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https://mathoverflow.net/questions/386989 | 6 | **Problem.** Let $a$ be a positive integer that is **not** a perfect cube. Show that the Diophantine equation $(xz+1)(yz+1)=az^{3}+1$ has no solutions in positive integers $x, y, z$ with $z > a^{2}+2a$.
If this result is true, we can prove that for a given integer $a$, $a \not= m^3$, there are **finitely** many Ferma... | https://mathoverflow.net/users/166404 | The Diophantine equation $ (xz+1)(yz+1)=az^{3}+1$ has no solutions in positive integers with $z > a^2+2a$ | The given equation $(xz+1)(yz+1)=az^3+1$ can be rewritten as $az^2-xyz-(x+y)=0$. We shall show that for any solution $(x,y,z)$, we have $z \le a^2+2a. \ $
Note that $z \ | \ x+y$, therefore $z \le x+y. \ $ Treating $x, y$ as constants, the only positive solution for $z \ $ is \begin{equation} z = \frac{xy+\sqrt{x^2y^2+... | 6 | https://mathoverflow.net/users/166404 | 388724 | 161,011 |
https://mathoverflow.net/questions/388640 | 1 | I am reading the paper [Ricci Curvature and Volume Convergence](https://www.jstor.org/stable/2951841?seq=1) written by Professor Colding. In section 2, they define Lipschitz functions $b\_j^+:M\to\mathbb R$ with $|\nabla b\_j^+|=1$ and set $$\Phi=(b\_1^+,...,b\_n^+):B\_1(p)\to B\_{1+\epsilon}(0)\subset\mathbb R^n.$$ Th... | https://mathoverflow.net/users/177798 | How can I check this volume comparison? | By approximation, we may assume that the map $\Phi$ is continuously differentiable. Then the absolute value of the Jacobian determinant of $\Phi$ is the volume of a parallelepiped with edge lengths $1$ and hence with volume $\le1$. So, the absolute value of the Jacobian determinant of $\Phi$ is $\le1$, and the volume c... | 2 | https://mathoverflow.net/users/36721 | 388735 | 161,012 |
https://mathoverflow.net/questions/387547 | 1 | I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$
\begin{equation}
\left\{ \begin{aligned}
x'(t) &= f(t, x(t)), \qquad t \in [a,b] \\
x(a) &= x\_0
\end{aligned} \right.
\end{equation}
By "global" I mean that the time interval is fixed, i.e. $[a... | https://mathoverflow.net/users/176090 | What is the most general Carathéodory-type global existence theorem? | (N.B. In the below I assume $[a,b] = [0,\infty]$, but the precise values don't matter and appropriate substitutions of $a,b$ into the discussion also gives you the same conclusion.)
Once you have a local existence theorem of the form
>
> For every compact set $K$ and compact subset $K\_0 \Subset K$, there exists ... | 3 | https://mathoverflow.net/users/3948 | 388741 | 161,015 |
https://mathoverflow.net/questions/387545 | 3 | I posted this over on MSE without much luck. Not sure if posting here is considered cross-posting but I can remove it if it is.
Let $X\sim\mathcal N(\sqrt 2,1/x^2)$. The expected value $\mathsf EX^{-1}$ is undefined; however, we can [assign it a value](https://math.stackexchange.com/questions/2368344/cauchy-principal... | https://mathoverflow.net/users/125801 | Using $\delta$-method to "estimate" undefined moments of a random variable? | $\newcommand\vp\varepsilon$
1. Of course, your divergent series for $EX^{-1}$ and $EX^{-2}$ should be understood as [asymptotic expansions](https://en.wikipedia.org/wiki/Asymptotic_expansion#:%7E:text=In%20mathematics%2C%20an%20asymptotic%20expansion,the%20function%20tends%20towards%20a). The delta method practically... | 4 | https://mathoverflow.net/users/36721 | 388743 | 161,016 |
https://mathoverflow.net/questions/388745 | 1 | Let we have an elementary function $f(W)$, applicable to a matrix.
Now consider the function
$g(x)=\operatorname{tr} f(W+x),$
where $x$ is scalar. Is $g(x)$ necessarily an elementary function?
Simple case: complex numbers (which as widely known can be represented as matrices).
$g(x)=\Re (f(z+x))$
where $x$ ... | https://mathoverflow.net/users/10059 | Is trace of a slice of an elementary function of a matrix also elementary? | Here is a partial answer to your question that addresses the case of split-complex numbers and similar commutative number structures:
If $A$ is a finite-dimensional commutative associative unital $\mathbb{R}$-algebra, then it splits into local direct summands
$$
A \cong \bigoplus\_{k=1}^N (A\_k, \mathfrak{m}\_k)
$$
w... | 3 | https://mathoverflow.net/users/1849 | 388753 | 161,020 |
https://mathoverflow.net/questions/388755 | 5 | On page 4 of Nitin Nitsure's paper [Construction of Hilbert and Quot Schemes](https://arxiv.org/abs/math/0504590), the author refers to the fact that Hilbert polynomials are indeed polynomials as
>
> a special case of Snapper's Lemma, see "An Intersection Theory for Divisors (preprint 1994)" by Steven Kleiman for a... | https://mathoverflow.net/users/70751 | Reference request: Kleiman's proof of Snapper's Lemma | A proof by Kleiman can be found in ‘‘[Toward a Numerical Theory of Ampleness](https://www.jstor.org/stable/pdf/1970447.pdf)’’. I suspect it's the intended proof, although the paper is from around 30 years before the cited 1994 preprint.
| 5 | https://mathoverflow.net/users/75344 | 388762 | 161,021 |
https://mathoverflow.net/questions/388729 | 7 | I am curious to know what is the standard way (in AMS style) to cite a journal article that has an article number and/or a page range.
For instance, here is a BibTeX entry from MathSciNet of an article published in Proceedings A:
```
@article {MR4212412,
AUTHOR = {Kallosh, Renata},
TITLE = {M-theory, blac... | https://mathoverflow.net/users/74033 | How to cite an article with an article number (instead of page range) | I suggest to format this bibliographic reference as follows:
Renata Kallosh.
M-theory, black holes and cosmology.
Proceedings of the Royal Society A 20200786 (2021).
[doi:10.1098/rspa.2020.0786](https://doi.org/10.1098/rspa.2020.0786),
[arXiv:2009.11339](https://arxiv.org/abs/2009.11339).
**Rationale**: The purpose... | 3 | https://mathoverflow.net/users/402 | 388770 | 161,024 |
https://mathoverflow.net/questions/388604 | 5 | "[Measure Theory and Probability Theory](https://link.springer.com/book/10.1007/978-0-387-35434-7)" by Athreya and Lahiri introduces Lebesgue–Stieltjes measure construction on $\mathbb{R}^n$ in general in the following way:
Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ satisfy the following properties:
1. Let $(x\_1,x... | https://mathoverflow.net/users/63923 | Lebesgue–Stieltjes measure construction on $\mathbb{R}^n$ | $\newcommand\R{\mathbb R}$Concerning your question 1: A complete proof of this result (based on a discrete approximation) can be found, for instance, in [Kallenberg's book "Foundations of Modern Probability"](https://www.springer.com/gp/book/9780387953137), where it is given as Corollary 3.26. Another kind of arguments... | 4 | https://mathoverflow.net/users/36721 | 388774 | 161,026 |
https://mathoverflow.net/questions/388761 | 9 | Let $f(x\_1,\ldots,x\_n)$ be a polynomial of degree $d$ with coefficients in the finite field $\mathbb F\_q$ and let $V(f)\subseteq\mathbb F\_q^n$ be its set of zeroes. Assume $d<n$. Then Chevalley proved that $V(f)$ cannot consist of one element alone. In the same issue, Warning showed the stronger statement that $\#V... | https://mathoverflow.net/users/89948 | Chevalley-Warning-Ax for double covers | **Congruences for rational points on fibers.**
Let $\mathbb{F}\_q$ be a finite field with $q$ elements.
**Definition.** A quasi-projective $\mathbb{F}\_q$-scheme has **congruence $1$**, respectively **congruence $0$** if for every positive integer $r$, the set of $\mathbb{F}\_{q^r}$-points has cardinality congruent t... | 9 | https://mathoverflow.net/users/13265 | 388783 | 161,030 |
https://mathoverflow.net/questions/382112 | 15 | In the 1986 book *[An Introduction to the Theory of Surreal Numbers](https://doi.org/10.1017/CBO9780511629143)*, Gonshor, on page 117, notes that it is an open problem whether $\omega(\sqrt{2}+1)+1$ is a prime, using the standard definition of integral number he introduces earlier in the chapter.
I'll give a little b... | https://mathoverflow.net/users/172917 | Is the surreal number $\omega(\sqrt{2}+1)+1$ a prime? | The answer is no. The factorization is $(\sqrt{\sqrt{2}+1}\omega^{1/2} - \sqrt{2\sqrt{\sqrt{2}+1}}\omega^{1/4}+1)(\sqrt{\sqrt{2}+1}\omega^{1/2} + \sqrt{2\sqrt{\sqrt{2}+1}}\omega^{1/4}+1)$
| 12 | https://mathoverflow.net/users/172917 | 388790 | 161,032 |
https://mathoverflow.net/questions/388775 | 7 | Is there a convex function $F$ that is not differentiable, but whose subdifferential admits a continuous selection, i.e. a continuous $g$ with $g(x) \in \partial F(x)$ for all $x$ in the domain?
In one dimension I think I can prove there is not: if $|\partial F(x)| > 1$ then it contains an open set, whose inverse ima... | https://mathoverflow.net/users/29697 | Nondifferentiable convex function whose subdifferential admits a continuous selection | The answer is no.
See Rockafellar's *Convex Analysis*, part V.
First, let $D$ be the set of points where $F$ is differentiable. *Theorem 25.5* proves that $D$ is dense in the interior of the domain of $F$, with measure zero complement. And that $\nabla F$ is a continuous mapping on $D$.
Next, if the domain of $F$... | 7 | https://mathoverflow.net/users/3948 | 388797 | 161,034 |
https://mathoverflow.net/questions/388791 | 2 | Is there a notation for the statement $H$ is isomorphic to a subgraph of $G$? I was thinking of using $H<G$, but I'd like to use standard notation if possible.
| https://mathoverflow.net/users/177875 | Notation for H is isomorphic to a subgraph of G | Inspired by the $\simeq$ notation for isomorphic structures, I would suggest $\lesssim$, $\prec$, $\precsim$, or [this symbol](https://tex.stackexchange.com/questions/108403/how-to-create-subset-with-sim-symbol).
The pair of symbols $\prec$ and $\precsim$ have the advantage of providing a symbol for both allowing and... | 1 | https://mathoverflow.net/users/158328 | 388800 | 161,035 |
https://mathoverflow.net/questions/374468 | 15 | I have noticed that in the literature on causality in general relativity one sees apparent counterexamples to the cosmic censorship hypothesis (somehow you have models for gravitational collapse which assume spherical symmetry and things like this so that naked singularities can in fact arise). Hawking conceded that th... | https://mathoverflow.net/users/119114 | Counterexamples to the Penrose Conjecture | Having thought about this more and discussed it with others, the answer seems to be that there are likely no counterexamples to the Penrose inequality, even if one allows for unphysical violations.
For recent numerical evidence of this, [a paper](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.103.064025) of K... | 4 | https://mathoverflow.net/users/119114 | 388802 | 161,036 |
https://mathoverflow.net/questions/388737 | 1 | This question is a follow-up to [Would Elliott-Halberstam conjecture follow from GRH?](https://mathoverflow.net/questions/153078/would-elliott-halberstam-conjecture-follow-from-grh)
Assuming any $\theta<1-\Lambda$ where $\Lambda$ is the de Bruijn-Newman constant is an exponent of distribution of the primes, which bou... | https://mathoverflow.net/users/13625 | Which gap between primes can be reached under $EH[0.8]$? | As I understand matters, the only way to get an explicit bound for gaps between consecutive primes (not strings of $m$ consecutive primes for some $m\geq 3$) using that particular level of distribution that is optimal relative to the method is to completely rework everything in
* DHJ Polymath, *Variants of the Selber... | 5 | https://mathoverflow.net/users/111215 | 388808 | 161,037 |
https://mathoverflow.net/questions/388815 | 8 | $\DeclareMathOperator\Th{Th}$The random poset is the Fraisse limit of the class of finite posets, just like the random graph is the Fraisse limit of the class of finite graphs? That is, the random poset is the unique (upto isomorphism) countable poset $P$ such that every finite poset embeds into $P$ and every order iso... | https://mathoverflow.net/users/5017 | What is the theory of the random poset? | It is confusing to call this the “random poset”, as it is very different from what is usually called random posets in the literature (in accordance with random graphs): e.g., random posets have height 3 with high probability. Unambiguous descriptions of this structure found in the literature are the countable universal... | 10 | https://mathoverflow.net/users/12705 | 388820 | 161,040 |
https://mathoverflow.net/questions/388825 | 1 | Let $M$ be a connected closed manifold of dimension $n$.
Suppose we have a subset $I\subset I\_n=\{1, \dots, n\}$ such that for any two continuous maps $f, g:M\to M$ if $f^\*=g^\*|\_{\oplus\_{i\in I}H^i(M, \mathbb{Z})}$ then $f^\*=g^\*|\_{\oplus\_{i\in I\_n}H^i(M, \mathbb{Z})}$.
Does it follow that $\oplus\_{i\in I... | https://mathoverflow.net/users/177900 | When given degrees generate the cohomology ring of a manifold? | I think $\mathbb{C}P^3$ is a counterexample. Since $\text{H}(\mathbb{C}P^3;\mathbb Z) = \mathbb Z[\alpha]/\alpha^4$ with $\alpha$ of degree $2$, we can take $I = \{6\}$, then if $f\_{\ast} \alpha^3 = g\_{\ast} \alpha^3$ we must have $f\_{\ast} \alpha = g\_{\ast} \beta$ and thus $f\_{\ast} = g\_{\ast}$ in all degrees. B... | 4 | https://mathoverflow.net/users/14233 | 388827 | 161,042 |
https://mathoverflow.net/questions/388830 | 0 | I am looking for an English version of the following paper:
[Е. Б. Дынкин, Оптимальный выбор момента остановки марковского процесса, Dokl. Akad. Nauk SSSR 150, 238-240 (1963)](http://www.mathnet.ru/links/90266b686b9cdc78a7808142cc8e51da/dan27932.pdf).
| https://mathoverflow.net/users/83682 | English version on Dynkin's 1963 paper on stopping | Dynkin, E. B.
The optimum choice of the instant for stopping a Markov process. (English. Russian original) Zbl 0242.60018
Sov. Math., Dokl. 4, 627-629 (1963); translation from Dokl. Akad. Nauk SSSR 150, 238-240 (1963).
| 4 | https://mathoverflow.net/users/100904 | 388833 | 161,043 |
https://mathoverflow.net/questions/388749 | -1 | It is true to write that
$W^{1,\infty}(]0,\infty[) \hookrightarrow C([0,\infty[)$ et $W^{1,1}(]0,\infty[) \hookrightarrow C([0,\infty[)$ ?
Thanks
| https://mathoverflow.net/users/175318 | Sobolev injections | Let $f$ be a fonction in $L^1(0,+\infty)$
with norm 1 and let us define for $x\ge 0$
$$
\phi(x)=\int\_x^1f(t) dt.
$$
Then $\phi$ is continuous since
$
-\phi(h)+\phi(0)=\int\_0^h f(t) dt,
$
which goes to 0 with $h$ from the Lebesgue Dominated Convergence Theorem.
| 0 | https://mathoverflow.net/users/21907 | 388836 | 161,045 |
https://mathoverflow.net/questions/388792 | 3 | Let $S$ be a finitely generated domain with the field of fractions $F.$ Let X be a smooth,
geometrically connected affine variety over $S.$ Let $A$ be an Azumaya algebra over $X.$
Assume that for all large enough primes $p,$ $A\_p$ splits over $X\_p$-the reduction modulo $p$ of $X.$ Does this assumption imply that $A\_... | https://mathoverflow.net/users/177878 | A local-to global principle for splitting of Azumaya algebras | Let $n$ be the order of $A$ in the Brauer group of $X$, then $A\_{\overline F}$ splits if and only if the corresponding class $[A\_{\overline F}]\in H^2\_{et}(X\_{\overline F}, \mu\_n)$ is 0. If $X$ is smooth and proper and $n$ is invertible on $S$ then by Deligne's theorem in SGA $4\frac{1}{2}$ the pushforward $R^2p\_... | 3 | https://mathoverflow.net/users/42606 | 388837 | 161,046 |
https://mathoverflow.net/questions/388831 | -1 | Let $M$ be a connected closed manifold of dimension $n\geq 2$.
Can it happen that for any $I\subsetneq I\_n=\{1, \dots, n\}$ there are continuous maps $f, g:M\to M$ such that $f^\*=g^\*|\_{\oplus\_{i\in I}H^i(M, \mathbb{Z})}$ yet $f^\*\neq g^\*|\_{\oplus\_{i\in I\_n}H^i(M, \mathbb{Z})}$?
| https://mathoverflow.net/users/177900 | Manifold for which you need to specify the action on cohomology in each degree | I believe this cannot happen: trivially, if $\text{H}^1(M;\mathbb Z) = 0$, you clearly won't find a example of $f, g$ as desired with $I = \{2, \dots, n\}$.
Suppose first that $M$ is orientable. Pick $0 \neq \alpha \in \text{H}^1(M)$ that is not a proper integral multiple of some other cohomology class. Then, by Poin... | 1 | https://mathoverflow.net/users/14233 | 388838 | 161,047 |
https://mathoverflow.net/questions/388767 | 15 | Let $ExDisc\_\kappa$ denote the category of $\kappa$-small extremally disconnected topological spaces (for now fix a strong limit cardinal $\kappa$). There's a functor $ExDisc\_\kappa \to \mathsf{RTop}$ (the $\infty$-category of $\infty$-topoi and geometric morphisms) given by $X\mapsto Sh(X)$ and the pushforward.
It... | https://mathoverflow.net/users/102343 | $\infty$-topoi versus condensed anima | So I think Dustin's comment and the linked comment thread answer most of the questions, if not all. I think it would nonetheless be good to have an account here so let me try to write a coherent answer here - all of it comes from the comment thread [here](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_... | 2 | https://mathoverflow.net/users/102343 | 388850 | 161,051 |
https://mathoverflow.net/questions/388854 | 1 | Consider a function $f:[0,1]\to[0,1]$ which is continuous on a co-meager set $C\subset[0,1]$ and discontinuous on $D=[0,1]\setminus C$. Suppose that $D\cap I$ is uncountable for every open interval $I\subseteq[0,1]$. Can $f$ be of Baire class 1?
| https://mathoverflow.net/users/167834 | Baire class 1 and (uncountably many) discontinuities | YES. In fact “$D\cap I$ *is uncountable for every nonempty open interval* $I \subseteq [0,1]$” can be strengthened to “$D\cap I$ *has Hausdorff dimension 1 for every nonempty open interval* $I\subseteq[0,1]$” (thus “uncountable” can be strengthened to “continuum many”, and much more) simultaneously with a strengthening... | 4 | https://mathoverflow.net/users/15780 | 388862 | 161,054 |
https://mathoverflow.net/questions/388863 | 7 | Let $G$ be a finite group, $g \geq 0, k\geq 1 $ integers, and $(C\_1,...,C\_k)$ a tuple of conjugacy classes of $G$. I am interested in proofs of the following identity
$$
\sum\_{(c\_1,...,c\_k) \in C\_1 \times \cdots \times C\_k} \sum\_{V\_\lambda} d(V\_\lambda)^{1-2g} \chi\_\lambda(c\_1^{-1}\cdots c\_k^{-1}) = |C\_... | https://mathoverflow.net/users/99414 | Proofs of a character identity? | $\newcommand\card[1]{\lvert#1\rvert}$I use the functional identity $\card{C\_1}\chi(c\_1)\chi(c\_2) = \chi(1)\sum\_{c\_1' \in C\_1} \chi(c\_1' c\_2)$, which identifies multiples of irreducible characters $\chi$, and which I'll re-write in the form
$$
\sum\_{c\_1 \in C\_1} \chi(c\_1)\chi(c\_2)
= \chi(1)\sum\_{c\_1 \in C... | 8 | https://mathoverflow.net/users/2383 | 388865 | 161,055 |
https://mathoverflow.net/questions/388847 | 3 | I've recently met with the [Temperley-Lieb algebra](https://en.wikipedia.org/wiki/Temperley%E2%80%93Lieb_algebra) in my work. I'm in no way a specialist, and it's seems like a pretty simple question, but nevertheless. I'm interested in the subalgebra generated by two neighbour generator $ U\_i, \, U\_{i+1}$. This subal... | https://mathoverflow.net/users/125724 | Subalgebras of the Temperley-Lieb algebra | As YCor mentions in the comments, this is (isomorphic to) the algebra $TL\_3$, so I will call its generators $U\_1,U\_2$. I will henceforth write $R := TL\_3$. I do not know what your ground ring is or what your parameter $\delta$ is. If $1-\delta^2$ is invertible, then $R$ contains a special idempotent $p$ named after... | 6 | https://mathoverflow.net/users/78 | 388868 | 161,056 |
https://mathoverflow.net/questions/388870 | 1 | A graph's circuit rank is the minimum number of edges that have to be removed for the graph to become a tree or forest. Is there a term that represents the minimum number of *vertices* that we must remove to get a tree or forest?
I am working on a project that involves reducing cyclic graphs to trees by removing vert... | https://mathoverflow.net/users/177938 | "Circuit rank" but for vertices | Feeding findstat with the first few values yields <https://www.findstat.org/StatisticsDatabase/St001331>, which in particular links to a wikipedia page on the concept.
| 2 | https://mathoverflow.net/users/3032 | 388880 | 161,060 |
https://mathoverflow.net/questions/388878 | 0 | I am tasked with randomly sampling from the following probability density function, which is a modified Erlang Function:
$$f(k,q,\nu)=\frac{(k q)^{k-1}}{[(k-1) !]^{v}} \quad \text { with } \quad q \equiv p \cdot e^{-p}$$
The ordinary Erlang function is given by
$$f(x ; k, \lambda)=\frac{\lambda^{k} x^{k-1} e^{-\l... | https://mathoverflow.net/users/177944 | Random sampling from modified Erlang distribution | $\newcommand\la\lambda$If $X\_1,X\_2,\dots$ are iid random variables (r.v.'s) each with the exponential distribution $E(\la)$ with parameter $\la$, [then](https://en.wikipedia.org/wiki/Erlang_distribution#Related_distributions)
$\sum\_1^n X\_i\sim E(k,\la)$.
In turn, if $U\_1,U\_2,\dots$ are iid r.v.'s each with the ... | 3 | https://mathoverflow.net/users/36721 | 388881 | 161,061 |
https://mathoverflow.net/questions/388891 | 2 | Let there be $m$ points in $\mathbb R^n$. Let $D$ be the longest distance between two of these points and let $d$ be the smallest. What is the smallest possible value of $\frac{D}{d}$ for each value of $n$ and $m$, and which configurations reach it?
It is clear when $n$ is $1$ the solution is reached when the points ... | https://mathoverflow.net/users/24478 | minimum ratio between the shortest and longest distances between $m$ points in $\mathbb R^n$ | This is only a partial answer. Though it hints that the situation is already complicated enough even asymptotically for $m$ large.
In the following we define $\lambda(n, m):=\min D/d$, where minimum is taken over all possible sets of $m$ points in $\mathbb{R}^n$.
Turns out, asymptotically, for $m$ large we have $\l... | 3 | https://mathoverflow.net/users/128741 | 388892 | 161,066 |
https://mathoverflow.net/questions/388874 | 3 | It follows from Whitney extension theorem that for every closed set $ C \subseteq \mathbb{R}^n $ and for every $ k \geq 1 $ there exists a function $ f \in C^k(\mathbb{R}^n) $ such that $ C = \{x : f(x)=0 \} $ and $ D^if(x) =0 $ for every $ x \in C $ and $ i = 1, \ldots , k $.
Is it possible to replace $ k $ with $ \... | https://mathoverflow.net/users/88920 | Smooth functions with zeros of infinite order on a closed set | Yes, for example, take an open cover of the complement of $C$
by countably many open balls of radius $r\_i$ centered at $v\_i$, and use a partition of unity $\{f\_i\}\_{i∈I}$ subordinate to this cover to glue bump functions
$\exp(-(\max(r\_i^2-‖x-v\_i‖^2,0))^{-2})$ into a globally smooth function
$$∑\_{i∈I}f\_i \exp(-(... | 6 | https://mathoverflow.net/users/402 | 388893 | 161,067 |
https://mathoverflow.net/questions/388859 | 4 | Let $C$ be a complete category, let $I$ be a small category, let $F,G:I\to C$ be functors, and let $W,U:C\to\mathrm{Set}$ be also functors, which we view as "weights".
The [weighted limits](https://ncatlab.org/nlab/show/weighted+limit) are then defined, $\lim\_W F$ and $\lim\_U G$.
Suppose now that we have natural tr... | https://mathoverflow.net/users/30366 | Functoriality of weighted limits | Let $X$ be an arbitrary object in $\mathcal{C}$.
I write $\{ W, F \}$ for the limit of $F$ weighted by $W$.
By definition,
$$\mathcal{C} (X, \{ W, F \}) \cong [\mathcal{I}, \textbf{Set}] (W, \mathcal{C} (X, F))$$
naturally in $X$.
If we have $U \Rightarrow W$ (note the direction!) and $F \Rightarrow G$ then functoriali... | 4 | https://mathoverflow.net/users/11640 | 388894 | 161,068 |
https://mathoverflow.net/questions/388897 | 6 | This question is superficially similar to a [previous question](https://mathoverflow.net/q/993/2363). Suppose I am given a permutation group $G \subseteq S\_n$. Is it always possible to find a set $X$ of $n$ points in $\mathbb{R}^n$, such that the isometry group of $X$ (together with its natural action on $X$) is equal... | https://mathoverflow.net/users/2363 | Is every permutation group on $n$ letters the symmetry group of a set of $n$ points in some euclidean space? | No. If $G$ is $2$-transitive $-$ or even transitive on pairs $-$ then
$X$ must be the set of vertices of a regular simplex,
which has isometry group $S\_n$. But there are plenty of
examples of $2$-transitive groups $G$ properly contained in $S\_n$
(such as the $ax+b$ group if $n$ is a prime power and $n>3$).
| 14 | https://mathoverflow.net/users/14830 | 388898 | 161,070 |
https://mathoverflow.net/questions/388846 | 7 | Let $A$ be a symmetric $k$-tensor over a real or complex vector field $W$. We may define its *spectral norm* $|A|$ by
$$|A| = \sup\_{v\in W} \frac{|\langle A,x^{\otimes k}\rangle|}{|x|\_2^k}.$$
(Alternatively, we may define it in a different, standard way and then show that the above inequality holds - that is a theore... | https://mathoverflow.net/users/398 | Strategies for bounding the spectral norm of a tensor? | I will show, with some non-rigorous steps, that a bound of this form that is valid for arbitrary tensors and useful for sparse tensors (fewer than $n^{k/2}$ nonvanishing entries) does not exist.
First note that there is a problem with using the $\ell^2$ norm to define the spectral norm for very sparse tensors $A$, re... | 6 | https://mathoverflow.net/users/18060 | 388899 | 161,071 |
https://mathoverflow.net/questions/388875 | 7 | The following inequality appeared in the analysis of a random approximation algorithm:
$$
\int\_u^{u+1} x^p\ \mathrm{dx} \leq \sqrt{u^p(u+1)^p}\text{, for } -1\leq p\leq 0, u\geq 1.
$$
This resembles the well-known Hermite-Hadamard inequality for convex functions
$$
\int\_a^b f(x)\ \mathrm{dx} \leq (b-a)\frac{f(a)+f(... | https://mathoverflow.net/users/177941 | A geometric mean form of the Hermite-Hadamard inequality, for negative powers | The inequality in question is a particular case (with $v=u+1$) of the inequality
$$\int\_u^v x^p\, dx \le(v-u)u^{p/2}v^{p/2}\tag{1}$$
for $v\ge u>0$, where without loss of generality (wlog) $p\in(-1,0]$. By the homogeneity in $(u,v)$, wlog $u=1$, and then (1) can be rewritten as
$$g(v):=g\_p(v):=v^{p+1}-1-(p+1)(v-1) v^... | 3 | https://mathoverflow.net/users/36721 | 388909 | 161,075 |
https://mathoverflow.net/questions/388734 | 4 | Let $X$ be a random variable in $\mathbb{R}^n$ with distribution $\mu$ and characteristic function $\varphi$ (i.e. $\varphi(t)=\mathbb{E} e^{i\langle t,X\rangle}$). The standard inversion formula asserts that
$$\mu \big(\{a<x<b\}\big) = \frac{1}{(2\pi)^n} \lim\_{T\_1\to\infty}\cdots\lim\_{T\_n\to\infty} \int\limits\_{-... | https://mathoverflow.net/users/24494 | On inverting characteristic functions | A general way to view these formulae is as a Parseval identity. If your measure were of the form $d\mu(x)=f(x)\,dx$, where $f\in L^2$, then you would have, for any set $E$ of finite positive measure,
$$
\mu(E)=\int\_{\mathbb{R}^n}\mathbf{1}\_Ef=\frac{1}{(2\pi)^n}\int\_{\mathbb{R}^n} \hat{\mathbf{1}}\_E\hat{\mathbf{f}}=... | 2 | https://mathoverflow.net/users/56624 | 388918 | 161,078 |
https://mathoverflow.net/questions/388822 | 1 | I have recently started reading Bruns-Herzog's 'Cohen Macaulay rings' and this is problem 1.4.27 in it.
We say that a module $M$ over a Noetherian ring $R$ is perfect if the projective dimension of $M$ is equal to the grade of $M$.
I am required to show that if an ideal $I$ is generated by a regular sequence of len... | https://mathoverflow.net/users/177896 | Quotient of ideal generated by regular sequence is a perfect module | By Theorem 1.1.8, $I^m/I^{m+1}$ is isomorphic as an $R/I$-module to the $m$th graded component of $R/I[X\_1, \ldots, X\_n]$, so $I^m/I^{m+1}$ is a free $R/I$-module of finite rank. Now consider the exact sequence
$$
0 \to I^m/I^{m+1} \to R/I^{m+1} \to R/I^m \to 0
$$
of $R$-modules. Now apply induction on $n$ to show th... | 1 | https://mathoverflow.net/users/14895 | 388921 | 161,079 |
https://mathoverflow.net/questions/388924 | 0 | Let $X$ be an irreducible projective variety over $\mathbb{C}$ admitting a morphism $\pi:X\rightarrow \mathbb{P}^1$ with connected fibers. We may assume that the general fiber of $\pi$ is smooth.
My question is the following: can we always find a Zariski open subset $U\subset\mathbb{P}^1$ such that the group $Aut(X\_... | https://mathoverflow.net/users/nan | Moving general fibers of a fibration | Definitely not; it may be that none of the fibres are isomorphic to each other.
The typical situation is that a given fibre is isomorphic to only finitely many others.
| 2 | https://mathoverflow.net/users/7653 | 388927 | 161,081 |
https://mathoverflow.net/questions/388919 | 16 | Fix an embedding $X\subset Y$ of *smooth* complex affine varieties, or Stein manifolds.
I would guess that in general there is no analytic neighbourhood $X\subset U\subset Y$ with a holomorphic retraction $U\to X$.
But does anyone know a counterexample? (If not, a proof of the existence of $U$?)
| https://mathoverflow.net/users/7653 | Affine (or Stein) tubular neighbourhood theorem | A theorem of Siu says there in fact exists such a retraction, see Corollary 1 in
[Every Stein Subvariety Admits a Stein Neighborhood](https://link.springer.com/article/10.1007/BF01390170), Inventiones math. 38, 89-100 (1976).
| 17 | https://mathoverflow.net/users/49151 | 388928 | 161,082 |
https://mathoverflow.net/questions/388896 | 3 | Recall that $I\Delta\_0$ is the theory in the language of arithmetic that consists of the axioms of $\mathsf{PA}$ with induction restricted to $\Delta\_0$ formulas (i.e., formulas where all quantifiers are bounded).
It is not too difficult to build a model $M$ of $I\Delta\_0$ with a proper cut $N \subset M$ such that... | https://mathoverflow.net/users/83901 | Consistency and consistency strength of certain special cuts in $I\Delta_0$ | $T$ is inconsistent. The argument below is due to Robert Solovay (it is attributed to a letter from Solovay to Nelson in Visser’s [Peano Basso and Peano Corto](https://dspace.library.uu.nl/handle/1874/253993), see Lemma 3.7).
Let $2^x\_n$ denote the iterated exponential function $2^x\_0=x$, $2^x\_{n+1}=2^{2^x\_n}$. I... | 2 | https://mathoverflow.net/users/12705 | 388931 | 161,083 |
https://mathoverflow.net/questions/388947 | 5 | I'm looking for references regarding an unpublished Deligne's manuscript "Une descrption de catégorie tressée (inspiré par Drinfeld)" and the subject it touches, that is described in the post title. If it's not available online maybe someone can orient me to some other references regarding the same connection or to the... | https://mathoverflow.net/users/177992 | Connection between braided tensor categories and local systems on moduli of stable marked genus zero curves | Welcome to MO.
One place where this idea is somewhat explained is Bezrukavnikov-Finkelberg- Schechtman, Factorizable sheaves and quantum groups (<https://arxiv.org/abs/q-alg/9712001>) but maybe this is where you heard about it.
I'd say nowadays this idea might be best understood as a particular case of the identifica... | 5 | https://mathoverflow.net/users/13552 | 388949 | 161,089 |
https://mathoverflow.net/questions/333964 | 17 | Peter Scholze formulated several conjectures about $q$-de Rham complex in the paper
>
> *Canonical $q$-deformations in arithmetic geometry*, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 5, pp 1163–1192, doi:[10.5802/afst.1563](http://dx.doi.org/10.5802/afst.1563), arXiv:[1606.01796](https://arxiv.org/abs/1606.... | https://mathoverflow.net/users/21620 | Conjectures of Peter Scholze about q-de Rham complex: examples | Good question! I wished I understood what this conjecture really means, concretely. First, I should say that in my paper with Bhatt on prismatic cohomology, much of the content of these conjectures has been proved, although the connection is not made very clear, and we only work after $p$-adic completion for some prime... | 14 | https://mathoverflow.net/users/6074 | 388951 | 161,090 |
https://mathoverflow.net/questions/387229 | 3 | Given a type one $C^\*$-algebra $A$, its center $Z$ acts by scalars on each irreducible representation space. Mapping a representation to its central character yields a continuous map from the structure space $\hat A$ to the structure space $\hat Z$.
Is this map open?
| https://mathoverflow.net/users/nan | Is restriction to the center an open map? | This map is not always open. Take for example
$$
A=\Big\{f\in C([0,1],M\_2(\mathbb C)): f(0)=\begin{pmatrix}\lambda &0\\ 0&\mu\end{pmatrix},\, f(1)=\begin{pmatrix}\lambda &0\\ 0&\lambda\end{pmatrix},\, \lambda,\mu\in \mathbb C\Big\}.
$$
(Continuous functions from $[0,1]$ to $M\_2(\mathbb C)$ with endpoint conditions.)
... | 3 | https://mathoverflow.net/users/13381 | 388957 | 161,092 |
https://mathoverflow.net/questions/388706 | 1 | Suppose $G$ is a locally contractible, metric, path-connected topological group. In my particular case, $G$ will be the group of orientation-preserving homeomorphisms of the plane, denoted $Aut(\mathbb{R}^2)$, in the compact-open topology. In this context, it's the same as the topology of compact convergence. $G$ is me... | https://mathoverflow.net/users/110965 | Do Locally Contractible, Path-Connected Groups have Accessible Bases? | For your restricted question the answer is affirmative. It follows from the following theorem of [Yagasaki](https://doi.org/10.1016/S0166-8641(99)00130-3):
**Theorem.** For any subpolyhedron $X$ in a connected 2-manifold $M$, the connected component of the group $H\_X(M)$ of homeomorphisms of $M$ that are identity on... | 2 | https://mathoverflow.net/users/61536 | 388966 | 161,097 |
https://mathoverflow.net/questions/388970 | 4 | Suppose I have $f\in S\_2(\Gamma\_0(N))$ a classical modular newform of level $N$. I want to understand what information (if any) is carried by its Atkin-Lehner eigenvalues for primes $p\mid N$, as defined [here](https://www.lmfdb.org/knowledge/show/cmf.atkin-lehner).
I know the Atkin-Lehner operator commutes with th... | https://mathoverflow.net/users/178005 | Meaning of Atkin-Lehner eigenvalues | Write the Fourier expansion of the newform $f\in S\_2(\Gamma\_0(N))$ as
$\displaystyle\sum\_{n=1}^{\infty}\lambda\_f(n)n^{1/2}e^{2\pi inz}$
so that the Deligne bound is $|\lambda\_f(n)|\leq d(n)$, where $d(n)$ is the divisor function. If for a prime $p|N$ we let $\lambda\_p$ denote the eigenvalue of the Atkin-Lehne... | 4 | https://mathoverflow.net/users/111215 | 388973 | 161,098 |
https://mathoverflow.net/questions/388766 | 2 | For $\mu=0,1,2,3$, let $\gamma^{\mu}$ the set of Dirac gamma matrices. What does it mean to say that $\{\gamma^{\mu}: \hspace{0.1cm} \mu=0,1,2,3\}$ is irreducible?
From my [previous question](https://mathoverflow.net/questions/386335/what-is-the-relationship-between-the-dirac-algebra-and-the-clifford-algebra), I know... | https://mathoverflow.net/users/150264 | Gamma matrices are irreducible | The irreducible representations of the Clifford algebra $C$ and the associated group $\Gamma$ (Dirac group) are worked out in [these notes.](https://arxiv.org/abs/hep-th/9811101)
In *even* $D$ space-time dimensions (in particular, for $D=3+1=4$) the group $\Gamma$ has $2^D+1$ inequivalent irreducible representations.... | 3 | https://mathoverflow.net/users/11260 | 388979 | 161,099 |
https://mathoverflow.net/questions/388989 | 1 | In section 2.4 (Summation of strictly stable random variables), page 54, of the book "chance and stability", Zolotarev, Uchaikin there is the following consideration :
>
> The general relation of
> equivalence for sums S n of independent identically distributed strictly stable
> r.v.s $Y\_i$ is of the form:
>
> ... | https://mathoverflow.net/users/174176 | Greater contribution in a sum of independent random variables | [Heyde](https://www.zbmath.org/?q=an%3A0182.22903) showed the following:
>
> Let $X\_1,X\_2,\dots$ be a sequence of iid random variables such that, for $S\_n:=\sum\_1^n X\_i$,
> $S\_n/B\_n$ converges in distribution to a stable law, whose support is the whole real line and which has index $\alpha\ne1$.
> Then for a... | 2 | https://mathoverflow.net/users/36721 | 388993 | 161,103 |
https://mathoverflow.net/questions/388992 | 1 | Let's consider the problem:
$$
\partial\_t u + \partial\_x(f(u)) = 0, (x,t)\in \mathbb R \times \mathbb R^+.\\
u|\_{t=0}=u\_0.
$$
I have seen three formulations for the entropy condition of this equation. The first two can be found in Evan's PDE book.
1. (Page 142.) Consider the characteristics of the above PDE, assu... | https://mathoverflow.net/users/121404 | Entropy condition for quasi-linear evolution equations | This is an extended comment. Dennis Serre is on this site and can weigh in on this better than I can, if he is up to it.
*"These definitions are certainly not equivalent, and some of them assume more things than others."* I somewhat disagree. They are equivalent in the sense that each condition guarantees uniqueness ... | 2 | https://mathoverflow.net/users/137457 | 388995 | 161,104 |
https://mathoverflow.net/questions/389014 | 1 | Let $A$ be an abelian variety over $\mathbb{F}\_p$.
Then of course for every natural number $i$, we have that $\# A(\mathbb{F}\_{p^i})$ divides $\# A(\mathbb{F}\_{p^{i+1}})$.
But MAGMA says this is false:
Here is my code:
```
P<x> := PolynomialRing((FiniteField(3)));
J := Jacobian(HyperellipticCurve(x^6 - 2 * x^5 ... | https://mathoverflow.net/users/159361 | The orders of $\mathbb{F}_{p^n}$- rational points of a fixed abelian variety and MAGMA computation | $\mathbf F\_{p^i}$ is only a subfield of $\mathbf F\_{p^j}$ when $i |j$ so you only have the divisibility of group orders for $i|j$ not for $j = i+1$.
| 7 | https://mathoverflow.net/users/23872 | 389016 | 161,108 |
https://mathoverflow.net/questions/389025 | 2 | What is the maximum number vectors of Hamming weight at most $w$ in a $d$-dimensional subspace of $\mathbb{F}\_p^n$, where $w,d,p$ are constant and $p$ is odd. (The Hamming weight is the number of nonzero coordinates.) In particular, does an upper bound of the form $d^{Cw}$ hold for some constant $C$ depending only on ... | https://mathoverflow.net/users/73811 | Low-Hamming weight vectors in low-dimensional subspaces of $\mathbb{F}_p^n$ | **Answer.** It is $\sum\_{k=0}^w (p-1)^k{d\choose k}$ (for a coordinate subspace, the equality holds).
**Proof.** Denote by $N(f)$, for $f=(x\_1,\ldots,x\_n)\in \mathbb{F}\_p^n\setminus 0$, the minimal $i$ for which $x\_i\ne 0$. By Gauss elimination we may find a basis $f\_1,\ldots,f\_d$ in our $d$-dimensional space ... | 4 | https://mathoverflow.net/users/4312 | 389026 | 161,112 |
https://mathoverflow.net/questions/389027 | 3 | I am curious about the existence of inner models of $\mathrm{ZFC}$ in conjunction with forcing axioms, under assumptions inconsistent with such theories. For example:
* can we prove under any extension of $\mathrm{ZF} + \mathrm{AD}$, that there is an inner model of $\mathrm{ZFC} + \mathrm{MA}$?
* can we prove under a... | https://mathoverflow.net/users/29231 | Existence of inner models of $\mathrm{ZFC} \ +$ forcing axioms, under incompatible assumptions | All you have to have is an inner model where some set that it thinks is large, is actually countable. To answer the first question, ZF+AD implies that $0^\sharp$ exists, so there exists a generic for a poset forcing MA over $L$. For the second question, the best known upper bound for PFA is a supercompact, and we don’t... | 10 | https://mathoverflow.net/users/11145 | 389032 | 161,115 |
https://mathoverflow.net/questions/388964 | 8 | In [this post](https://mathoverflow.net/questions/348782/aspherical-manifold-with-superperfect-fundamental-group-and-non-trivial-center), it is discussed how a Brieskorn homology sphere $\Sigma(a\_1,a\_2,a\_3)$ with $\displaystyle \frac{1}{a\_1}+ \frac{1}{a\_2}+ \frac{1}{a\_3} < 1$ is an aspherical manifold with a supe... | https://mathoverflow.net/users/97207 | Outer automorphism group of Brieskorn homology sphere? | The answer is that the outer automorphism group is $\mathbb{Z}\_2$. (Compare Ian Agol's answer to [What is the order of the isotopy group of the Brieskorn homology 3-sphere?](https://mathoverflow.net/questions/153953/what-is-the-order-of-the-isotopy-group-of-the-brieskorn-homology-3-sphere)). This is obtained by cobbli... | 7 | https://mathoverflow.net/users/3460 | 389041 | 161,117 |
https://mathoverflow.net/questions/389036 | 0 | In the study of superconformal indices for certain quantum field theories, one encounters the elliptic $\Gamma$ function, which can be expressed as:
$$ \log \Gamma(z;\tau,\sigma)=\sum^{\infty}\_{l=1}\frac{1}{l}\frac{x^l-(x^{-1}pq)^l}{(1-p^l)(1-q^l)}, \quad q=e^{2\pi i\tau},p=e^{2\pi i\sigma},x=e^{2\pi i z}$$
I am inter... | https://mathoverflow.net/users/41130 | Analytic continuation of a periodic function on the real line | Yes, of course. Since $g(\lfloor z \rfloor)$ is analytic on $k < \text{Re}(z) < k+1$, and agrees with $f(z)$ for $z \in (k,k+1)$, it is the unique analytic continuation of $f$ from $(k,k+1)$ to the strip $k < \text{Re}(z) < k+1$ by the identity theorem for analytic functions.
| 3 | https://mathoverflow.net/users/13650 | 389043 | 161,118 |
https://mathoverflow.net/questions/389046 | 4 | Let $S(\mathbb R^n)$ denote the space of all [Schwartz functions](https://en.wikipedia.org/wiki/Schwartz_space) on $\mathbb R^n$ equipped with the topology induced by the usual Schwartz semi-norms. Let $S(\mathbb R^n)^\*$ denote its dual.
**My question.** Suppose that $A, B\in S(\mathbb R^n)^\*$ both satisfy the same... | https://mathoverflow.net/users/129831 | Uniqueness of distributional solutions to the Poisson equation | You can conclude that $A - B$ is a harmonic polynomial: If you take the Fourier transform, you see that $|\xi|^2 (\hat{A} - \hat{B}) = 0$, so $\hat{A} - \hat{B}$ is supported at the origin. (All of this makes sense at the level of tempered distributions.) Therefore, $A - B$ is a polynomial. (This required the classific... | 6 | https://mathoverflow.net/users/137457 | 389049 | 161,119 |
https://mathoverflow.net/questions/389017 | 2 | I am aware of the following statement of the lifting theorem. For $i\in \{1,2\}$ let $B\_i$ be a contraction on a Hilbert space $H\_i$ and let $A\_i$, acting on the Hilbert space $K\_i$, be the minimal unitary dilation of $B\_i$. Let $P\_i$ be the orthogonal projection of $K\_i$ onto $H\_i$. Then an operator $X$ from $... | https://mathoverflow.net/users/178049 | Lifting theorem for n operators | This will not happen in general:
Consider the example: $$C=\left[\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right], D =\left[\begin{matrix}0 & 1/\sqrt2 \\ 0 & 0\end{matrix}\right], \ and \ X = \left[\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right] $$
Then $CX = XC$ and $DX=XD$.
Now the minimal unitary dilation of $C$ ... | 2 | https://mathoverflow.net/users/76593 | 389059 | 161,122 |
https://mathoverflow.net/questions/388972 | 12 | On the [Stanford Encyclopedia of Philosophy article on alternative axiomatic set theories](https://plato.stanford.edu/entries/settheory-alternative/), it is stated without reference that bare $\mathsf{NFU}$ (i.e., $\mathsf{NFU}$ without the axiom of infinity) is weaker in consistency strength than $\mathsf{PA}$. In [th... | https://mathoverflow.net/users/83901 | What is the proof-theoretic ordinal of bare $\mathsf{NFU}$? | As in the question, I will use $\mathsf{NFU}$ for "bare" $\mathsf{NFU}$, i.e., the result of weakening the extensionality axiom in Quine's $\mathsf{NF}$ so as to allow urelements. Let $\mathsf{NFU}^{-\infty}$ be $\mathsf{NFU} \cup \{\lnot \mathsf{Infinity} \}$, where $\mathsf{Infinity}$ is the axiom of infinity.
In 2... | 9 | https://mathoverflow.net/users/9269 | 389065 | 161,124 |
https://mathoverflow.net/questions/389072 | 11 | Let $\bf Cat'$ be the category that has as objects small categories $A, B...$, and as arrows functors $F:A\to B$ that are either covariant or contravariant. The identity on $A\in\bf Cat'$ is the usual identity functor; the composition of a covariant and a contravariant functor is contravariant and the composition of tw... | https://mathoverflow.net/users/166165 | What is the category of covariant and contravariant functors? | $\mathbf{Cat'}$ can be thought of as a semi-direct product. There is an action $G=(\mathbb{Z}/2\mathbb{Z})$ on $\mathbf{Cat}$ given be the oposite category endofunctor and $\mathbf{Cat'}$ is isomorphic to the semi-direct product $G \ltimes \mathbf{Cat}$.
In general the semi-direct product $G \ltimes C$ of a category ... | 17 | https://mathoverflow.net/users/22131 | 389077 | 161,127 |
https://mathoverflow.net/questions/389052 | 8 | I've been reading about the large sieve inequality in Serre's "Lectures on the Mordell-Weil theorem", which states it in the following setting, which I've simplified a bit here:
Suppose $\Lambda \cong \Bbb Z^n$ is a lattice and we have a norm $\| \cdot \|$ on $\Lambda\_{\Bbb R} = \Lambda \otimes \Bbb R \cong \Bbb R^n... | https://mathoverflow.net/users/422 | How does the bound in the large sieve depend on the norm on the lattice? | One approach is to reduce to the parallelepiped case. Let $\Lambda'$ be the dual lattice of $\Lambda$ with the dual norm $$|| y|| = sup\_{x \in \Lambda} \frac{ x \cdot y}{ || x||}.$$ Let $\lambda\_1,\dots, \lambda\_n$ be the successive minima of $\Lambda'$. Let $w\_1,\dots, w\_n$ be the associated shortest vectors, for... | 4 | https://mathoverflow.net/users/18060 | 389080 | 161,130 |
https://mathoverflow.net/questions/389061 | 2 | Let $\mathcal{F}$ be a coherent sheaf on a variety $X$, and assume $\mathcal{F}$ has generic rank $n$. I expect (see e.g. [here](https://www.uni-due.de/%7Ehm0002/Artikel/CohNote_v1a.pdf)) that this actually puts no conditions on its Chern classes $c\_1(\mathcal{F}),c\_2(\mathcal{F}),...$ . Constrast this with the vecto... | https://mathoverflow.net/users/119012 | Making coherent sheaves with nonvanishing higher Chern classes | Take any affine variety $X$ of dimension $k$ which has a vector bundle $E$ of rank $k$ with $c\_k(E)\neq 0$. If $n\geq k$ take $F=E\oplus O\_X^{n-k}$. If $n<k$ take $F$ to be the quotient of $E$ by $k-n$ general sections of $E$.
| 3 | https://mathoverflow.net/users/9502 | 389081 | 161,131 |
https://mathoverflow.net/questions/389082 | 2 | It is well-known that the singular homology of the classifying space of a group $G$ is isomorphic to the group homology of $G$ with coefficients in the trivial $G$-module $\mathbb{Z}$, i.e. $H\_\*(BG,\mathbb{Z})\cong H\_\*(G,\mathbb{Z})$. The question is whether there exists a chain level map (naturally defined) $C\_\*... | https://mathoverflow.net/users/178096 | Group homology and singular homology | Indeed, the answer is given in Eilenberg-MacLane's 1945 paper "Relations between homology and homotopy groups of spaces", as pointed out by Chris Gerig.
| 1 | https://mathoverflow.net/users/178096 | 389087 | 161,133 |
https://mathoverflow.net/questions/389060 | 4 | Every finitely generated group is a colimit of a sequence of finitely presented groups. If every group in the sequence is residually finite what can one say about the colimit?
More precisely:
Is the colimit of a sequence of residually finite groups again residually finite?
| https://mathoverflow.net/users/38190 | Colimits of residually finite groups | (a) There exists a finitely generated (f.g.) group that is an inductive limit of a sequence of epimorphisms of f.g. residually finite groups, and is not residually finite. The first such example is due to B.H. Neumann (1937) and I describe it below.
Write $Y=\{(i,j)\in\mathbf{Z}\times\mathbf{N}: |i|\le j\}$. View it ... | 7 | https://mathoverflow.net/users/14094 | 389098 | 161,137 |
https://mathoverflow.net/questions/389039 | 11 | One of the most incredible results in modern set theory, due to Jensen, is that given any model of $\sf ZFC$, there is a class forcing which adds a real number $r$ and in the extension $V=L[r]$. Moreover, if we assume some basic things like $\sf GCH$, then cardinals are not collapsed, so in particular things like inacc... | https://mathoverflow.net/users/7206 | Coding the universe into a real over better core models | For measurable cardinals, the answer is yes and is due to Sy Friedman. See [Coding Over a Measurable Cardinal](https://www.jstor.org/stable/2274810?origin=crossref&seq=1).
There is some difficulty to extend the result to the context of Woodin cardinals, see [Genericiy and large cardinals](https://www.worldscientific.... | 11 | https://mathoverflow.net/users/11115 | 389102 | 161,139 |
https://mathoverflow.net/questions/388714 | 2 | I have encountered the following problem.
Let $\chi:=\chi\_{B(0,1/2)}$ be characteristics function i.e it take $1$ if $x\in B(0,1/2)$ otherwise $0$.
$\nabla\cdot ((1+\chi\_{B(0,1/2)})\nabla u )=0 $ in $B(0,1)$ with $u(1,\theta)=f(\theta)$.
I wanted to find $(\nabla u\cdot \nu)|\_{B(0,1)} $ from above data where $... | https://mathoverflow.net/users/119011 | How to find $\nabla u\cdot \nu|_{B(0,1)} $ where $u$ is solution of given conductivity equation? | It is a separable problem. First you notice that the constant part of $f$ will play no role, since it would lead to a constant potential, so you may assume
$$
\int\_0^{2\pi} f(\theta)d\theta =0.
$$
If you write
$$
f(\theta) = \sum\_{n=1} a\_n \cos n\theta + b\_n\sin n\theta,
$$
then you notice that the solution $u$ ca... | 1 | https://mathoverflow.net/users/40120 | 389107 | 161,140 |
https://mathoverflow.net/questions/389096 | 0 | I'm a math undergraduate and I'm about to finish my second year at UCL. It's time to consider my graduate school, and I have some questions about it. In the UK, many universities provide math PhD directly after 3-year-undergrad, and they provide MSc for undergraduates as well.
Take Oxford as an example. It says that ... | https://mathoverflow.net/users/178105 | Applying Phd directly after undergraduate | These sorts of questions are really best asked of a trusted advisor, who knows your skill level, life goals, etc. That said, I think Zhen Lin gives a great answer in the comments, that you should do 4 years of university studies before starting a PhD in the UK. If your goal is to be a professor, it's not wise to enter ... | 4 | https://mathoverflow.net/users/11540 | 389118 | 161,143 |
https://mathoverflow.net/questions/389103 | 5 | In Hakim's book "Topos annelés et schémas relatifs", Chap. III, Def. 2.3 states that a ringed topos $(X,A)$ is a locally ringed topos when two equivalent conditions are satisfied:
(i) For each $U \in X$ and each section $s \in A(U)$ one has $U = U\_s \cup U\_{1-s}$.
(ii) For each $U \in X$ and each family $(s\_i)\_... | https://mathoverflow.net/users/2841 | Hakim's definition of a locally ringed topos | Agreed. $(ii)$ is equivalent to "$(i)$ and $( 0 = 1$ in $A(U) ) \Rightarrow U= \emptyset$".
*Remark:* I haven't looked at Hakim's convention, by $U= \emptyset$ I mean "as a sheaf", that is, if we are talking about an object of a site and not an object of a topos it means that the empty sieve is a covering of $U$.
T... | 4 | https://mathoverflow.net/users/22131 | 389124 | 161,144 |
https://mathoverflow.net/questions/389100 | 1 | I am trying to understand if the following claim is true:
>
> Let $P$, $Q$ be probability measures on $\mathcal{X}$. For any $\sigma$-algebra $\mathcal{G}$, with countably many atoms (sets with $\emptyset$ as their only subset in $\mathcal{G}$), let
> $$D(P||Q|\mathcal{G}) = \sum\_{A\in atom(\mathcal{G})} P(A)\log\... | https://mathoverflow.net/users/178108 | KL-divergence and sub-$\sigma$-algebras | $\newcommand{\G}{\mathcal G}\newcommand{\HH}{\mathcal H}\newcommand{\A}{\mathcal A}\newcommand{\X}{\mathcal X}$First here, a *nonempty* member $A$ of a sigma-algebra $\G$ is an atom of $\G$ if the only *proper* subset of $A$ in $\G$ is empty.
Let now $\A(\G)$ denote the set of all atoms of $\G$. Without the condition... | 0 | https://mathoverflow.net/users/36721 | 389129 | 161,147 |
https://mathoverflow.net/questions/389123 | 13 | **Notation:** We say a sequence of real numbers diverges if it does not converge to a finite limit. We say a sequence $f\_n$ of real valued functions on $[0, 1]
$ are equibounded if $\sup\_{n \in \mathbb N}\sup\_{x \in [0, 1]} |f\_n (x)| < \infty$.
**Some motivation:**
The Arzela Ascoli theorem for $C[0, 1]$ says ... | https://mathoverflow.net/users/173490 | Anti Arzela-Ascoli | Under the stated conditions, there always exists a subsequence that Cesaro converges almost everywhere. This was a question of Steinhaus, solved by Revesz [1].
More generally, it suffices that the sequence $f\_n$ be uniformly bounded in $L^1$; This is a striking Theorem of Komlos [2] which in particular implies the Kol... | 23 | https://mathoverflow.net/users/7691 | 389135 | 161,149 |
https://mathoverflow.net/questions/388860 | 3 | In the 1-category of 2-categories, with objects being categories enriched over Cat, and morphisms being 2-functors, is there an explicit way to describe a pullback of two functors $G:E\to D$ and $F:C\to D$? In particular I am interested in the case of $(2,1)$-categories, i.e. categories enriched over groupoids, but the... | https://mathoverflow.net/users/137822 | Explicit description of a pullback of $(2,1)$-categories | For a strict pullback of strict 2-functors, the 0-, 1-, and 2-cells of the pullback are precisely the pullbacks of the underlying sets of cells.
Strict pullbacks of pseudofunctors do not generally exist. For instance, if $C$ and $E$ are the ordered set with 3 elements $x\xrightarrow{f} y\xrightarrow{g} z$, and $D$ is... | 5 | https://mathoverflow.net/users/49 | 389141 | 161,151 |
https://mathoverflow.net/questions/389117 | 20 | Consider the following diagram of algebraic varieties:
$$\mathbb{A}^0 \to \mathbb{A}^1 \rightrightarrows \mathbb{A}^2$$
Here the first arrow is the inclusion of the origin into the line, and the next two are the inclusion of the line into the plane as the X and the Y axes.
>
> Does this diagram have a colimit i... | https://mathoverflow.net/users/4707 | Can I glue the X axis to the Y axis? | We can rewrite the coequalizer as the pushout of the diagram
$$
\begin{array}{ccc}
X & \to & \mathbb A^2 \\
\downarrow & & \\
\mathbb A^1 & & \\
\end{array}
$$
where $X$ is the union of the $x$- and $y$-axis, and the vertical map quotients by the involution swapping the two components.
The category of affine schem... | 29 | https://mathoverflow.net/users/1310 | 389154 | 161,153 |
https://mathoverflow.net/questions/389152 | 4 | I'm now interested in the modular representation of symmetric groups.
It is well-known that for a fixed prime $p$, there is a bijection between the irreducible representations of $S\_{n}$ over a field of characteristic $p$ and $p$-regular partitions of $n$.
And it seems that the sign representation of $S\_{n}$ exis... | https://mathoverflow.net/users/123226 | What is the $p$-regular partition corresponding to the sign representation of $S_{n}$ over a field of characteristic $p$? | The answer is the $p$-regularisation of $(1^n)$: this is $(1^n)$ if $n < p$ and otherwise the partition
$$ (r^a, (r-1)^{p-1-a}) $$
with $p-1$ parts, where $r$ and $a \in \{0,1,\ldots, p-1\}$ are defined uniquely by
$$(r-1)(p-1) + a = n, $$
taking $a = 0$ if $p-1$ divides $n$.
For more on this see G. D. James, *On the... | 10 | https://mathoverflow.net/users/7709 | 389160 | 161,156 |
https://mathoverflow.net/questions/389069 | 3 | I am trying to calculate the average degree of a complex network, which requires me to solve for the following integral:
$$\int \mathrm{d} x \frac{\exp{\left[-x -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right]}}{1 + a\exp{(-x)}},$$
where $\mu$, $\sigma$, and $a$ are known constants. Any advice on how to tackl... | https://mathoverflow.net/users/163859 | Integrals involving fractions of exponentials | We can obtain power series in $a$ by expanding in geometric series. Since the specific treatment depends on the integration region, let's introduce lower and upper limits, $y\_0 <y$,
$$
I = \int\_{y\_0 }^{y} dx\, \frac{\exp \left[ -x -\frac{1}{2} \left( \frac{x-\mu }{\sigma } \right)^{2} \right] }{1+a\exp (-x)}
$$
For ... | 3 | https://mathoverflow.net/users/134299 | 389161 | 161,157 |
https://mathoverflow.net/questions/389138 | 5 | Let $X$ be a compact Riemann surface of genus $g\geq 2$. By the Riemann-Roch theorem, $X$ is a branched cover of the sphere, branched over finitely many branched values. What is the smallest number of such values, for general $X$ of genus $g$?
Since moduli space has complex dimension $3g-3$, and branched covers branc... | https://mathoverflow.net/users/3651 | Branched covers of the sphere branched over few points | Let me post my comment as an answer. Take a Weierstrass point on $X$, that is, a point $P$ for which there exists a meromorphic function $f$ with a pole of order $k\leq g$ at $P$ (there always exists such a point). Then apply the argument in the post: by Riemann-Hurwitz the number of critical points of $f$, counted wit... | 7 | https://mathoverflow.net/users/40297 | 389163 | 161,159 |
https://mathoverflow.net/questions/389151 | 7 | There is an elementary theory of subsets of $\Bbb{R}^n$ of measure zero, namely one defines the volume of a cube in the obvious way and one says that a subset $A$ has measure zero if given any $\epsilon>0$ there exists a countable number of cubes that cover $A$ and such that the sum of the volumes of the cubes is $\leq... | https://mathoverflow.net/users/2985 | Elementary proof that an open subset of $\Bbb{R}^n$ does not have measure zero? | The first part of my argument is borrowed from Iosif Pinelis, and the second part is different.
Every open set contains a closed cube, so it suffices to show the closed cube does not have measure $0$. In fact, we will show that for a cube of side length $r$, the total volume of open cubes covering it is at least $r^n... | 16 | https://mathoverflow.net/users/18060 | 389169 | 161,162 |
https://mathoverflow.net/questions/389159 | 3 | Consider the tensor product of bounded operators. Does this tensor product satisfy the universal property of the tensor product, i.e., for any bilinear map $F: B(\mathcal H\_1)\times B(\mathcal H\_2)\to B(\mathcal H)$, there is a unique map $\hat F: B(\mathcal H\_1)\otimes B(\mathcal H\_2)\to B(\mathcal H)$ such that $... | https://mathoverflow.net/users/101775 | Universal property of tensor products of bounded operators | This is false in general even for very "well-behaved" classes of maps. For instance, the multiplication map on $B(H)$, which is bounded bilinear as a map $B(H)\times B(H) \to B(H)$, does not extend continuously to $B(H\otimes H)\to B(H)$ if $H$ is infinite-dimensional.
In fact, the problem is worse thane one might th... | 5 | https://mathoverflow.net/users/763 | 389175 | 161,167 |
https://mathoverflow.net/questions/388657 | 3 | Is there a function $f$ with the following properties
1. $f$ meromorphic at the upper half plane $\mathfrak h$,
2. $f$ is of weight $k$ under a congruence subgroup of $\operatorname{SL}\_2(\mathbb Z)$,
3. $f$ has an essential singularity at $\infty$.
Modular forms or functions are defined to have good behaviour at ... | https://mathoverflow.net/users/122104 | Modular form not meromorphic at $\infty$ | [expanding on my comment to convert it to an answer]
An example is $e^j \varphi$ where $j$
is the $j$-invariant and $\varphi$ is any nonzero form of weight $k$.
In general a holomorphic (or meromorphic) function on the upper half-plane
that is invariant under ${\rm SL}\_2({\bf Z})$ is the same as an entire
holomorp... | 8 | https://mathoverflow.net/users/14830 | 389181 | 161,169 |
https://mathoverflow.net/questions/389172 | 12 | Given $\lambda$ an [integer partition](https://en.wikipedia.org/wiki/Partition_(number_theory)) of $n$, let $h\_{ij}(\lambda)$ denote the [hook length](https://en.wikipedia.org/wiki/Hook_length_formula#Definitions_and_statement) of cell $(i,j)$ in the [Young diagram](https://en.wikipedia.org/wiki/Young_tableau) of $\la... | https://mathoverflow.net/users/66131 | Alternating sum of hook lengths: Part I | The hook length $h\_{ij}(\lambda)$ counts the number of boxes directly below or directly to the right of box $(i,j)$. (I picture the Young diagram of $\lambda$ as having the corner $(0,0)$ located in the upper left.) With this in mind, the alternating sum $\sum\_{(i,j)\in \lambda}(-1)^{i+j}h\_{ij}(\lambda)$ can be expa... | 21 | https://mathoverflow.net/users/2384 | 389183 | 161,170 |
https://mathoverflow.net/questions/389168 | 1 | If $q = [q\_0;q\_1 \dots]$, say $q\_i$ is the $i$-th partial quotient of $q$. My question is the following:
>
> Can one construct an explicit example of irrational $r,s > 0$ such that
>
>
> 1. $\{ 1,r,s\}$ is $\mathbb{Q}$-linearly independent,
> 2. $r,s$ have bounded, aperiodic partial quotients, and
> 3. $r+s$ h... | https://mathoverflow.net/users/128941 | Bounded, aperiodic irrationals with bounded, aperiodic sum | Theorem 3.1 page 972 in [1] ensures that every number in a certain interval I is a sum of two real numbers with partial quotients bounded by 4. If you consider numbers in I with partial quotients bounded by 5 (Call this set $I \cap F(5)$), they could not all be obtained via periodic summands or linearly dependent summa... | 5 | https://mathoverflow.net/users/7691 | 389186 | 161,172 |
https://mathoverflow.net/questions/387354 | 3 | I posted this question on MSE but got no answer even after putting a bounty on it, so I figured I can try to ask here.
Let $(A, \Delta: A \to A \otimes A)$ be bialgebra (unital and counital) such that the map
$$T: A \otimes A \to A \otimes A: a \otimes b \mapsto \Delta(a)(1 \otimes b)$$
is surjective.
We write $\De... | https://mathoverflow.net/users/nan | Show that a certain element is a linear combination of tensors | I think your argument shows that there exist $a,b,c$ with $$\Delta(a) = \sum\_i a^i\_{(1)} \otimes a^i\_{(2)}$$
$$\Delta(b) = \sum\_j b^j\_{(1)} \otimes b^j\_{(2)}$$
$$x \otimes y \otimes 1 = \sum\_i \sum\_j a\_{(1)}^i \otimes b\_{(1)}^i \otimes a\_{(2) }^ib\_{(2)}^jc $$
I don't really understand your notation but ... | 0 | https://mathoverflow.net/users/18060 | 389189 | 161,173 |
https://mathoverflow.net/questions/354191 | 2 | Consider the following two dimensional Laplace equation on the unit disk $D$ with homogeneous Robin boundary condition:
$$\Delta u = 0, ~~\frac{\partial u}{\partial n} = b(x) u(x)~~ \forall x \in \partial D.$$
Here $n$ denotes the outer normal direction to the bounary $\partial D$. Assume that $b(x)$ is **piecewisely ... | https://mathoverflow.net/users/114951 | Laplace equation on the disk with Robin boundary condition | The normal trick is to set it up as an eigenvalue problem, namely to look instead at
$$\Delta u = 0, ~~\frac{\partial u}{\partial n} = \lambda b(x) u(x)~~ \forall x \in \partial D. (1)$$
You first establish that this corresponds to an eigenvalue problem on $L^2(\partial \Omega)$ for a compact operator, thanks to the co... | 2 | https://mathoverflow.net/users/40120 | 389209 | 161,182 |
https://mathoverflow.net/questions/389207 | 2 | **Set-up:** Consider the action of $\mathbb{C}^\*$ on $\mathbb{C}^4$ defined as follows: $(t,(x,y,z,w))=(tx,ty,t^{-1}z,t^{-1}w)$. I know that the affine GIT quotient is equal to $$\phi: \mathbb{C}^4 \to \mathbb{C}^4//\mathbb{C}^\*=Z(XW-YZ),$$ $$(p\_1,p\_2,p\_3,p\_4)\to (p\_1p\_3,p\_1p\_4,p\_2p\_3,p\_2p\_4).$$
**Quest... | https://mathoverflow.net/users/177964 | Geometric quotients obtained by throwing away limits | Throwing away a subset of codimension two you will not give any more invariants. In particular, points of the form $(p\_1,p\_2,0,0)$ will not be separated.
Nevertheless, the geometric quotient exists. It is just not an affine variety anymore. There are two basically equivalent ways to see this:
1. The ad-hoc way: S... | 2 | https://mathoverflow.net/users/89948 | 389215 | 161,184 |
https://mathoverflow.net/questions/382178 | 24 | I come today to mathoverflow to showcase some genuine confusion about the motivic world. I want to ask some questions before actually starting to study the subject, to build some sense of direction.
I take Grothedieck's orginal idea of motives to be that of an abelian category through which every good cohomology shou... | https://mathoverflow.net/users/139854 | What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences? | There is no need a priori to define these categories of motives starting from correspondences. The stable homotopy theory of schemes $SH$ may be characterized by a universal property saying that it is the universal setting in which one may define the six operations; see [Drew and Gallauer](https://arxiv.org/abs/2009.13... | 15 | https://mathoverflow.net/users/1017 | 389222 | 161,186 |
https://mathoverflow.net/questions/389192 | 2 | Let $G$ be a non-uniform lattice Fuchsian group and let $P$ be a Dirichlet region for $G$. In particular $G$ has parabolic elements, $P$ is not compact and has finite area. We are in the unit disc. Is the following statement true? Is the proof correct? Errors? Counterexamples? Thanks.
STATEMENT: If $G$ is a free grou... | https://mathoverflow.net/users/178149 | Dirichlet region of a free group | The statement is false. Here is a counter-example. Let $T$ be an ideal triangle (say in the unit disk model). Let $S$ be the surface obtained by *doubling* $T$ across it’s boundary: that is, take two copies and glue by the identity on the boundary. Let $x$ be the centre of $T$. The Dirichlet domain based at $x$ has six... | 2 | https://mathoverflow.net/users/1650 | 389228 | 161,187 |
https://mathoverflow.net/questions/389223 | 4 |
>
> Let $P(x) \in \mathbb{Z}[x]$ be a monic polynomial of degree $n$, with no integer roots, such that $\upsilon\_2(P(m))$ can assume $n$ different positive values, where $m \in \mathbb{Z}$ and $\upsilon\_2(P(m))$ is the $2$-adic order of $P(m)$. Hence $\upsilon\_2(P(m)) < \dfrac{3n^2}{2}$ for all integer $m$.
>
>
... | https://mathoverflow.net/users/70464 | $2$-adic order bound for $P(x)$ | Let $P$ be a monic polynomial of degree $n$ over $\mathbb Z$. Let $\alpha\_1,\dots, \alpha\_n$ be its roots, which we view as elements of $\overline{\mathbb Q\_p}$. Then all $p$-adic valuations of $P$ are at most
$$ n \left( k -1 + \frac{p}{(p-1)^2}\right)$$
and this is close to sharp.
---
Proof: Let $m$ be a... | 5 | https://mathoverflow.net/users/18060 | 389230 | 161,188 |
https://mathoverflow.net/questions/389225 | 6 | I would like to ask a question which may look strange at the first sight nevertheless I find it interesting. Let $H$ be a separable Hilbert space: for any *separable* $C^\*$-algebra $A$ one can embed $A$ into $B(H)$. Very often one can do even better and many nonseparable $C^\*$-algebras still embed into $B(H)$, for ex... | https://mathoverflow.net/users/24078 | Embedding of $C(X)$ into $B(H)$ where $H$ is separable | No, not necessarily. If $C(X)$ embeds into $B(H)$ then $X$ must be [ccc](https://en.wikipedia.org/wiki/Countable_chain_condition), but there exist compact Hausdorff spaces of cardinality $\mathfrak{c}$ which are not ccc. (One example is to take $\mathfrak{c}$ with its discrete topology and form the one-point compactifi... | 7 | https://mathoverflow.net/users/4832 | 389231 | 161,189 |
https://mathoverflow.net/questions/389234 | 3 | Let $F/ \mathbb{Q}$ be a biquadratic field of Galois group $C\_2 \times C\_2$. Then I know that the Dedekind Zeta function of $F$ can be factored into $L$-functions as; $$\zeta\_F(s) = \zeta(s) L(s, \chi\_1) L(s, \chi\_2) L(s, \chi\_3)$$ where $\chi\_i$'s are the primitive real characters associated with the quadratic ... | https://mathoverflow.net/users/167999 | Dedekind Zeta functions of Biquadratic fields | You could view a biquadratic field $\mathbf Q(\sqrt{a},\sqrt{b})$ as the top of a tower of two quadratic extensions: it is quadratic over $\mathbf Q(\sqrt{b})$, which in turn is quadratic over $\mathbf Q$. Then show for a quadratic extension of number fields $E'/E$ that
$$
\zeta\_{E'}(s) = \zeta\_{E}(s)L(s,\chi),
$$
wh... | 5 | https://mathoverflow.net/users/3272 | 389243 | 161,193 |
https://mathoverflow.net/questions/389221 | 8 | Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x\_2,\dots,x\_d]]/(f)$, where $f\in(x,y,x\_2,\dots,x\_d)^2$, $f\neq0$. By results of Buchweitz-Greuel-Schreyer and Knörrer, $R$ has finite Cohen-Macaulay representation type if and only if $R\cong k[[x,y,x\_2,\dot... | https://mathoverflow.net/users/106706 | Class group of hypersurfaces of finite representation type | When $d\geq 3$, these are isolated hypersurface singularities of dimension at least $4$, so are UFD by the Grothendieck's local Lefschetz Theorem.
When $d=2$ and the field has characteristic $0$, the class group is $\mathbb Z^{r-1}$ where $r$ is the number of branches of $g$. See 2.2.6 of Kollár's paper ["Flip, flops... | 3 | https://mathoverflow.net/users/2083 | 389244 | 161,194 |
https://mathoverflow.net/questions/389245 | 5 | Let $Gr(1,n)$ be the Grassmannian of lines in $\mathbb{P}^n$, and $f:\mathbb{P}^1\rightarrow Gr(1,n)$ a morphism of degree two. The pull-back $f^{\*}S$ of the tautological bundle $S$ on $Gr(1,n)$ splits as $f^{\*}S = \mathcal{O}\_{\mathbb{P}^1}(a)\oplus \mathcal{O}\_{\mathbb{P}^1}(b)$. If $f$ is general in the moduli s... | https://mathoverflow.net/users/14514 | Jumping conics in Grassmannians | These conics are exactly conics contained in some
$$
\mathbb{P}^{n-1} \subset Gr(1,n),
$$
that parameterizes all lines in $\mathbb{P}^n$ passing through a fixed point.
| 8 | https://mathoverflow.net/users/4428 | 389252 | 161,195 |
https://mathoverflow.net/questions/389266 | 1 | I have a known distribution $f(x)$ (in fact, I can safely assume that $f(x)$ is the Maxwell-Boltzmann distribution, i.e. $f(x)\propto x^2 \exp(-x^2)$). I take $N$ samples from the distribution, but am only interested in a subset of these samples: samples that are larger than $x\_0$ (a fixed and known value). I'll denot... | https://mathoverflow.net/users/178203 | Distribution of a two-part sampling process | For each $x\_i$ of the $N$ samples you perform simultaneously two checks: Is $x\_i$ larger than $x\_0$ (with probability $p\_{x\_0}$ and then your Bernoulli trial with probability $p$. Assuming independence of this trial and the check $x\_i > x\_0$ the probability is $p \cdot p\_{x\_0}$ that you have a successful trial... | 1 | https://mathoverflow.net/users/100904 | 389268 | 161,200 |
https://mathoverflow.net/questions/389270 | 1 | I've tried asking this question on Mathematics site, but I only got an upvote and no answer. I've searched online, tried to find something about this topic, but I haven't found much (and the things I have found I don't understand).
Let $X$ be a compact metric space and $f:X\to X
$ homeomorphism. Let $V:X\to \mathbb{... | https://mathoverflow.net/users/146092 | Gradient-like dynamical systems | Let $A = V^{-1}([a, \infty))$. Both $A$ and $f^{-1}(A)$ are compact.
Because $V$ is a strict Lyapunov function and $a \not\in V(Fix(f))$, it follows $\min\_{f^{-1}(A)} V > a$.
So you can set $\delta = \min\_{f^{-1}(A)} V - a$, and you have $V(x) < a + \delta$ implies $V(f(x)) < a$.
| 2 | https://mathoverflow.net/users/1227 | 389276 | 161,204 |
https://mathoverflow.net/questions/389284 | 3 | Let $k$ be a number field and denote by $\Omega \_k$ the set of places of $k$, by $\Omega \_\infty$ the set of archimedean places of $k$, and by $S$ a nonempty finite subset of $\Omega \_k$ such that $\Omega \_\infty \subset S$.
Let $X$ be a variety over $k$. We define an *integral model* $\mathcal{X}$ of $X$ to be a... | https://mathoverflow.net/users/172132 | Integral models and adelic points | for your first question, the idea is basically what MikhailBorovoi said but of course you have to be careful to get a model outside $S$: add all the prime divisors of dominators of the equations defining $X$ to $S$ to obtain a (faithfully flat)model for $X$ outside $S$. In general you want to glue several affine variet... | 1 | https://mathoverflow.net/users/65846 | 389289 | 161,209 |
https://mathoverflow.net/questions/389269 | 10 | Let $G$ be a nonabelian finite simple group all of whose Sylow subgroups of odd order are cyclic.
If we further assume that its Sylow $2$-subgroup is dihedral, then due to Suzuki, we know that $G\cong \operatorname{PSL}(2,p)$ for a prime $p>3$.
Without any further assumption, what is the list of finite nonabelian s... | https://mathoverflow.net/users/47344 | Finite simple groups all of whose Sylow subgroups of odd order are cyclic | Such simple groups are a subset of the finite simple "thin" groups, and the latter have been classified (by Michael Aschbacher in 1976 and 1978). A $2$-local subgroup of a finite group is the normalizer of some non-trivial $2$-subgroup. A finite simple group $G$ is said to be "thin" if all its $2$-local subgroups have ... | 17 | https://mathoverflow.net/users/14450 | 389293 | 161,210 |
https://mathoverflow.net/questions/389298 | -4 | Could you please give a detailed definition (or construction)of tautological vector bundle of Grassmannian over arbitrary base scheme? Thank you in advance!
| https://mathoverflow.net/users/173314 | Definition of tautological vector bundle | Ravi Vakil, [The Rising Sea](http://math.stanford.edu/%7Evakil/216blog/), 16.7 *The Grassmannian as a moduli space*, p. 442.
| 1 | https://mathoverflow.net/users/13268 | 389299 | 161,212 |
https://mathoverflow.net/questions/389249 | 2 | Let $X$ be a Banach space; $X^\*$ be its dual; and $g:X^\*\to\mathbb R\cup\{\infty\}$ be a proper, convex, weak${}^\*$-lower semicontinuous function with weak${}^\*$-compact effective domain.
Question: Is there a known characterization of when $g$ is/isn't the convex conjugate of some (proper, convex, lower semiconti... | https://mathoverflow.net/users/145424 | When is a function a convex conjugate? | If I recall correctly, every proper, convex, weak\*-lower semi-continuous function is a conjugate. In fact, it is the conjugate of its *pre-conjugate*
$$
f(x) = \sup\_{x^\*\in X^\*} \langle x^\*,x\rangle - g(x^\*).
$$
| 3 | https://mathoverflow.net/users/9652 | 389309 | 161,213 |
https://mathoverflow.net/questions/383604 | 1 | How many non-isomorphic associative algebras of dimension 2 over the field F\_{p^k} are there?
As much as I have searched, I have not found any results that answer my question; not even for k = 1,2.
| https://mathoverflow.net/users/168671 | How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there? | The answer is: 8 isomorphism classes. The classification up to absolute isomorphism yields: 7.
To show this in a greater and more natural generality, let in general $K$ be a field. I claim that there are 7 isomorphism classes of 2-dimensional $K$-algebras that are not (commutative) fields. Denote by $0\_i$ the $i$-di... | 2 | https://mathoverflow.net/users/14094 | 389320 | 161,218 |
https://mathoverflow.net/questions/389311 | -2 | Let $x\_i \in\mathbb{R}^d$ and $a\_i\in [0,1]$ for $i = 1,\dots,k$. How to compute the following derivative?
$$
\frac{d}{da\_j}\log \det\left(\sum\_{i = 1}^k a\_ix\_ix\_i^\top\right).
$$
| https://mathoverflow.net/users/156139 | Derivative of log determinant | \begin{align\*}
&\frac{d}{da\_j}\log \det\left(\sum\_{i = 1}^k a\_ix\_ix\_i^\top\right)
\\
&= \frac1{\det\left(\sum\_{i = 1}^k a\_ix\_ix\_i^\top\right)}\det\left(\sum\_{i = 1}^k a\_ix\_ix\_i^\top\right)\operatorname{Tr}\left(\left(\sum\_{i = 1}^k a\_ix\_ix\_i^\top\right)^{-1}x\_jx\_j^\top\right)
\\&=
\operatorname{Tr}... | 2 | https://mathoverflow.net/users/26935 | 389325 | 161,219 |
https://mathoverflow.net/questions/389150 | 1 | Let $X$ be a compact metric set and $\mathcal{C}(X)$ be the set on continuous real functions over $X$ endowed with the supremum norm $\|\cdot\|\_{\infty}$. Let $\Omega\_1$ and $\Omega\_2$ be two Markov generators on $\mathcal{C}(X)$ (using the definition of Liggett) with domains $\mathcal{D}\_1$ and $\mathcal{D}\_2$ su... | https://mathoverflow.net/users/159940 | Is a linear combination of Markov generator a Markov generator? | The answer to your question is positive under certain conditions. For example, by the "product formula" of H. Trotter, (<https://www.ams.org/journals/proc/1959-010-04/S0002-9939-1959-0108732-6/S0002-9939-1959-0108732-6.pdf>), if $\mathcal D\_1\subset\mathcal D\_2$, then the set of $\mu\ge 0$ such that $\Omega\_1+\mu\Om... | 2 | https://mathoverflow.net/users/42851 | 389333 | 161,221 |
https://mathoverflow.net/questions/389302 | 0 | Let $A$ be an $n\times n$ symmetric positive definite matrix with eigenvalues and eigenvectors $\lambda\_1\ge\lambda\_2\ge\cdots\ge\lambda\_n>0$ and $v\_1,v\_2,\cdots,v\_n$ respectively.
We know that the largest eigenvalue of $A$ can be obtained by "trace maximization" or "Rayleigh quotient maximization". I also note... | https://mathoverflow.net/users/157997 | Is there a specific name for this optimization problem? | It's the symmetric version of the [low-rank approximation](https://en.wikipedia.org/wiki/Low-rank_approximation) problem.
| 3 | https://mathoverflow.net/users/1898 | 389334 | 161,222 |
https://mathoverflow.net/questions/389282 | 4 | I'm interested in the regularity of solutions to Monge-Ampère equations in a bounded convex domain $\Omega\subset\mathbb{R}^n$. It seems that the following statement can be deduced from known results:
>
> **Proposition.** Let $u\in C^0(\overline{\Omega})$ be a convex function satisfying the following conditions:
> ... | https://mathoverflow.net/users/17294 | Regularity of Aleksandrov solution to Monge-Ampère equation with infinite slope at boundary | The proposition is true. One can argue as follows: for $x \in \Omega$, let $L$ be a supporting linear function to $u$ at $x$. Then $L < u$ on $\partial \Omega$ by (b). For $h > 0$ small we conclude that $S\_h := \{u < L + h\}$ is compactly contained in $\Omega$, that $u|\_{\partial S\_h}$ is affine, and that $\det D^2u... | 4 | https://mathoverflow.net/users/16659 | 389336 | 161,224 |
https://mathoverflow.net/questions/389275 | 0 | I am reading GTM 175 An introduction to knot theory by Lickorish and have some questions on the proof of Lemma 4.5 given.
For (a), it says "Suppose that $C$ is amongst the $n$ components of $F\cap S\_+$ that do not bound disc components of $F\cap B\_+$."; I was wondering how could $C$, as a component of $F\cap S\_+$,... | https://mathoverflow.net/users/174967 | Questions on the proof Lemma 4.5 GTM 175, Lickorish | For (a), $C$ does bound a disk in $B^+$ which is later called $\Delta'$, but by assumption $C$ does not bound a disk in $F\cap B^+$. A priori, the components of $F\cap B^+$ are subsurfaces of the sphere $F$; they could be disks, annuli, pairs of pants, etc.
For (b), the surface $F$ could intersect a bubble in any num... | 2 | https://mathoverflow.net/users/126206 | 389337 | 161,225 |
https://mathoverflow.net/questions/389343 | 6 | For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ associate a sequence $\text{seq}(f))$ defined recursively by
* $\text{seq}(f)\_1 = f(1)$, and
* $\text{seq}(f)\_{k+1} = f(\text{seq}(f)\_k)$ for all $k\in\mathbb{N}$.
Eventuall... | https://mathoverflow.net/users/8628 | Expectation of period length of functions $f:\{1,\ldots,n\}\to \{1,\ldots,n\}$ | The standard references here include *An Introduction to the Analysis of Algorithms* by Sedgewick and Flajolet.
The first significant reference is probably "Probability distributions related to random mappings", Bernard Harris, *Ann. Math. Statist.* 31 (1960), 1045-1062, linked to in another answer. A summary is also... | 9 | https://mathoverflow.net/users/17773 | 389347 | 161,228 |
https://mathoverflow.net/questions/389345 | 5 | Let $R = k[x\_1 , \dots , x\_n]$ be a polynomial ring over a field and $I$ a monomial ideal in $R$. Then, is it true that the Koszul homology of $R/I$ is always generated by elements of the form
$$r e\_{i\_1} \wedge \cdots \wedge e\_{i\_k} \quad \textrm{where} \ x\_{i\_\ell} r \in I \ \textrm{for all} \ 1 \leq \ell \le... | https://mathoverflow.net/users/73780 | Is Koszul homology of a monomial ideal always generated by the "obvious" things? | This holds for $n\leq 3$ but may fail for $n=4$ and higher. See Proposition 2.6 and Example 2.9 in the paper ["On monomial Golod ideals"](https://arxiv.org/abs/1902.00806) (but probably known to experts before).
| 7 | https://mathoverflow.net/users/2083 | 389348 | 161,229 |
https://mathoverflow.net/questions/388768 | 3 | Consider a finite field $\mathbb{F}\_q$ such that $q \equiv 1 \pmod3$ (i.e., $\omega \mathrel{:=} \sqrt[3]{1} \in \mathbb{F}\_q$ for $\omega \neq 1$) and an element $b \in (\mathbb{F}\_q^\*)^2 \setminus (\mathbb{F}\_q^\*)^3$ (a quadratic residue and cubic non-residue in $\mathbb{F}\_q$). Further, I am interested in the... | https://mathoverflow.net/users/69852 | Mordell–Weil rank of some elliptic $K3$ surface | Let $E\_0$ be the elliptic curve $y^2 = x^3 + 1$,
and choose $\beta$ in $k = {\bf F}\_q$ so that $b = \beta^2$.
Then $W = W\_b$ has $\rho=18$ unless the elliptic curve
$$
E\_\beta : Y^2 = X^3 + \beta \, (3X + (\beta+1)^2)^2
$$
is isogenous to $E\_0$, in which case $\rho(W)$ is
$20$ ("singular") or $22$ ("supersingular"... | 4 | https://mathoverflow.net/users/14830 | 389349 | 161,230 |
https://mathoverflow.net/questions/389327 | 7 | Let $G$ be some outdegree-regular directed graph with $n$ vertices and let $H$ be the Laplacian of $G$, so that the rows of $H$ correspond to chip-firing moves. I’m interested in linear functions $f$ from $\mathbb{Z}^n$ to $\mathbb{Z}/k\mathbb{Z}$ with the property that performing a chip-firing move on a vector in $\ma... | https://mathoverflow.net/users/3621 | Chip-firing clocks | For two such functions $f$ and $g$, the difference $f-g$ is invariant under chip-firing, i.e. it factors through a function from the cokernel of the Laplacian to $\mathbb Z/k$. Conversely, for such a function $f$, adding any linear function from the cokernel of the Laplacian to $\mathbb Z/k$ produces another such funct... | 3 | https://mathoverflow.net/users/18060 | 389356 | 161,232 |
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