newcommand{\C}{\mathcal{C}}\newcommand{\mor}{\mathrm{mor}}\mor(\C)$ is $\newcommand{\ob}{\mathrm{ob}}U$-small” — this is what the authors of SGA4 clear... | 6 | https://mathoverflow.net/users/2273 | 440169 | 177,746 |
https://mathoverflow.net/questions/440181 | 28 | I was thinking about the idea that succession, addition, multiplication, exponentiation, tetration and so on form a sequence of operations where each is defined as a repeated self application of the previous one.
And then it struck me that the first 2 operations in this sequence are commutative but this breaks at exp... | https://mathoverflow.net/users/757 | Addition and multiplication are commutative but exponentiation and tetration are not. Do we know why? | Not really an answer, but too long for a comment: it's worth noting that if we assume that
* $f$ is associative,
* $g$ is associative,
* $g$ is cancellative for at least one $a$, meaning that $g(a,u)=g(a,v)$ implies $u=v$ for this particular $a$,
then $f$ and $g$ *must* be addition and multiplication.
Indeed, let... | 34 | https://mathoverflow.net/users/17064 | 440184 | 177,750 |
https://mathoverflow.net/questions/440179 | 9 |
>
> Find all continuous and bounded functions $g$
> with :
> $$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$
>
>
>
I have posted this question [here](https://math.stackexchange.com/questions/4630194/x-ens-functional-equation), but received no answer.
| https://mathoverflow.net/users/110301 | How may I find all continuous and bounded functions g with the following property? | $\newcommand\de\delta$Considering $g$ a distribution (in the generalized-function sense), let $\hat g$ be the Fourier transform of $g$. Then your functional equation yields
$$4\hat g(t)=e^{it}\hat g(t)+e^{-it}\hat g(t)+e^{i\pi t}\hat g(t)+e^{-i\pi t}\hat g(t),$$
or
$$(\cos t+\cos\pi t-2)\hat g(t)=0,$$
for real $t$.
T... | 18 | https://mathoverflow.net/users/36721 | 440186 | 177,751 |
https://mathoverflow.net/questions/440150 | 12 | In the paper [The Continuumproblem](https://www.pnas.org/doi/abs/10.1073/pnas.24.2.101), Felix Bernstein introduces a new axiom and uses it to conclude the continuum hypothesis.
(1) As the paper is relatively old and the writing style is somehow informal, I am wondering if there is a more exact and concrete proof of ... | https://mathoverflow.net/users/11115 | Bernstein's proof of the continuum hypothesis | This is really a long comment. This paper has been reviewed twice by zbMATH: [one by H. B. Curry](https://zbmath.org/0019.00903), which is not informative; [another by W. Ackermann](https://zbmath.org/64.0035.02), which is in German. The following is the (manipulated) translation of part of his review by DeepL.
>
>... | 11 | https://mathoverflow.net/users/38866 | 440190 | 177,755 |
https://mathoverflow.net/questions/440185 | 6 | The objects of the desired category are epimorphisms of sets $E \to B$ (in what follows, the notation $E/B$ will be used instead of the arrow). Is it possible to naturally define morphisms such that:
1. $E/B \cong (E \times A) / (B \times A)$ for each inhabited $A$ and there are no other non-trivial isomorphisms (non... | https://mathoverflow.net/users/148161 | Is there a topos of quotients of sets? | I'll argue Requirement (1) and (2) together are impossible - at least not without making some highly unnatural construction. To be honest the main problem is with (1) alone.
Informally, the idea is that being of the form $E \times A \to B \times A$ is a structure, there are many way of being of this form, and each su... | 8 | https://mathoverflow.net/users/22131 | 440191 | 177,756 |
https://mathoverflow.net/questions/431429 | 14 | It was shown by [Hamilton](https://www.degruyter.com/document/doi/10.1515/9781400882571-013/html#:%7E:text=AN%20ISOPERIMETRIC%20ESTIMATE%20FORTHE%20RICCI%20FLOWON%20THE%20TWO-SPHERE,%E2%80%94%20RQij%20where%20R%20is%20the%20scalar%20curvature.) in the 1990s that the isoperimetric ratio $C\_H$ on the $2$-sphere improves... | https://mathoverflow.net/users/119114 | Does the Cheeger constant satisfy a heat-type equation? | Answer to (2) is negative. For a counterexample, consider a $2$-sphere and blow up the radius. The isoperimetric ratio $C\_H$ remains the same, but the Cheeger constant $h$ shrinks, essentially because $C\_H$ is a dimensionless quantity which is independent of the scaling, whereas $h$ is not.
Actually in Lemma 5.85 o... | 3 | https://mathoverflow.net/users/119114 | 440193 | 177,758 |
https://mathoverflow.net/questions/440196 | 12 | **Introduction**
This question is about Picard spectra for the symmetric monoidal $\infty$-category of spectra. We say that a spectrum $X$ is invertible if there is another spectrum $Y$ such that $X\wedge Y\simeq S^0$. It is well-known that any such $X$ is equivalent to $S^n$ for some $n\in\mathbb{Z}$, and that the s... | https://mathoverflow.net/users/10366 | Why are ordinary spheres not strictly invertible? | An $E\_{\infty}$ structure extending the $E\_1$ structure on $R(n)$ in particular yields maps $(\mathbb{S}^{2n})^{\otimes p}\_{hC\_p} \to \mathbb{S}^{2pn}$ splitting the inclusion of the bottom cell. The left hand spectrum can be viewed as homotopy orbits of the $C\_p$ action on the representation sphere $\mathbb{S}^{2... | 14 | https://mathoverflow.net/users/39747 | 440199 | 177,760 |
https://mathoverflow.net/questions/440210 | 4 | For a semisimple complex Lie algebra $\frak{g}$ it is well known that irreducible finite-dimensional representation are not characterised by their dimension.
More formally, let us define an equivalence relation on dominant weights by $\lambda ~ \mu$, for $\lambda, \mu \in \mathcal{P}^+$, is it holds that
$$
\mathrm{d... | https://mathoverflow.net/users/378228 | Number of representations of a semisimple Lie algebra of any given dimension | For $\mathfrak{sl}\_2\times \mathfrak{sl}\_2$, the number of irreps of dimension $n$ is the number of factorizations $n=n\_1n\_2$ (you tensor the irreps of the two $\mathfrak{sl}\_2$'s), so there's no upper bound.
For $\mathfrak{sl}\_3$, the Weyl dimension formula says that these dimensions are $\frac{1}{2}(n\_1+1)(n... | 6 | https://mathoverflow.net/users/66 | 440220 | 177,765 |
https://mathoverflow.net/questions/440223 | 6 | I checked some relations between primes, here $1<n<10^5$ and $p\_n$ is the $n$th prime.
$a) p\_n^{1/3} - p\_{n-1}^{1/3}<1/2$
$b) p\_n^{1/n} - p\_{n-1}^{1/n}<1/n $
$c) (\log p\_n)^{1/2} - (\log p\_{n-1})^{1/2} < 1/4$
$d) (\log p\_n)^{1/n} - (\log p\_{n-1})^{1/n} < 1/n^X, n\geq7,X=2$
In $d)$ I tried to find a l... | https://mathoverflow.net/users/126334 | Some conjectures about prime gaps | The first three statements are true for $n$ sufficiently large. The fourth statement is equivalent to a very strong bound on prime gaps (which has a chance to hold but is right on the edge and hopeless to prove or disprove at present), while any $X>2$ produces a false bound for all $n>n\_0(X)$.
Regarding a), a classi... | 9 | https://mathoverflow.net/users/11919 | 440227 | 177,767 |
https://mathoverflow.net/questions/440114 | 9 | Consider a system of $n$ linear equations with $n$ unknowns, all of whose coefficients and right hand sides are nonnegative integers, with a unique solution consisting of nonnegative rational numbers. Is it always possible to solve the system using restricted subtraction-moves that only let us subtract one equation fro... | https://mathoverflow.net/users/3621 | Solving systems of linear equations without introducing negative numbers | Here is a proof that it is always possible by keeping at most $n+1$ equations throughout.
Suppose the system $Ax=b$ has a unique solution $c \in \mathbb{Q}\_{\geq 0}^n$, where $A \in \mathbb{Z}\_{\geq 0}^{n \times n}$ and $b \in \mathbb{Z}\_{\geq 0}^n$. Note that the equation $x\_1=c\_1$ can be written as a linear co... | 4 | https://mathoverflow.net/users/2233 | 440239 | 177,770 |
https://mathoverflow.net/questions/435153 | 1 | If I sample three points independently, uniformly at random on an $n$-dimensional sphere of radius $R$, what is the probability density function of their [polar sine](https://en.wikipedia.org/wiki/Polar_sine)?
More generally, for $k<n$ if I sample $k$ points independently uniformly at random on an $n$-dimensional sph... | https://mathoverflow.net/users/489157 | Probability density function for the polar sine of uniformly distributed points on the sphere | For $k=3$, $0\le s\le 1$ it is
$$c\_ns^{n-3}\left(\frac\pi2-\arcsin(s)\right)$$
where
$$c\_n=\begin{cases}\frac2\pi(n-2)\frac{2^{n-2}\left(\frac{n-2}2\right)!}{(n-2)!}&\text{ for $n$ even}\\
\frac{(n-2)!}{2^{n-3}\left(\frac{n-3}2\right)!}&\text{ for $n$ odd}.\end{cases}$$
Using polyspherical coordinates, the cumulati... | 0 | https://mathoverflow.net/users/489157 | 440241 | 177,771 |
https://mathoverflow.net/questions/440235 | 3 | I am trying to figure out whether or not the following property is first-order expressible in the language of groups.
$$\text{$G$ has a subgroup $H$ with which it forms a Frobenius pair $(H,G)$.}$$
My search online yielded no answers. Even if there is a positive or negative result, it seems to be not very well-known.
... | https://mathoverflow.net/users/498986 | Is having a Frobenius pair first-order expressible in the language of groups? | I'll answer here positively the natural variant of the question:
>
> does there exist a 1st order sentence such that for every finite group $G$, the group $G$ is Frobenius over some subgroup iff the given sentence holds in $G$.
>
>
>
In a sense, this sound to me like a more natural question, since Frobenius pa... | 4 | https://mathoverflow.net/users/14094 | 440254 | 177,775 |
https://mathoverflow.net/questions/440250 | 0 | Let $\lambda \in \mathcal{P}^+$ be a dominant weight for $\frak{sl}(n,\mathbb{C})$. When does it hold that the zero weight space, of the associated finite-dimensional $L(\lambda)$, is non-trivial?
What about the same question for the other seires?
| https://mathoverflow.net/users/378228 | Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules | I'm not quite sure this question rises to the level of MathOverflow, which is why I initially posted only a comment, but at the request of the question-asker I am converting my comment to an answer.
For any semisimple Lie algebra $\mathfrak{g}$ and dominant weights $\mu,\lambda \in P^+$, the condition for the $\mu$-w... | 2 | https://mathoverflow.net/users/25028 | 440260 | 177,776 |
https://mathoverflow.net/questions/440236 | 1 | [Originally posted at [math.stackexchange](https://math.stackexchange.com/q/4626210/573047) without answer]
Consider a family $\mathcal{F}$ of $n=|\mathcal{F}|$ sets, $\emptyset \not\in \mathcal{F}$ and an universe $U(\mathcal{F})$ of $q=|U(\mathcal{F})|$ elements.
It is known that at least a fraction $r\binom{n}2$... | https://mathoverflow.net/users/136218 | Improving a lower bound for the minimum of the maximum frequency of an element in a family of sets | I do not think the bound can be improved without further assumptions. For example, consider the family $\mathcal{F}$ consisting of all singleton subsets of $\{1, \dots, n-1\}$ together with $\{1, \dots, n-1\}$. Here $q=n-1$, and $r\binom{n}{2}=n-1$. Thus, the smallest $m$ for which $$\binom{m}{2} \geq \frac{r\binom{n}{... | 2 | https://mathoverflow.net/users/2233 | 440263 | 177,777 |
https://mathoverflow.net/questions/440237 | 7 | Has Pontryagin duality been extended to condensed abelian groups? The obvious approach being to define $\hat M$ as the internal hom to the circle group. Is it true that $\hat{\hat M}=M$ with this definition?
| https://mathoverflow.net/users/473423 | Condensed Pontryagin duality | This cannot be true for all condensed abelian groups. Indeed, in [this answer](https://mathoverflow.net/a/356261/82179) to [Are there (enough) injectives in condensed abelian groups?](https://mathoverflow.net/questions/352448/are-there-enough-injectives-in-condensed-abelian-groups), Scholze explains that there are no n... | 7 | https://mathoverflow.net/users/82179 | 440272 | 177,781 |
https://mathoverflow.net/questions/438912 | 2 | Let $\mathbb{N}\_+$ denote the set of positive integers and let $\mathbb{N}\_0 = \mathbb{N}\_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}\_+$ we let the *approximation radius* of $n$ be $$\text{rad}\_\alpha(n) = \min\Big\{\Big|\alpha-\frac{x}{n}\Big|:x\in\mathbb{N}\_0\text{ and } x \leq n\... | https://mathoverflow.net/users/8628 | Measuring how "close" $\alpha\in[0,1]\setminus\mathbb{Q}$ is to being rational | I think the answer is yes.
We will assume that $\text{appr}\_{\alpha}$ is defined for $\alpha\in\mathbb{Q}$ also but $\text{appr}\_{\alpha}(n) = \infty$ for all large $n$. Consider another function $\text{Appr}\_{\alpha}(n)\colon \mathbb{N}\to\mathbb{N}\_+^2$ which will return the pair $(x, \text{appr}\_{\alpha}(n))$... | 1 | https://mathoverflow.net/users/498423 | 440277 | 177,784 |
https://mathoverflow.net/questions/400602 | 9 | In the known paper *On the reconstruction of topological spaces from their group of homeomorphisms* by Matatyahu Rubin several deep reconstruction theorems of the form "if $X$ and $Y$ are topological spaces in a broad class of spaces $K$ and there is an isomorphism between $\mathrm{Homeo}(X)$ and $\mathrm{Homeo}(Y)$, t... | https://mathoverflow.net/users/49381 | On a result by Rubin on elementary equivalence of homeomorphism groups and homeomorphisms of the underlying spaces | There's a new paper that proves something even stronger for compact manifolds: <https://arxiv.org/abs/2302.01481>. I don't know about noncompact manifolds, though, and it seems neither do the authors.
| 4 | https://mathoverflow.net/users/499018 | 440283 | 177,787 |
https://mathoverflow.net/questions/440264 | 0 | I am learning local cohomology from Hartshorne’s Local Cohomology book.
My question is about the notion of essentially zero inverse system of abelian groups, which is defined to be an inverse system of abelian groups $(M\_{m})$ indexed by the non-negative integers, such that for every $m$, there is some integer $m’\geq... | https://mathoverflow.net/users/477848 | Essentially zero inverse system of abelian groups | I don't like indexing with primes, so let's consider an exact sequence of inverse systems:
$$ 0 \to (A\_m) \to (B\_m) \to (C\_m) \to 0 $$
Now we are assuming that $(A\_m)$ and $(C\_m)$ are essentially zero inverse systems, which means that for a given $m$ we can find a larger $m'$ so that both $A\_{m'} \to A\_m$ an... | 3 | https://mathoverflow.net/users/184 | 440284 | 177,788 |
https://mathoverflow.net/questions/440176 | 3 | $\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}$Recall that $n$-simplices of the classifying space (simplicial set) $BG$ of a group are orbits of the diagonal action by shifts
on $n+1$-tuples of its elements. But what if we take elements from some other set the group acts on ?
That is, the same definition... | https://mathoverflow.net/users/494312 | Defining the classifying space of a group acting on a set | As in Tom Goodwillie's comment, if you take the construction that you discuss for a $G$-set $X$, the equivariant homotopy type of the space obtained before you quotient out by the action of $G$ is called a classifying space for a family of subgroups of $G$; the family being the subgroups that fix some point of $X$.
... | 4 | https://mathoverflow.net/users/124004 | 440296 | 177,794 |
https://mathoverflow.net/questions/440271 | 5 | Let $u \in W^{1,2}(\mathbb{R}^2)$ be a given function satisfying $$\frac{1}{|B\_r|}\int\_{B\_r(x)} |\nabla u|^2 dy \leq \frac{1}{r^{\delta}}$$ for all $r \leq 1$, $x \in \mathbb{R}^2$ and some fixed $\delta \in (0,1)$.
Hueristically, this looks like that the gradient cannot go to infinity too quickly. In this sense, ... | https://mathoverflow.net/users/100801 | Higher integrability for Sobolev functions | Since the [OP asked for a discussion of features](https://mathoverflow.net/questions/440271/higher-integrability-for-sobolev-functions#comment1135653_440271), I provide one by way of an explanation of Christian Remling's counterexample:
Holder's inequality states that
$$ | \int fg | \leq \| f\|\_p \|g\|\_q $$
if $p^{... | 7 | https://mathoverflow.net/users/3948 | 440301 | 177,797 |
https://mathoverflow.net/questions/440319 | 3 | Let $C$ be a smooth projective geometrically connected curve of genus 2 defined over a number field $k$. Here are some definitions:
The *index* $I$ of a curve $C$ is the greatest common divisor of all effective divisors $D \in \mathrm{Div}(C)$. Equivalently, it is the greatest common divisor of the degrees $[L:k]$, w... | https://mathoverflow.net/users/172132 | If a genus 2 curve has no $k$-rational points, can it have a $k$-rational divisor class of degree 1? | Rummaging a bit through the LMFDB turns up the curve
<https://www.lmfdb.org/Genus2Curve/Q/129600/b/129600/1>
with equation $y^2 = -(2x^3+3x-2)(2x^3+4x^2+x-2)$
with no rational points (indeed trivial Mordell-Weil group)
but a degree-3 divisor $2x^3+3x-2 = y = 0$.
| 7 | https://mathoverflow.net/users/14830 | 440321 | 177,803 |
https://mathoverflow.net/questions/440304 | 15 | The triangle inequality seems much stronger than necessary for a lot of analysis. So I will define a "loose metric" on a set $X$ to be a function $d \colon X \times X \to [0,\infty)$ with the following properties:
* $d(x,y)=d(y,x)\ $ for all $x,y \in X;$
* every $\, x,y \in X\ $ has $\ d(x,y)=0\ $ if and only if $\ x... | https://mathoverflow.net/users/15570 | Is the topology generated by this weaker notion of a metric necessarily metrisable? | For a loose metric $d$ as above, we can consider the function
$$d\_1(x,y):=\sup\{|d(x,z)-d(y,z)|;z\in X\}.$$
It is easy to verify that $d\_1$ is a metric, and $d(x,y)\leq d\_1(x,y)\leq\rho(d(x,y))$ for all $x,y$, so $d\_1$ and $d$ generate the same topology.
Edit: As mentioned in the comments, we can let $d\_2=\min... | 17 | https://mathoverflow.net/users/172802 | 440322 | 177,804 |
https://mathoverflow.net/questions/440312 | 0 | Let $g: \mathbb R \to \mathbb R$ be a function of locally bounded variation, and $f$ a locally integrable function with respect to $dg$, the Lebesgue–Stieltjes measure associated with $g$.
Let $\eta$ be a smooth, compactly supported function. Define
$$F(x) := \int\_0^x f(s) \, dg(s).$$
**Question:** Is it true th... | https://mathoverflow.net/users/173490 | Does convolution commute with Lebesgue–Stieltjes integration? | If (say) $g$ is absolutely continuous (with an almost-everywhere derivative $g'$), then the left-hand side of your identity is
$$L(\eta):=(F\*\eta)(x)=\int dy\,\eta(x-y)\int\_0^y ds\,g'(s) f(s)$$
and its right-hand side is
$$R(\eta):=\int\_0^x dg(s)\,(f\*\eta)(s)=\int\_0^x ds\,g'(s)\,\int dt\,f(t)\eta(s-t).$$
We see th... | 3 | https://mathoverflow.net/users/36721 | 440327 | 177,806 |
https://mathoverflow.net/questions/440307 | 0 | Consider $p(u,x)=(4\pi u)^{-d/2}e^{-\frac{|x|^2}{4u}},u>0,x\in \mathbb{R}^d.$
Prove that there exists $C>0$ such that for all $0<u\leq v,r\in[0,u[,$ $$\int\_{\mathbb{R}^d}|p(v-r,x)-p(u-r,x)|\, dx \leq C\frac{v-u}{u-r}$$
Any ideas how to prove it?
| https://mathoverflow.net/users/138491 | $\int_{\mathbb{R}}|p(v-r,x)-p(u-r,x)|\,dx \leq C\frac{v-u}{u-r}$ | $\newcommand\R{\mathbb R}$Letting $s:=u-r$ and $t:=v-r$, rewrite the inequality in question as
\begin{equation\*}
\int\_{\R^d}dx\,|p(t,x)-p(s,x)| \le C\Big(\frac ts-1\Big) \tag{0}\label{0}
\end{equation\*}
given that $0<s\le t<\infty$.
Note that
\begin{equation\*}
|p(t,x)-p(s,x)|\le\int\_s^t dw\,|D\_w p(w,x)|,
\end... | 1 | https://mathoverflow.net/users/36721 | 440334 | 177,810 |
https://mathoverflow.net/questions/440345 | 1 | Imagine to have a set of random variables $\{ X\_i \}\_{i=1}^{n}$ independent (**Non** identically distributed). In these scenario, if the Lindeberg's condition hold we can extend the result of the CLT, *i.e.*, calling $\mu\_i = \mathbb{E}(X\_i)$ , $s\_n^2 = \sum\_i \left(\mathbb{E}(X\_i^2) - \mathbb{E}(X\_i)^2\right)$... | https://mathoverflow.net/users/174176 | Hypothesis to guarantee Lindeberg's condition | $\newcommand\ep\varepsilon$No, of course the Lindeberg condition is not necessary for the CLT.
E.g., for each natural $i$ let
\begin{equation\*}
X\_i:=Z\_i+Y\_i,
\end{equation\*}
where $Z\_i\sim N(0,1)$, $P(Y\_i=i!)=2^{-i-1}=P(Y\_i=-i!)$, $P(Y\_i=0)=1-2^{-i}$, and the random variables (r.v.'s) $Z\_1,Y\_1,Z\_2,Y\_2,\... | 2 | https://mathoverflow.net/users/36721 | 440362 | 177,823 |
https://mathoverflow.net/questions/440326 | 3 | I apologize in advance if the answer to this question is well-known to experts.
So let $F$ be a $p$-adic field, and $G$ a reductive group over $F$. Let $P$ be an $F$-parabolic subgroup of $G$ and $M$ the Levi of $P$. (For convenience, I just use the same notations for an algebraic group and a group of rational points... | https://mathoverflow.net/users/149922 | $L$-parameters and parabolic induction | I think what you say is a part of the local Langlands Conjecture. See Conjecture 4.1 (7)(8)(10)in Kaletha and Taibi's Lecture notes on LLC for IHES 2022. The local Langlands conjectures for different groups should be compatible with parabolic inductions. As a special case, you can consider spherical representations and... | 3 | https://mathoverflow.net/users/168680 | 440368 | 177,824 |
https://mathoverflow.net/questions/440353 | 3 | Let $S$ be the set of real matrices with at least one real logarithm. For some couple of its elements, for example those with at least (one real logarithm each with submultiplicative norm smaller than $\ln 2/4$), the product of the couple can be proven to lie in $S$ via the Baker-Campbell-Hausdorff formula.
My questi... | https://mathoverflow.net/users/113020 | Is the set of real matrices with at least one real logarithm closed under multiplication? | This is already not true for $2$-by-$2$ matrices: Consider
$$
A = \begin{pmatrix}2 & 0 \\0 &\frac12\end{pmatrix}\quad
\text{and}\quad
B = \begin{pmatrix}-1 & 0 \\0 &-1\end{pmatrix}.
$$
$A = \exp\bigl(\ln(2) K\bigr)$ and $B = \exp\bigl(\pi J\bigr)$ for
$$
K= \begin{pmatrix}1 & 0 \\0 &-1\end{pmatrix}\quad\text{and}\quad ... | 13 | https://mathoverflow.net/users/13972 | 440373 | 177,828 |
https://mathoverflow.net/questions/440318 | 6 | Suppose $A,G,H$ are finitely generated groups and $A$ is quasi-isometrically embedded into $G$ and $H$. Does it follow that the natural embeddings of $G$ and $H$ into $G\*\_AH$ are quasi-isometric?
I can more or less see that this is true if $A$ is finite (in particular, for the free product), or, more generally, if ... | https://mathoverflow.net/users/54415 | Are the canonical embeddings into $G*_AH$ quasi-isometric? | In the setting when $G, H$ are hyperbolic groups and $A$ is almost malnormal in $G, H$ (the one you are actually interested in, per your comments), the positive answer is given in
*Kapovich, Ilya*, [**The combination theorem and quasiconvexity**](http://dx.doi.org/10.1142/S0218196701000553), Int. J. Algebra Comput. 1... | 6 | https://mathoverflow.net/users/39654 | 440374 | 177,829 |
https://mathoverflow.net/questions/440377 | 1 | I'm stuck with the following problem:
In Petit's work "[Faster Algorithms for Isogeny Problems using Torsion Point Images](https://eprint.iacr.org/2017/571)", p. 8, he says that we can deduce $\ker \psi\_{N\_2}$ knowing the action of $\psi = \psi\_{N\_1'}\circ\psi\_{N\_2}$ on $E[N\_2],$ where $\psi\_{N\_1'}$ and $\ps... | https://mathoverflow.net/users/497497 | Deduce kernel of isogeny from action on torsion points | The key details here are
1. $N\_2$ is "smooth by assumption", so solving discrete logarithms in $E[N\_2]$ is supposed to be easy (using Pohlig-Hellman; how easy this is depends on how smooth $N\_2$ is).
2. $\gcd(N\_1,N\_2) = 1$, so the kernel of $\psi\_2$ is equal to the kernel of the restriction of $\psi$ to $E[N\_2... | 2 | https://mathoverflow.net/users/156215 | 440386 | 177,833 |
https://mathoverflow.net/questions/440379 | 2 | I was wondering whether anything is known on the following: Let
$h\_k (x)= (-1)^k e^{x^2/2} \frac {d^k}{dx^k} \, e^{-x^2/2}$, $k \geq 0$, be the classical
Hermite polynomials ($h\_0(x) = 1$, $h\_1(x) = x$,
$h\_2(x) = x^2 -1$, $h\_3(x) = x^3 - 3x$, ...).
Is there anything known on the asymptotic distributional behavior ... | https://mathoverflow.net/users/499080 | Random variables with density distributions given by squared Hermite polynomials | This is related to the probability density of the [quantum harmonic oscillator.](https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator) (The Hermite polynomials are the eigenstates of that problem.) For large $k$ the density
$$P\_k(x)=\frac{1}{\sqrt{2\pi}k!} [h\_k(x)]^2 \, e^{-x^2/2},$$
normalized to unity, has the... | 0 | https://mathoverflow.net/users/11260 | 440397 | 177,836 |
https://mathoverflow.net/questions/440365 | 4 | Let $A$ be a commutative ring, and $M$ be an $A$-module, and $M^\*$ be $\mathrm{Hom}\_A(M,A)$. Let $f$ be the map from $M \otimes\_A M^\*$ to $\mathrm{Hom}\_A(M,M)$, such that, for all $x=\sum\_i a\_i \otimes b\_i \in M \otimes\_A M^\*$, $f(x)$ is the homomorphism $y \in M \mapsto \sum\_ib\_i(y)a\_i \in M$.
Is it true ... | https://mathoverflow.net/users/456131 | Tensor product and homomorphism | Here is how I would start.
Let $S$ be the class of all pairs of $A$-modules $(M,N)$ such that the canonical map
$$M \otimes N^\* \to \hom(N,M), ~ m \otimes \omega \mapsto (n \mapsto \omega(n) \cdot m)$$
is a monomorphism. Both sides are additive functors in both variables. It follows formally that $S$ is closed under... | 8 | https://mathoverflow.net/users/2841 | 440399 | 177,838 |
https://mathoverflow.net/questions/440394 | 5 | Consider the space $\mathcal{S}'$ of functions $\mathbb{R}^n\to\mathbb C$ that are (real-)analytic and with exponential decay at infinity. This is an analogue of [Schwartz space](https://en.wikipedia.org/wiki/Schwartz_space), but real-analytic rather than smooth. (Exponential decay [suffices](https://mathoverflow.net/q... | https://mathoverflow.net/users/499084 | Real-analytic analogue of Schwartz functions | I'll assume that $n=1$.
I'd say that the natural choice is to consider the space $A$ of functions $f:\Bbb{R\to C}$ such that for some $c>0$, $f$ extends analytically to the strip $|\Im(z)|< c$ and $f(x+iy)=O(e^{-c|x|})$ in that strip.
Due to the Cauchy integral theorem this implies that the same holds for $\hat{f}$... | 4 | https://mathoverflow.net/users/84768 | 440403 | 177,839 |
https://mathoverflow.net/questions/440404 | 0 | I have problems to understand a proof in [this paper](https://www.pierrickdartois.fr/homepage/wp-content/uploads/2022/04/seminar_report.pdf) by Pierrick Dartois on Abelian varieties:
**Theorem 1.13 (rigidity lemma).** Let $ \varphi: X \times\_k Y \to Z$ be a morphism of $k$-schemes. Assume that $X$ is proper and geom... | https://mathoverflow.net/users/108274 | Proof of rigidity lemma | If $\varphi\_{\bar{k}}(X\_{\bar{k}} \times\_{\bar{k}} Y\_{\bar{k}})=\{z\_0\}$ then since $\varphi(X\times Y) \subset \varphi\_{\bar{k}}(X\_{\bar{k}} \times\_{\bar{k}} Y\_{\bar{k}})$ we also have $\varphi(X\times Y)\subset\{z\_0\}$.
| 2 | https://mathoverflow.net/users/327 | 440408 | 177,841 |
https://mathoverflow.net/questions/440251 | 2 | $\DeclareMathOperator\codim{codim}$Let $X=V(I)$, $Y=V(J)$ be two affine varieties. I'd like to know possibile strategies to understand when their schematic intersection, i.e. $X\cap Y=V(I+J)$, is reduced.
I am aware that reduceness is equivalent to Serre's conditions $R\_0$+$S\_1$ (at least under Noetherian hypothesis)... | https://mathoverflow.net/users/95513 | How to show that the intersection of two certain affine varieties is reduced? | It seems to me that if there are no $Z$'s, then this works. Indeed, both $X$ and the $Y\_i$ are linear spaces, so the intersection $X\cap Y$ is just a union of pairwise different linear spaces, so it is reduced. I think one could write down this with equations.
However, adding the $Z\_i$'s is problematic. I think the... | 1 | https://mathoverflow.net/users/10076 | 440426 | 177,844 |
https://mathoverflow.net/questions/440289 | 4 | $C/ \Bbb{Q}: 3X^3 + 4Y^3 + 5Z^3 = 0$ is known to be a nontrivial element of the Tate–Shafarevich group of the elliptic curve $E/\Bbb{Q}:X^3 + Y^3 + 60Z^3 = 0$. It is also an example of an abelian variety for which finiteness of Sha is known. In fact, $|\mathrm{III}(E/\Bbb{Q})| = 3^2$.
But I have never seen the proof ... | https://mathoverflow.net/users/144623 | Tate–Shafarevich group of Jacobian of Selmer curve $3X^3 + 4Y^3 + 5Z^3 = 0$ | $\DeclareMathOperator{\sha}{Ш}$
I am not sure that the proof that Sha has order 9 is anywhere spelled out in full. Here the ideas how to do it.
First, that the order of $C$ is three in the $\sha$ is just saying that it has a point over a field of degree 3 (index=period), which is obvious, and none of degree 1, which ... | 6 | https://mathoverflow.net/users/5015 | 440441 | 177,846 |
https://mathoverflow.net/questions/440348 | 1 | Let $\frak{g}$ be a complex semisimple Lie algebra with root lattice $Q$ and positive weight space $P^+$. Let $\lambda, \mu \in Q \cap P^+$, with corresponding respective fin-dim irreducible representations $V\_{\lambda}$ and $V\_{\mu}$. Form the tensor product $V\_{\mu} \otimes V\_{\lambda}$ and then decompose it into... | https://mathoverflow.net/users/378228 | Tensoring irreducible representations corresponding to root lattice elements | Just to summarize what was mentioned in the comments ([1](https://mathoverflow.net/questions/440348/tensoring-irreducible-representations-corresponding-to-positive-root-lattice-ele#comment1136048_440348) [2](https://mathoverflow.net/questions/440348/tensoring-irreducible-representations-corresponding-to-positive-root-l... | 3 | https://mathoverflow.net/users/25028 | 440449 | 177,851 |
https://mathoverflow.net/questions/440439 | 1 | I was working more on the topic on my previous question when I have to know whether the following statement is true to circumvent the "exception" caused by division by singular matrices; again, long story short, the statement follows:
If two singular matrices $A, B$ exist s.t. the determinant of $EA-B$ is identically... | https://mathoverflow.net/users/113020 | Singularity of matrix pencil-like expression | No.
The first condition is satisfied if (and only if) there is some vector in the kernel of $A$ that is also in the kernel of $B$.
The second condition is satisfied (if and) only if the kernel of $A$ is contained in the kernel of $B$ or the kernel of $B$ is contained in the kernel of $A$.
To make a counterexample... | 2 | https://mathoverflow.net/users/18060 | 440450 | 177,852 |
https://mathoverflow.net/questions/440412 | 0 | Let $k$ be an algebraically closed field. Let $X\subset \mathbb{A}^n\_{k}$ be a conical closed subvariety. In other words,
$\mathcal{O}(X)=k[x\_1,\cdots, x\_n]/I$, where $I$ is generated by homogeneous polynomials. Assume also that $X$ is a normal variety. In this general setting, is there anything known about the Pica... | https://mathoverflow.net/users/481692 | Picard group of a normal conical affine variety | Let $X$ be the affine cone over a normal projective variety $Y$.
Let
$$
\pi \colon \tilde{X} \to X
$$
be the blowup of $X$ at the vertex. Then $\tilde{X}$ comes with a projection
$$
p \colon \tilde{X} \to Y
$$
that identifies $\tilde{X}$ with the total space of a line bundle on $Y$ (the restriction to $Y$ of the tautol... | 2 | https://mathoverflow.net/users/4428 | 440453 | 177,853 |
https://mathoverflow.net/questions/440443 | -1 | Let $(X,D)$ be a pair and $f:Y\rightarrow X$ a log resolution. Write
$$
K\_Y + \widetilde{D} = f^{\*}(K\_X) + \sum\_{i}a\_iE\_i
$$
where $\widetilde{D}$ is the strict transform of $D$. I found the following definition:
the pair $(X,D)$ is $(t,c)$ is $X$ is terminal and $(X,D)$ is canonical meaning that $a\_i\geq 0$ f... | https://mathoverflow.net/users/14514 | Definition of canonical pair | You are using the wrong equation to compute discrepancies. It should be
$$ K\_Y = f^\*(K\_X + D) + \sum a\_E(X,D) E $$
where the $E$ are not all necessarily exceptional.
For example if $(X,D)$ is already smooth and simple normal crossing then
$$ K\_X = \operatorname{id}\_X^\*(K\_X + D) - D $$
and so every irreducible... | 2 | https://mathoverflow.net/users/104695 | 440456 | 177,854 |
https://mathoverflow.net/questions/440491 | 6 | The motivation for this question is the startling fact that there is an order-preserving injective map (embedding) from $\mathbb{R}$ into ${\cal P}(\omega)$. (Think [Dedekind cuts](https://en.wikipedia.org/wiki/Dedekind_cut).) I am wondering how "large" a linear order can become and still be embeddable in ${\cal P}(\om... | https://mathoverflow.net/users/8628 | Smallest ordinal $\mu$ not embeddable in ${\cal P}(\omega)$ | This is just $\omega\_1$. The naive argument ("pick the least new thing") that no uncountable linear order embeds into $\mathcal{P}(\omega)$ actually establishes that no uncountable *well*-order embeds into $\mathcal{P}(\omega)$: if $f$ is an injection of an ordinal $\theta$ into $\mathcal{P}(\omega)$, the map $$\hat{f... | 12 | https://mathoverflow.net/users/8133 | 440492 | 177,863 |
https://mathoverflow.net/questions/440480 | 4 | Let $a\_n$ be a sequence of strictly positive real numbers such that $\lim\_{n \to \infty}a\_n=0$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ that admit primitives (i.e. there exists a function $F:\mathbb{R} \to \mathbb{R}$ such that $\frac{dF(x)}{dx}=f(x), \forall x \in \mathbb{R}$) and satisfy the following eq... | https://mathoverflow.net/users/480453 | Solving the functional equation $2f(x)=f(x+a_n)+f(x-a_n)$ | $\newcommand{\De}{\Delta}$This problem can be solved by using the Fourier transform -- cf. [this previous answer](https://mathoverflow.net/a/440186/36721).
Let us present here an elementary solution:
Letting $G\_n(x):=F(x+a\_n)+F(x-a\_n)-2F(x)$, we get $G'\_n(x)=f(x+a\_n)+f(x-a\_n)-2f(x)=0$ for all $x$. So,
\begin{... | 3 | https://mathoverflow.net/users/36721 | 440497 | 177,865 |
https://mathoverflow.net/questions/440544 | 3 | Let $s\_1, s\_2, \dotsc$ be a real sequence and define
$$\sigma\_n = \frac{s\_1 + s\_2 + \dotsb + s\_n}{n}.$$ The inequality
$$\operatorname{lim sup}\sigma\_n \leq \operatorname{lim sup} s\_n$$
is well known and trivially proved.
Consider a real valued continuous function $f(x)$ defined on the positive real line and ... | https://mathoverflow.net/users/156678 | Integral analog of an inequality for the Cesàro mean of a sequence | $\newcommand\si\sigma$Note that
$$\si(T)=\frac1T\,\int\_0^T dx f(x)\,\int\_x^T dt
=\frac1T\,\int\_0^T dt\,\int\_0^t dx\,f(x)
=\frac1T\,\int\_0^T dt\,s(t).$$
Take any real $L>\limsup\_{T\to\infty} s(T)$ (if such $L$ exists, that is, if $\limsup\_{T\to\infty} s(T)<\infty$) and then take any real $A>0$ such that $s(t)\le ... | 7 | https://mathoverflow.net/users/36721 | 440546 | 177,875 |
https://mathoverflow.net/questions/440549 | 0 | If $p \in \mathbb Z[x]$ has non-negative coefficients $\le n$ and if $q$ is a proper divisor of $p$, are the absolute values of the (integer) coefficients of $q$ bounded by some function of $n$; if so, what is a good bound for the case $n=1$?
| https://mathoverflow.net/users/499203 | Factorising single variable polynomials with non-negative integer coefficients | Cyclotomic polynomials divide $p=1+x+\cdots+x^m$ but the (absolute values of) coefficients of cyclotomic polynomials grow unboundedly: see e.g. “ON THE SIZE OF THE COEFFICIENTS OF THE CYCLOTOMIC POLYNOMIAL” by Bateman, available online at <https://www.jstor.org/stable/44165422>
So there is no bound even in the case $... | 2 | https://mathoverflow.net/users/25028 | 440550 | 177,877 |
https://mathoverflow.net/questions/440475 | 2 | Here exponentially localized can be thought in a non-rigorous manner as a measure that is mostly supported on a sparse number of nodes.
Some intuition can gained by thinking about a diffusion process, e.g., we know that in presence of a "confining potential" e.g., $V(x)\approx x^2/2$, the Fokker-Planck equation in 1D... | https://mathoverflow.net/users/30684 | When is a stationary measure of a Markov chain "exponentially localized"? | I will give some answers in terms of diffusion processes, since the examples are easiest for me to describe in that context. There are more general examples which follow the same pattern, but typically require additional care to state rigorously.
Suppose that you are interested in the Markov process
$$\mathrm{d} X ... | 2 | https://mathoverflow.net/users/121692 | 440566 | 177,881 |
https://mathoverflow.net/questions/440500 | 1 | Suppose I have an extension of fields $L/K$, a group scheme $G\_K$ over $\operatorname {Spec} K$. Let $G\_L$ denote the pullback of $G\_K$ to $\operatorname{Spec} L$. Then, under what conditions on the extension $L/K$ can one say that we have an equality of $L$-algebras of the form $\operatorname{Lie} (G\_L) = \operato... | https://mathoverflow.net/users/499148 | Lie algebras and pulled back group schemes | As the link in Erica's [comment](https://mathoverflow.net/questions/440500/lie-algebras-and-pulled-back-group-schemes#comment1136246_440500) shows, you can find this in SGA3 Exp. 2, but it is not so easy to extract from the very general language. Here is a rough guide: From Definition 3.9.0, the Lie algebra of $G$ over... | 3 | https://mathoverflow.net/users/121 | 440575 | 177,885 |
https://mathoverflow.net/questions/440560 | 2 | $\DeclareMathOperator\Coh{Coh}\DeclareMathOperator\ev{ev}\DeclareMathOperator\cone{cone}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Ext{Ext}$This is a specific question concerning a statement in [Symplectic structures on moduli spaces of sheaves via the Atiyah class](https://www.sciencedirect.com/science/article/... | https://mathoverflow.net/users/nan | Homomorphism between Ext induced by the left mutation functor | To say that mutation is a functor, we must define its action on morphisms. It is defined as follows: any morphism $f \colon F\_1 \to F\_2$ in the derived category induces $H^\bullet(f) \colon H^\bullet(F\_1) \to H^\bullet(F\_2)$ and there is a unique morphism $\mathbb{L}(f)$ that fits into a commutative diagram
$$
\req... | 0 | https://mathoverflow.net/users/4428 | 440580 | 177,887 |
https://mathoverflow.net/questions/435640 | 1 | Let $(X, \mathcal A, \mu)$ be a $\sigma$-finite measure space and $(E, |\cdot|)$ a Banach space. Here we use the [Bochner integral](https://math.stackexchange.com/questions/4298588/dominated-convergence-theorem-for-banach-space). Let $p \in [1, \infty)$ and $q \in (1, \infty]$ such that $p^{-1}+q^{-1}=1$. In an attempt... | https://mathoverflow.net/users/99469 | Does an isometric automorphism of $L_p (X,\mu, E)$ preserve pointwise convergence? | Point 1) does not hold for the Fourier transform on $L^2({\bf R}, {\bf C})$, which is an isometry for a well chosen normalization.
Consider the sequence $$f\_n(x) = n {\bf 1}\_{[-1/n,1/n]}(x)$$ which converges to 0 for all $x\neq 0$. The Fourier transform of $f\_n$ is proportional to $${sin(x/n) \over {x/n}}$$ which ... | 2 | https://mathoverflow.net/users/6129 | 440589 | 177,890 |
https://mathoverflow.net/questions/440525 | 4 | Is there any known example of a one-ended finitely presented group with exponential growth that does not contain a quasi-isometric copy of the hyperbolic plane?
This question is motivated by the following question of Papasoglu mentioned in the paper *'Quasi-hyperbolic planes in hyperbolic group'* by Bonk–Kleiner whic... | https://mathoverflow.net/users/429294 | Groups that don't contain quasi-hyperbolic plane | It is a result of Buyalo and Schroeder [BS, Corollary 1.2] that for every $n\ge 2$ there is no QI embedding of the hyperbolic space $\mathbb{H}^n\_\mathbf{R}$ into any product of $n-1$ trees with a Euclidean space.
In particular, the 1-ended group $F\_2\times\mathbf{Z}$ (as well as $F\_2\times\mathbf{Z}^d$ for arbitr... | 7 | https://mathoverflow.net/users/14094 | 440590 | 177,891 |
https://mathoverflow.net/questions/440594 | 1 | Suppose that $(X,d)$ is a locally compact metric space and $\mu$ is a $\sigma$-finite Radon measure on the Borel sigma-algebra of this space. I am aware that if $(X,d)$ is separable and $\mu$ has full support then $L^2(X,\mu)$ is separable i.e. it admits a dense countable subset.
My question is the following: does th... | https://mathoverflow.net/users/327983 | On the existence of a countable dense family in "increasing" pointwise convergence | Yes, this is true. First note that if there is a sequence $f\_n$ that converges to $f$ from below, $f\_n \leq f$, then there is an increasing sequence converging to $f$ given by $\tilde{f}\_n = max(f\_1,...,f\_n)$.
You can add such functions to your dense countable set without changing its cardinal.
So we are left to... | 1 | https://mathoverflow.net/users/6129 | 440600 | 177,898 |
https://mathoverflow.net/questions/440588 | 6 | Let $S\_g$ be a closed orientable surface of genus $g>1$.
*How can one prove that its mapping class group $\mathrm{Mod}(S\_g)$
is not generated by two Dehn twists?*
A pair of simple closed curves in $S\_g$ may be very complicated
(e.g. there are *filling* pairs of curves, such that they are in minimal position
and ... | https://mathoverflow.net/users/76500 | Generate $\mathrm{Mod}(S_g)$ by two Dehn twists | In
Humphries, Stephen P.
Generators for the mapping class group. Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 44–47,
Lecture Notes in Math., 722, Springer, Berlin, 1979.
Humphries proves that no collection of less than or equal to 2g Dehn twists generates the mapping c... | 8 | https://mathoverflow.net/users/317 | 440605 | 177,900 |
https://mathoverflow.net/questions/440597 | 2 | Consider the analytic function $f : (0,\infty) \to (0,\infty)$ given by
$$
f(x) = \bigg( \sum\_{i=1}^n a\_i b\_i^{1/x} \bigg)^x
$$
where $n\in\mathbb N$, $a\_i>0$ and $0<b\_1<\ldots<b\_n<1$. I am interested in analytic continuations of $f$ to the right half plane $\mathbb H\_+ = \{z\in\mathbb C : \mathrm{Re}(z)>0\}$.
... | https://mathoverflow.net/users/485160 | Analytic continuation of a function on the half line | Partial answer:
$g(z)=\sum\_{k=1}^n a\_kb\_k^{1/z}$ is analytic on the punctured plane and clearly $f(z)=e^{z\log g(z)}$ defines $f$ analytic outside the zeroes of $g$ - as usual locally and then we can stitch together a global $f$ with appropriate cuts.
Now $g >0$ on the real axis outside the origin by definition,... | 3 | https://mathoverflow.net/users/133811 | 440607 | 177,901 |
https://mathoverflow.net/questions/440583 | 4 | **Question.** Is there an entire function $F$ satisfying first two or all three of the following assertions:
* $F(z)\neq 0$ for all $z\in \mathbb{C}$;
* $1/F - 1\in H^2(\mathbb{C}\_+)$ -- the classical Hardy space in the upper half-plane;
* $F$ is bounded in every horizontal half-plane $\{z\colon \text{Im}(z) > \delt... | https://mathoverflow.net/users/498423 | Existence of nonzero entire function with restrictions of growth | There is a zero-free entire function bounded in every left half-plane, and such that $f-1$ is in $H^2$ in every left half-plane.
Let $\gamma$ be the boundary of the region $$D=\left\{ x+iy: |y|<2\pi/3, x>0\right\}
.$$ Consider the function
$$g(z)=\int\_\gamma \frac{\exp e^\zeta}{\zeta-z}d\zeta,\quad z\in {\mathbf{C}}... | 5 | https://mathoverflow.net/users/25510 | 440610 | 177,902 |
https://mathoverflow.net/questions/440427 | 1 | Given only an arc of a circle, we can easily reconstruct it fully without any use of analytic geometry - indeed using only compass and straightedge. Note that by "given only an arc", we mean that "from a circle that was originally drawn on the plane, everything except an arc got erased".
**Question:** Given an arc of... | https://mathoverflow.net/users/142600 | Reconstructing an ellipse from an arc, synthetically | Five points in general position lie on a unique conic. With the help of Pascal's theorem one can construct arbitrarily many points on the same conic.
EDIT. Answering the question about construction of the foci: yes, it is possible. If 5 points on a conic are given, then the coefficients of the corresponding quadratic... | 2 | https://mathoverflow.net/users/98590 | 440611 | 177,903 |
https://mathoverflow.net/questions/440606 | 4 | I have been working on my master's thesis which is about the equivalence of the Hardy-Littlewood conjecture on primes represented by quadratic polynomials and the Lang-Trotter conjecture for CM elliptic curves. Although I haven't yet worked out all the details but based on some recent papers by Wan/Xi and H. Qin it loo... | https://mathoverflow.net/users/483436 | Recent works on the Hardy-Littlewood conjecture on primes represented by quadratic polynomials | For the convenience of the reader, I have written Hardy and Littlewood's conjecture from their paper [linked in the comments above](https://link.springer.com/article/10.1007/BF02403921):
Suppose that $a,b,c$ are integers and $a$ is positive; that $\gcd(a,b,c) = 1$; that $a+b$ and $c$ are not both even; and that $D = ... | 9 | https://mathoverflow.net/users/10898 | 440612 | 177,904 |
https://mathoverflow.net/questions/387536 | 3 | Let $n\geq 3$ be an integer and $0<\alpha\_1, \dots ,\alpha\_{n-2}<1$. Let's say a tuple of positive numbers $(e\_1,\dots, e\_n)$ is *nice* if there is a convex $n$-gon $A\_1\dots A\_n$ such that $\hat A\_i=\pi\alpha\_i$ and edge lengths $\overline{A\_iA\_{i+1}}=e\_i$.
(The convexity condition probably will make thin... | https://mathoverflow.net/users/2083 | Generalizing Pythagorean Theorem: equations defining edges of a (convex) $n$-gon with $n-2$ prescribed angles? | One has $$\overrightarrow{A\_nA\_1} + \cdots + \overrightarrow{A\_{n-2}A\_{n-1}} = \overrightarrow{A\_nA\_{n-1}}.$$
Taking the squared norm of both sides one arrives at
$$\sum\_{i=1}^{n-1} e\_{i-1}^2 + 2 \sum\_{1\le i < j\le n-1}e\_{i-1}e\_{j-1}\cos\sum\_{k=i}^{j-1}(\pi-\alpha\_i) = e\_{n-1}^2,$$
which is the equation ... | 2 | https://mathoverflow.net/users/98590 | 440617 | 177,905 |
https://mathoverflow.net/questions/440618 | 5 | By $MRDP$ resolution of Hilbert's tenth, we infer, counting number of solutions to Diophantine equations is undecidable.
Is parity of number of solutions to Diophantine equations undecidable?
| https://mathoverflow.net/users/10035 | Parity of number of solutions to Diophantine equations | While it’s unclear what you mean by “parity” when the number of solutions is infinite, all such questions have been answered by M. Davis, *On the number of solutions of Diophantine equations*, Proc. AMS 35 (1972), 552–554, [doi link](https://doi.org/10.2307/2037646):
>
> **Theorem:** Fix $\varnothing\subsetneq A\su... | 16 | https://mathoverflow.net/users/12705 | 440621 | 177,907 |
https://mathoverflow.net/questions/394390 | 2 | Given $4$ distinct points on the Riemann sphere, thought of as the sphere at infinity of hyperbolic $3$-space $H^3$, one may define the cross-ratio in the usual way. Note that the cross-ratio is the ratio of products of $2$ pairwise differences (using an affine coordinate on the Riemann sphere), so in this sense it has... | https://mathoverflow.net/users/81645 | How to express this hyperbolic extension of the cross-ratio in terms of hyperbolic distances and volumes? | The quantity $C(p\_1,p\_2,p\_3,p\_4)$ does not change if $p\_1$ is replaced by any other point on the ray $p\_2p\_1$. It also does not change if $p\_4$ is replaced by any other point on the ray $p\_3p\_4$. Thus this is an invariant of a triple of oriented lines $\ell\_1 = p\_1p\_2, \ell\_2 = p\_2p\_3, \ell\_3 = p\_3p\_... | 1 | https://mathoverflow.net/users/98590 | 440624 | 177,908 |
https://mathoverflow.net/questions/440625 | 0 | I am wondering why a standard notation for open sets is $G$ and that for closed sets is $F$. I mean, $F$ precedes $G$ in the alphabet, whereas open sets are usually introduced before closed ones.
| https://mathoverflow.net/users/36721 | Notations for open and closed sets | I see that @LSpice has already provided an answer in their comment (I see that Emil has added a clarification, so to speak). Mine will compliment the comment by @LSpice a little.
Historically, closed sets were before the open sets (I believe so). Kazimierz Kuratowski defined topology (of general $T\_1$-spaces} via th... | 2 | https://mathoverflow.net/users/110389 | 440626 | 177,909 |
https://mathoverflow.net/questions/435445 | 12 | $\DeclareMathOperator\SO{SO}$I'm not asking for a proof but for some hint that might be helpful to understand this "anomaly" in 4 dimensions. I'm aware of the parallelism with the $A\_4$ finite group case as shown in the [answer](https://mathoverflow.net/a/145066) by @Benoit Kloeckner which I would also like to see ela... | https://mathoverflow.net/users/118787 | Why is $\operatorname{SO}(4)$ not a simple Lie group? | Here's a different approach than the ones that have been suggested so far. For any $n, k$ we can consider the action of $SO(n)$ on the exterior power $\Lambda^k(\mathbb{R}^n)$. These representations are almost all irreducible, *except* when $n = 2k$ and $k \ge 2$ is even; in this case the [Hodge star](https://en.wikipe... | 7 | https://mathoverflow.net/users/290 | 440634 | 177,911 |
https://mathoverflow.net/questions/440638 | 5 | As far as I know, there are two versions of Waldspurger's formula (classical and adelic), which can be vaguely stated as follows
* (Classical version) Let $f$ be a half-integral weight modular form of weight $k/2$ (where $k$ is odd), level $N$ and character $\chi$ and $F = \operatorname{Sh}(f)$ be its Shimura corresp... | https://mathoverflow.net/users/95471 | Waldspurger's formula and toric periods — classical and adelic versions | These are two separate theorems, proved in different papers of Waldspurger (I think in 1980/1981 and 1985, respectively), so you shouldn't conflate them. The first theorem can be viewed as an "$L$-value correspondence between automorphic representations of GL(2) and Mp(2) (Shimura correspondence), and the second betwee... | 4 | https://mathoverflow.net/users/6518 | 440640 | 177,913 |
https://mathoverflow.net/questions/440609 | 0 | Recently I got interested in the following property of topological spaces:
$(X,\mathcal{T})$ satisfies (P) if the following holds:
For any nonempty closed subsets $F$ and $G$ with $F\ne G$, there are closed subsets $F'\subseteq F$ and $G'\subseteq G$ satisfying the following conditions:
* $F'$ has nonempty interi... | https://mathoverflow.net/users/499248 | Property stronger than $T_1$ and weaker than regularity | 1. Hausdorff spaces need not have this property. Consider [Bing's Countable connected Hausdorff space](https://doi.org/10.1090/S0002-9939-1953-0060806-9) (Example 75 in Steen and Seebach's *Counterexamples in Topology*); it has the property that for every pair of nonempty open sets $U$ and $V$ the closures $\overline U... | 1 | https://mathoverflow.net/users/5903 | 440642 | 177,914 |
https://mathoverflow.net/questions/440582 | 2 | Assume that $\boldsymbol{v}\_{\boldsymbol{1}}, \ldots, \boldsymbol{v}\_{\boldsymbol{n}} \in \mathbb{R}^n$ satisfy $\forall i, j \in[n],i \neq j,\left\langle\boldsymbol{v}\_{\boldsymbol{i}}, \boldsymbol{v}\_{\boldsymbol{j}}\right\rangle=0,\left\|\boldsymbol{v}\_{\boldsymbol{i}}\right\|=1$. Let $\mathcal{L}=\left[\boldsy... | https://mathoverflow.net/users/482299 | A problem about matrix | Nice question!
First of all, notice that the most we can hope for is to get $v\_i$ up to a permutation and $\pm 1$. Besides these obvious obstructions, the answer is yes.
---
The $x\_i$ are independent and identically distributed, and thus all the information about the $x\_i$ comes from taking functions $f : \mat... | 3 | https://mathoverflow.net/users/88679 | 440644 | 177,916 |
https://mathoverflow.net/questions/440602 | 2 | It is a well-known result that there is a bijective correspondence between harmonic sections and cohomology classes of an elliptic complex in a Riemannian/Hermitian manifold. Now consider Riemannian/Hermitian metrics on the vector bundles of a differential complex, that might be non-elliptic. Is this still true if we d... | https://mathoverflow.net/users/494777 | Hodge decomposition for non-elliptic complexes | This is false. Equip $T^2 = S^1 \times S^1$ with the Lorentzian metric $g = -ds^2 + dt^2$. For concreteness, regard $S^1 = \mathbb{R} / \mathbb{Z}$. Consider the de Rham complex. The Hodge Laplacian on functions is now the wave operator $\Box = -\frac{\partial^2}{\partial s^2} + \frac{\partial^2}{\partial t^2}$. Its ke... | 5 | https://mathoverflow.net/users/121820 | 440646 | 177,918 |
https://mathoverflow.net/questions/439894 | 8 | For $n\in\omega+1$ let $\mathsf{ZFC}\_n$ be $\mathsf{ZC}$ + $\{\Sigma\_k$-$\mathsf{Rep}: k<n\}$. Let $\widehat{\mathsf{ZFC}}$ be the strongest *consistent* theory $\mathsf{ZFC}\_n$ (so if $\mathsf{ZFC}$ is consistent then $\widehat{\mathsf{ZFC}}=\mathsf{ZFC}$). Note that this definition is entirely "internal" - for exa... | https://mathoverflow.net/users/8133 | Modal logic of "mostly-satisfiability" | First point is that this logic is simply the provability logic of formalized $\widehat{\mathsf{ZFC}}$-provability, i.e. the provability logic of the provability predicate:
$$\mathsf{Prv}\_{\widehat{\mathsf{ZFC}}}(x)\colon\;\;\;\;\; \exists y(\mathsf{Con}(\mathsf{ZFC}\_y)\land \mathsf{Prv}\_{\mathsf{ZFC}\_y}(x)).$$
This... | 3 | https://mathoverflow.net/users/36385 | 440659 | 177,922 |
https://mathoverflow.net/questions/440639 | 6 | **Deep Choice**:
$\forall X \ [\forall a,b \in X \, ( a \neq \emptyset \land (a \neq b \to a \cap b = \emptyset)) \to \\ \exists Y \exists f \,(f: X \rightarrowtail Y \land \forall x \in X \,( f(x) \in \operatorname {tc}(x)))]$
In English: for every family $X$ of pairwise disjoint nonempty sets, there is an injective... | https://mathoverflow.net/users/95347 | Can Deep Choice entail Axiom of Choice? | It proves AC. For this, recall it's enough to see that for every ordinal $\alpha$, $\mathcal{P}(\alpha)$ is wellorderable, and for that it's enough to see that $\mathcal{P}(\mathcal{P}(\alpha))\backslash\{\emptyset\}$ has a choice function. Supposing this fails for some $\alpha$, let $\alpha$ be the least such; then $\... | 8 | https://mathoverflow.net/users/160347 | 440668 | 177,925 |
https://mathoverflow.net/questions/440655 | 2 | I asked a question a few days ago and got a response
But my follow-up question was not answered (maybe my email was not sent successfully)
[A question about computability and Turing machines](https://mathoverflow.net/questions/440338/a-question-about-computability-and-turing-machines)
My quesion is:
1. If $E$ is ... | https://mathoverflow.net/users/499050 | A question about computability and Turing machines Part 2 | I believe you refer to "well founded" not "well based".
You are asking about a relation $E$ on $\omega$ and a function $F$ for which
$$F(n)=\{F(m)\mid m\mathrel{E} n\}.$$
Such a function $F$ would have the property that
$$m\mathrel{E} n\implies F(m)\in F(n)$$
And from this it follows that $E$ must be well-founded, si... | 4 | https://mathoverflow.net/users/1946 | 440669 | 177,926 |
https://mathoverflow.net/questions/440670 | 26 | Let ${\mathcal A}$ be the family of subsets $A$ of the natural numbers ${\mathbf N}$ which are co-infinite (i.e., their complement is infinite). We partially order this family by set inclusion. A [cofinal](https://en.wikipedia.org/wiki/Cofinal_(mathematics)) subset of ${\mathcal A}$ is a subcollection ${\mathcal A}'$ s... | https://mathoverflow.net/users/766 | What is the cofinality of the co-infinite subsets of ${\bf N}$? | Every such cofinal family $\mathcal{A}'$ must have size continuum. The reason is that there is an almost disjoint family $\mathcal{D}$ of size continuum, a family of infinite co-infinite sets $A\subseteq\mathbb{N}$ for which any two have finite intersection. To construct such an almost disjoint family, label the nodes ... | 36 | https://mathoverflow.net/users/1946 | 440671 | 177,927 |
https://mathoverflow.net/questions/440664 | 1 | Let $n \in \mathbb N^\* , \alpha \in \mathbb R$, and $w\_i \in \mathbb R$ for all $i=1, \ldots, n$. Consider the map $f:\mathbb R\_{\ge 0} \to \mathbb R$ defined by
$$
f(x) := \sum\_{i=1}^n w\_i x^{\color{red}{1/i}}.
$$
I'm interested in the solutions of $f(x)=\alpha$. I would like to search on Google Scholar, but I ... | https://mathoverflow.net/users/477203 | The solutions of $\sum_{i=1}^n w_i x^{1/i} = \alpha$ | Setting $x=e^t$ we obtain an exponential sum. Exponential sums were much studied, from various points of view. One reference is Pólya–Szegő, Problems and theorems in Analysis, vol. 2, part V, Chap 1, section 6, where they are studied in the real domain. Also notice that exponential sums are solutions of linear differen... | 4 | https://mathoverflow.net/users/25510 | 440672 | 177,928 |
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