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https://mathoverflow.net/questions/369856 | 5 | Take [second order logic](https://plato.stanford.edu/entries/logic-higher-order/), weaken the [comprehension axiom schemata](https://plato.stanford.edu/entries/logic-higher-order/#AxioSecoOrdeLogi) to using only FIRST order formulas; that is, $\phi(x\_1,..,x\_n)$ in the referred article is restricted to be a first orde... | https://mathoverflow.net/users/95347 | IF we weaken comprehension of second order logic to first order formulas, would the resulting system be a conservative extension of FOL? | Preliminary remark: after some clarification in the comments, I am posting my comment as an answer.
Let $\varphi$ be a first-order formula which is proved in your Hilbert style system $S$. Let $M$ be a first-order structure (adapted to the appropriate first-order language). From the completeness of first-order logic,... | 5 | https://mathoverflow.net/users/9825 | 370079 | 154,844 |
https://mathoverflow.net/questions/370021 | 1 | Let $K\_{p,q}$ be a $(p,q)$-cable of the non-trivial knot $K$ in $S^3$.
Is there a closed formula for the Jones polynomial for $K\_{p,q}$ as in the case of [Alexander polynomial](https://mathoverflow.net/questions/123141/vassilliev-invariants-of-knots-and-their-cables) or [Seifert matrices](https://mathoverflow.net/q... | https://mathoverflow.net/users/nan | Jones polynomial of cable knots | As Ian Agol mentioned, if there were a closed formula for the Jones polynomial $V(K\_{p,q})$ in terms of $V(K)$, this would give a closed formula for the colored Jones polynomials $V\_n(K)$ in terms of the original Jones polynomial $V(K) = V\_2(K)$.
However, this makes me think that there is no such simple formula. I... | 1 | https://mathoverflow.net/users/113402 | 370080 | 154,845 |
https://mathoverflow.net/questions/370084 | 1 | Under what conditions is there a strictly positive hyperreal probability measure on a group $G$? This would be a finitely-additive non-negative function $\mu$ from the powerset of $G$ to a hyperreal field $^\*R$ such that $\mu(A)>0$ for all $A\ne\varnothing$, $\mu(G)=1$ and $\mu(gA)=\mu(A)$ for all $g\in G$ and $A\subs... | https://mathoverflow.net/users/26809 | Invariant strictly positive hyperreal probability measures on groups | Local finiteness is not only sufficient but necessary for the existence of the requisite hyperreal measure. Not having a subset equidecomposable with a proper subset is clearly necessary. But this implies local finiteness [by a theorem of Scarparo](https://arxiv.org/pdf/1606.08262.pdf). In particular, supramenability+t... | 1 | https://mathoverflow.net/users/26809 | 370098 | 154,851 |
https://mathoverflow.net/questions/370095 | 4 | It is easy to prove that
$\lim\_{p\rightarrow 1} m\_p = (x\_1 + \cdots + x\_n)/n$. The following fact about the derivative of $m\_p$ with respect to $p$ is also elementary:
$$m'\_p =\frac{dm\_p}{dp}
=\frac{1}{p \log p}\cdot\Big[\frac{x\_1p^{x\_1}+\cdots+x\_np^{x\_n}}{p\_1^{x\_1}+\cdots+p\_n^{x\_n}}-m\_p\Big].$$
My inte... | https://mathoverflow.net/users/140356 | Remarkable limit involving $m_p=\log_p(p^{x_1} + \cdots + p^{x_n})-\log_p(n)$ | $\newcommand\bar\overline$
Letting $t:=\ln p$, we see that the limit in question is the limit of
$$d(t):=\frac1t\Big(\sum\_1^n x\_j e^{tx\_j}\Big/\sum\_1^n e^{tx\_j}-m\_{e^t}\Big)$$
as $t\to0$.
Next, letting
$\bar x:=\frac1n\,\sum\_1^n x\_j$, $\bar{x^2}:=\frac1n\,\sum\_1^n x\_j^2$, and $s^2=\bar{x^2}-\bar x^2$, we have... | 10 | https://mathoverflow.net/users/36721 | 370099 | 154,852 |
https://mathoverflow.net/questions/370100 | 0 | Consider some algebraically independent polynomials $f\_1,\ldots, f\_n\in\mathbb{C}[x\_1,\ldots, x\_n]$.
Is it possible that $I\subseteq\mathbb{C}[f\_1,\ldots, f\_n]\subsetneq\mathbb{C}[x\_1,\ldots, x\_n]$ for some not trivial ideal $I$ of $\mathbb{C}[x\_1,\ldots, x\_n]$?
| https://mathoverflow.net/users/100359 | Big polynomial subalgebra of polynomials | It is not possible.
First, note that in the situation, $0\neq I\subset A=\mathbb{C}[f\_1,\ldots, f\_n]\subset B=\mathbb{C}[x\_1,\ldots, x\_n]$ with $I$ an ideal of $B$, one must have $A$ and $B$ birational. To see this, let $0\neq p\in I$. Then, $x\_ip\in I$ and thus, $x\_i=x\_ip/p$.
Next, assume that $x\_i\not\in ... | 0 | https://mathoverflow.net/users/9502 | 370108 | 154,855 |
https://mathoverflow.net/questions/370087 | 4 | Does every short exact sequence $0\to \mathbb Z\to A \to \mathbb R \to 0$ split in the category of Abelian groups?
| https://mathoverflow.net/users/164280 | Short exact sequence $0\to \mathbb Z\to A \to \mathbb R \to 0$ | The calculation of $\text{Ext}^1(\mathbb{Q}, \mathbb{Z})$ can be found in [this MO answer](https://mathoverflow.net/a/283804/290); in terms of just its isomorphism type the conclusion is that it's an uncountable-dimensional vector space over $\mathbb{Q}$, abstractly isomorphic to $\mathbb{R}$. It can also be written as... | 6 | https://mathoverflow.net/users/290 | 370109 | 154,856 |
https://mathoverflow.net/questions/370092 | 10 | The *Dyck language* is defined as the language of balanced parenthesis expressions on the alphabet consisting of the symbols $($ and $)$. For example, $()$ and $()(()())$ are both elements of the Dyck language, but $())($ is not. There is an obvious generalisation of the Dyck language to include several different types... | https://mathoverflow.net/users/120914 | Why is the Dyck language/Dyck paths named after von Dyck? | Diekert and Lange, in [Variationen über Walther von Dyck und Dyck-Sprachen](http://www2.informatik.uni-stuttgart.de/fmi/ti/veroeffentlichungen/pdffiles/DiekertLange2009dyck.pdf), quote a personal communication from Chomsky that attributes the name Dyck language to Schützenberger's 1962 paper on [Certain elementary fami... | 8 | https://mathoverflow.net/users/11260 | 370110 | 154,857 |
https://mathoverflow.net/questions/369628 | 2 | I encountered the following hypergeometric function in my research: $${}\_3F\_2(2,1+n,1+n;1,2+n;z)$$ where $0<z<1$. I'm interested in its behavior for large $n$. Semilog plot suggests exponential increase in $n$, however, I'm having trouble deriving the expression for the asymptotic expansion. There are lots of referen... | https://mathoverflow.net/users/18910 | Asymptotic expansion of hypergeometric function ${}_3F_2$ for large parameters | Below is a closed-form evaluation in terms of simple functions. Let $y=z/(z-1).$ Then
$${}\_3F\_2(2,n,n;1,n+1;z)=n(-z)^{-n}\Big((n-1)\big(\log(1-y)+\sum\_{k=1}^{n-1}\frac{y^k}{k} \big) + y^n \Big)$$
I proved this by using Pochhammer symbol properties, which gets me to a linear combination of ${}\_2F\_1.$ Then I used a ... | 1 | https://mathoverflow.net/users/121836 | 370122 | 154,860 |
https://mathoverflow.net/questions/370117 | 5 | Is there an exhaustive list of conditions satisfied by rational surgery coefficients assigned to the components of the Hopf link in $S^3$ such that the resulting 3-manifold by Dehn surgery acting on $S^3$ along this Hopf link is again $S^3$?
| https://mathoverflow.net/users/149888 | Dehn surgery on $S^3$ along a Hopf link with rational surgery coefficients | If you're doing surgery with coefficients $p/q$ and $p'/q'$, the condition is that ${pp}'-{qq}' = \pm 1$.
In order to see this, let's think of what we're doing: we're drilling out of $S^3$ two solid torus neighbourhoods of the two components, obtaining $M$, and then we're gluing in solid tori, $T$ and $T'$. Since $M$... | 5 | https://mathoverflow.net/users/13119 | 370128 | 154,863 |
https://mathoverflow.net/questions/370137 | 4 | In his plenary lecture "L-functions and Automorphic Representations" at the Seoul ICM James Arthur makes the following remark (Proceedings of the ICM 2014, vol. 1, p. 173):
>
> Riemann conjectured that the only zeros of L(s) lie on the vertical line Re(s) = 1/2. This is the famous Riemann hypothesis, regarded by ma... | https://mathoverflow.net/users/164298 | Fourier relationship between primes and zeros of L functions | This follows simply from contour integration.
See
<https://en.wikipedia.org/wiki/Explicit_formulae_for_L-functions>
| 4 | https://mathoverflow.net/users/164302 | 370139 | 154,868 |
https://mathoverflow.net/questions/370152 | 2 | Let $G$ be a group and $x\_1,\ldots,x\_n,y\_1,\ldots,y\_n \in G$ involutions
such that
* $G = \langle x\_1, \ldots , x\_n \rangle = \langle y\_1 , \ldots ,
y\_n \rangle$
* $g:=x\_1 \cdots x\_n = y\_1 \cdots y\_n$ is of finite order
Now assume that there exists $1 \leq k < n$ such that
$$ g = x\_1 \cdots x\_k y\_{... | https://mathoverflow.net/users/91107 | A property for a group generated by involutions | No, this is false even when $G$ is abelian and finite. For instance take
$$G = \langle (1,2), (3,4), (5,6), (7,8) \rangle \le \mathrm{Sym}\_8.$$
Define $x\_1,x\_2,x\_3,x\_4,y\_1,y\_2,y\_3,y\_4$ as indicated by
$$G = \bigl\langle (1,2)(5,6), (3,4)(5,6), (5,6), (7,8) \bigr\rangle. $$
and
$$G = \bigl\langle (1,2),... | 9 | https://mathoverflow.net/users/7709 | 370153 | 154,872 |
https://mathoverflow.net/questions/370155 | 4 | Apparently, there is a paper
M. Sablik, [Final part of the answer to a Hilbert's question.](https://link.springer.com/chapter/10.1007/978-1-4757-5288-5_18) Functional Equations - Results and Advances. Edited by Z. Daróczy and Zs. Páles, Kluwer Academic Publishers 2002, 231-242.
however I struggle to understand what... | https://mathoverflow.net/users/164326 | What does this paper have to do with Hilbert's fifth question? | Here's a translation of part of the address dealing with the Fifth Problem ([source](https://mathcs.clarku.edu/%7Edjoyce/hilbert/problems.html)):
>
> For infinite groups the investigation of the corresponding question is, I believe, also of interest. Moreover we are thus led to the wide and interesting field of fun... | 4 | https://mathoverflow.net/users/4177 | 370160 | 154,875 |
https://mathoverflow.net/questions/369957 | 5 | I'm looking for a seemingly natural generalization of a Chernoff bound.
In many scenarios, we have a distribution $D$ with support $\mathsf{Supp}(D)$, and some event $E \subset \mathsf{Supp}(D)$ telling us whether a property of a sample from $D$ holds (i.e. $a \in E$ iff $a\sim D$ has the property we want). Denoting ... | https://mathoverflow.net/users/119263 | Chernoff-style concentration inequality for k-tuples | Theorem 2 in [1] gives a bound of $1-\frac{4e^{-\delta^2n/8}}{\delta}$. I think you can incorporate $p$ in the bound since the proof of that theorem uses the standard Chernoff bound.
[1] Yakov Babichenko, Siddharth Barman, Ron Peretz (2017) Empirical Distribution of Equilibrium Play and Its Testing Application. Mathe... | 4 | https://mathoverflow.net/users/85550 | 370161 | 154,876 |
https://mathoverflow.net/questions/370159 | 2 | Let $X=(X\_t)\_{t\in I}$ ($I\subset\mathbb{R}$ an interval) be a stochastic process with continuous sample paths and such that $X\_t$ admits a continuous Lebesgue density $\chi\_t\in C(\mathbb{R}^d)$ for each $t\in I$.
Are you aware of (minimal) conditions on $X$ which guarantee that the function $(t,x)\mapsto \chi\_... | https://mathoverflow.net/users/160714 | Continuity of the densities of a stochastic process | The truly minimal condition on $X$ that guarantees that the function $(t,x)\mapsto p\_t(x):=\chi\_t(x)$ is continuous is tautological: $p\_t(x)$ is continuous in $(t,x)$ if and only $p\_t(x)$ is continuous in $(t,x)$. As far as the minimality is concerned, I don't think you can do much better than this.
However, one ... | 2 | https://mathoverflow.net/users/36721 | 370163 | 154,878 |
https://mathoverflow.net/questions/370130 | 6 | It is natural to ask if it is possible for the mapping cone $X\cup\_\alpha CA$
to be homeomorphic to the mapping cone $X\cup\_\beta CB$ with $A$ and $B$
nonhomeomorphic. Is there a standard go-to example for this?
I have vague memories that there are manifolds $M$ and $N$ that are not homeomorphic, but
$M\times \math... | https://mathoverflow.net/users/3634 | Nonhomeomophic spaces with homeomorphic mapping cones | The [double suspension theorem](https://en.wikipedia.org/wiki/Double_suspension_theorem) says that if $Y$ is a homology $3$-sphere, then its double suspension $\Sigma^2 Y$ is homeomorphic to $S^5$. If we take $Y$ to be the Poincaré sphere, then $\Sigma Y$ is not a topological manifold, since the suspension points are n... | 6 | https://mathoverflow.net/users/8103 | 370168 | 154,879 |
https://mathoverflow.net/questions/370173 | 8 | Does anyone know an English reference for the original proof of Hilbert's syzygy theorem? The three proofs that I know use either:
* the theory of projective dimension and change of rings (plus a step to go from projective to free resolutions)
* the symmetry of the Tor functors
* Groebner bases
None of these tools ... | https://mathoverflow.net/users/828 | Original proof of Hilbert's syzygy theorem | See [Theory of Algebraic Invariants](https://books.google.de/books?hl=de&lr=&id=xCneO62aHiIC&oi=fnd&pg=PR7&dq=Hilbert:+Ueber+die+Theorie+der+algebraischen+Formen+englisch&ots=FlulibiSC-&sig=FsB1P8lQZJKadWwt4l2HkZqldg4#v=onepage&q&f=false)
| 11 | https://mathoverflow.net/users/161310 | 370179 | 154,883 |
https://mathoverflow.net/questions/370182 | 0 | Are there any rational numbers $x, y, z$ with $xyz \neq 0$ such that $y^2 = x^3 + z^{4k}$ for some $k \in \mathbb{Z}\_{>1}$ ?
| https://mathoverflow.net/users/480516 | On the elliptic curve $y^2 = x^3 + z^{4k}$ | For $k=1$, the surface $y^2=x^3+z^4$ is a rational surface, so it has lots of rational points. The substitution $x=zu$ and $y=z^2v$ leads to $z=u^3/(v^2-1)$, so for almost all $u,v\in\mathbb Q$, the point
$$
\left( \frac{u^4}{v^2-1},\; \frac{u^6v}{(v^2-1)^2},\; \frac{u^3}{v^2-1} \right)
$$
satisfies $y^2=x^3+z^4$. (The... | 9 | https://mathoverflow.net/users/11926 | 370184 | 154,886 |
https://mathoverflow.net/questions/370134 | 3 | Let $\mathcal{H}\_{n,p,h}=(V,E)$ be a random $h$-uniform hypergraph on $[n]$, sampled according to the usual binomial distribution. We known that with high probability, the number of edges in $\mathcal{H}\_{n,p,h}$ is
$$m = (1+o(1))\binom{n}{h}p$$
Let $\ell$ be given. I would like to delete some edges in order to
*... | https://mathoverflow.net/users/118450 | making a random uniform hypergraph linear | **Note:** in order to understand the proof, it was key (at least for me) to see that a cycle of length $t$ in a $k$-uniform hypergraph is set of $t$ edges $(e\_1,\ldots,e\_t)$ such that (viewing each edge as a $k$-set of vertices)
$$ \left\vert \bigcup\_{i=1}^t e\_i \right\vert \leq (k-1)t$$
Following @LouisD comment... | 2 | https://mathoverflow.net/users/118450 | 370210 | 154,893 |
https://mathoverflow.net/questions/370174 | 6 | I think my question is more suitable for Mathematics Stack Exchange than to MathOverflow but I've already posted two related questions there and I got even more confused, so maybe I can clarify things here. I'm studying spectral theory by myself as part of my research activity and the following question arose.
Let $H... | https://mathoverflow.net/users/152094 | Representation theorem for quadratic form on Hilbert space | There is indeed a simple proof using the Riesz representation theorem. First note that replacing $x$ by $\lambda^{-1}x$ and $y$ by $\lambda y$ in $\lvert \Psi(x,y)\rvert\leq K(\lVert x\rVert^2+\lVert y\rVert^2)$, you get $\lVert \Psi(x,y)\rvert\leq K(\lambda^{-2}\lVert x\rVert^2+\lambda^2\lVert y\rVert^2)$. With $\lamb... | 7 | https://mathoverflow.net/users/95776 | 370226 | 154,901 |
https://mathoverflow.net/questions/370227 | 4 | Let $B\leftarrow A\to C$ be a diagram of commutative rings, and let $\mathcal{D}(A)$ be the derived $\infty$-category of $A$-modules (as in Lurie's "Higher Algebra"). Then is there an equivalence
$$\mathcal{D}(B\otimes\_A^LC):=\operatorname{Mod}\_{B\otimes\_A^LC}(\mathrm{Sp})\simeq\mathcal{D}(B)\otimes\_{\mathcal{D}(A)... | https://mathoverflow.net/users/29322 | Does formation of the derived $\infty$-category preserve pushouts? | A hands-on explanation: Relative tensor products like $B\otimes\_AC$ are computed as the colimit of the simplicial object $B\otimes A^{\otimes \bullet} \otimes C$. The functor $\mathsf{Mod}\_{(-)}: \mathsf{Alg}(\mathsf{Sp}) \to \mathsf{Pr}^{L, \mathrm{st}}\_{\mathsf{Sp}/}$ preserves all colimits and is symmetric monoid... | 13 | https://mathoverflow.net/users/6936 | 370236 | 154,905 |
https://mathoverflow.net/questions/370212 | 6 | Let $\mathbb{N}$ denote the set of positive integers. For $\alpha\in \; ]0,1[\;$, let $$\mu(n,\alpha) = \min\big\{|\alpha-\frac{b}{n}|: b\in\mathbb{N}\cup\{0\}\big\}.$$ (Note that we could have written $\inf\{\ldots\}$ instead of $\min\{\ldots\}$, but it is easy to see that the infimum is always a minimum.)
Is there ... | https://mathoverflow.net/users/8628 | Approximating by incrementing positive integers | There is no such $\alpha$.
If $\alpha\in\mathbb Q$, there is $n$ such that $\mu(n,\alpha)=0$, thus $\mu(n+1,\alpha)<\mu(n,\alpha)$ is impossible.
If $\alpha\notin\mathbb Q$, a classical result in Diophantine approximation says that there are infinitely many $n$ such that
$$\mu(n,\alpha)<\frac1{\sqrt5n^2}.$$
If then... | 6 | https://mathoverflow.net/users/12705 | 370241 | 154,907 |
https://mathoverflow.net/questions/370215 | 1 | We consider the following function
$$f:(\mathbb S^n)^N \rightarrow \mathbb R^{n+1} \text{ such that } f(x\_1,...,x\_N)= \sum\_{i=1}^N x\_i.$$
This function can be written in Cartesian coordinates as $f(x)=(f\_1(x),..,f\_{n+1}(x))$ and I would like to know if one can find a simple expression for the derivative
$$\... | https://mathoverflow.net/users/119875 | Directional gradient on sphere | $\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$Take any $(x\_1,x\_2,\dots,x\_N)\in(\S^n)^N$.
Let $(-1,1)\ni t\mapsto X\_1(t)\in\S^n$ be any smooth curve such that $$X\_1(0)=x\_1.$$ For any $t\in(-1,1)$, let
$$X(t):=(X\_1(t),x\_2,\dots,x\_N)$$ and $$S(t):=f(X(t))=X\_1(t)+x\_2+\dots+x\_N[\in\R^{n+1}],$$
so that $X(0)=... | 2 | https://mathoverflow.net/users/36721 | 370253 | 154,913 |
https://mathoverflow.net/questions/370254 | -3 | The super square root of $n$ is the solution/solutions to $x^x=n$. Is the super square root of $2$ irrational?
| https://mathoverflow.net/users/164393 | Is the super square root of $2$ irrational? | Yes.
If $x$ is rational, say $x=p/q$ with coprime integers $p,q$. We know $x$ cannot be an integer, so $q\neq 1$. Then $(p/q)^{p/q}=2$, or $(p/q)^p=2^q$. But the left-hand side cannot be an integer because its simplest fraction representation is $p^p/q^p$ with denominator $q^p\neq 1$. (Note that $p^p$ and $q^p$ are a... | 4 | https://mathoverflow.net/users/44352 | 370257 | 154,915 |
https://mathoverflow.net/questions/370216 | 5 | So I got an email from one of my colleagues on the Collatz Conjecture with a link to the article *[Computer Scientists Attempt to Corner the Collatz Conjecture](https://www.quantamagazine.org/can-computers-solve-the-collatz-conjecture-20200826/)* by Kevin Hartnett in Quanta Magazine.
On digging through the conjecture... | https://mathoverflow.net/users/10035 | Does this prove Collatz is a $\Sigma_1$ problem? | It seems I was wrong - see Emre Yolcu's comment below.
------------------------------------------------------
---
My understanding is that that has **not** been accomplished (although the Quanta article is pretty vague so I could be misunderstanding the situation).
The Quanta article describes the following pro... | 12 | https://mathoverflow.net/users/8133 | 370265 | 154,917 |
https://mathoverflow.net/questions/370269 | 18 | L. Carlitz has a paper, [Classes of pairs of commuting matrices over a finite field](https://www.jstor.org/stable/2312894), that computes the number of simultaneous similarity classes of of pairs of commuting matrices in $\operatorname{Mat}\_n(\mathbb F\_q)$. Two pairs $(A,B)$ and $(A',B')$ are called **simultaneously ... | https://mathoverflow.net/users/44352 | Is Carlitz's paper correct about the number of similarity classes of commuting matrices? | It is wrong. See [the correction published in AMM **71** (1964), issue 8, page 900](https://sci-hub.se/https://www.tandfonline.com/doi/abs/10.1080/00029890.1964.11992349). (You have to scroll down to the bottom of the last page to find this correction.)
Unfortunately this was published in the "Mathematical Notes" sec... | 17 | https://mathoverflow.net/users/2530 | 370286 | 154,921 |
https://mathoverflow.net/questions/355703 | 1 | The torsion of a link complement $S^3 \setminus L$ is defined in terms of the twisted chain complex $C\_\*(S^3 \setminus L; \rho)$, where $\rho : S^3 \setminus L \to \operatorname{GL}\_k(\mathbb{k})$ is a representation of the link complement into the group of matrices over a field $\mathbb{k}$. (For the usual Reidemei... | https://mathoverflow.net/users/113402 | Equivalence of conditions for torsions of links to be defined | I'm not sure that $\det(\rho(x)-I\_k) \neq 0$ is sufficient for $C\_\*(S^3 \setminus L;\rho)$ to be acyclic. Write $X\_L:=S^3 \setminus L$. Let $\omega \in S^1 \setminus \lbrace 1 \rbrace$ and take the one-dimensional representation $\rho \colon \pi\_1(X\_L) \to \mathbb{C}^\times$ that maps each meridian to $\omega$. T... | 2 | https://mathoverflow.net/users/36098 | 370293 | 154,922 |
https://mathoverflow.net/questions/368722 | 15 | I'm a master student and I have read "Monotonicity of the volume of intersection of balls" by Gromov and it was a great experience. When trying to fill the gaps, I often end up finding some very beautiful ideas. I want to keep reading Gromov because he inspires me and was delighted by the ideas that he show on that art... | https://mathoverflow.net/users/62218 | Gromov's articles suitable for master students | I agree with Alexandre Eremenko in that most can be hard to read. But I think you can get a lot out of trying to understand them, as long as you're willing to black-box certain parts which may be inaccessible. Here's a summary of a few major articles. At the least, they all have significant parts which may be understan... | 16 | https://mathoverflow.net/users/156492 | 370299 | 154,924 |
https://mathoverflow.net/questions/370296 | 0 | Let $n>3$ be an integer. What are the rational solutions of
$$y^2 = 4x^n + z^{n-1}$$
?
| https://mathoverflow.net/users/480516 | What are the rational solutions of $y^2 = 4x^n + z^{n-1}$? | We consider two cases: $n$ is odd or $n$ is even. If $n=2k+1$ then put
$$
x=uz, \quad y=vz^{k},
$$
where $u, v$ are non-zero rational parameters. Thus $v^2z^{2k}=4u^{2k+1}z^{2k+1}+z^{2k}$ and after division by $z^{2k}$ we get linear equation in $z$ which can be easily solved, i.e., $z=(v^2-1)/(4u^{2k+1})$. We thus get
... | 10 | https://mathoverflow.net/users/164119 | 370307 | 154,928 |
https://mathoverflow.net/questions/370266 | 8 | Recall that a commutative square of commutative rings
$$\begin{matrix}
A&\to&B\\
\downarrow &&\downarrow\\
A^\prime&\to&B^\prime\end{matrix}$$
is called a *Milnor square* if the vertical maps are surjective and the square is both a pullback and pushout of rings.
It has been shown that Milnor squares give rise to ... | https://mathoverflow.net/users/1353 | Milnor excision for algebraic stacks | In upcoming (now on [arXiv:2205.08623](https://arxiv.org/abs/2205.08623)) joint work with Jarod Alper, Jack Hall and Daniel Halpern-Leistner:
*Artin algebraization for pairs with applications to the local structure of stacks and Ferrand pushouts*
we prove more generally the existence of pushouts of affine morphisms... | 9 | https://mathoverflow.net/users/40 | 370320 | 154,933 |
https://mathoverflow.net/questions/370319 | 12 | I am a masters student. I am interested in **short articles** which have counter examples and very few references. I want to write a short and interesting article.
**For example**; One of the best known shortest and best academic paper articles I read is *Counterexample to Euler's Conjecture on Sums of Like Powers* b... | https://mathoverflow.net/users/150970 | Short research articles | [a bit too long for a comment]
I understand from the question that the aim is to find a research project based on the search for a counterexample. By construction, this will mean showing that some existing paper in the literature is mistaken. That is typically not a productive way to start a project in a new field, s... | 15 | https://mathoverflow.net/users/11260 | 370321 | 154,934 |
https://mathoverflow.net/questions/370297 | 3 | Let $T$ be a triangle in $\mathbb{R}^2$ defined by $y = \alpha x$, $y = \beta$ and $x = \gamma$ where
$\alpha, \beta, \gamma \in \mathbb{R}\_{>0}$. I am interested in obtaining an estimate for the number of integral points inside (either within or on the boundaries) of $T$ which I denote $N$. A simple computation yield... | https://mathoverflow.net/users/84272 | number of integer points inside a triangle and its area | The bound on $|E|$ can certainly be improved: $|E|\le|E|$; so, we get the "perfect" (but completely useless) upper bound, $|E|$, on $|E|$.
To make the problem meaningful, we need to specify the terms in which the upper bound is to be expressed. Counting the unit squares intersecting the boundary of the triangle, it i... | 5 | https://mathoverflow.net/users/36721 | 370322 | 154,935 |
https://mathoverflow.net/questions/370306 | 3 | I ran into the following "simple" question and I am wondering whether there are any references, which might help me. I am coming from statistics, so I am not so aware which branch of math could have already dealt with this question.
Let $S\subset\mathbb{R}^D$ be compact and assume that $f,g: S\rightarrow \mathbb{R}$ ... | https://mathoverflow.net/users/156091 | Number of critical points of sum of two functions | Is the following example helpful? This is inspired by the fact that no bound on the "degree" of the two functions $f$ and $g$ is assumed. As far as I understand Bezout's theorem, this would make a bound difficult even for polynomial functions.
Here let $A,B > 1$ be large and $0 < \epsilon < 1$ be small. Let $D = 1$, ... | 5 | https://mathoverflow.net/users/103792 | 370325 | 154,937 |
https://mathoverflow.net/questions/370336 | 2 | [MathWorld](https://mathworld.wolfram.com/SylvestersDeterminantIdentity.html) presents the following two versions of Sylvester's determinant identity, relating to an $n\times n$ matrix $\mathbb{A}$:
First:
$$
|\mathbb{A}||A\_{r\,s,p\,q}| = |A\_{r,p}||A\_{s,q}| - |A\_{r,q}| |A\_{s,p}|
$$
where $r$ and $s$ ($p$ and $... | https://mathoverflow.net/users/141969 | Two versions of Sylvester identity | For the special case that $r,s,p,q$ are single elements, it is shown in [these notes](http://www.dm.unibo.it/~morigi/prova1_file/Talks_Web/MRRusso.pdf) (page 7) how the first identity (known as the [Desnanot-Jacobi identity](https://en.wikipedia.org/wiki/Dodgson_condensation#Desnanot%E2%80%93Jacobi_identity_and_proof_o... | 5 | https://mathoverflow.net/users/11260 | 370338 | 154,939 |
https://mathoverflow.net/questions/370333 | 9 | Let $R$ be a reduced root system, $W$ the associated Weyl group, and $w\_0 \in W$ the longest element of $W$. In general $w\_0$ admits more than one reduced decomposition into a product of reflections, a number which we denote by $d\_R$. Where can one find a list of values of $d\_R$ for low-dimensional root systems?
... | https://mathoverflow.net/users/121660 | Number of reduced decompositions of the longest element of the Weyl group | This is easy to do in [SageMath](https://www.sagemath.org/). E.g. the following code
```
G = WeylGroup("F4")
w = G.long_element_hardcoded()
print(w)
rw = w.reduced_words()
len(rw)
```
outputs 2144892. If you want to look at some of these reduced words just examine the list rw. To create a list for classical types... | 5 | https://mathoverflow.net/users/6818 | 370347 | 154,942 |
https://mathoverflow.net/questions/370337 | 5 | Let $K$ be a field of characteristic 0 (maybe it works for more general fields) and $K[x\_1,...,x\_n]$ the polynomial ring in $n$ variables. Let $e\_1,e\_2,...,e\_n$ denote the elementary symmetric polynomials in $n$ variables (see for example <https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial> ). (One migh... | https://mathoverflow.net/users/61949 | Frobenius algebras from symmetric polynomials | In this commutative situation, a Frobenius algebra is the same as an
artinian Gorenstein ring. In general, if $\theta\_1,\dots,\theta\_n$ are
homogeneous elements of positive degree of $A=K[x\_1,\dots,x\_n]$ and if
$K[x\_1,\dots,x\_n]/(\theta\_1,\dots,\theta\_n)$ is artinian (i.e., a
finite-dimensional vector space in ... | 5 | https://mathoverflow.net/users/2807 | 370350 | 154,943 |
https://mathoverflow.net/questions/370132 | 4 | I would like to justify the "one can see that" statement in Page 477 of [Wang - Stability estimates of an inverse problem for the stationary transport equation](http://www.numdam.org/item/AIHPA_1999__70_5_473_0) on the stationary transport equation. Let $(x,v)\in (\Omega, V)$, where $\Omega\subset\mathbb{R}^n$, $n = 2,... | https://mathoverflow.net/users/148510 | Inner product of velocity and gradient of backwards escape is 1 | The time $\tau\_-(\vec{x},\vec{v})$ is the time it takes a particle at $\vec{x}$ to reach the boundary while moving in the direction $-\vec{v}$. Let $\vec{x}\_-$ be the boundary point reached by that particle,
$$\vec{x}\_-\equiv\vec{x}-\tau\_-(\vec{x},\vec{v})\vec{v}.$$
If we vary $\vec{x}$ in the direction $-\vec{v}$ ... | 1 | https://mathoverflow.net/users/11260 | 370352 | 154,944 |
https://mathoverflow.net/questions/370360 | 1 | I have done some search many times on web to find any approximation of $\log|(\zeta'(s))|$ in [Dirichlet polynomial](https://encyclopediaofmath.org/wiki/Dirichlet_polynomial) but I didn't got it, Probably that $\log(|\zeta'(s)|$ dosn't have a Dirichlet polynomial approximation yield probably for $\log(|\zeta'(\frac{1}{... | https://mathoverflow.net/users/51189 | What is the approximation of $\log(|\zeta'(\frac{1}{2}+it)|)$ in Dirichlet polynomial if it is exists? | A Dirichlet polynomial is a function of the form
$\sum\_{1\le n \le X} a\_n n^{-s}$, where $a\_n$ are complex numbers and
$s = \sigma + i t$ with $\sigma$ and $t$ real. It is an analytic
function of $s$.
You ask whether certain functions can be approximated by a Dirichlet
polynomial. The specific functions you mentio... | 4 | https://mathoverflow.net/users/10220 | 370367 | 154,946 |
https://mathoverflow.net/questions/370368 | 12 | Let $p>3$ be a prime.
We set $R=\{x\in\mathbb{Z}: (x/p)=1\}$, where $(\cdot/p)$ is the Legendre symbol. When $p\equiv3\pmod4$, by class formulae of imaginary quadratic fields $\mathbb{Q}(\sqrt{-p})$, we can easily obtain that
$$A\_p:=\sum\_{0<x<p/2,x\in R}x=(p^2-1)/16,\ \text{if}\ p\equiv7\pmod8,$$
and that
$$A\_p=\sum... | https://mathoverflow.net/users/nan | On sums of quadratic residues | By standard formulas for values of L functions at negative integers, for
$p\equiv1\pmod4$ one has
$$A\_p=(p^2-1)/16+aL(\chi\_p,-1)\;,$$
with $a=3/4$ if $p\equiv1\pmod8$ and $a=5/4$ if $p\equiv5\pmod8$
where $\chi\_p(x)=(x/p)$.
| 13 | https://mathoverflow.net/users/81776 | 370375 | 154,948 |
https://mathoverflow.net/questions/370313 | 2 | Is there any known result of decomposing multivariable power series over $p$-adic field into product of single variable power series ?
For example, consider the following power series in $n$ variables:$$ f(x\_1,~x\_2, \cdots, x\_n)=\sum\_{j\_1,~j\_2,\cdots, j\_n=0}^{\infty} a\_{j\_1,~j\_2, \cdots, j\_n} \prod\_{k=1}^... | https://mathoverflow.net/users/122445 | When can we decompose a multivariable p-adic power series into product of single variable power series? | Some night thoughts on your question which are too long for a comment. For simplicity, I will look at the two variable case. Firstly, there is a very simple discrete criterion for when a function of two variable splits in the way you are interested in: Let $f$ be a function from $X \times Y$. Then is can be represented... | 2 | https://mathoverflow.net/users/131781 | 370378 | 154,950 |
https://mathoverflow.net/questions/370389 | 7 | The Gelfand transform on the commutative Banach \*-algebra $L^1(\mathbb{R})$ is just the Fourier transform.
Q. What can we say concerning the Laplace transform?
| https://mathoverflow.net/users/84390 | Where does the Laplace transform come from? | Set $A = L^1([0, \infty))$, equipped with the structure of a Banach $\*$-algebra via convolution. The spectrum of this algebra is the half plane $\text{Re}(z) \geq 0$, and the Gelfand transform is the Laplace transform.
| 21 | https://mathoverflow.net/users/4362 | 370390 | 154,955 |
https://mathoverflow.net/questions/370187 | 33 | Mac Lane homology is a homology theory for (not necessarily commutative) rings. Given a ring $A$, Eilenberg and Mac Lane define its *cubical construction* $QA$ to be a certain connective chain complex, whose homology is isomorphic to the stable homology of Eilenberg-Mac Lane spaces: $$H\_i(QA)\cong H\_{i+j}(K(A,j)) \te... | https://mathoverflow.net/users/159143 | Equivalence of topological Hochschild homology and Mac Lane homology via an equivalence $QA\simeq HA \wedge_{\mathbb{S}} H\mathbb{Z}$ | The answer to your question is yes, these two $H\mathbb{Z}$-algebras are equivalent (and if $A$ is commutative, they are equivalent as commutative $H\mathbb{Z}$-algebras). I am in fact in the process of writing this up with Maxime Ramzi. Hopefully the paper will be finished in a week or 2.
**Update :** The paper has ... | 15 | https://mathoverflow.net/users/10707 | 370395 | 154,956 |
https://mathoverflow.net/questions/370396 | 3 | one of the most prominent functions of the first $n$ natural numbers is the factorial $n!$ that denotes their product.
Today however I wondered whether the least common multiple $\mathrm{lcm}(n):=\mathrm{lcm}(\{i\in\mathbb{N}\,|\,1\leqq i\leqq n\})$ of the first $n$ natural numbers has already been the subject of mat... | https://mathoverflow.net/users/31310 | Name and properties of $\mathrm{lcm}(\{1,\,\cdots,\,n\})$ | The quantity $\operatorname{lcm}(n)$ is equal to $e^{\psi (n)}$, where $\psi$ is the second [Chebyshev function](https://en.wikipedia.org/wiki/Chebyshev_function). This function is well studied, and the prime number theorem is equivalent (indeed, usually proved using this equivalence) to the fact that $\psi (x)\sim x$.... | 11 | https://mathoverflow.net/users/88679 | 370400 | 154,957 |
https://mathoverflow.net/questions/370387 | 1 | The harmonic majorization for a subharmonic function $h$ is well-known for bounded regions $\Omega \subset \mathbb{C}$:
$$h \le 0 \text{ in }\partial \Omega \Longrightarrow h \le 0 \text{ in }\Omega.$$
I know this is related to maximum principle.
I need a reference for the same result for unbounded regions of the com... | https://mathoverflow.net/users/151918 | Subharmonic function in unbounded regions | Yes, this is called the Phragmen-Lindelof Principle: For every region on the Riemann sphere, if $h$ is subharmonic and bounded from above, and
$$\limsup\_{z\to\zeta}h(z)\leq 0$$ for all $\zeta\in\partial\Omega$, except finitely
many points, then $h\leq 0$ in $\Omega$. If your domain $\Omega$ is an unbounded domain in $... | 4 | https://mathoverflow.net/users/25510 | 370401 | 154,958 |
https://mathoverflow.net/questions/369825 | 5 | Consider the following two definitions of the natural numbers:
* The natural numbers are the algebraic structure $\mathbb{N}\_1$ generated by one constant, $0$ and one unary function, $S$ (and no relations).
* The natural numbers are the monoid $(\mathbb{N}\_2, 0, +)$ with presentation $\langle 1 \mid \rangle$.
The... | https://mathoverflow.net/users/5736 | "Tietze-like transformations" for defining interesting bijections between algebraic structures | Tietze transformations for arbitrary algebraic theories (with respect to their presentations) have been considered in Malbos–Mimram's [Homological Computations for Term Rewriting Systems](https://hal.archives-ouvertes.fr/hal-01678175/document), in the context of rewriting systems (that is, equations are considered dire... | 3 | https://mathoverflow.net/users/152679 | 370404 | 154,960 |
https://mathoverflow.net/questions/370397 | 2 | I'm new to K-Theory for $C^{\*}$-algebra and $C^{\*}$-algebra of groups.
If $X$ is the group of finite support bijections of natural numbers then what is the K-Theory of $C^{\*}(X)$?
I was planning to calculate $C^{\*}(X)$ first and then its projections and their homotopy classes but I failed to determine $C^{\*}(X... | https://mathoverflow.net/users/137242 | K-Theory of $C^{*}(X)$ | The group you describe should be the infinite symmetric group $S\_{\infty}$. The $K$-theory of its $C^\*$-algebra has been determined by Kerov and Vershik in
The K -functor (Grothendieck group) of the infinite symmetric group
<https://link.springer.com/article/10.1007/BF02104985>
The main result can be summarised a... | 9 | https://mathoverflow.net/users/3995 | 370405 | 154,961 |
https://mathoverflow.net/questions/370399 | 8 | The point of Shimura varieties, as far as I've understood it, is that for a given Shimura datum $(G,D)$, there exist models, by which I mean that for congruence subgroups $\Gamma$ there exists a Shimura variety $X(\Gamma)$ defined over some number field. Hence we get a action of the absolute galois group $G\_{\mathbb{Q... | https://mathoverflow.net/users/152554 | Artin reciprocity via Shimura varieties | If $K$ is 'everything 1 mod N' for some N, then the canonical model of $\mathbf{Q}^\times\_+ \backslash \mathbf{A}^\times\_{\mathrm{f}} / K$ is exactly $\mu\_N / \mathbf{Q}$, the scheme of $N$-th roots of unity. Any open compact $K$ will contain one of these, so $GL\_1 / \mathbf{Q}$ Shimura varieties all look like quot... | 10 | https://mathoverflow.net/users/2481 | 370407 | 154,963 |
https://mathoverflow.net/questions/370207 | 4 | Let $n$ be a natural number. Let $U\_n = \{d \in \mathbb{N}\mid d\mid n \text{ and } \gcd(d,n/d)=1 \}$ be the set of unitary divisors, $D\_n$ be the set of divisors and $S\_n=\{d \in \mathbb{N}\mid d^2 \mid n\}$ be the set of square divisors of $n$.
The set $U\_n$ is a group with $a\oplus b := \frac{ab}{\gcd(a,b)^2}$... | https://mathoverflow.net/users/nan | The action of the unitary divisors group on the set of divisors and odd perfect numbers | Here are some general comments:
1. You don't need to bring these actions of abelian groups on various sets of divisors. The identity
$$\sigma(n)=\sum\_{d^2|n}d\sigma^{\*}(\frac{n}{d^2})$$
is easy to check directly, without appeal to anything fancy.
2. Let's call $\alpha(n)$ the number of prime divisors of $n$ which a... | 3 | https://mathoverflow.net/users/2384 | 370408 | 154,964 |
https://mathoverflow.net/questions/370339 | 4 | One up-vote, 35 views, and no comments and no answers have resulted from this reference request that [I posted](https://math.stackexchange.com/questions/3804232/reference-request-where-does-this-proposition-about-ellipses-appear-in-the-lite) on math.stackexchange.com . This was actually inspired by a probability exerci... | https://mathoverflow.net/users/6316 | Reference request on a characterization of ellipses | Yes, this characterization is a theorem proven by Blaschke in "Kreis und Kugel" (1916). The theorem has a higher dimensional version, characterizing ellipsoids as the unique strongly convex bodies with the property that all centroids of codimension 1 sections that are parallel to a fixed plane lie on a line. See the pa... | 5 | https://mathoverflow.net/users/2384 | 370410 | 154,965 |
https://mathoverflow.net/questions/370151 | 2 | For $x\in\mathbb R$ let
$$
u(x)=\begin{cases}
|x|^{2s-1}-1 &\mbox{if } |x|>1,\\
0 & \mbox{otherwise}.
\end{cases}
$$
Is it possible to calculate explicitly the fractional Laplacian $(-\Delta)^{s} u(x)$ for a fixed $s\in (0, 1/2)$?
| https://mathoverflow.net/users/127663 | explicit computation of fractional Laplacian of a function | Yes, it is, as long as you are OK with special functions. See Corollary 3(ii) in my paper with B. Dyda and A. Kuznetsov titled *Fractional Laplace operator and Meijer G-function*, [DOI:10.1007/s00365-016-9336-4](https://doi.org/10.1007/s00365-016-9336-4). Here one needs to apply this result twice, with $d = 1$, $\alpha... | 0 | https://mathoverflow.net/users/108637 | 370426 | 154,969 |
https://mathoverflow.net/questions/370437 | 1 | Let $X, Y, Z$ be measurable spaces with measures $\mu\_X, \mu\_Y, \mu\_Z$ respectively. Let $\pi\_Y : Y \times Z \rightarrow Y$ be the projection on $Y$ and $\pi\_Z : Y \times Z \rightarrow Z$ the projection on $Z$. Let $\psi : X \rightarrow Y \times Z$ be a **surjective** map such that $(\pi\_Y \circ \psi)\_{\*} \mu\_... | https://mathoverflow.net/users/143701 | Pushforward of measure under surjective map | The answer to the first part is no. The point is that surjectivity is not as strong a condition in this context as one might wish for.
Let $X=Y=Z=[0,1]$ with the Borel $\sigma$-algebra and $\mu\_X=\mu\_Y=\mu\_Z$ be the uniform distribution. Consider the non-surjective function $x\mapsto (x,x)$. It's push-forward is c... | 2 | https://mathoverflow.net/users/35357 | 370438 | 154,973 |
https://mathoverflow.net/questions/370355 | 7 | Let $G$ be a complex reductive group acting on a complex affine variety $X$ and let $X // G = \operatorname{Spec}\mathbb{C}[X]^G$ be the GIT quotient.
Is there a relationship between the singular locus of $X$ and that of $X // G$?
Of course, $X//G$ can be highly singular while $X$ is smooth. But, for example, I was... | https://mathoverflow.net/users/123207 | GIT and singularities | This is an answer to the revised question. It is the simplest counterexample that I can think of where the reductive group is smooth and connected, where $X$ is normal and affine, and where $Y=X//G$ is smooth, even though there are fibers of the quotient map that are contained in the singular locus of $X$.
Let $Y$ be... | 6 | https://mathoverflow.net/users/13265 | 370439 | 154,974 |
https://mathoverflow.net/questions/370434 | 12 | Can anyone provide any suggestion if the following is true for natural number $x$?
$$\limsup\_{x\to\infty}\dfrac{\sum\limits\_{d|3^x-1}{\frac{1}{d}}}{\sum\limits\_{p<x}\dfrac{1}{p}}<\infty$$ where $p$ runs over primes.
**P.S** I know that
the bound ${\sum\limits\_{d|n}{\frac{1}{d}}}<e^{\gamma}\log\log n +O(1)$is true... | https://mathoverflow.net/users/164499 | Is $\limsup_{x\to\infty}\big(\sum\limits_{d|3^x-1}{1/d}\big)/\big(\sum\limits_{p<x}1/p\big)<\infty$? | The OP's first display is true. Indeed, [Erdős (1971)](https://users.renyi.hu/%7Ep_erdos/1971-14.pdf) proved that
$$\sum\_{d\mid 2^x-1}\frac{1}{d}\ll\log\log x\ll\sum\_{p<x}\frac{1}{p},$$
and he noted that the proof works equally well for the divisors of $a^x-1$.
| 17 | https://mathoverflow.net/users/11919 | 370440 | 154,975 |
https://mathoverflow.net/questions/370430 | 6 | If $\mathcal C$ is a $\kappa$-accessible 1-category, then the category of morphisms $Mor \mathcal C$ is a $\kappa$-accessible 1-category, with the $\kappa$-presentable objects being those morphisms whose domains and codomains are each $\kappa$-presentable.
In the context of $\infty$-categories, the best result I know... | https://mathoverflow.net/users/2362 | If $\mathcal C$ is a $\kappa$-accessible $\infty$-category, then is $Mor \mathcal C$ $\kappa$-accessible? | Marc Hoyois answered in the comments: the answer is affirmative, by HTT 5.3.5.15.
| 2 | https://mathoverflow.net/users/2362 | 370451 | 154,981 |
https://mathoverflow.net/questions/370445 | 2 | Do there exist functions $F(x) \! : \, \mathbb R \to \mathbb R$ which are non-zero and bounded:
$$ \mathrm {Range} (F) = [l, u] \, , \quad \mathrm {where} \quad l, u \in \mathbb R \land u > l \, ; \tag 1 $$
continuous; differentiable at the origin; and compactly supported:
$$ \mathrm {supp} (F) = (a, b) \, , \quad \mat... | https://mathoverflow.net/users/164521 | Non-zero, bounded, continuous, differentiable at the origin, compactly supported functions with everywhere non-negative Fourier transforms | Credit to @fedja who answered my question in a comment ten minutes after I posted it. Thanks!
>
> As many as you want: just take any smooth even real-valued compactly supported function and convolve with itself.
>
>
>
This works because convolving the function with itself squares the Fourier transform. We get ... | 4 | https://mathoverflow.net/users/164521 | 370454 | 154,983 |
https://mathoverflow.net/questions/370452 | 5 | In his 1969 paper "On projective modules of finite rank", Wolmer Vasconcelos writes
>
> Let $M$ be a projective $R$-module... The trace of $M$ is defined to be the image of the map $M \otimes\_R \operatorname{Hom}\_R(M, R) \to R$, $m \otimes f \to f(m)$; it is denoted by $\tau\_R(M)$. If $M \oplus N = F$ (free), it... | https://mathoverflow.net/users/828 | Trace ideal of a projective module | The confusion is linguistic, as identified in the comments.
**Lemma.** *Let $M$ be a projective $R$-module, and suppose $M \oplus N \cong F$ is free on a basis $\mathcal B$. For $b \in \mathcal B$, write $\varepsilon\_b \colon F \to R$ for the 'dual' element taking $b$ to $1$ and all other basis elements to $0$. Then... | 7 | https://mathoverflow.net/users/82179 | 370459 | 154,985 |
https://mathoverflow.net/questions/370464 | 2 | Let $p$ be a prime, and let $\mathcal{U}$ be a family of finite $p$-groups such that
1. Any group isomorphic to a group in $\mathcal{U}$ is also in $\mathcal{U}$
2. Any product of groups in $\mathcal{U}$ is also in $\mathcal{U}$
3. Any subgroup of a group in $\mathcal{U}$ is also in $\mathcal{U}$.
Is it automatical... | https://mathoverflow.net/users/10366 | Family of $p$-groups closed under products and subgroups: closed under quotients? | No. Fix an odd prime $p$. Let $H\_p$ be a non-abelian group of order $p^3$ and exponent $p$ (this is unique to isomorphism).
Let $\mathcal{C}\_p$ be the class of $p$-groups not containing any subgroup isomorphic to $H\_p$. Then $\mathcal{C}\_p$ is stable under taking subgroups (obvious) and direct products (easy beca... | 5 | https://mathoverflow.net/users/14094 | 370466 | 154,986 |
https://mathoverflow.net/questions/370467 | 6 | I am reading Francis Borceux’s “Handbook of Categorical Algebra I” and on page 135 it says
>
> In particular a finite version of 4.2.5 does not hold: a finitely complete and well-powered category certainly admits finite intersections of subobjects (see 4.2.3), but not in general finite unions of subobjects. Finite ... | https://mathoverflow.net/users/164542 | A meet-semilattice with top element that is not a lattice? | It is well-known that a finite meet-semi-lattice with a maximum element is a lattice. The reason is that we can define $a \vee b := \wedge \{c\colon \textrm{$c$ is an upper bound for $a,b$}\}$, where this set is non-empty (since we have a maximum) and finite (since the poset is finite), and finite meets exist by suppos... | 11 | https://mathoverflow.net/users/25028 | 370468 | 154,987 |
https://mathoverflow.net/questions/370471 | 10 | In equations (20) - (25) of Mathworld's [article](https://mathworld.wolfram.com/BinomialSums.html) on binomial sums, identities are given for sums of the form $$\sum\_{k=0}^{n} k^{p}{n \choose k}, $$ with $p \in \mathbb{Z}\_{\geq 0}$. I wonder whether identities also exist for the alternating counterparts: $$\sum\_{k=0... | https://mathoverflow.net/users/93724 | Are there any identities for alternating binomial sums of the form $\sum_{k=0}^{n} (-1)^{k}k^{p}{n \choose k} $? | A rewrite of formula (10) on [MathWorld](https://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html) (replacing the summation index $k-i\mapsto i$) gives the desired formula:
$$\sum\_{k=0}^{n} (-1)^{k}k^{p}{n \choose k} =(-1)^n n! S\_2(p,n),$$
where $S\_2(p,n)$ is the [Stirling number of the second kind](https://... | 12 | https://mathoverflow.net/users/11260 | 370472 | 154,989 |
https://mathoverflow.net/questions/355274 | 9 | Let $A$ be an $n\times n$ skew-symmetric matrix of rank $r$.
Given subsets $X$ and $Y$ of row and column indices respectively, let $A\_{X,Y}$ denote the submatrix of $A$ obtained by only keeping rows with indices in $X$ and columns with indices in $Y$.
Prove that for any subsets $X, Y\subseteq \{1, 2, \ldots, n\}$ ea... | https://mathoverflow.net/users/138628 | Minors of low rank skew-symmetric matrix | Since no one else has posted a complete answer so far, let me give one.
Note that it is only complete in the sense of answering the OP's question;
several other questions arise that I cannot easily address.
In Section 1, I will prove the main result (Theorem 1), which is more general
than the OP's equality. In Sectio... | 5 | https://mathoverflow.net/users/2530 | 370491 | 154,998 |
https://mathoverflow.net/questions/370480 | 10 | For the purpose of this question, you may assume that we are working over the complex numbers.
Given a connected reductive group $G$, one can choose a maximal torus $T$, and then let $T$ act on the Lie algebra $\mathfrak{g}$ of $G$. One can use this action to define the root datum, which in turn is invariant of the c... | https://mathoverflow.net/users/155667 | Why are root data a natural candidate for classifying connected reductive groups? | I can't give you a very deep reason for why root data appear in this context (because, let's face it, root systems spring out everywhere), but there are some very elementary reasons to why the action in question is very natural with regard to the classification.
Let me start with the following two considerations:
*... | 9 | https://mathoverflow.net/users/5018 | 370495 | 155,000 |
https://mathoverflow.net/questions/370490 | 23 | I'm wondering if the function $$f(x)=\prod\_{k \in \mathbb{N}}\left(1-\frac{x^3}{k^3}\right)$$ has a name, or if there are any properties (especially about derivatives of $f$) have studied so far.
I got inspired by proof of the Basel problem ($\frac{\pi^2}{6}=\sum \frac{1}{k^2}$) using the product form of $$\frac{\si... | https://mathoverflow.net/users/164547 | About the function $\prod_{k \in \mathbb{N}}(1-\frac{x^3}{k^3})$ | If one starts with the Weierstrass factorisation
$$ \Gamma(z) = \frac{e^{-\gamma z}}{z} \prod\_{k=1}^\infty (1 + \frac{z}{k})^{-1} e^{z/k}$$
of the [Gamma function](https://en.wikipedia.org/wiki/Gamma_function), applied to $z = -x, -\omega x, -\omega^2 x$ (where $\omega = e^{2\pi i/3}$ is the cube root of unity), and m... | 40 | https://mathoverflow.net/users/766 | 370496 | 155,001 |
https://mathoverflow.net/questions/370498 | 9 | I'm a student of mathematics and I need know about the status of the Milnor conjecture (if there are partial results or if someone solved that). The statement is:
>
> A complete Riemannian manifold with non-negative Ricci curvature has a finitely generated fundamental group.
>
>
>
If someone can help me with r... | https://mathoverflow.net/users/100268 | Information about Milnor conjecture | According to David Roberts comment and the following paper it is open for dimensions $n\geq 4$.
*Pan, Jiayin*, [**A proof of Milnor conjecture in dimension 3**](http://dx.doi.org/10.1515/crelle-2017-0057), J. Reine Angew. Math. 758, 253-260 (2020). [ZBL1432.53053](https://zbmath.org/?q=an:1432.53053).
There is a ni... | 13 | https://mathoverflow.net/users/90655 | 370505 | 155,003 |
https://mathoverflow.net/questions/369486 | 2 | Just want to double check if the lemma on page 9 of this slides is correct:
[http://www.math.leidenuniv.nl/~avdvaart/talks/09hilversum.pdf](http://www.math.leidenuniv.nl/%7Eavdvaart/talks/09hilversum.pdf)
Lemma: $N(\epsilon,\cal F,||\cdot||)\leq N\_{[]}(2\epsilon,\cal F,||\cdot||). $
Proof: If $f$ is in the $2\epsi... | https://mathoverflow.net/users/163923 | bracketing number vs covering number | Your elaboration is essentially right, except the brackets themselves are not $\|\cdot\|$-balls.
If $[l,u]$ is a $2\epsilon$-bracket, then it is contained in the $\|\cdot\|$-ball of radius $\epsilon$ centered at $(l+u)/2$, since $l \le f \le u$ implies
$$\|f - (l+u)/2\| \le \frac{1}{2} \|f-l\| + \frac{1}{2} \|f - u\|... | 2 | https://mathoverflow.net/users/36614 | 370508 | 155,004 |
https://mathoverflow.net/questions/370456 | 13 | Let $\mathcal C$ be an accessible category. For $C \in \mathcal C$, define the *presentability rank* $rk(C)$ of $C$ to be the minimal regular $\kappa$ such that $C$ is $\kappa$-presentable. Following [Lieberman, Rosicky, and Vasey](https://arxiv.org/abs/1902.06777), say that $C$ is *filtrable* if it is the colimit of a... | https://mathoverflow.net/users/2362 | In a locally presentable category, is every object (a retract of) the colimit of a chain of smaller objects? | The last Remark in my joint paper gives a positive answer to the Question.
| 11 | https://mathoverflow.net/users/73388 | 370512 | 155,005 |
https://mathoverflow.net/questions/370330 | 4 | Certain Galois toposes can be written as $\lim\_{i \in I} \mathbf{PSh}(G\_i)$ where $(G\_i)\_{i \in I}$ is an inverse system of discrete groups. (The limit is a strict limit in the 2-category of Grothendieck toposes and geometric morphisms.)
What are the objects and morphisms in this topos?
If all groups $G\_i$ in ... | https://mathoverflow.net/users/37368 | Objects and morphisms in inverse limits of toposes? | The question was solved thanks to comments by Marc Hoyois.
The classifying topos associated to a pro-group is described by Grothendieck and Verdier in SGA4, Exposé IV, 2.7 ([link](http://library.msri.org/books/sga/sga/pdf/sga4-1.pdf)). Grothendieck and Verdier assume that the transition maps $\pi\_{ij} : G\_j \to G\_... | 1 | https://mathoverflow.net/users/37368 | 370519 | 155,007 |
https://mathoverflow.net/questions/370514 | 0 | Given a vector $Q \in \mathbb{R}^{S\times A}$ where $S$ and $A$ are sets of finite cardinality, for $0<\gamma<1$ define the function $H\_{w}:\mathbb{R}^{S\times A} \rightarrow \mathbb{R}^{S\times A}$ as $H\_{w}Q(s,a) := w\left(r(s,a)+ \gamma \sum\_{s' \in S}p(s'|s,a)\displaystyle\max\_{a \in A} Q(s',a)\right)+(1-w)\dis... | https://mathoverflow.net/users/145023 | Is the following function Lipschitz? | In view of your Note 1, it appears that the definition of $H\_w Q(s,a)$ has to corrected as follows:
$$H\_w Q(s,a):=\gamma w\left(r(s,a)+\sum\_{s'\in S}p(s'|s,a)\max\_{a\in A}Q(s',a)\right)
+(1-w)\max\_{a\in A}Q(s,a).$$
(Your definition is missing $\gamma$.)
Now it is clear that $H\_w Q$ is not Lipschitz in $(w,Q)$, ... | 2 | https://mathoverflow.net/users/36721 | 370531 | 155,011 |
https://mathoverflow.net/questions/370411 | 11 | I have to say that after the two last posts by Timothy Chow on Forcing I got so intrigued that I am trying to rethink the little I know about this formidable chapter of mathematics.
I have also to add that, although aware of the new field of **set-theoretic geology**, I am far from having a full grasp of it, so pre-e... | https://mathoverflow.net/users/15293 | Set-theoretic geology: controlled erosion? | What a fantastic question, and thanks to Asaf and Mirco for the great discussion in comments! I love the idea of “removing” a given set from a model of ZFC, to obtain a smaller model of ZFC - some kind of inner model method analogous to the outer model method of forcing. This may not be a complete answer, but I think t... | 9 | https://mathoverflow.net/users/10671 | 370536 | 155,012 |
https://mathoverflow.net/questions/370481 | 21 | I have already asked this question here in a different form, but really need an answer.
Let $L(s)$ be a "standard" $L$-function, say with Euler product, functional equation, etc...
(Selberg class if you like), of order 1, and let $\Lambda(s)$ be the completed $L$-function
with gamma factors. We thus have $\Lambda(k-s)=... | https://mathoverflow.net/users/81776 | Hadamard factorization of L-functions | I believe you are correct and $b$ is zero, although I find it inexplicable why this is not better known (certainly I didn't know it before). Let's stick to a primitive Dirichlet character $\mod q$, but what follows should be applicable in general. If we take logarithmic derivatives, then
$$
\frac{\Lambda^{\prime}}{\La... | 14 | https://mathoverflow.net/users/38624 | 370538 | 155,013 |
https://mathoverflow.net/questions/370530 | 3 | I know that by using Hodge decomposition and the fact that Schubert cells are Hodge cycles you can compute the Hodge numbers of Grassmanian but is there a more elementary way to compute sheaf cohomology $H^i(\Omega\_{Gr(m,n)}^j$)?
| https://mathoverflow.net/users/65846 | Elementary way to compute Hodge numbers of Grassmanian | One of the ways is to use the projective bundle theorem that says that if $X = \mathbb{P}\_Y(E)$ is a projectivization of a rank $r$ vector bundle $E$ over $Y$ then
$$
H^\bullet(X) = H^\bullet(Y) \oplus H^\bullet(Y)[-2] \oplus \dots \oplus H^\bullet(Y)[2-2r],
$$
where $[s]$ stands for the shift of grading.
Now consid... | 9 | https://mathoverflow.net/users/4428 | 370540 | 155,014 |
https://mathoverflow.net/questions/370539 | 2 | Let $\Lambda$ be a manifold and $p:H\to\Lambda$ a continuous Hilbert bundle with $H(\lambda):=p^{-1}(\lambda)$. Suppose $\Gamma\_0^0(\Lambda)$ is the space of continuous sections vanishing at infinity of $H$. I proved that $\Gamma\_0^0(\Lambda)$ has the structure of a $C\_0(\Lambda)$ module (with $C\_0$ being the space... | https://mathoverflow.net/users/164586 | Hilbert module over $C_0(\Lambda)$ as space of continuous sections of HIlbert bundle | The reference I cited in my book is Fell and Doran, *Representations of ${}^\*$-Algebras, Locally Compact Groups, and Banach ${}^\*$-Algebraic Bundles*, vol. 1 (1988). Did you check there? I don't have a copy handy but I remember this treatment being very complete.
If that doesn't work, you could look at my paper wit... | 3 | https://mathoverflow.net/users/23141 | 370543 | 155,015 |
https://mathoverflow.net/questions/370542 | 0 | As far as I understood, the knot equivalence (ambient isotopy) is a $3$-dimensional phenomenon for knots while the knot concordance is a $4$-dimensional phenomenon.
The knot concordance does not imply the knot equivalence. For example, we consider the unknot and the stevedore knot. They are concordant but not equival... | https://mathoverflow.net/users/nan | Knot equivalence and concordance | Knot equivalence implies knot concordance.
Specifically, let $K\_0$ and $K\_1$ be equivalent, smoothly embedded knots in $S^3$. Then there is a smooth ambient isotopy $f: [0,1] \times S^3 \to S^3$ where each map $f(t, -)$ is a smooth diffeomorphism $S^3 \to S^3$, and $f(1, -)$ sends $K\_0$ to $K\_1$.
To define a sm... | 2 | https://mathoverflow.net/users/43158 | 370545 | 155,016 |
https://mathoverflow.net/questions/370222 | 0 | Let $\mathcal{F}$ be a $P$-filter on $\omega$. Denote by $\Omega=\bigsqcup \omega\_i$ where $\omega\_i=\omega$. Consider the $P$-filter $\mathcal{S}$ on $\Omega$ whose base is as follows
$(\bigsqcup\_i F\_i, F\_i\in \mathcal{F})$.
$\mathcal{F}$ filter is isomorphic to $\mathcal{S}$ filter ?
| https://mathoverflow.net/users/112417 | P-filter property? | After the clarifications in comments of OP (August 28 and today), here's a counterexample. Let $\mathcal F$ be the filter consisting of only $\omega-\{0\}$ and $\omega$. Then $\mathcal S$ (as defined in the question) contains a set whose complement is infinite, whereas $\mathcal F$ does not. So they cannot be isomorphi... | 3 | https://mathoverflow.net/users/6794 | 370546 | 155,017 |
https://mathoverflow.net/questions/370541 | 8 | The plethysm $s\_{\nu}[s\_{\mu}]$ of two symmetric functions is the character of the composition of Schur functors $S^{\nu}(S^{\mu}(V))$. We know that this operation is linear and multiplicative in its first argument. But is there a way to develop
1. $s\_{\nu}[s\_{\mu} + s\_{\lambda}]$;
2. $s\_{\nu}[s\_{\mu}s\_{\lamb... | https://mathoverflow.net/users/73667 | Interaction of plethysm with other operations | In principle one can develop (1) using the coproduct in the ring of symmetric functions. By the Littlewood–Richardson rule, $\Delta(s\_\nu) = \sum\_{\alpha}\sum\_\beta c^\nu\_{\alpha\beta} s\_\alpha \otimes s\_\beta$ where $c^\nu\_{\alpha\beta}$ is a Littlewood–Richardson coefficient, and correspondingly
$$s\_\nu[s\_... | 6 | https://mathoverflow.net/users/7709 | 370551 | 155,018 |
https://mathoverflow.net/questions/369692 | 4 | Let $\alpha$ be a differential form on the torus $T^n$ whose support $\mathrm{supp}(\alpha)$ is contained in a small neighborhood of the subtorus $T^k\equiv T^k\times \{0\}^{n-k}$.
>
> **Question:**
> Suppose $\alpha$ is closed or even harmonic with respect to some metric. I was wondering if the de Rham cohomology ... | https://mathoverflow.net/users/69190 | A differential form whose support is in a tubular neighborhood of $T^k\times \{0\}^{n-k}\subset T^n$ | By compactness, $\operatorname{supp}(\alpha)\subset T^k\times B^{n-k}$,
where $B^{n-k}\subset T^{n-k}$ is a small open ball.
So $[\alpha]$ is in the image of $H^\*\_{dR}(T^n,T^n\setminus T^k\times B^{n-k})\to H^\*\_{dR}(T^n)$.
By the Künneth formula and excision,
$$ H^\*\_{dR}(T^n,T^n\setminus T^k\times B^{n-k})
\cong ... | 3 | https://mathoverflow.net/users/70808 | 370554 | 155,019 |
https://mathoverflow.net/questions/370552 | 14 | Differential graded algebras, or DGAs, are a basic object of study in many areas of modern mathematics. While they were present (implicitly at least) since the start of modern differential geometry, I would like to know where the abstract definition of a DGA was first written down, and by whom?
| https://mathoverflow.net/users/121660 | Who introduced the abstract definition of a DGA? | Search for the earliest appearance of "differential graded algebra" and "DGA" on MathSciNet. The earliest hit is DGA in a review of a 1954 paper of Cartan where DGA (or more precisely, the redundant term "DGA-algebra") is defined: "Sur les groupes d'Eilenberg-Mac Lane $H(\Pi,n)$ I. Méthode des constructions," Proc. Nat... | 7 | https://mathoverflow.net/users/3272 | 370557 | 155,021 |
https://mathoverflow.net/questions/370570 | 4 | I am a graduate student in differential geometry and would like to learn more about algebraic geometry recently. Are there any recommended textbook/reference/lecture notes which is easier for a differential geometer to approach? Or are there any standard textbooks that can build my way up to algebraic geometry instead ... | https://mathoverflow.net/users/164604 | Algebraic geometry reference for differential geometer | I started some years ago from this lecture note: [Notes Algebraic Geometry 2 by by Karen Smith](https://www.google.com/url?sa=t&source=web&rct=j&url=https://pdfs.semanticscholar.org/de3b/129ed932f10e02004f36a2a5a9026e294102.pdf&ved=2ahUKEwihmKC0rsfrAhX4AxAIHXQuDwoQFjAAegQIARAB&usg=AOvVaw0WrJCd2Ib0fU98MAQLTKgD) which is... | 4 | https://mathoverflow.net/users/nan | 370577 | 155,024 |
https://mathoverflow.net/questions/370596 | 1 | Let $Z=(X,Y) : \Omega\rightarrow\mathbb{R}^2$ be a Borel-measurable random vector and $U\subset\mathbb{R}$ be open. Suppose that $Z$ is absolutely continuous with Lebesgue density $\zeta$.
I was wondering about the following **question:** If we assume that $\zeta$ factorizes on the square $U\times U$, i.e. that the r... | https://mathoverflow.net/users/160714 | If a joint density factorizes on a square, does this imply that the marginal random variables are locally independent? | The answer is no. E.g., let $U=(0,1)$ and
$$\zeta=\frac12\times 1\_{(0,1)^2}+\frac12\times 1\_{(1,2)^2};$$
that is, the joint distribution of $(X,Y)$ is the half-and-half mixture of the uniform distributions on the the squares $(0,1)^2$ and $(1,2)^2$. Then
$$P\_{X,Y}(U\times U)=\frac12\ne\frac12\times\frac12=P\_X(U)P\_... | 1 | https://mathoverflow.net/users/36721 | 370598 | 155,028 |
https://mathoverflow.net/questions/370601 | 10 | When the answer to the Lüroth problem is affirmative, the genus (for curves) or Castelnuovo's criteria (for separably unirational surfaces) give computable invariants which decide if a given variety is unirational.
I am interested in the simplest cases where non-rational examples are known: namely surfaces over $\ove... | https://mathoverflow.net/users/154157 | Is unirationality decidable? | This is a widely studied problem, and I think with the current technology it looks unlikely this will be answered. Let me stick to characteristic $0$ for simplicity; the story gets much richer in positive characteristic (even if you add the adjective 'separably' everywhere).
Unirational varieties are rationally conne... | 6 | https://mathoverflow.net/users/82179 | 370606 | 155,031 |
https://mathoverflow.net/questions/370605 | 4 | This question was originally posted on [MSE](https://math.stackexchange.com/q/3801328/272127).
But I would like to post it here to see whether anyone could recommend some reference for me.
I am currently reading the paper "Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds" by G... | https://mathoverflow.net/users/164604 | Reading material for an analytical aspect of Kähler Geometry | I'll give an answer that is specifically tailored towards the Kahler-Ricci flow. Hopefully other answers can give some good materials for geometric analysis on Kahler manifolds more generally. For KR flow, I found the following manuscripts to be very informative. All of them can be found for free online as well, which ... | 4 | https://mathoverflow.net/users/125275 | 370613 | 155,033 |
https://mathoverflow.net/questions/368680 | 1 | All the definition and results which I am using here has been taken from the Garry Chartrand book $\textbf{The Introduction of Graph Theory}$ and the paper $\textbf{On the Commuting Graph of Dihedral Group}$. Here I am denoting the set N$(v)$ is the collection of all vertices which are adjacent to $v$ in a simple graph... | https://mathoverflow.net/users/51105 | Question related to complete vertex, interior vertex and boundary vertex of a graph | I suspect the issue is with the definition of interior you are using.
This should say that $v$ is an interior vertex of $\Gamma$ if for each $u\neq v$, there exists a vertex $w$ and a $u-w$ **geodesic** containing $v$.
With this definition, the path $u\sim v\sim w$ in $K\_n$ does not cause any problems, because it is... | 2 | https://mathoverflow.net/users/163516 | 370622 | 155,037 |
https://mathoverflow.net/questions/370617 | 4 | I read on [this n-lab page](https://ncatlab.org/nlab/show/full+and+faithful+functor) that a fully faithful functor $F: C\to D$ reflects all limits and colimits by the universal property.
On the other hand, I think a fully faithful functor does not always preserve limits and colimits since a "testing object" in $D$ is... | https://mathoverflow.net/users/24965 | Does a fully faithful functor always preserve limits and colimits? | The forgetful functor from abelian groups to groups is fully faithful, and does not preserve coproducts. For example, in abelian groups, $\mathbb Z\coprod \mathbb Z=\mathbb Z\times \mathbb Z$, but in groups $\mathbb Z\coprod \mathbb Z$ is the free group on two generators.
| 20 | https://mathoverflow.net/users/164546 | 370624 | 155,039 |
https://mathoverflow.net/questions/370612 | 12 | Let $\alpha$ be irrational. A famous theorem of Vinogradov says that $\{ \alpha p\}$ is equidistributed in $[0,1]$ as $p$ runs over all primes.
Let $a,q$ be natural numbers with $\gcd(a,q) = 1$. Then is the sequence $\{ \alpha p\}$ equidistributed in $[0,1]$, as $p$ runs over primes with $p \equiv a \bmod q$?
Almos... | https://mathoverflow.net/users/5101 | Equidistribution of $\{\alpha p\}$ for $p$ in an arithmetic progression | I think the sought result follows from Vinogradov's theorem. By Weyl's criterion and the orthogonality of Dirichlet characters, the sought result can be reformulated as follows. For every nonzero integer $k$, and for every Dirichlet character $\chi$ modulo $q$, we have
$$\sum\_{p<x}\chi(p)e(k\alpha p)=o(\pi(x))\qquad\t... | 14 | https://mathoverflow.net/users/11919 | 370638 | 155,044 |
https://mathoverflow.net/questions/370642 | 2 | If $X$ denotes a $m \times n$ random matrix whose entries are independent identically distributed random variables with mean $\mu$ and $\sigma^2 < \infty$, let
$$Y = X X^T$$
with $X^T$ the transpose of $X$. Let $\lambda\_1 , \lambda\_2 ,\ldots, \lambda\_m$ be the eigenvalues of $Y$ (viewed as random variables).
I... | https://mathoverflow.net/users/161614 | Limit law of eigenvalue of random matrix with mean different to 0 | The addition of the same constant $\mu$ to all elements of $X$ (so that their mean becomes $\mu$) is a rank-one perturbation of the matrix, which has no effect on the distribution of the eigenvalues of $XX^T$ in the limit $n,m\rightarrow\infty$ at fixed $n/m$ --- this limiting distribution remains the Marchenko-Pastur ... | 4 | https://mathoverflow.net/users/11260 | 370659 | 155,049 |
https://mathoverflow.net/questions/370662 | 3 | Is it true, that $\forall x \in \mathbb N, \pi(x+200)-\pi(x) \leq 50 $ ?
$$\pi(x)=\text{card}(\{n \in [0,x] \cap \mathbb N, n\text{ is prime}\})$$
| https://mathoverflow.net/users/110301 | $\pi(x+200)-\pi(x)\leq 50$? | Yes.
Up to $207$ there are $46$ primes. Hence, the inequality is true for $x \le 7$.
Let $$\pi\_{210}(x) = \textrm{card}(\{n \in [0,x] \cap \mathbb{N}, \, \gcd(n,210)=1\}).$$
For $x>7$, $\pi(x+200)-\pi(x) \le \pi\_{210}(x+200)-\pi\_{210}(x)$. Since $\pi\_{210}$ is $210$-periodic, it is enough to verify that $\pi\_{... | 12 | https://mathoverflow.net/users/31469 | 370664 | 155,052 |
https://mathoverflow.net/questions/370643 | 3 | I'm trying to build an example of a rational homology 3-sphere $M$ (that is not an integral homology 3-sphere) with an irreducible genus 2 Heegaard splitting so that $M$ is not a lens space or connect sum of lens spaces.
I haven't been successful trying to draw a Heegaard diagram for such a 3-manifold, so I suspect t... | https://mathoverflow.net/users/149240 | Genus 2 Heegaard decompositions of rational homology 3-spheres | Suppose that $K$ is the figure eight knot in the three-sphere $S^3$. Let $n(K)$ be a small open neighbourhood of $K$. Let $X = S^3 - n(K)$. You can find a Heegaard diagram for $X$ as follows.
Let $N = N(K)$ be a closed neighbourhood of $K$, slightly larger than $n(K)$. Draw both in a knot diagram of $K$. Add to $N$ a... | 5 | https://mathoverflow.net/users/1650 | 370667 | 155,053 |
https://mathoverflow.net/questions/331248 | 16 |
>
> Why are [Thompson's groups](https://en.wikipedia.org/wiki/Thompson_groups) called $F$, $T$ and $V$?
>
>
>
I never saw Thompson's unpublished notes, in which he introduces these groups; maybe an explanation can be found there?
| https://mathoverflow.net/users/122026 | Why are Thompson's groups called $F$, $T$ and $V$? | The "$F$" actually stands for "free homotopy idempotent", since $F$ is the universal group encoding a free homotopy idempotent (the endomorphism sending each standard generator $x\_i$ to $x\_{i+1}$ is idempotent up to conjugation). The universality was proved by [Freyd–Heller](https://doi.org/10.1016/0022-4049(93)90088... | 26 | https://mathoverflow.net/users/164670 | 370677 | 155,055 |
https://mathoverflow.net/questions/370423 | 2 | I was reading a paper on Euclidean ideals by [H Graves and M. Ram Murthy](https://www.dropbox.com/s/ftnjf08c7pxcp4p/HESTER%20GRAVES%20AND%20M.%20RAM%20MURTY.pdf?dl=0). I have a problem in understanding one of the claims.
**setup**
Let $K$ be a number field and $H(K)$ is its Hilbert class field. Suppose $H(K)/\Bbb Q... | https://mathoverflow.net/users/131448 | What are conditions to satisfied by rational prime p so that every prime lying above p is a prime of order 1 and generates class group? | First of all, as they observe, the assumption that $H(K)/\mathbb{Q}$ is abelian ensures that $H(K)\subseteq\mathbb{Q}(\zeta\_{f(K)})$ and not only $K\subseteq\mathbb{Q}(\zeta\_{f(K)})$. Also, they work under the assumption that the class group of $K$ is cyclic, and I have added this to your question.
Now, Artin recip... | 1 | https://mathoverflow.net/users/18238 | 370679 | 155,056 |
https://mathoverflow.net/questions/370621 | 6 | Let $R=\mathbb C\{x\_1,...,x\_n\}\subset S=\mathbb C [[x\_1,...,x\_n]]$ denote the ring of convergent, respectively formal, power series over $\mathbb C$.
Suppose $f\in R$ is irreducible in $R$. Does it remain irreducible in $S$?
| https://mathoverflow.net/users/157954 | Factorization in formal power series versus in convergent power series over the complexes | As Arno Fehm points out, this follows from results in Nagata's [Some Remarks on Local Rings II](https://projecteuclid.org/euclid.kjm/1250777424). Both $R$ and $S$ are UFD's, so $f$ is irreducible, in $R$ or $S$ respectively, if and only if the ideal it generates, in $R$ or $S$ respectively, is prime. At the bottom of p... | 4 | https://mathoverflow.net/users/297 | 370681 | 155,057 |
https://mathoverflow.net/questions/370682 | 12 | Let $X$ be a quasi-projective scheme over a field $k$.
Let $G$ be a finite group acting on $X$ whose order is invertible in $k$.
If $X$ is Cohen-Macaulay, can we conclude that the subscheme of fixed points $X^G$ is Cohen-Macaulay?
| https://mathoverflow.net/users/110362 | Fixed point scheme of finite group Cohen-Macaulay? | Here is a simpler example than the one I left before, using the same strategy. Let
$$X = \{ x\_1 x\_3 = x\_1 x\_4 = x\_1 x\_5 = x\_2 x\_4 = x\_2 x\_5 = x\_3 x\_5 = 0 \} \subset \mathbb{C}^5.$$
This is the reduced union of four $2$-planes. Here is a projective picture, where $j$ represents the point where $x\_j$ is the ... | 13 | https://mathoverflow.net/users/297 | 370688 | 155,059 |
https://mathoverflow.net/questions/370687 | 15 | I have been reading about transfinite diameter and its applications to number theory and have been hunting for the following paper for quite a while:
Cantor D.: On an extension of the definition of transfinite diameter and some
applications, J. reine Angew. Math., vol. 316 (1980), pp. 160-207.
Unfortunately, I am a... | https://mathoverflow.net/users/157984 | Looking for a paper on transfinite diameter by David Cantor | Hopefully this works:
[Cantor D.: On an extension of the definition of transfinite diameter and some applications](https://gdz.sub.uni-goettingen.de/id/PPN243919689_0316?tify=%7B%22pages%22:%5B164,165%5D,%22panX%22:0.966,%22panY%22:0.416,%22view%22:%22scan%22,%22zoom%22:0.464%7D)
---
Since you said that you had... | 20 | https://mathoverflow.net/users/22971 | 370689 | 155,060 |
https://mathoverflow.net/questions/370710 | 3 | Let $\mathbb{Z}/p^{\infty}$ denote the Prufer group. By $p$-completion properties, it follows that $(B\mathbb Z/p^{\infty})^{\wedge}\_p\simeq K(\mathbb{Z}^{\wedge}\_p,2)\simeq(BS^1)^{\wedge}\_p$. But, why does the inclusion $\mathbb{Z}/p^{\infty}\hookrightarrow S^1$ induce such a homotopy equivalence?.
| https://mathoverflow.net/users/112348 | $(B\mathbb Z/p^{\infty})^{\wedge}_p\rightarrow (BS^1)^{\wedge}_p$ induced by inclusion is an equivalence | Think of the Prüfer group as $\mathbb{Q}/\mathbb{Z}$ and of $S^1$ as $\mathbb{R}/\mathbb{Z}$. Here $\mathbb{Q}$ is discrete, $\mathbb{R}$ has its usual topology and is contracible. The map $\mathbb{Q} \to\mathbb{R} $ becomes an equivalence after $p$-completion.
| 7 | https://mathoverflow.net/users/39747 | 370711 | 155,065 |
https://mathoverflow.net/questions/370714 | 7 | In Elliot's book "Probabilistic Number Theory", there seems to be an inaccuracy. The author defines, for any sequence $a\_n$, the quantity
$$V(p)=\sum\_{r=0}^{p-1}\left|\sum\_{\substack{n=1 \\n \equiv r\mathrm{mod}(p)}}^N a\_n-p^{-1}\sum\_{n=1}^N a\_n
\right|^2$$
He then asserts that, if $a\_n$ assumes only the val... | https://mathoverflow.net/users/159298 | A possible error in Elliot's book "Probabilistic Number Theory" | You are right that the second display is false in general (Elliott might impose some conditions). The following version is well-known, and a consequence of Selberg's optimized large sieve inequality:
$$\sum\_{p\leq Q}pV(p)\leq (N+Q^2-1)\sum\_{n=1}^N|a\_n|^2.$$
This holds for any complex numbers $a\_n$. See (23) in [Mon... | 8 | https://mathoverflow.net/users/11919 | 370717 | 155,066 |
https://mathoverflow.net/questions/370674 | 5 | Let $X\_1,...,X\_n$ be points on $\mathbb S^1.$
We then define the expectation value $E(X)=\frac{1}{n}\sum\_{i=1}^n X\_i.$
Let $\frac{dS(X\_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i.e. $X\_i$ are uniformly distributed random variables on the circle.
I am curious to know:
How does
$$\int... | https://mathoverflow.net/users/150564 | Spherical average of $\frac{1}{x}$ | The probability distribution $P(R)$ of $R=n|E(X)|$ was calculated by [Kluyver](https://www.dwc.knaw.nl/DL/publications/PU00013859.pdf) (1906), it is given by
$$P(R)=\frac{1}{2\pi}\int\_0^\infty [J\_0(x)]^n J\_0(rx)x\,dx.$$
For $n\gg 1$ one has a Rayleigh distribution (here is [derivation](https://projecteuclid.org/down... | 4 | https://mathoverflow.net/users/11260 | 370740 | 155,072 |
https://mathoverflow.net/questions/370722 | 1 | **Context:**
------------
When studying percolation in finite sized systems, there exist various definitions and criteria for determining when a given system is percolating, i.e., given a definition for connectivity, it contains a system-spanning cluster which mimics that of an infinite cluster in the limit of infini... | https://mathoverflow.net/users/115841 | Understanding the wrapping criterion in percolation theory | **Q1:** Here is an image that shows a wrapping cluster [[source](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.64.016706)]. So yes, the wrapping condition means that the cluster would extend out to infinity if the lattice is repeated periodically in all directions. Just imagine tiling the plane with the image,... | 2 | https://mathoverflow.net/users/11260 | 370741 | 155,073 |
https://mathoverflow.net/questions/30637 | 6 | Let $X$ be a smooth variety defined over $\mathbb{Q}$. If we want to check that $X$ is locally soluble at a prime $p$, then it suffices to find a non-singular $\mathbb{F}\_p$-point, which can be lifted to a $\mathbb{Q}\_p$-point by Hensel's lemma.
However, it might happen that $X$ does not have any non-singular $\mat... | https://mathoverflow.net/users/5101 | Checking local solubility of varieties at "bad" primes | As mention in the comments by Pete Clark, it is a theorem that when $n$ is sufficiently large, you will either be able to apply Hensel's Lemma to get a solution in the valuation ring, or you will find that there are no (primitive) solutions $\pmod{p^n}$.
In practice, I will assume your variety $X$ is affine and it is... | 8 | https://mathoverflow.net/users/158462 | 370750 | 155,077 |
https://mathoverflow.net/questions/370733 | 5 | There have been a lot of information published about bounds on lcm of polynomials and other types of sequences evaluated at consecutive naturals.
Moreover it’s known that $\operatorname{lcm}(1,2,3,\dotsc, n)>2^n$, and it behaves asymptotically as $e^n$.
However I wanted to ask two questions.
1. Does there exist a... | https://mathoverflow.net/users/164499 | Lower bound on the cardinality of set $A\in \{1,2,3,\dotsc,n\}$ with $\operatorname{lcm}(A)>\phi^n$ and asymptotic of number of such subsets | In the paper [The least common multiple of random sets of positive integers](https://arxiv.org/abs/1112.3013), Cilleruelo, Rué, Šarka, Zumalacárregui prove that if each subset of
$\{1,\dotsc,n\}$ is chosen with equal probability (equivalently each point $a\in \{1,\dotsc,n\}$ is included in $A$ with probability $\delta=... | 4 | https://mathoverflow.net/users/17773 | 370751 | 155,078 |
https://mathoverflow.net/questions/370645 | 7 | Let $f : X^m \to Y^n$ be an algebraic fiber space (between projective manifolds) whose discriminant locus is denoted by $E$. Let $U$ be a polydisk in $\mathbb{C}^n$ (with coordinates $(y\_1, ..., y\_n)$) such that $U \backslash E \simeq (\Delta^{\ast})^{\ell} \times \Delta^{n-\ell}$, where $1 \leq \ell \leq n$. Let $x\... | https://mathoverflow.net/users/105103 | What is the meaning of the monodromy theorem in Hodge theory? | I'm not completely sure what you're after, but perhaps it's simply some insight. So let make a few remarks. For simplicity, assume $Y$ is a curve. (In general, you need to assume that $E$ has normal crossings, which you did implicitly.) Replace it by a small disk. So $f:X\to D$ becomes a $C^\infty$ fibre bundle when re... | 6 | https://mathoverflow.net/users/4144 | 370756 | 155,080 |
https://mathoverflow.net/questions/360021 | 4 | Researching a question related to closed walks on graphs, I have come across the following problem. Let $G$ be a connected graph on $n$ vertices and $k=O(\log(n))$. Pick a random closed walk on $G$ as follows: Pick a uniformly random vertex $v$ of $G$ and then pick a uniformly random closed walk $w$ of length $2k$ star... | https://mathoverflow.net/users/152267 | Support of random closed walk in arbitrary graph | Such a bound for regular graphs valid for $\alpha<1/4$ was recently obtained in "Support of Closed Walks and Second Eigenvalue Multiplicity of Regular Graphs"
<https://arxiv.org/pdf/2007.12819.pdf> Theorem 1.3
| 5 | https://mathoverflow.net/users/7691 | 370774 | 155,084 |
https://mathoverflow.net/questions/370786 | 0 | I'm reading Soundararajan's <https://arxiv.org/pdf/0705.0723.pdf>, and on page 5, one has
$$\sum\_{n\leq x} \frac{\Lambda(n)}{n^z} \log (x/n) = -\frac{\zeta'}{\zeta}(z)\log x - \Big(\frac{\zeta'}{\zeta}(z) \Big)' -\sum\_{\rho} \frac{x^{\rho-z}}{(\rho-z)^2} + O(1/T),$$
where $\Lambda$ denotes the von Mangoldt function... | https://mathoverflow.net/users/480516 | On Soundararajan's explicit formula | The formula you quote is on page 8 (not page 5). The sum is over all zeros $\rho$, not just those with $|\Im(\rho)|\leq T$. On the other hand, there is the assumption $\Im(z)\in[T,2T]$.
| 9 | https://mathoverflow.net/users/11919 | 370787 | 155,086 |
https://mathoverflow.net/questions/370782 | 10 | Let $M$ be an integer matrix with determinant equal to one (or maybe also minus one, but I did not do any tests for this case) and assume that $M$ is periodic, that is $M^n$ is the identity matrix for some $n$. Let $p\_M$ denote the characteristic polynomial of $M$.
>
> Question 1: Is it true that then $p\_M(1) \ge... | https://mathoverflow.net/users/61949 | Sum of the coefficients of the characteristic polynomial of periodic matrices | **Q1:** This was already given in the comments, but: a matrix $M \in GL\_k(\mathbb{Z})$ of finite order $n$ must have [rational normal form](https://en.wikipedia.org/wiki/Frobenius_normal_form#A_rational_normal_form_generalizing_the_Jordan_normal_form) a block-diagonal matrix with blocks the [companion matrices](https:... | 10 | https://mathoverflow.net/users/290 | 370793 | 155,088 |
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