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https://mathoverflow.net/questions/337530 | 2 | Let $X$ be a non-compact manifold and let $C\_0(X)$ act on $L^2(X)$ by pointwise multiplication.
We say $T\in\mathcal{B}(L^2(X))$ has *finite propagation* if there exists an $r>0$ such that: for all $f,g\in C\_c(X)$ with supports separated by a distance greater than $r$, we have $fTg=0$.
**Question:** Suppose $S\i... | https://mathoverflow.net/users/78729 | Does the square root of a finite propagation operator have finite propagation? | Greg Kuperberg and I studied operators of this type in our AMS Memoir *A von Neumann Algebra Approach to Quantum Metrics*. We called them *finite displacement operators*.
The definition you give is ill-posed because it refers to distance and you have not specified a metric (unless "manifold" means "Riemannian manifol... | 2 | https://mathoverflow.net/users/23141 | 337553 | 144,070 |
https://mathoverflow.net/questions/337556 | 3 | Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose that $M$ is left proper combinatorial, then the Bousfield localisation $M\_{\bar{S}}$ is a model topos [Prop.6.2.1.2, HT... | https://mathoverflow.net/users/124163 | Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers |
>
> are the fibrant objects of sPre(C)\_Nis the same as sPre(C)\_Nis?
>
>
>
Yes. This follows from the characterization of fibrant objects
in a left Bousfield localization as fibrant objects in the original
model structure that are also local.
The closure of S under homotopy base changes does not change the cl... | 2 | https://mathoverflow.net/users/402 | 337564 | 144,073 |
https://mathoverflow.net/questions/337566 | 10 | Let $G$ be a finite abelian group. Write $G\approx \mathbb{Z}/p\_1^{i\_1}\mathbb{Z}\times\dots \mathbb{Z}/p\_m^{i\_m}\mathbb{Z}$, with $m\ge 0$, $p\_1,\dots,p\_m$ primes (not necessarily distinct) and $i\_k\ge 1$ for all $k$.
For $n\ge 0$, is it true that there is an injective homomorphism $G\rightarrow S\_n$ if and... | https://mathoverflow.net/users/nan | A criterion for finite abelian group to embed into a symmetric group | Yes, this is true (provided we add the hypothesis $i\_1, \ldots, i\_n \ge 1$), and not hard to prove by considering centralizers of products of cycles in symmetric groups.
According to <http://www-history.mcs.st-andrews.ac.uk/Biographies/Povzner.html>, the minimum degree of a permutation representation of an arbitrar... | 13 | https://mathoverflow.net/users/7709 | 337567 | 144,074 |
https://mathoverflow.net/questions/337544 | 5 | I am studying the generalisation of the Selberg's integral by using Jack polynomial as outlined by Forrester and Warnaar ([arXiv: 0710.3981](https://arxiv.org/abs/0710.3981)). They write down the *symmetric* Jack polynomial as
\begin{equation}
P\_\lambda^{(\alpha)} = m\_\lambda + \sum\_{\mu < \lambda}a\_{\lambda \mu} ... | https://mathoverflow.net/users/143942 | Jack polynomial and Selberg integral | Try this set of notes on the Jack symmetric functions / polynomials: <https://tcjpn.wordpress.com/2016/11/27/a-note-on-the-jack-symmetric-functions-polynomials/>
| 0 | https://mathoverflow.net/users/12178 | 337572 | 144,076 |
https://mathoverflow.net/questions/337551 | 9 |
>
> Is there a way to prove, that $\lim\_{n \to \infty} \frac{\text{the number of all } 2 \text{-generated groups of order less than }n}{\text{the number of all groups of order less than } n} = 0$?
>
>
>
This statement is implied by a well known conjecture:
>
> $\lim\_{n \to \infty} \frac{\text{the number of... | https://mathoverflow.net/users/110691 | Is there a way to prove, that $2$-generated groups are rare among finite groups? | Lubotzky proved that the number of $2$-generator groups of order at most $n$ (or order exactly $n$) is bounded by $n^{A \log n}$ for some constant $A$. The number of $2$-groups of order $2^m$ with $n \ge 2^m > n/2$ is (**corrected, see Will Sawin's comment**) $n^{B \log^2 n}$ for some explicit constant $B$. So you win!... | 12 | https://mathoverflow.net/users/143960 | 337576 | 144,077 |
https://mathoverflow.net/questions/337598 | 2 | Call a pair of integers $(a,b)$ *trivial* if it satisfies some simple divisibility condition, like for some prime $p$ we have $p$ divides both $a-1$ and $b+1$, or that $p$ divides both $a$ and $b$. This implies that $a^n+b$ is always divisible by $p$, like $7^n+2$ by $3$. But there can be more complicated conditions, l... | https://mathoverflow.net/users/955 | Primes in shifted geometric sequence | The answer is No.
We know that $78557$ is a [Sierpinski number](https://en.wikipedia.org/wiki/Sierpinski_number) with the covering set $S = \{ 3, 5, 7, 13, 19, 37, 73 \}$, i.e., every integer of the form $78557\cdot 2^n+1$ is divisible by an element of $S$. Set $a:=2$ and $b:=128100173$, which form a non-trivial pair... | 4 | https://mathoverflow.net/users/7076 | 337610 | 144,087 |
https://mathoverflow.net/questions/337608 | 2 | Frullani's theorem is a deep theorem in real analysis with applications, see the Wikipedia [*Frullani integral*](https://en.wikipedia.org/wiki/Frullani_integral) and other uses and contexts (see [2]). I wrote two imaginative examples of what can be deduced using this theorem, I add these as the first comment, as illust... | https://mathoverflow.net/users/142929 | Does exist a variant of Frullani's theorem valid for $f(x)=\pi(x)/x$ or $f(x)=\psi(x)/x$, where the numerators are prime-counting functions? | As you mentioned, there is an explicit formula for $\psi(x)$ of the form
$$
\psi(x) = x - \sum\_\rho \frac{x^\rho}\rho - \log2\pi - \frac12\log(1-x^{-2})
$$
(if we define the left-hand side suitably at its discontinuities). Assuming the Riemann hypothesis, this implies that $\psi(x)-x$ is equal to $\sqrt x$ times a fun... | 4 | https://mathoverflow.net/users/5091 | 337617 | 144,091 |
https://mathoverflow.net/questions/337581 | 4 | Let $M$ be a manifold, and $U$ be the forgetful functor from the category of Lie groups to the category of manifolds. My question is whether there is a universal arrow $(G, i)$ from $M$ to $U$? More precisely, whether there is a Lie group $G$ and a smooth map $i: M \rightarrow U(G)$ such that, for all smooth map $\phi:... | https://mathoverflow.net/users/83349 | Existence of universal arrow from manifolds to forgetful functor of Lie groups | Such arrow exists if and only if $Dim(M)=0$. In this case $G$ is the free group $F(M)$, considered as a $0$-dimensional manifold.
If $Dim(M)>0$, you can cook up a smooth map $M\rightarrow {\mathbb R}^n$, such that $\{ \alpha e\_k | \alpha \in (-1,1)\}$ is in the image for each standard basis element $e\_k$. This mean... | 4 | https://mathoverflow.net/users/5301 | 337630 | 144,095 |
https://mathoverflow.net/questions/337614 | 5 | Let $rS^{d-1}$ denote the sphere of radius $r$ in dimension $d$ (centered at the origin). I'm interested in the number of lattice points on the sphere (not inside).
More precisely, let $$
N(r,d):=\text{number of lattice points on the sphere of raduis } r=\#\{x\in rS^{d-1}: x\in \mathbb{Z}^d\}.
$$
I'm especially int... | https://mathoverflow.net/users/4519 | Lower bound for the number of lattice points on high dimensional spheres | My answer to [this MO question](https://mathoverflow.net/questions/217698/many-representations-as-a-sum-of-three-squares) contains the answer to your question, especially if you take into account that $L\left(1,\left(\frac{D}{\cdot}\right)\right)$ can be estimated unconditionally (i.e. without GRH):
$$|D|^{-\varepsilon... | 4 | https://mathoverflow.net/users/11919 | 337633 | 144,096 |
https://mathoverflow.net/questions/337631 | 7 | In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement:
>
> Suppose $\{X\_n\}\_{n=2}^\infty$ is a sequence of independent random variables, such that $X\_n \sim Bern(\frac{1}{\ln(n)})$.
> Then $\lim\_{n \to \infty} \sup |\frac{... | https://mathoverflow.net/users/110691 | A theorem by Harald Cramér? | This is a special case of a general law of the iterated logarithm for non-iid random variables (r.v.'s), which states the following:
>
> Suppose that $Y\_1,Y\_2,\dots$ are independent zero-mean r.v.'s, $S\_n:=\sum\_1^n Y\_i$, $B\_n:=Var\, S\_n\to\infty$, $|Y\_n|\le M\_n\in(0,\infty)$, and $M\_n=o((B\_n/\ln\ln B\_n... | 11 | https://mathoverflow.net/users/36721 | 337635 | 144,097 |
https://mathoverflow.net/questions/337644 | 4 | I have two $\mathbb{C}$-linear semisimple tensor categories $C$ and $D$. Let $K(C)$ and $K(D)$ be their Grothendieck groups. I have a specific ring homomorphism $f \colon K(C) \to K(D)$ that I would like to realize as coming from a monoidal functor $F \colon C \to D$. What I know is that for every simple object $X \in ... | https://mathoverflow.net/users/321 | Lifting ring homomorphism of Grothendieck rings to functor of semisimple categories | Without the symmetry it’s easy to produce oodles of examples where a map of fusion rings doesn’t lift to the category level. The simplest example is probably $\mathrm{Vec}(G,w)$ the category of G-graded vector spaces with associator given by a 3-cocycle $w$. The fusion ring has an automorphism for any automorphism of G... | 9 | https://mathoverflow.net/users/22 | 337650 | 144,103 |
https://mathoverflow.net/questions/337661 | 2 | The number $2$ has the interesting property that whenever $n>1$ is an integer, then $n \nmid (2^n-1)$. (It's a good exercise to prove this statement.)
Let's call a positive integer $b$ *$2$-like* if for all integers $n>1$ we have $n\nmid (b^n-1)$, and let's call it *almost $2$-like* if for all integers $n>1$ *except ... | https://mathoverflow.net/users/8628 | Integers $b$ such that $n \nmid (b^n-1)$ for $n>1$ | $b=2$ is the only almost 2-like number. Indeed, if $n\mid (b^n-1)$ and $p$ is a prime divisor of $(b^n-1)/n$, then $np\mid (b^{np}-1)$. That is, existence of one $n>1$ dividing $b^n-1$ implies existence of infinitely many of them.
Also, for $b>2$, there exist at least one such $n$, e.g., $n=b-1$.
| 13 | https://mathoverflow.net/users/7076 | 337667 | 144,106 |
https://mathoverflow.net/questions/337480 | 4 | Consider the polynomial ring $R[x\_1,x\_2,\ldots,x\_n]$, where $R$ be an algebraically closed field (preferably $\mathbb{C}$) and the ideal $J=\langle m\_1, m\_2,\ldots,m\_n\rangle$ generated by monomials . The monomials are homogeneous and each variable has maximum degree $1$. Let $|m\_i|$ denotes the number of variab... | https://mathoverflow.net/users/100231 | Condition for a monomial to belong to a particular ideal | Such a value $d$ does not necessarily exist. For $J=(xy,xz)$ we have $(xyz)^{d-1} \notin J^d$ for any $d$. It’s even worse (but maybe a little degenerate) if $J$ is principal.
A necessary and sufficient condition for existence of such a $d$ is that the monomials generating $J$ don't have a common factor. To be explic... | 4 | https://mathoverflow.net/users/88133 | 337680 | 144,108 |
https://mathoverflow.net/questions/337672 | 1 | Please see the below link for the complete description. I already have an answer shown in the link, based on many Excel simulations ($n=4$ to $100$, $x\_i$ generated by RAND() function of Excel). I want to know if the answer $\frac{1}{n^2}\left(1+\frac{1}{\pi}\right)$ has a theoretical basis. If no, how to derive as a ... | https://mathoverflow.net/users/42700 | Expected value of square[X/sigmaX] = 1/n^2(1+1/pi)? | Your conjecture is obviously false for $n=1$, and I think it is false for any $n$. Using the [Irwin-Hall formula](https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall_distribution), one can write an explicit but very complicated expression for the expectation in question, and that expression will not involve $\pi$.
Also... | 6 | https://mathoverflow.net/users/36721 | 337683 | 144,109 |
https://mathoverflow.net/questions/337670 | 27 | I was watching some lectures by Huisken where he mentioned that one-loop renormalization group flow was in some analogous to mean curvature flow. I have tried reading up the exact definition of what this flow actually is, but could not find anything suitable and was wondering if anyone could explain it to me.
I have ... | https://mathoverflow.net/users/119114 | Formal mathematical definition of renormalization group flow | The renormalization group (RG) as a geometric flow (like the Ricci flow) is a very
special case of the RG, namely, the one corresponding to the nonlinear sigma-model (NLSM) in two dimensions with values in a Riemannian manifold.
Now the RG is much more general and applies to all sorts of models, not just the NLSM.
In o... | 19 | https://mathoverflow.net/users/7410 | 337692 | 144,112 |
https://mathoverflow.net/questions/337647 | 4 | Let $(W,S)$ be a Coxeter system and $\beta$ a positive root in it. Is there a good way to compute a reduced expression for the reflection across the hyperplane with normal $\beta$? References please.
| https://mathoverflow.net/users/143995 | Reduced expressions for reflections in a Coxeter group | Depending on what you mean by a "good way", maybe there is and maybe there isn't. If you want to do this all in terms of the combinatorics of reduced words, probably the following is the best you can do:
Since $\beta$ is a root, there is an element $w\in W$ and a simple root $\alpha$ such that $\beta=w\alpha$. If $s\... | 8 | https://mathoverflow.net/users/5519 | 337694 | 144,113 |
https://mathoverflow.net/questions/337677 | 9 | **Some initial clarifications**
By [lattice](https://en.wikipedia.org/wiki/Lattice_(group)) I mean an additive subgroup of $\mathbb R^n$ which is isomorphic to $\mathbb Z^n$ and has full rank (i.e. spans $\Bbb R^n$ when considered as set of vectors). A lattice $\mathcal L$ is *integral* if $\langle v,w\rangle\in\math... | https://mathoverflow.net/users/108884 | Does every positive-definite integral lattice admit an angle-preserving homomorphism into $\Bbb Z^n$ for some $n$? | Yes, this is true. There's some fancy number theory that one can apply (the Hasse-Minkowski invariant and embedding of quadratic forms), but one can see this directly without number-theoretic machinery.
First, notice that one can choose a basis for the lattice which is orthogonal. Just start with any basis and apply... | 24 | https://mathoverflow.net/users/1345 | 337695 | 144,114 |
https://mathoverflow.net/questions/337681 | 3 | Recently I've started studying the theory of singular values for entire functions so I'm far from being a specialist in this field. In the literature I came across the following results:
In [1] Gross constructed an entire function for which every point in the complex plane is an asymptotic value.
In [2] Heins prove... | https://mathoverflow.net/users/47862 | Singular set of entire functions | Gross's example and the second example of Heins are locally univalent. (About the second result of Heins, you can check yourself: Annals is available. I just checked). If I remember correctly, Heins's first example is also locally univalent. Proc. of Scandiavian congresses are indeed not available online, but many libr... | 4 | https://mathoverflow.net/users/25510 | 337698 | 144,115 |
https://mathoverflow.net/questions/337518 | 25 | When it comes to numbering results in a mathematical publication, I'm aware of two methods:
1. Joint numbering: *Thm. 1, Prop. 2, Thm. 3, Lem. 4, etc.*
2. Separate numbering: *Thm. 1, Prop. 1, Thm. 2, Lem. 1, etc.*
Every piece of writting advice I have encountered advocates the use of 1. over 2., the rationale bei... | https://mathoverflow.net/users/14988 | Why would one number theorems, propositions and lemmas separately? | This is a slight elaboration of François Dorais's comment. If you have a small number of theorems/lemmas/propositions—let's say, small enough that readers can reasonably be expected to hold all the theorems in their head at once—then the second method of numbering can help readers grasp the flow of the paper and can ev... | 4 | https://mathoverflow.net/users/3106 | 337704 | 144,120 |
https://mathoverflow.net/questions/337558 | 52 | Warning: I am only an amateur in the foundations of mathematics.
My understanding of [this Wikipedia page about Tarski's axiomatization of plane geometry](https://en.wikipedia.org/wiki/Tarski%27s_axioms) (and especially the discussion about decidability) is that "plane geometry is decidable".
The [2019 Internation... | https://mathoverflow.net/users/1384 | Automatically solving olympiad geometry problems | Arguably, the so-called "[area method](https://hal.archives-ouvertes.fr/hal-00426563/PDF/areaMethodRecapV2.pdf)" of Chou, Gao and Zhang represents the state of the art in the field of machine proofs of Olympiad-style geometry problems. Their book [Machine Proofs in Geometry](http://www.mmrc.iss.ac.cn/~xgao/paper/book-a... | 39 | https://mathoverflow.net/users/3106 | 337705 | 144,121 |
https://mathoverflow.net/questions/319542 | 0 | Let $K\_2^+(W)$ be the following theory in the language $L(\in,W)$ with the constant symbol $W$.
1. **Extensionality:** $\forall x (x \in a \leftrightarrow x \in b) \to a=b$
$\mathcal{Define:} \ set(x) \iff \exists y (x \in y)$
2. **Class comprehension:** if $\varphi$ is a formula in which $x$ is not free, then: $\... | https://mathoverflow.net/users/95347 | What is the consistency status of this theory? | This doesn't add any strength, because you only asserted first order elementarity. If $(M,E,W)\vDash K\_2(W)$, then $(2^M,E,W)\vDash K\_2^+(W)$. This can be verified one axiom at a time, because comprehension doesn't effect any other axiom. Additionally, if you added $W\prec\_1 V$, you would get inconsistency, because ... | 2 | https://mathoverflow.net/users/141402 | 337706 | 144,122 |
https://mathoverflow.net/questions/337709 | 8 | [Rabin and Shallit](https://doi.org/10.1002%2Fcpa.3160390713) have a randomized polynomial-time algorithm to express an integer $n$ as a sum of four squares $n=a^2+b^2+c^2+d^2$ (in time $\log(n)^2$ assuming the Extended Riemann Hypothesis).
I'm wondering why this does not give an efficient factorization algorithm? H... | https://mathoverflow.net/users/1345 | Representing a number as a sum of four squares and factorization | I think the reason is that there are $p+1$ distinct ways of writing an odd prime $p$ as the sum of four squares up to sign changes; these correspond to the same number of elements of the Lipschitz order up to units. If you take two different Lipschitz elements of reduced norm $p$ up to units, their greatest common divi... | 12 | https://mathoverflow.net/users/4433 | 337710 | 144,123 |
https://mathoverflow.net/questions/337346 | 4 | Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$.
Let $W$ be the associated Weyl group and let $\Phi$ be its root system.
We write $\Phi^+$ for the set of positive roots in $\Phi$.
Fix a subset of simple roots $I$ and let $W\_I$ be the corresponding stand... | https://mathoverflow.net/users/110229 | Verma module and vanishing of extension groups | The answer is No.
We need the following lemma:
>
> Lemma: $W\_I\cap W\_J=W\_{I\cap J}$.
>
>
>
to prove the following proposition:
>
> Proposition: $e\in {}^IW^{\Sigma\_\mu}\iff I\cap \Sigma\_\mu=\emptyset$.
>
>
>
Proof: Recall that ${}^IW^{\Sigma\_\mu} : = \{w\in {}^IW: w<ws\_\alpha\in {}^IW\ \text{fo... | 0 | https://mathoverflow.net/users/110229 | 337720 | 144,125 |
https://mathoverflow.net/questions/337666 | 2 | Let $(X,d)$ be a metric space and $f:X\rightarrow X$ be continuous. Then, is there any meaning/research done on the metric
$$
D(x,y)\triangleq \sum\_{n \in \mathbb{N}} \frac1{2^n} d(f^n(x),f^n(y));
$$
where $f^0(x)=x$?
| https://mathoverflow.net/users/36886 | Orbit-based metric | In the linear case (for a bounded operator $T$ on a Banach space), the analogous construction for equivalent norms, possibly with a different ratio in place of $1/2$, is often named *adapted norm* (see e.g. *Global stability of dynamical systems*, by Michael Shub). Its main feature is that gives $T$ an operator norm ar... | 2 | https://mathoverflow.net/users/6101 | 337725 | 144,126 |
https://mathoverflow.net/questions/337738 | 15 | What are some other classes of word-hyperbolic groups other than the finite groups, fundamental groups of surfaces with Euler characteristics negative and virtually free groups?
| https://mathoverflow.net/users/125490 | Examples of hyperbolic groups | Below are some sources of hyperbolic groups. Of course, the list is far from being exhaustive.
1. Groups defined by generators and relations:
* Finitely generated free groups, as their Cayley graphs are simplicial trees.
* If $\varphi$ is an atoroidal automorphism of a free group $\mathbb{F}\_n$, then the extens... | 29 | https://mathoverflow.net/users/122026 | 337748 | 144,131 |
https://mathoverflow.net/questions/337457 | 28 |
>
> Let $x>0$ and $n$ be a natural number. Prove that:
> $$\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1.$$
>
>
>
This question is very similar to many contests problems, but I think it's much more harder than contest problem and it's just impossible to solve this problem during ... | https://mathoverflow.net/users/135040 | Prove that $\left(\frac{x^n+1}{x^{n-1}+1}\right)^n+\left(\frac{x+1}{2}\right)^n\geq x^n+1$ | **Corrected proof, see GH from MO's comment and answer:** A generalization of the inequality gives more flexibility for variations of parameters, which eventually yields a proof. One observation is $\frac{x+1}{2}=\frac{x^b+1}{x^{b-1}+1}$ for $b=1$. If one wants to achieve $(\frac{x^a+1}{x^{a-1}+1})^n+(\frac{x^b+1}{x^{b... | 20 | https://mathoverflow.net/users/18739 | 337753 | 144,132 |
https://mathoverflow.net/questions/337768 | 4 | EDIT: Let $g$ be a smooth Riemannian metric on $S^2$ which is invariant under the antipodal involution $x\mapsto -x$. Assume it has Gauss curvature at least $-1$ and diameter $D$.
>
> Does there exist a constant $c(D)>0$ (but independent of the metric) such that $\sup\_x dist(x,-x)\geq c(D)$?
>
>
>
| https://mathoverflow.net/users/16183 | Riemannian metrics on 2-sphere invariant under antipodal involution | The answer is "yes".
Let $g$ be the induced Riemannian metric on $\mathbb R\mathrm P^2=\mathbb S^2/\mathbb Z\_2$.
Denote by $\ell$ the systole of $(\mathbb R\mathrm P^2,g)$ and let $\gamma$ be the corresponding closed geodesic.
Let us cut $(\mathbb R\mathrm P^2,g)$ along $\gamma$.
We obtain a disc $\Delta$ with diam... | 7 | https://mathoverflow.net/users/1441 | 337778 | 144,142 |
https://mathoverflow.net/questions/337782 | 0 | For a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$ do we have that
$$\sum\limits\_{k\in\mathbb{Z}}\int\_{\mathbb{R}}|f(x+k)f'(x)|dx$$
converges?
| https://mathoverflow.net/users/61204 | Proof of $\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx<\infty$ for Schwartz function $f$ | Trivially, you have the pointwise bound
$$\sum\_{k\in\mathbb{Z}} |f(x+k)f'(x)| \leq |f'(x)|\sup\_{y\in\mathbb{R}} \sum\_{k\in\mathbb{Z}} |f(y+k)|$$
By translation invariance,
$$\sup\_{y\in\mathbb{R}}\sum\_{k\in\mathbb{Z}} |f(y+k)| \leq \sup\_{y\in [0,1)} \sum\_{k\in\mathbb{Z}} |f(y+k)|,$$
and since $f$ is Schwartz ... | 4 | https://mathoverflow.net/users/54316 | 337783 | 144,143 |
https://mathoverflow.net/questions/337433 | 2 | **Update**: I removed what I thought was unecessary and tried to be more straightforward in the hope to get an answer.
**Context:**
Suppose I have a $\mathbb{G}\_m$-gerbe $\mathcal{G}$ over a scheme $X$ with the fppf (or lisse-etale) topology. Because $\mathcal{G}$ is a $\mathbb{G}\_m$-gerbe, there is an fppf (or eta... | https://mathoverflow.net/users/141653 | Local question and descent category for a quasi-coherent sheaf on $\mathbb{G}_m$-gerbe | I'm not sure how you have chosen to define "quasicoherent sheaf on the stack $\mathcal{G}$". One way to make a definition is to construct the fibered category $QCoh$ over affine schemes, whose objects are pairs $(X, \mathcal{F})$, where $X$ is an affine scheme, and $\mathcal{F}$ is a quasicoherent sheaf on $X$. Morphis... | 1 | https://mathoverflow.net/users/121 | 337785 | 144,145 |
https://mathoverflow.net/questions/337665 | 2 | Suppose I have two smooth projective varieties $X$ and $Y$ in $\mathbb{P}^n$, that intersect along a smooth subvariety $Z$. Is there a formula to compute the degree of the join variety $J(X,Y)$ of $X$ and $Y$? Of course if $Z$ is $\emptyset$ then we have the classical formula, but what if $Z$ is non-empty?
| https://mathoverflow.net/users/4096 | Join of two intersecting varieties | You can find a lot of material on the intersection theory of join varieties (including all that I write in this answer) in the conference paper [2] of Flenner. At the end of the second page (p.130) you find the statement of the general formula for the degree of joins:
$$\deg X \deg Y = \deg V + \deg \pi \deg J,$$
The n... | 4 | https://mathoverflow.net/users/58242 | 337789 | 144,146 |
https://mathoverflow.net/questions/337791 | 1 | I have a question about the following paper:
**One-dimensional asymptotic classes of finite structures**
by Macpherson and Steinhorn [(link at Trans. AMS website)](https://www.ams.org/journals/tran/2008-360-01/S0002-9947-07-04382-6/S0002-9947-07-04382-6.pdf).
---
Let $\mathbf{K}$ be a one-dimensional asymptot... | https://mathoverflow.net/users/141388 | On asymptotic classes of finite structures | At risk of triviality: An infinite ultraproduct (of members of $\mathcal{C}$) is a structure which is (a) infinite, and (b) an ultraproduct (of members of $\mathcal{C}$).
So, for example, Lemma 2.5 reads: "Let $\mathcal{C}$ be a class of finite $\mathcal{L}$-structures, and suppose that every infinite ultraproduct of... | 4 | https://mathoverflow.net/users/2126 | 337796 | 144,148 |
https://mathoverflow.net/questions/337806 | 0 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) such that $\bigcup E = V$. For $D\subseteq V$ we set $N\_D = \bigcup\{e\in E: D\cap e\neq \emptyset\}$. We say that $D\subseteq V$ is *dominating* if $N\_D = V$.
Hypergraphs [need not have minimal dominating sets](https://mathoverflow.net/a/26... | https://mathoverflow.net/users/8628 | Dominating vertex sets in hypergraphs | Let $V$ be the set of positive integers, and edges be the sets of the form $\{n,n+1,\dots\}$. Then any infinite set is dominating and any finite set is not.
| 1 | https://mathoverflow.net/users/4312 | 337809 | 144,151 |
https://mathoverflow.net/questions/337805 | -1 | Does this statement correct? if it does how we can prove it. In Banach spaces
a map is local diffeomorphism if and only if it is a Fredholm map of index zero with no critical points?
| https://mathoverflow.net/users/136096 | Are local diffeomorphisms Fredholm maps with index zero? | If a map is a local diffeomorphism, then it is differentiable and has a differentiable inverse, so by the chain rule the derivative is a linear isomorphism of topological vector spaces, and therefore has 0 kernel and 0 cokernel, and is therefore Fredholm with index zero. This answers the question in the title. On the o... | 5 | https://mathoverflow.net/users/13268 | 337810 | 144,152 |
https://mathoverflow.net/questions/337800 | 4 | Let $\chi\_{q}$ be a primitive Dirichlet character with modulus $q$ [(see definition at wikipedia )](https://en.wikipedia.org/wiki/Dirichlet_character).
For example for $q=5$ we have
\begin{equation}
\begin{aligned}
\chi\_{5,1}&=(1, 1, 1, 1, 0),\\
\chi\_{5,2}&=(1, i, -i, -1, 0),\qquad\qquad \text{(1)}\\
\chi\_{5,3}&=... | https://mathoverflow.net/users/33672 | The sign of an interesting sum involving a Dirichlet character | (3) and (4) are false in general, even if we weaken $>$ to $\geq$. Let $\zeta:=e^{i\pi/8}$ be a primitive $16$-th root of unity, and let $\chi$ be the unique primitive Dirichlet character modulo $17$ satisfying $\chi(3)=\zeta^5$. Then $\chi(-1)=-1$, and
$$(\chi(1),\chi(2),\chi(3),\chi(4),\chi(5),\chi(6),\chi(7),\chi(8)... | 8 | https://mathoverflow.net/users/11919 | 337811 | 144,153 |
https://mathoverflow.net/questions/337820 | 3 | Let $K$ be an imaginary quadratic field and let $\mathcal O\_f$ be an order in $K$ with conductor $f$. Let $\chi$ be a proper character of $\operatorname{Cl}(\mathcal O\_f)$ (non-principal if $f=1$). Define
$$L\_f(s,\chi)=\sum\_{(\mathfrak a,f)=1} \chi(\mathfrak a)N(\mathfrak a)^{-s}.$$
Here the sum is over all $\mathc... | https://mathoverflow.net/users/122104 | $L$-series and the $\zeta$-function of ideal classes modulo $f$ | Decompose the original sum according to the ideal class of $\mathfrak{a}$. Fixing the ideal class of $\mathfrak{a}$, we can write $\mathfrak{a}=\mathfrak{b}^{-1}\gamma$ where $\mathfrak{b}$ is a fixed representative of the inverse class, and $\gamma\in\mathfrak{b}$. The map $\gamma\mapsto\mathfrak{a}$ is $2$-to-$1$, be... | 3 | https://mathoverflow.net/users/11919 | 337821 | 144,156 |
https://mathoverflow.net/questions/337822 | 3 | Let $X$ be a small site. Let $\aleph$ be an infinite cardinal, such that $|Ob(X)|\leq \aleph$ and $|Mor(X)|\leq \aleph$, where $Mor(X)$ is the set of all morphisms.
We define the size of a presheaf $F$ of sets (or abelian groups, or modules, or ...) as
$$
|F|=\left|\bigsqcup\_{U\in X} F(U)\right|
$$
In [Flat Cover... | https://mathoverflow.net/users/82627 | The size of sheafification | *I'm going to write "$\kappa$" for your "$\aleph$," to more consistently match set-theoretic usage.*
While $2^\kappa$ appears a sharper bound than $\kappa^\kappa$, they are in fact the same (for infinite $\kappa$ at least, and I don't think finite $\kappa$ are important here). $2^\kappa\le\kappa^\kappa$ is clear. For... | 7 | https://mathoverflow.net/users/8133 | 337824 | 144,157 |
https://mathoverflow.net/questions/337836 | 9 | It is well known that the unit interval $[0,1]$ cannot be decomposed as a countable union of pairwise disjoint closed (nonempty) subsets. See for instance this [math.stackexchange question.](https://math.stackexchange.com/questions/6314/is-0-1-a-countable-disjoint-union-of-closed-sets) The proof using Baire category th... | https://mathoverflow.net/users/41274 | Uncountable disjoint closed coverings of $[0,1]$ | This is question has a long and interesting history, which is discussed in Arnie Miller's paper cited below. The first construction of a model of ZFC + $\aleph\_1 < 2^{\aleph\_0}$ where $[0,1]$ can be partitioned into $\aleph\_1$ pairwise disjoint nonempty closed sets is due to Jim Baumgartner (unpublished).
Early on... | 14 | https://mathoverflow.net/users/2000 | 337851 | 144,160 |
https://mathoverflow.net/questions/337549 | 1 |
>
>
> >
> >
> > >
> > > **My question:** Are the conjectures as follows correct?
> > >
> > >
> > >
> >
> >
> >
>
>
>
Given a positive integer $P>1$, let its prime factorization be written
$$P=p\_1^{a\_1}p\_2^{a\_2}p\_3^{a\_3}...p\_k^{a\_k}$$.
Define the functions $h(P)$ by $h(1)=1$ and $h(P)=min(a\... | https://mathoverflow.net/users/122662 | A generalization of Lander, Parkin, and Selfridge conjecture | The conjectures could not be true as stated,
due to simple counterexamples such as $3^8+3^8+3^8+2^9=2^8+2^8+3^9$.
One could exclude such constructions by conjecturing,
in the spirit of Schmidt's Subspace Theorem, that:
>
> if $n<d$, and $A\_i$ ($1 \leq i \leq n$) are nonzero integers
> with $\gcd(A\_1,\ldots,A\_... | 6 | https://mathoverflow.net/users/14830 | 337866 | 144,164 |
https://mathoverflow.net/questions/337857 | 1 | Consider the exponentiated Riemann-Zeta function $\zeta(s)^p$. If it is represented as
$$\zeta(s)^p = \sum\_{n=1}^\infty\frac{a\_n}{n^s}$$
Is there any upper bound we can put on $|a\_n|$ in terms of $n$ and $p$.
For example, not that when $p = 2$, we get the divisor function which can be bounded above by $O(n^{... | https://mathoverflow.net/users/75293 | Bounding Coefficients of Dirichlet Series | Let me rename $p$ to $z$, because $p$ usually stands for prime numbers in the subject. I will assume that $z\geq 2$, but I will not assume that $z$ is an integer. The Dirichlet coefficients of $\zeta(s)^z$ form a generalized divisor function:
$$\zeta(s)^\nu=\sum\_{n=1}^\infty\frac{\tau\_z(n)}{n^s},\qquad \Re(s)>1.$$
Th... | 7 | https://mathoverflow.net/users/11919 | 337877 | 144,168 |
https://mathoverflow.net/questions/337859 | 2 | I am trying to prove three inequalities that would help me solve the proof of a larger theorem.
Let $P(X,Y)$ be a discrete bivariate distribution and
$$
I(X;Y) = \sum\_{i,j} p(x\_i, y\_j) \log \frac{p(x\_i, y\_j)}{p(x\_i)p(y\_j)}
$$
the mutual information between $X$ and $Y$.
Let's call $\bar{P}(X,Y)$ the function ... | https://mathoverflow.net/users/101100 | Mutual information inequality | First, in general $I^a(X;Y) \geq 0$ does not hold. One can find easy counterexamples with just two states.
The other part of inequality (1) does hold.
For inequality (2), the reverse does actually hold. And with that, inequality (3) is trivially true.
We show that $\frac{\partial I^a}{\partial a} \geq 0$ for $a \le... | 1 | https://mathoverflow.net/users/106046 | 337887 | 144,171 |
https://mathoverflow.net/questions/337890 | 7 | For any Riemannian manifold are the exp and log maps (from a predetermined base) conformal? If not, are there some manifolds where they are and others where they aren't?
| https://mathoverflow.net/users/143645 | Are the exp and log maps of Riemannian geometry conformal | See Robert Bryant's answer to [Complex manifolds in which the exponential map is holomorphic](https://mathoverflow.net/questions/67903/complex-manifolds-in-which-the-exponential-map-is-holomorphic) in which he proves that the surfaces for which the exponential map is conformal are precisely the flat ones.
If the expo... | 6 | https://mathoverflow.net/users/13268 | 337893 | 144,172 |
https://mathoverflow.net/questions/337897 | 2 | Define $\pi(x)$ to be the prime counting function and Li(x) the logarithmic integral. Let $I\_s$ be defined as above.
Is $I\_s$ known to be convergent for any real number $s<1$ ?
| https://mathoverflow.net/users/nan | On the integral $I_s =\int_{1}^{\infty} (\pi(x)-Li(x))x^{-s-1} dx$ | I claim that no such $s$ is known to exist. Indeed, define $\sigma\_c$ to be the abscissa of convergence of $I$. Then
$$\sigma\_c = \limsup\_{x\rightarrow \infty} \frac{\log|\pi(x)-Li(x)|}{\log x}.$$ Since we do not know of any absolute $\theta<1$ such that $|\pi(x)-Li(x)|\ll x^{\theta}$, the claim follows.
| 7 | https://mathoverflow.net/users/480516 | 337899 | 144,173 |
https://mathoverflow.net/questions/337898 | 0 | Consider Lebesgue measure $m$ on $[0, 1]$. Fix a countable sequence $a\_i, 0 < a\_i < 1$ such that $\sum\_i a\_i = 1$. Is there a sequence of disjoint measurable subsets of $[0, 1]$, $E\_i$ whose measure in every open interval $I$ respectively is $a\_i m(I)$?
| https://mathoverflow.net/users/132446 | Existence of sequence of measurable sets with prescribed densities | There is no measurable subset $E$ of $[0,1]$ such that $m(E\cap I)=m(I)/2$ for every open interval $I\subseteq [0,1]$.
Indeed, assume there is such $E$. Then $m(E)=1/2$, so there is an open set $U$, $E\subseteq U \subseteq [0,1]$ such that $m(U)=3/4$. But $U$ is a union of a sequence of pairwise disjoint open interv... | 3 | https://mathoverflow.net/users/95282 | 337905 | 144,174 |
https://mathoverflow.net/questions/337869 | 7 | Using a "geometrical" argument of dimension, like the one [here](https://math.stackexchange.com/questions/499264/projective-space-is-not-affine), one can show that the projective space is not affine.
I am interested in showing that, but using a categorical argument, i.e. I want to show that $\mathbb{P}^n:\operatorna... | https://mathoverflow.net/users/142626 | The projective functor $\mathbb{P}^n: \operatorname{CRing} \to \operatorname{Set}$ is not representable: categorical argument | $\def\PP{\mathbb{P}}\def\AA{\mathbb{A}}\def\GG{\mathbb{G}}\def\Spec{\mathrm{Spec}}$This is probably going to sound too classical to satisfy, but it seems straightforward to me. Let $\PP^{n}\_{charts}$ be the functor represented by the scheme which is normally called projective $n$-space. In other words, $\PP^{n}\_{char... | 5 | https://mathoverflow.net/users/297 | 337909 | 144,176 |
https://mathoverflow.net/questions/337912 | -1 | How do I evaluate the following finite sum over $k$
$1+\frac{1}{2^n}+\frac{1}{2^n3^n}+\frac{1}{2^n3^n4^n}+\cdots+\frac{1}{2^n3^n\cdots k^n}$
or if there is an expression of this sum in terms of other known numbers ??
| https://mathoverflow.net/users/144164 | How do I calculate this sum $\sum_k(k!)^{-n}$? | $${}\_1F\_n(1;2,2,\ldots 2; 1) - \frac{{}\_1F\_{n}(1;2+k,2+k,\ldots 2+k; 1)}{((k+1)!)^n} $$
| 0 | https://mathoverflow.net/users/13650 | 337914 | 144,178 |
https://mathoverflow.net/questions/334672 | 3 | I'm trying to understand a more basic version about Gromov's theorem about the compactly supported symplectomorphisms of $D^2 \times D^2$ being contractible.
Consider the dimensional disk $D^2 \subset \mathbb{R}^2$. Let $\omega\_0$ be the standard symplectic form on $\mathbb{R}^2$. Let $\omega$ be any symplectic for... | https://mathoverflow.net/users/92483 | Compactly supported symplectomorphisms of $D^2$ | Mike is right. I guess that you assume that $\omega$ and $\omega\_0$ have the same total area $\pi$. You have to put things in the right order. Moser's lemma assures that there is a self-diffeomorphism $\psi$
of $D^2$ (everything is understood to be the identity close to $\partial D^2$), such that $\omega\_0=\psi^\*(\o... | 3 | https://mathoverflow.net/users/105095 | 337923 | 144,181 |
https://mathoverflow.net/questions/337920 | 3 | I have an interest in the set
$$A= \bigg\{\frac{ab+c}{(2a+1)b+c}\,\bigg|\, a \in {\mathbb Z}^+, b\in{\mathbb Z}^+~\text{is \((a+1)\)-smooth}, 0\leq c\leq ab\bigg\}.$$ In particular, is $A$ dense in the interval $\big[\frac 13,\frac 12\big)$?
The question is pretty much self-explanatory (given I mean by ${\mathbb Z}^... | https://mathoverflow.net/users/128140 | Is this set of fractions dense in the interval $\big[\frac 13,\frac 12\big)$? | Take $a=1$, so $b = 2^k$, and let $c = t 2^k$ where $t$ is a dyadic rational in $[0,1]$.
Then $$ \frac{ab+c}{(2a+1)b+c} = \frac{t+1}{t+3}$$
The dyadic rationals are dense in $[0,1]$ and the function $f:\; t \mapsto (t+1)/(t+3)$ is continuous from $[0,1]$ onto $[1/3, 1/2]$, so these values are indeed dense in $[1/3, 1/2... | 5 | https://mathoverflow.net/users/13650 | 337924 | 144,182 |
https://mathoverflow.net/questions/337889 | 0 | Let $X,Y$ be Banach spaces. Suppose that $X^{\*\*\*}$ has the metric approximation property. Let $T:X^{\*\*}\rightarrow Y$ be a finite-rank operator and let $\epsilon>0$. Is there a finite-rank operator $S:X\rightarrow X$ such that $\|S\|\leq 1+\epsilon$ and $\|T-TS^{\*\*}\|<\epsilon$?
Furthermore, I want to know whe... | https://mathoverflow.net/users/41619 | A question on the metric approximation property | No. The operator $T$ can vanish on $X$ while any such $S^{\*\*}$ necessarily has its range contained in $X$.
| 2 | https://mathoverflow.net/users/2554 | 337928 | 144,183 |
https://mathoverflow.net/questions/337621 | 7 | I am trying to find the next rank 8 curve with the torsion subgroup Z/6 using Kihara's family as described in <https://arxiv.org/pdf/1503.03667.pdf>. Meanwhile, I came across a curve generated by $t=629/3287$ (or $t=6202/8089$, $t=-8089/1772$, $t=-23009/1258$).
Magma Calculator (<http://magma.maths.usyd.edu.au/calc/... | https://mathoverflow.net/users/95511 | One more generator needed for a Z/6 elliptic curve | Yes. A 7th generator has $x$-coordinate
$$
181265389257356655988118224516379188326810855287159053664052560/3919647209484520988422390115383428889.
$$
Knowing the $6$ generators Magma finds (let's say they are P1, P2, P3, P4, P5, P6), the Magma command
```
twocovers := TwoDescent(E : RemoveTorsion := true, RemoveGens ... | 7 | https://mathoverflow.net/users/48142 | 337934 | 144,184 |
https://mathoverflow.net/questions/337931 | 6 | Let $C$ be a category and $A\in\mathrm{ob}(C)$. A *relation* is a subobject $q:Q\hookrightarrow A^{\times 2}$ and the quotient is defined as the coequalizer
$$A/Q:=\mathrm{coeq}\left(Q\stackrel{q}{\hookrightarrow} A^{\times 2}\rightrightarrows A\right)$$
where the two maps are the two projections. Moreover, we can defi... | https://mathoverflow.net/users/124042 | Equivalence relations in arbitrary categories | **Short answer** :Yes, assuming $\overline{Q}$ exists and $C$ has kernel pairs (for example if it has finite limits).
**For more details:** The relation $\overline{Q}$ do not always exists, you need some assumption on the underlying categories, and there are various type of assumption that can work.
For example if... | 6 | https://mathoverflow.net/users/22131 | 337940 | 144,185 |
https://mathoverflow.net/questions/337935 | 2 | Let $f : AC[0, 1] \to R$ be defined by $ f(x(.)) := \int\_{0}^{1} F ( x(t)) \; dt$. Where, $AC[0, 1]$ is the set of absolutely continuous functions with the norm $W^{1,1}$, and $F: R^n \to R$ is continuously differentiable function. My question is that, Is $f$ Fréchet differentiable? If not is it at least Gateaux diffe... | https://mathoverflow.net/users/108824 | Is $ f(x(.)) := \int_{0}^{1} F ( x(t)) \; dt$ differentiable? | From the context, here $AC[0, 1]$ should be the set of all absolutely continuous functions **from $[0,1]$ to $\mathbb R^n$** with the norm $W^{1,1}$.
Note that for all $x=(x\_1,\dots,x\_n)\in AC[0, 1]$ we have $\|x\|\_\infty:=\sup\{|x\_i(t)|\colon i=1,\dots,n,0\le t\le 1\}\le2\|x\|\_{1,1}$; see the lemma at the end ... | 3 | https://mathoverflow.net/users/36721 | 337944 | 144,186 |
https://mathoverflow.net/questions/337927 | 1 | This is probably known already, but I could not find a quick argument.
Let $M$ be an $n\times m$ binary matrix with iid Bernoulli$(1/2)$ entries, and $n>m$. Tikhomirov recently settled that the probability that an $m\times m$ such matrix is singular is $(1/2+o(1))^m$.
My question is: What is a good lower bound on... | https://mathoverflow.net/users/127150 | Probability that random Bernoulli matrix is full rank | Apparently, the paper "ON THE PROBABILITY THAT A RANDOM ±1-MATRIX IS SINGULAR" by Kahn, Komlos and Szemeredi (Corollary 4 therein) answers my question, and states that it is $(1+o(1))2\binom{m}{2}/2^n$, thereby improving the bound from exponential in $m$ to polynomial in $m$.
<https://www.ams.org/journals/jams/1995-0... | 1 | https://mathoverflow.net/users/127150 | 337946 | 144,187 |
https://mathoverflow.net/questions/337939 | 10 | I look for a reference for the following problem.
Given an integer $k$, find a set $A\subset\mathbb{N}$ with $|A|=k$
that maximizes $t$ such that $\left[0,1,..,t\right]\subset A+A$.
| https://mathoverflow.net/users/71090 | Sets A such that A+A contains the largest set [0,1,..,t] | A table of values for these $t$ are given in the introduction Graham and Sloane's [On Additive Bases and Harmonius Graphs](https://epubs.siam.org/doi/pdf/10.1137/0601045) (your sequence corresponds to $n\_\beta(k)$ in their notation). Graham and Sloane also give some references to previous work with this sequence, both... | 10 | https://mathoverflow.net/users/405 | 337953 | 144,190 |
https://mathoverflow.net/questions/337949 | 3 | Given a compact Hausdorff topological group $G$ and probability measures $\mu$ and $\tau$ on the Borel sets of $G$, their convolution is the probability measure
$(\tau\*\mu)(A)=\int\int1\_A(xy)d\tau(x)d\mu(y)$.
A useful interpretation of this is that $G$ is a permutation group of some deck of cards, $\mu$ describes... | https://mathoverflow.net/users/58787 | Central limit type theorems for compact Hausdorff topological groups? | For instance, in the recent [paper by Harremoes, page 12](https://arxiv.org/abs/0901.0015) one finds this:
>
> **Corollary 20.** Let $P$ be a probability measure on the compact group $G$ with Haar probability measure
> $U$. Assume that the support of $P$ is not contained in any coset of a proper subgroup of $G$. ... | 4 | https://mathoverflow.net/users/36721 | 337956 | 144,191 |
https://mathoverflow.net/questions/337965 | 3 | Could someone will verify my statement: For every locally finite Borel measure on metric space and closed set $F$ with finite measure, there exists open set $U$ such that $F \subset U$ and $U$ has finite measure ?
| https://mathoverflow.net/users/143974 | Is every closed subset of finite measure contained in an open subset of finite measure? | No, not in general.
My metric space is the disjoint union of uncountably many copies of $\mathbb R$.
$$X = \bigsqcup\_{t \in T} X\_t$$
where $T$ is uncountable and $X\_t = \mathbb R$ for all $t$. The metric: two points in the same $X\_t$ have distance $\min(|x-y|,1)$, two points in different $X\_t$ have distance $1$.... | 8 | https://mathoverflow.net/users/454 | 337968 | 144,194 |
https://mathoverflow.net/questions/337951 | 4 | Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous functions with the following properties:
1. $f(x)$ and ${g(x)}/x$ are bounded;
2. ${g(x)}/{\left(1+x^2\right)}\in L^1\left(\mathbb{R}\right)$;
3. $\lim\_{x\to0}f(x)/x^2=1$;
and also
$$\int\_{-\infty}^\infty\frac{f(x)}{\left(1+x^2\right)^p}dx=p\int\_{-\infty}^{\infty}\... | https://mathoverflow.net/users/117091 | Functions orthogonal to powers of $1/{\left(1+x^2\right)}$ | Let $\tilde{f}(x) = f(x)+f(-x)$, and $\tilde{g} (x) = g(x)+g(-x)$, and let $F(x) = \int\_{0}^{x} \tilde{f}(t)dt$. Then your condition can be rewritten as (after integration by parts in the left hand side)
$$
\int\_{0}^{\infty} \frac{2xF(x)-\tilde{g}(x) (1+x^{2})}{1+x^{2}} (1+x^{2})^{-p}dx =0
$$
for all $p\geq 1$.
N... | 2 | https://mathoverflow.net/users/50901 | 337976 | 144,199 |
https://mathoverflow.net/questions/337941 | 5 | Schur functions are irreducible characters of the unitary group $U(N)$. This implies
$$ \int\_{{U}(N)}s\_\lambda(u)\overline{s\_\mu(u)}du=\delta\_{\lambda\mu},$$
where the overline means complex conjugation.
My question is what is the result of the same integral performed over $SU(N)$ instead,
$$ \int\_{{SU}(N)}s\_\la... | https://mathoverflow.net/users/83671 | Integral of Schur functions over $SU(N)$ instead of $U(N)$ | The desired integral is given in equation (13) of [arXiv:1812.06069](https://arxiv.org/abs/1812.06069):
$$\int\_{{SU}(N)}s\_\lambda(u)\overline{s\_\mu(u)}du=\sum\_{q=-\infty}^\infty\prod\_{i=1}^N\delta\_{\lambda\_i,\mu\_i+q},$$
where $\lambda=(\lambda\_1,\lambda\_2,\ldots\lambda\_N)$ and $|\lambda|=\sum\_{i}\lambda\_... | 6 | https://mathoverflow.net/users/11260 | 337992 | 144,204 |
https://mathoverflow.net/questions/337938 | 6 | In Weibel's *An introduction to homological algebra* he defines a cone as an explicit chain complex associated to the given one -i.e. for a chain $C=(C\_i, d)$ he defines $Cone(C)=\left(C\_{i-1} \oplus C\_i, \begin{bmatrix}
-d & 0 \\
-id & d
\end{bmatrix} \right)$.
It is good and it is trivial that we have a monomor... | https://mathoverflow.net/users/127260 | Why does every chain complex have a map into its cone? | This is really just rephrasing Simon Henry's comment, but there is a natural transformation $F\to\text{Hom}(C,-)$ given by $(f,s)\mapsto f$, and so if $\text{Cone}(C)$ represents $F$ then by Yoneda's lemma this natural transformation is induced by a map $C\to\text{Cone}(C)$.
You can also see that this map is a monomo... | 7 | https://mathoverflow.net/users/22989 | 338004 | 144,207 |
https://mathoverflow.net/questions/337767 | 2 | Given a compact homogeneous space $M = K/L$, consider its de Rham complex $(\Omega^\*,d)$. Will every cohomology class $[\omega] \in ker(d)/im(d)$ contain a representative $\nu$ which is invariant with respect to the left $K$-action on $\Omega^\*$?
| https://mathoverflow.net/users/143172 | Coinvariant representative of homogeneous space cohomology | Yes, assuming that $K$ is a connected compact Lie group.
Indeed, fix $n$ such that $0\le n\le d={\rm dim}(M)$.
The group $K$ acts on the integral cohomology group $H^n(M,\Bbb Z)$ trivially, because $K$ is connected, while $H^n(M,\Bbb Z)$ is discrete.
Therefore, $K$ acts trivially on the de Rham cohomology group
$... | 1 | https://mathoverflow.net/users/4149 | 338007 | 144,209 |
https://mathoverflow.net/questions/337995 | 3 | Let $G=(V,E)$ be a simple, undirected graph with $\bigcup = E$ (that is, there are no isolated vertices). We say that $C\subseteq E$ is an *edge cover* of $G$ if $\bigcup C = V$. For any edge cover $C$ of $G$ we define the set of multiply covered vertices by $$\text{m}(C) = \big\{v\in V: |\{e\in C: v\in e\}|>1\big\}.$$... | https://mathoverflow.net/users/8628 | Avoiding multiply covered vertices in graph edge coverings | For $n \in \mathbb N$ Let $a\_n, b\_n$ be a pair of vertices connected by an edge. For every finite subset $M \subset \mathbb N$ take an additional vertex $v\_M$ and connect it to every vertex $a\_n$ With $n \notin M$.
Any edge cover of this graph must contain all edges $a\_nb\_n$, otherwise $b\_n$ would be uncovered... | 4 | https://mathoverflow.net/users/97426 | 338014 | 144,213 |
https://mathoverflow.net/questions/337749 | 5 | Suppose $G$ is a finite group. We will say, that it force solvability if any finite group $H$, such that $G$ is isomorphic to its maximal proper subgroup, is solvable. Does there exist some sort of classification of such groups?
On one hand, all such groups have to be solvable. On the other hand, there are several la... | https://mathoverflow.net/users/110691 | Is there some sort of classification of finite groups that force solvability? | I make a remark in the opposite direction. Let $p \geq 17$ be a Fermat or Mersenne prime, so that $X = {\rm PSL}(2,p)$ has a dihedral Sylow $2$-subgroup $D$ which is maximal. Let $d >1$ be a power of $2$, and let $Q$ be a transitive $2$-subgroup of $S\_{d}$. Let $A= {\rm Aut}(X)$ and let $T$ be a Sylow $2$-subgroup of ... | 4 | https://mathoverflow.net/users/14450 | 338030 | 144,217 |
https://mathoverflow.net/questions/337986 | 3 | There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: <http://www.math.brown.edu/~deigen/chern.pdf>
**Any isometry between two closed smooth convex surfaces (equipped with the induced path metrics) in the Euclidean space $\mathbb{R}^3$ is established by an isometry of $\mat... | https://mathoverflow.net/users/16183 | Pogorelov's rigidity theorem vs Cohn-Vossen rigidity theorem | Below is the answer from the comments. First some terminology.
A *convex surface* is the boundary of a compact convex body in $\mathbb R^3$.
Each convex surface comes with two metrics: the path-metric and the metric obtained by restricting the distance function on $\mathbb R^3$, which we call *intrinsic* and *extri... | 6 | https://mathoverflow.net/users/1573 | 338046 | 144,223 |
https://mathoverflow.net/questions/337818 | 3 | Let $C$ be a smooth projective curve in a surface $S$. Suppose $E$ is a vector bundle of rank $r$ on $C$. Then what is the total Chern class of the sheaf $i\_\*E$, where $i$ is an embedding of $C$ in $S$ ?
| https://mathoverflow.net/users/130022 | Chern class of direct image sheaf | The following is the way I like to do this computation. It is entirely equivalent to using GRR, and ends up being longer, but is a little more elementary. The final answer will be $c\_1(i\_\*E)=rC$ and $c\_2(i\_\*E)=\frac{1}{2}r(r+1)C^2 -d$ where $r=rk(E)$ and $d=deg(E)$.
We may assume for this sort of computation t... | 4 | https://mathoverflow.net/users/9617 | 338047 | 144,224 |
https://mathoverflow.net/questions/337570 | 1 | Let $p,q \in \mathbb{P}$, $p \geq 3$ and $q$ is the next prime to $p$.
For $b \in \mathbb{P}$ Consider : $N\_b = \displaystyle{\small \prod\_{\substack{a \leq b \\ \text{a prime}}} {\normalsize a}}$
Let $n \in \mathbb{N}$,
$n$ is **q-point** iff $n = q \alpha$ with $\gcd(\alpha, N\_p)=1$
>
> **My Conjecture:*... | https://mathoverflow.net/users/164630 | Prove the existence of this number | This is almost surely false. The size of the largest gap between numbers coprime to $N=N\_p$ grows at a faster rate than do the primes. There may even be an example with q less than 1000 where q is the least prime factor of the numbers c and d, and every number in between c and d has a smaller least prime factor. Such ... | 3 | https://mathoverflow.net/users/3402 | 338052 | 144,227 |
https://mathoverflow.net/questions/338048 | 3 | I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been great, so I'm hoping the (higher) category theory people here can help me out!
**Categories Of Functors:** Let $\cal{C}$ b... | https://mathoverflow.net/users/123015 | Proving a Kan-like condition for functors to model categories? | I think the following works, if I’ve followed your terminology conventions correctly: Define each $T\_i$ (first the missing codim-1 face, then the main simplex itself) as the *fibrant* replacement of the colimit of sub-faces you tried, so that the map $\varinjlim (S\_x) \to T\_i$ is an acyclic cofibration? Being acycli... | 2 | https://mathoverflow.net/users/2273 | 338063 | 144,231 |
https://mathoverflow.net/questions/338077 | 8 | I am looking for a reference for the proof of the following question following Theorem 5 in Mazur's *Rational Isogenies of Prime Degree*.
**Theorem 5** There is a constant $C$ such that every elliptic curve $E\_{/\mathbb{Q}}$ is isogenous (over $\mathbb{Q}$) to at most $C$ (mutually nonisomorphic) elliptic curves.
... | https://mathoverflow.net/users/4306 | Max order of an isogeny class of rational elliptic curves is 8? | M. Kenku, [On the number of $\mathbf{Q}$-isomorphism classes of elliptic curves in each $\mathbf{Q}$-isogeny
class](https://core.ac.uk/download/pdf/82233672.pdf), J. Number Theory **15**, 199 (1982):
*It is shown that there are at most eight $\mathbf{Q}$-isomorphism classes of elliptic curves in each $\mathbf{Q}$-is... | 13 | https://mathoverflow.net/users/11260 | 338078 | 144,236 |
https://mathoverflow.net/questions/338075 | 12 | Let $X$ be either Euclidean space $\mathbb{R}^n$, the sphere $\mathbb{S}^n$, or hyperbolic space $\mathbb{H}^n$.
>
> I would like to have a classification of all diffeomorphisms $X\to X$ which map every geodesic line to a geodesic line.
>
>
>
In the first two cases, the group of all such transformations is st... | https://mathoverflow.net/users/16183 | Geodesic preserving diffeomorphisms of constant curvature spaces | For $\mathbb{R}^n$: the fundamental theorem of projective geometry (proof:
<https://www3.nd.edu/~andyp/notes/FunThmProjGeom.pdf>) says that the bijections of $\mathbb{R}^n$ taking lines to lines are the affine maps $x\mapsto Ax+b$ for an invertible matrix $A$ and a constant vector $b$.
For $S^n$: a theorem by the sa... | 16 | https://mathoverflow.net/users/13268 | 338080 | 144,237 |
https://mathoverflow.net/questions/338068 | 6 | I was expressed by how Mendelson describes models in his Introduction to mathematical logic. Now I am looking for a nice model theory guide. The book (video source, etc.) must:
1. Include the concrete methods with their proofs and must answer the following questions:
1.1. how to know if a theory has a model
1.2... | https://mathoverflow.net/users/144230 | Simple book on model theory | The best book for you is probably *A Shorter Model Theory* by Hodges.
Some comments on your question, though: First, you should be aware that the model theory of finite structures and the model theory of infinite structures have extremely different characters - so much so that finite model theory is essentially a sep... | 9 | https://mathoverflow.net/users/2126 | 338083 | 144,238 |
https://mathoverflow.net/questions/338074 | 3 | Given a sequence of complex numbers $\{a\_n\}\_n$, one says that this sequence admits $a$ as a *sequential density* if
$$\underset{N\_s\to\infty}{\lim}\frac{1}{N\_s}\sum\_{n=1}^{N\_s} a\_n = a$$
where $N\_s = 2^{2^s}$ for instance. The sequence admits $a$ as a *logarithmic density* if
$$\underset{N\to\infty}{\lim}\fr... | https://mathoverflow.net/users/122199 | Logarithmic vs sequential density of a sequence | The answer is "no" even if we assume that the $a\_n$ are bounded. For example, take $a\_n=1$ if the fractional part of $\log\_2\log\_2 n$ is between $0$ and $\frac12$ (equivalently, if $n$ is between $2^{2^k}$ and $2^{2^{k+1/2}}$ for some integer $k$), and $a\_n=0$ otherwise. Then the sequential limit will equal $0$, b... | 3 | https://mathoverflow.net/users/5091 | 338086 | 144,239 |
https://mathoverflow.net/questions/337919 | 0 | I would like to know if there exist a Harnack type inequality for the non-local operator of the form $$(-\Delta)^s u= f \text{ in } B\subset \mathbb R^N$$ with $0<s<1$ and $N \geq 1.$ Here $B$ is a unit ball and $f\not \equiv 0$ is bounded in $\mathbb R^N$ and $u$ is non-negative and smooth function in $\mathbb R^N$. I... | https://mathoverflow.net/users/127663 | Harnack inequaliity for the fractional Laplacian | The classical Harnack's inequality $\sup\_{B\_{1/2}} u\_1 \le C \inf\_B u$ for non-negative solutions of $(-\Delta)^s u\_1 = 0$ in $B$ goes back to M. Riesz's 1938 seminal paper.
The bound $\sup\_B |u\_2| \le C \|f\|\_\infty$ for solutions of $(-\Delta)^s u\_2 = f$ in $B$ with $u\_2 = 0$ in $B^c$ follows from compari... | 0 | https://mathoverflow.net/users/108637 | 338101 | 144,243 |
https://mathoverflow.net/questions/338073 | 10 | I previously asked [In which topological spaces does the existence of a loop not contractable to a point imply there is a non-contractable simple loop also?](https://mathoverflow.net/questions/338070/in-which-topological-spaces-does-the-existence-of-a-loop-not-contractable-to-a-p)
Given the broad scope of this questi... | https://mathoverflow.net/users/7113 | In a subset of $\mathbb{R}^2$ which is not simply connected does there exist a simple loop that does not contract to a point? | One-dimensional metric spaces and planar sets do have the property that you're interested in. To explain why this works out in such generality requires a combination of planar topology, continuum theory, and shape theory.
**One-Dimensional Case:** This is pretty classical, going back to work of Curtis and Fort in the... | 12 | https://mathoverflow.net/users/5801 | 338104 | 144,244 |
https://mathoverflow.net/questions/338099 | 9 | Are there general ways for given rational coefficients $a,b,c$ (I am particularly interested in $a=3,b=1,c=8076$, but in general case too) to answer whether this equation has a rational solution or not?
| https://mathoverflow.net/users/4312 | Diophantine equations $ax^4+by^2=c$ in rational numbers | Your curve is a genus 1 curve, usually expressed as
$$ y^2= -abx^4+bc$$
(just by multiplying everything by $b$ and changing $y$ by $by$).
The curve has local points everywhere, but since it looks like has no rational points, you can try to do a 2-descent. This can be done easily with the TwoCoverDescent algorithm a... | 8 | https://mathoverflow.net/users/24442 | 338106 | 144,245 |
https://mathoverflow.net/questions/338105 | 4 | Let $A$ be a commutative ring and $I \subseteq A$ a finitely generated ideal (I am **not** assuming that $A$ is Noetherian).
Assume that $A$ is derived $I$-complete, meaning, let's say, that $\mathrm{Hom}\_A(A\_f, A)=\mathrm{Ext}^1\_A(A\_f, A)=0$ for all $f \in I$, or equivalently that the map $A^{\times \mathbb{N}}... | https://mathoverflow.net/users/60903 | Jacobson radical of a derived $I$-complete ring | Never mind, I figured it out. Let me leave it here for potential benefit of other people trying to learn about derived completions:
Given $f \in I,$ there is a unique preimage $(y\_n)\_n$ to the element $(1, 1, 1, \dots)$ under the map $A^{\times \mathbb{N}} \rightarrow A^{\times \mathbb{N}}, (x\_n)\_n \mapsto (x\_n-... | 8 | https://mathoverflow.net/users/60903 | 338109 | 144,247 |
https://mathoverflow.net/questions/337603 | 4 | For the two dimensional anti de-sitter space $\mathrm{AdS}\_2$ one can consider the Poincaré-coordinates $\mathrm{d}s^2\_P = -r^2 \mathrm{d}t^2 + \frac{1}{r^2} \mathrm{d}r^2$ which covers only half of the spacetime. Most of the time though one uses the global coordinates $\mathrm{d}s^2\_{G1} = -(1+r^2) \mathrm{d}t^2 + ... | https://mathoverflow.net/users/142501 | Transformation Poincaré-coordinates to global coordinates in $\mathrm{AdS}_2$ | The coordinate transformation from $(r,t)$ to $(\rho,\tau)$ given by
$$\rho= \cosh r \cos t+\sinh r,\;\;\tau= \frac{\cosh r \sin t }{\cosh ^2 r \sin ^2 t-1}(\sinh r-\cosh r \cos t),$$
$$\text{in the range}\;\;r>0, \;-\pi/2<t<\pi/2\Leftrightarrow\rho>0,\;-\infty<\tau<\infty,$$
converts the global metric
$$ds^2 = - \cosh... | 2 | https://mathoverflow.net/users/11260 | 338126 | 144,252 |
https://mathoverflow.net/questions/338118 | 1 | This question is essentially a followup of [this question](https://mathoverflow.net/questions/337370/how-should-i-think-about-concrete-functors-and-in-particular-about-concrete-isom). But before going into the question let me introduce the relevant definitions as given in [*The Joy of Cats*](http://katmat.math.uni-brem... | https://mathoverflow.net/users/nan | Understanding the reason for the particular formulation of the definition of a concrete reflector (as stated in The Joy of Cats) | I suppose that there are possibly many different answer to this question. Here is the one I got.
Being a reflector is equivalent to being an inclusion that has a left adjoint.
In general being a concrete inclusion, thus an inclusion of categories, that is reflective ensure that the inclusion, as a functor of categ... | 1 | https://mathoverflow.net/users/14969 | 338128 | 144,254 |
https://mathoverflow.net/questions/338124 | 3 | Let $K$ be a global field with ring of integers $O$, and let $f$ some integer valued function whose domain is the set of ideals of $O$ (e.g. $f(I)=|O:I|$). Extending the ordinary definition from the case $K=\mathbb Q$, one says that $f$ is *multiplicative* if $\newcommand{\mfp}{\mathfrak{p}}f(\mfp\_1\mfp\_2)=f(\mfp\_1)... | https://mathoverflow.net/users/14443 | Reference request for Euler products in positive characteristic | It's better to just work with effective divisors. An effective divisor is simply a formal sum with nonnegative integer coefficients of finitely many valuations of $K$ (= closed points of the curve that $K$ is the function field of). The prime ideals of any ring of integers of $K$ will be naturally in bijection with thi... | 6 | https://mathoverflow.net/users/18060 | 338129 | 144,255 |
https://mathoverflow.net/questions/338125 | 1 | Let $B$ be a commutative, Jacobson semi-simple unital Banach algebra and take an invertible element $x$ in $B$. We may then compute the infimum of the Gelfand transform:
$\delta = \inf |f(x)|$
where the infimum is taken over all charaters on $B$. Can we estimate $\|x^{-1}\|$ from above by a function of $\|x\|$ and ... | https://mathoverflow.net/users/144254 | Bounding the norm of the inverse in a commutative Banach algebra from above | This does not always work, although for some algebras where it does not work, one has a weaker form of controlled inversion where one fixes $\delta$ to be greater than some threshold, and can then bound $\Vert f^{-1}\Vert$ from above by some function depending only on $\delta$ (assuming $\inf\_x \vert f(x)\vert \geq\de... | 3 | https://mathoverflow.net/users/763 | 338159 | 144,265 |
https://mathoverflow.net/questions/338115 | 3 | I've had some difficulty using Sage/Singular to compute the decomposition of an ideal in a reasonable time, so I've developed a probabilistic algorithm to do this, and I am wondering if anybody has already done something like this.
For the particular example that I've been working with, I've got a ring over the ratio... | https://mathoverflow.net/users/78871 | Probabilistic algorithm for decomposition of a variety | I think it's fair to say that people have done something like this, though I'm not aware of anyone using the exact sequence of steps you've described.
There is an active community of people (full disclosure--including me) using numerical methods to study algebraic varieties. These are most effective for questions ove... | 4 | https://mathoverflow.net/users/124323 | 338160 | 144,266 |
https://mathoverflow.net/questions/338188 | 0 | Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). If $\kappa$ is a cardinal, we say that a map $c:E\to \kappa$ is an *edge coloring* if whenever $e\_1,e\_2\in E$ with $e\_1\cap e\_2\neq \emptyset$ then $c(e\_1)\neq c(e\_2)$. The smallest cardinal $\kappa$ such that there is an edge coloring $c:... | https://mathoverflow.net/users/8628 | Edge coloring in dense linear hypergraphs | Take a finite projective plane $π=\{P,L\}$ of order $n$, and remove all the points from a line $l$. Let the removed points be $p\_1,p\_2,...p\_{n+1}$. The hypergraph $H$ defined by such an incidence structure has $n^2$ vertices, and every edge has $n$ elements.
Property 1 and 2 are obvious.
Property 3 follows from ... | 2 | https://mathoverflow.net/users/125498 | 338189 | 144,274 |
https://mathoverflow.net/questions/338200 | 1 | I am looking for a specific [matroid](https://en.wikipedia.org/wiki/Matroid). I found a source that claimed to discuss these matroids, but then, only discusses [geometric lattice](https://en.wikipedia.org/wiki/Geometric_lattice). Even more, in that paper, the geometric lattice that seems to be the right one was describ... | https://mathoverflow.net/users/108884 | From Steiner systems to geometric lattices to matroids | Let $B$ be the subsets of $\{1,2,...,22\}$ of size $4$ which are not contained in any block of $S(3,6,22)$. $B$ satisfies the basis exchange property.
Proof:
Let $X,Y\in B$, $a\in X$. $X \setminus a$ is a 3-element set, so it is contained in exactly one block of $S(3,6,22)$.
If $∀y\in Y :(X \setminus a) \cup y ... | 2 | https://mathoverflow.net/users/125498 | 338203 | 144,278 |
https://mathoverflow.net/questions/337524 | 4 | Per the title, what are some of the oldest books on algebraic curves out there with unsolved exercises? Maybe there are some hidden gems from before the 20th century out there.
| https://mathoverflow.net/users/126532 | Reference request: Oldest books on algebraic curves with unsolved exercises? | Zeuthen’s *Lehrbuch der abzählenden Methoden der Geometrie* ([1914](//zbmath.org/?q=an:45.0799.01)) has *Übungsaufgaben* at the end of most sections (§§35, 48, 54, 59, 92, 97, 115, 141, 157, 166, 178, 185, 205).
Magnus’ *Sammlung von Aufgaben und Lehrsätzen aus der analytischen Geometrie* ([1833, 1837](//gdz.sub.uni-... | 3 | https://mathoverflow.net/users/19276 | 338204 | 144,279 |
https://mathoverflow.net/questions/338207 | 4 | $\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$
Let $B$ be a (maybe nonseparable) Banach space equipped with the Borel $\sigma$-algebra $\mathscr{B}(B)$. Let $R:B\to \mathbb{R}$ be a bounded linear operator.
Let $(\Omega,\mathcal{F},P)$ be a probability space. Let $F$ be a $\mathscr{B}(B)|\mathcal{F}$-measurable mapping ... | https://mathoverflow.net/users/7646 | Basic properties of expectation in non-separable Banach spaces | $\newcommand{\E}{\operatorname{\mathsf{E}}}$
You do not need the separability of $B$ to define $\E F$ for a random vector $F\colon\Omega\to B$; however, you need to assume that $F$ is strongly measurable, in the sense that there is a sequence of finitely-valued random vectors $F\_n$ in $B$ such that $\|F\_n(\omega)-F(\... | 7 | https://mathoverflow.net/users/36721 | 338209 | 144,280 |
https://mathoverflow.net/questions/338174 | 5 | Working on my current research problem, Teichmuller spaces for surfaces with cone points have come into play. It's fairly easy to formulate some of the definitions, a few basic results, etc. To be precise, I am looking to understand the Teichmuller space of a closed surface $M$ with genus $g$ and $b$ cone points with f... | https://mathoverflow.net/users/135446 | Teichmuller space for surface with cone points | Here are some recent papers:
Rafe Mazzeo, Hartmut Weiss
arXiv:1509.07608
Teichmüller theory for conic surfaces
arXiv:1710.09781 Rafe Mazzeo and Xuwen Zhu,
Conical metrics on Riemann surfaces, I: the compactified configuration space and regularity.
| 2 | https://mathoverflow.net/users/25510 | 338214 | 144,281 |
https://mathoverflow.net/questions/338213 | 9 | This is inspired by [this old question](http://mathoverflow.net/questions/219315/aligned-roots-of-irreducible-polynomials), which may provide a bit more background. But the two present questions seem somewhat more fundamental to me.
Let $p$ be an irreducible polynomial with integer coefficients.
>
> 1. Is it poss... | https://mathoverflow.net/users/29783 | Collinear Galois conjugates | The answer to Q1 is yes. For example, $p(x) = x^6 + 45x^4 + 122x^3 + 504x^2 + 1740x + 2213$ is a polynomial with three roots on the line $y = 2x+3$ (and the other three on the line $y = -2x-3$). Take your favorite irreducible cubic with real roots $f(x)$ (mine is $x^{3} - 3x + 1$) and let $\alpha\_{1}$, $\alpha\_{2}$, ... | 11 | https://mathoverflow.net/users/48142 | 338215 | 144,282 |
https://mathoverflow.net/questions/338252 | 5 | For Lebesgue integrals one has the triangle inequality saying that for continuous functions let's say
$$\left\vert\int\_0^t f(s) \ ds\right\vert \le \Vert f \Vert\_{\infty} \int\_0^t \ ds$$
Now if we consider an Ito integral, then
$$\left\vert\int\_0^t f(s) \ dW(s)\right\vert \le \Vert f \Vert\_{\infty} \vert \i... | https://mathoverflow.net/users/119875 | Triangle inequality for Ito integral? | If the function $f$ is indeed deterministic, with $M:=\|f\|\_\infty$ and $\sigma^2:=\int\_0^t f(s)^2\,ds$, then $X:=\int\_0^t f(s)\,dW(s)\sim N(0,\sigma^2)$, whereas $Y:=MW(t)\sim N(0,M^2t)$, and $k^2:=\sigma^2/(M^2t)\le1$. So, $X$ equals $kY$ in distribution, and so, $P(|Y|\ge a)\ge P(|X|\ge a)$ for all real $a$; that... | 2 | https://mathoverflow.net/users/36721 | 338255 | 144,294 |
https://mathoverflow.net/questions/338191 | 1 | Given a category $\mathbf{C}$, we can consider monomorphisms into it. These are the faithful and injective-on-objects functors (this violates the [principle of equivalence](https://ncatlab.org/nlab/show/subcategory#VariantsInAccordWithThePrincipleOfEquivalence)). The idea is to try to get a lattice of subobjects.
Pul... | https://mathoverflow.net/users/45365 | Structure of a poset of subcategories | Like Kevin said in the comments, your definition of join should work just fine in any category with finite coproducts and a good notion of "image." The issue with $\textbf{Cat}$ is that you won't have a *distributive* lattice (which is specifically what the nLab says a coherent category will give you).
Here's an exam... | 3 | https://mathoverflow.net/users/132451 | 338264 | 144,297 |
https://mathoverflow.net/questions/338262 | 4 | Let $X$ be a locally compact, second countable and Hausdorff space, must there be a Radon measure on $X$ whose support is $X$?
The motivation for this question comes from Anton Deitmar's paper [On Haar systems for groupoids](https://arxiv.org/abs/1605.08580), in which he construct a groupoid with open range map admit... | https://mathoverflow.net/users/49381 | Must a locally compact, second countable, Hausdorff space support a Radon measure? | At the OP‘s request——consider $\sum \lambda\_n \delta\_{t\_n}$ where $(\lambda\_n)$ is a sequence of positive scalars which sum to $1$ and $(t\_n)$ is a dense sequence.
| 7 | https://mathoverflow.net/users/131781 | 338268 | 144,298 |
https://mathoverflow.net/questions/338224 | 6 | I would like an easy example of an infinite simple group along with an embedding into a finitely presented group. I know that there are infinite simple finitely presented group such as Thompson’s group $V$ and I know that any group with solvable word problem embeds into a simple subgroup of a finitely presented group, ... | https://mathoverflow.net/users/123905 | Easy example of an infinite simple group with an embedding into a finitely presented group | I think D. L. Johnson's article *[Embedding some recursively presented groups](https://www.cambridge.org/core/books/groups-st-andrews-1997-in-bath/embedding-some-recursively-presented-groups/A5CF8D51C4641ECB739C7A8CBDC98531)* should answer your question. The abstract is:
>
> We seek to illustrate the Higman Embeddi... | 7 | https://mathoverflow.net/users/122026 | 338269 | 144,299 |
https://mathoverflow.net/questions/338274 | 9 | Empirical evidence suggests that, for each positive integer $n$, the following equality holds:
\begin{equation\*}
\prod\_{s=1}^{2n}\sum\_{k=1}^{2n}(-i)^k\sin\frac{sk\pi}{2n+1}=(-1)^n\frac{2n+1}{2^n},
\end{equation\*}
where $i=\sqrt{-1}$.
Is it a known equality? If it is true, would you please give me some insights... | https://mathoverflow.net/users/120597 | An equality about sin function? | We have
$$
\sum\_{k=0}^{2n}(-i)^k\sin\frac{sk\pi}{2n+1}=\frac{h(s)-h(-s)}{2i},\quad\text{where}\\
h(s)=\sum\_{k=0}^{2n}e^{i(-\pi/2+\frac{\pi s}{2n+1})k}=\frac{1-e^{-i\pi(2n+1)/2+i\pi s}}{1-e^{i(-\pi/2+\frac{\pi s}{2n+1})}}=\frac{1+i(-1)^{n+s}}{1+ie^{i\frac{\pi s}{2n+1}}}.
$$
The numerators for $s$ and $-s$ are the same... | 26 | https://mathoverflow.net/users/4312 | 338279 | 144,303 |
https://mathoverflow.net/questions/338281 | 10 | I was trying to find (and failed) the original author of either
* the concept of Monoid (set with binary associative operation and identity)
* the name (which sounds french ? and also [Dioid](https://fr.m.wikipedia.org/wiki/Dio%C3%AFde) (for what seems to be a semiring) is exclusively french wiki article)
Question:... | https://mathoverflow.net/users/119441 | Who invented Monoid? | The name "monoid" was first used in mathematics by Arthur Cayley [\*] for a surface of order $n$ which has a multiple point of order $n-1$.
In the context of semigroups the name is due to Bourbaki [[source](https://www.researchgate.net/publication/226480216_The_Early_Development_of_the_Algebraic_Theory_of_Semigroups)... | 14 | https://mathoverflow.net/users/11260 | 338282 | 144,305 |
https://mathoverflow.net/questions/338253 | 6 | Let $\left(W\text{, }S\right)$ be a Gromov hyperbolic Coxeter system and denote by $\partial W$ the corresponding Gromov boundary. For $z\in\partial W$ let $\alpha$, $\beta$ be infinite geodesic paths with $z=\left[\alpha\right]=\left[\beta\right]$ and assume that there exists $w\in W$ such that $\left|w^{-1}\alpha\_{n... | https://mathoverflow.net/users/64444 | Starting letters of equivalent infinite geodesic paths of hyperbolic Coxeter groups | This is false. The archetypical family of hyperbolic Coxeter groups are the [hyperbolic triangle groups](https://en.wikipedia.org/wiki/Triangle_group#The_hyperbolic_case)
$$
T(l,m,n) = \langle a,b,c \mid a^2=b^2=c^2=(ab)^l=(bc)^m=(ca)^n=1\rangle
$$
where $l,m,n\geq 2$ and $(1/l)+(1/m)+(1/n)<1$. Such a group acts as a g... | 4 | https://mathoverflow.net/users/6514 | 338289 | 144,308 |
https://mathoverflow.net/questions/338287 | 2 | As I mentioned in my [previous question](https://mathoverflow.net/questions/337791/on-asymptotic-classes-of-finite-structures), I am reading the following paper:
[**One-dimensional asymptotic classes of finite structures.**](https://www.ams.org/journals/tran/2008-360-01/S0002-9947-07-04382-6/S0002-9947-07-04382-6.pdf... | https://mathoverflow.net/users/141388 | On Asymptotic classes of finite structures (2) | The answer to your second question is "no", but this doesn't have anything to do with $1$-dimensional asymptotic classes specifically, just ultraproducts.
Suppose $\mathcal{C}$ is a class of finite structures and $\mathcal{M}=\prod\_{\mathcal{U}}C\_i$ is an infinite ultraproduct of members of $\mathcal{C}$, where $\m... | 4 | https://mathoverflow.net/users/38253 | 338293 | 144,310 |
https://mathoverflow.net/questions/338242 | 6 | Take ZFC, remove axiom of Power set, and put instead of it the following axiom:
**Axiom of Successor Cardinals:** $\forall \kappa\, \exists x \, \forall \alpha \, ( \alpha \leq \kappa \to \alpha \in x)$
where "$\leq$" refers to "cardinal smaller than or equal" relation, and $\kappa, \alpha$ range over von Neumann o... | https://mathoverflow.net/users/95347 | Is the power set axiom essential for constructing L? | KP alone - which is vastly weaker than the theory in question - proves the sentence "For every ordinal $\alpha$, $L\_\alpha$ exists," since it is strong enough to enable effective transfinite recursion. (We're passing to an unnecessarily weak subtheory, but it's worth noting.) The proof of this can be found e.g. in Bar... | 12 | https://mathoverflow.net/users/8133 | 338313 | 144,315 |
https://mathoverflow.net/questions/338315 | 9 | I will formulate my question first; below is the relevant background information and notation. I have asked this question on stack, but I think the odds are better that I actually get an answere here (but do correct me if I'm wrong to post it here).
WHY is the Shapovalov form on a Verma module **symmetric**?
The fa... | https://mathoverflow.net/users/103852 | Shapovalov form on Verma modules | There is alternative way to define the Shapovalov form which makes the symmetry easy to see. By the PBW theorem you can write each element $X$ of $\mathfrak{U(g)}$ as $X = f\_{i\_1} \cdots f\_{i\_m} \cdot h\_{j\_1} \cdots h\_{j\_n} \cdot e\_{k\_1} \cdots e\_{k\_o}$. Now denote by $\pi: \mathfrak{U(g)} \to \mathfrak{U(h... | 6 | https://mathoverflow.net/users/6818 | 338319 | 144,317 |
https://mathoverflow.net/questions/338303 | 4 | I originally [posted this on Maths SE](https://math.stackexchange.com/questions/3322212/motivation-intuition-behind-the-definition-of-delta-functors-and-related-concept), but then realised that the question probably fits MO better, as my objective was to gain different perspectives regarding the matter.
1. Why are $\... | https://mathoverflow.net/users/143390 | Motivation/intuition behind the definition of delta-functors and related concepts | I am not an expert on the subject but looking at the definition I would dare to say that these are basically the axioms that characterize an homology theory, if it weren't for the fact the $\delta$ maps usually go in the opposite directions.
The family of functors play the role of the various homology functors (one f... | 5 | https://mathoverflow.net/users/14969 | 338341 | 144,323 |
https://mathoverflow.net/questions/338324 | 2 | Given a $3\times3$ matrix $M$, if we would like to get the closest $\mathrm{SO}(3)$ matrix $R$ that minimizes
\begin{equation}
\|R-M\|\_F
\end{equation}
then $R$ = $UV^{T}$ where $U$ and $V^{T}$ are the orthogonal matrices from the singular value decompositon of $M$. i.e. $M = U\Sigma V^{T}$ as explained in this [an... | https://mathoverflow.net/users/144388 | Finding the closest special orthogonal matrix in Frobenius norm sense | By right-multiplying both $R$ and $M$ by $V$ and left-multiplying by $U^T$ you are leaving the objective invariant. Now the new $M$ is diagonal (actually, it's $\Sigma$), and it's not hard to convince yourself that so should the new $R$ be. If $\det(UV^T) = -1$, then the constraint means $R$ is a diagonal matrix with $... | 0 | https://mathoverflow.net/users/20186 | 338346 | 144,325 |
https://mathoverflow.net/questions/338219 | 2 | edit: I added conjecture 2 that looks much more accessible.
Here is the elementary combinatorial translation of the problem (read below for the homological background):
Let $n \geq 2$.
A Nakayama algebra is a list $[c\_0,c\_1,...,c\_{n-1}]$ of $n$ integers with $c\_i \geq 2$, $c\_{i+1} \geq c\_i-1$ and $c\_{n-1}=c\... | https://mathoverflow.net/users/61949 | Combinatorial problem on periodic dyck paths from homological algebra | I think you have a typo in the definition of a Nakayama algebra list and it should read "$c\_{i+1}\geq c\_i-1$." If this is the case then conjecture 2 has a simple proof:
The condition for a module $M=(i,k)$ to be weird is that $n\le k\le c\_i-n$. We have to show that this implies that $\Omega^1(M)=(i+k,c\_i-k)$ is a... | 3 | https://mathoverflow.net/users/2384 | 338351 | 144,326 |
https://mathoverflow.net/questions/338325 | 6 | In [Question 337879](https://mathoverflow.net/questions/337879), I conjectured that for any prime $p\equiv3\pmod4$ the equation $$3x^2+4\left(\frac p3\right)=py^2\tag{1}$$
always has integer solutions, where $(\frac p3)$ is the Legendre symbol. Motivated by this, here I pose the following conjectures.
**Conjecture 1.... | https://mathoverflow.net/users/124654 | Is the diophantine equation $3x^2+1=py^2$ always solvable for each prime $p\equiv 13\pmod{24}$? | **Updated on 2019/08/21:** I prove Conjectures 1-3 below. I will use the usual terminology and notations for binary quadratic forms. In particular, I will use the first two pages of Pall: Discriminantal divisors of binary quadratic forms, J. Number Theory 1 (1969), 525-533.
**Proof of Conjecture 1.**
Consider the fun... | 20 | https://mathoverflow.net/users/11919 | 338355 | 144,328 |
https://mathoverflow.net/questions/338368 | 2 | Let $A$ be a unital $\*$-algebra, and $B$ a unital $\*$-subalgebra of $A$. In addition, assume that there exists a $B$-$B$-sub-bimodule $C \subset A$, such that
$$
A \simeq B \oplus C,
$$
where $\simeq$ means an isomorphism of $B$-$B$-bimodules. If $B$ is endowed with a pre-$C^\*$-norm $\|\*\|$, then is it always poss... | https://mathoverflow.net/users/128876 | Extending $C^*$-norms from $*$-subalgebras | Ignoring the P.S. the answer is an easy no: $A$ could be any unital $\*$-algebra with a codimension $1$ $\*$-ideal $C$. Then let $B = \mathbb{C}\cdot e$ where $e$ is the unit of $A$, so a $B$-$B$ bimodule is just a vector space. Of course $B$ is endowed with a C\*-norm but $A$ is arbitrary so it need not have a C\*-nor... | 2 | https://mathoverflow.net/users/23141 | 338370 | 144,331 |
https://mathoverflow.net/questions/338357 | 3 | In his [2008 paper](https://www.worldscientific.com/doi/pdf/10.1142/S0218216508006452),
*Tanaka, Toshifumi*, [**The colored Jones polynomials of doubles of knots**](http://dx.doi.org/10.1142/S0218216508006452), J. Knot Theory Ramifications 17, No. 8, 925-937 (2008). [ZBL1149.57023](https://zbmath.org/?q=an:1149.5702... | https://mathoverflow.net/users/116808 | Easy lemma for trivalent graphs in colored Jones polynomial | Rewrite each edge of the graph, labeled by $k$, as $k$ strands with the $k$-th JW idempotent in the middle. Make a similar modification at the vertices. Expand the sums appearing to one side of the $2n$ strand. Each of the resulting diagrams will have a strand which leaves the $2n$ idempotent at position $i$ and return... | 3 | https://mathoverflow.net/users/284 | 338380 | 144,333 |
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