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https://mathoverflow.net/questions/412416 | 0 | $\DeclareMathOperator\Orb{Orb}\newcommand\abs[1]{\lvert#1\rvert}$The Collatz or the $3n+1$ conjecture is open.
1. Are there non-trivial polynomials $f(x)\in\mathbb Z[x]$ and $g(x)\in\mathbb R[x]$ having unbounded $\abs{f(n)}$, $\abs{g(n)}$ as $n$ grows in $\mathbb N$ and an integer $x\_0\in\mathbb N$ satisfying $\abs... | https://mathoverflow.net/users/10035 | Polynomials, $3^x$ and the Collatz conjecture | Point 1. is easy to satisfy even with nontrivial polynomials, since if you know $t$ modulo $2^n$ then you know for sure what the first n transformations will be.
For example. I could take $f=2^{20}x+174762$ and $g=6x+2$. Indeed I choose $f$ such that $3(2f(n)+1)+1)$ is always divisible by $2^{20}$, meaning that the f... | 1 | https://mathoverflow.net/users/23501 | 412461 | 168,292 |
https://mathoverflow.net/questions/412463 | 1 | Let $\{a\_k\}(k\ge 0)$ be a sequence of nonzero real numbers which changes signs infinitely often. Suppose $|a\_k|\to 0 $ and $|a\_k|$ decreases fast. Let $n$ be a positive integer. What's the relation between
$$\sum\_{k\ge 0}\binom {n+k}{k}a\_k $$ and $$\sum\_{k\ge 0}\binom {n+k}{k}\frac{a\_k}{k}. $$ For the asymptoti... | https://mathoverflow.net/users/159935 | Relation between $\sum_{k\ge 0}\binom {n+k}{k}a_k $ and $\sum_{k\ge 0}\binom {n+k}{k}\frac{a_k}{k}$ | Since the second sum cannot start at $k=0$, I assume that both sums start at $k=1$.
Consider a particular example: $a\_k = k\alpha^k$ with $|\alpha|<1$. Then
$$\sum\_{k\geq 1} \binom{n+k}k \frac{a\_k}k = (1-\alpha)^{-(n+1)}-1$$
and
$$\sum\_{k\geq 1} \binom{n+k}k a\_k = (n+1)(1-\alpha)^{-(n+2)}\alpha.$$
Then the ratio... | 3 | https://mathoverflow.net/users/7076 | 412466 | 168,295 |
https://mathoverflow.net/questions/412384 | 5 | It is a well-known fact that given a first-order sentence $\psi$ in prenex normal form $\forall x\_1 \exists y\_1 \forall x\_2 \exists y\_2 \dots \forall x\_n \exists y\_n \theta(x\_1,\dots,x\_n,y\_1,\dots,y\_n)$ (i.e., $\theta$ is quantifier free or at least $\Delta^0\_0$ in the appropriate sense), the truth of $\psi$... | https://mathoverflow.net/users/83901 | Which arithmetical sentences have no counterexamples in the sense of Kreisel? | In fact all true arithmetical sentences have weak classical realizers. Namely, a true arithmetical sentence $\psi$ $$\forall x\_1\exists y\_1 \ldots \forall x\_n\exists y\_n \theta(x\_1,\ldots,x\_n,y\_1,\ldots,y\_n)$$
is weakly realized by $(e\_1,\ldots,e\_n)$, where $\Phi\_{e\_i}^{\vec{g}}(y\_1,\ldots,y\_{i-1})$ perfo... | 3 | https://mathoverflow.net/users/36385 | 412470 | 168,296 |
https://mathoverflow.net/questions/412433 | 8 | I would like to know what motivated or led Thom to think that the (un)oriented cobordism groups would correspond with the homotopy groups of some structure (Thom spectum), or with the coefficient groups of a cohomology theory.
| https://mathoverflow.net/users/167503 | What motivated Thom to relate the cobordism groups with some homotopy groups? | I did not know Thom so I can't speak to all his personal motivations. But I can speak as someone that has read much of his work carefully and I think I have some insights into this.
The primary motivation for his theorem boils down to thinking carefully about the implicit function theorem, and asking when a subset of... | 11 | https://mathoverflow.net/users/1465 | 412475 | 168,297 |
https://mathoverflow.net/questions/412474 | 7 | Let $G \subseteq \mathbb F\_p^\*$ be a subgroup. Then $G$ is called almost trivial if $G \cap (2-G)$ consists of the element 1.
Then I am wondering how big $G$ can be in terms of $p$. If $G$ is a random set containing 1 then according to the birthday paradox one expects that $G$ and $2-G$ have nontrivial intersection... | https://mathoverflow.net/users/23501 | Subgroups of the multiplicative group of a finite field satisfying a certain additive property | If we let $S$ be the set of characters of $\mathbb F\_p^\times$ trivial on $G$ then $$\sum\_{\chi \in S} \chi(g) = \begin{cases} \frac{p-1}{|G|} & g\in G \\ 0 & g\notin G \end{cases}$$
so $$\sum\_{\chi\_1,\chi\_2\in S} \sum\_{ g \in \mathbb F\_p \setminus \{0,1,2\} } \chi\_1(g) \chi\_2(2-g) = 0$$ if $G$ is almost tri... | 13 | https://mathoverflow.net/users/18060 | 412477 | 168,299 |
https://mathoverflow.net/questions/412241 | 7 | Let $G\_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant–Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by
$$
\tag{1}\label{1}
\lambda\_k = \min\_{V \in G\_{n-k+1,n}} R(A,V),
$$
where $R(A,V):= \max\_{x \in V,\,\lVert x\rVer... | https://mathoverflow.net/users/78539 | What does $\mathbb E_V \max_{x \in V,\,\|x\|=1} x^T Ax$ evaluate to when $V$ is random $k$-dim suspace of $\mathbb R^n$ and $A$ is fixed psd matrix? | In another answer, @dohmatob establishes the lower bound
$$
\alpha\_k(A)\geq\frac{1}{n}\max\Big\{\operatorname{tr}(A),k\lambda\_{\max}(A) \Big\}.
$$
In what follows, we show that this bound is near-optimal for general $A$ in the regime where $k$ is at most a fraction of $n$.
Let $\Lambda$ denote the diagonal matrix o... | 2 | https://mathoverflow.net/users/29873 | 412481 | 168,302 |
https://mathoverflow.net/questions/412480 | -3 | Suppose that $f\_{4}(x)$ is a polynomial of degree 4 with no multiple roots, $C$ is the curve defined by $y^{2}=f\_{4}(x)$, I want to show that there is a polynomial $f\_{3}(x)$ of degree 3 with no multiple root such that $C$ is birational equivalent to the curve defined by $y^{2}=f\_{3}(x)$, I completely don't know ho... | https://mathoverflow.net/users/206621 | Show that a polynomial of degree 4 is birational equivalent to a polynomial of degree 3 | As Alexandre pointed it out, just use the linear-fractional transformation to send one point to infinity. Explicitly, denote the four roots of $f\_4$ to be $a,b,c,d$, then there is a unique linear-fractional transformation
$$\sigma:\mathbb P^1\to \mathbb P^1,$$
sending $a,b,c$ to $0,1,\infty$. Denote $\lambda$ to be ... | 5 | https://mathoverflow.net/users/74322 | 412484 | 168,304 |
https://mathoverflow.net/questions/412485 | 1 | Let $M \in \mathbb{N}$ and let $\pi \in S\_{M}$ be an involution with at least one fixed point. I'm interested in finding a latin square $A$ of order $M$ such that $A\_{i,j} = \pi(A\_{j,i})$ for each $i,j \in \{1,\ldots, N^2\}$.
A $B\_1$-type latin square is a latin square $A$ indexed by $\mathbb{Z}\_{n,1} = \mathbb{... | https://mathoverflow.net/users/409086 | Existence of latin squares with an involutory symmetry | To achieve your goal, we can use the "tensor product construction" for Latin squares. It is carried out as follows: Let $X\_a$ be a latin square indexed by $a$, i.e. $X\_a$ is a function $a \times a \rightarrow a$ and $X\_b$ similarly. Then $X\_a \otimes X\_b : (a \times b) \times(a \times b) \rightarrow (a \times b) $... | 2 | https://mathoverflow.net/users/125498 | 412486 | 168,305 |
https://mathoverflow.net/questions/412490 | 2 | We got experimental evidence against the semi strong perfect graph theorem
and would like to learn what is wrong with it.
From [Recognizing the P4-structure of bipartite graph](https://reader.elsevier.com/reader/sd/pii/S0166218X99001043?token=95BAE4CC0839E24CC218B933184ED467786AF8799730EAA190CC2E2E1412A9F12C57EA719ED... | https://mathoverflow.net/users/12481 | What is wrong with the experimental evidence against the semi strong perfect graph theorem? | This is not a counterexample. $H$ is the $P\_4$-structure of $G$, a certain 4-uniform hypergraph, and $G$ is perfect. All you can derive from the theorem is that any graph with the same $P\_4$-structure $H$ must be perfect.
Besides, $H$ is a hypergraph. While one may try to somehow create a graph out of it, and check... | 2 | https://mathoverflow.net/users/11100 | 412493 | 168,308 |
https://mathoverflow.net/questions/412471 | 3 | I am working on elliptic curves with torsion group $\mathbb{Z}/14\mathbb{Z}$ over quadratic fields. The curves are constructed using the model $E\_1=[0,a,0,b,0]$ following the formulas on p. 13 of
>
> L. Halbeisen, N. Hungerbuehler, M. Voznyy, A. S. Zargar, *A geometric approach to elliptic curves with torsion grou... | https://mathoverflow.net/users/95511 | Rationalizing and minimizing elliptic curve coefficients | If I understand correctly: you are starting with $a = 10656\sqrt{-23} +13600$, and you want to choose $f$ in the same number field such that $af^2$ is "as nice as possible", ideally lying in $\mathbf{Z}$; and you have observed that you can get it down to $a' = -28\sqrt{-23} + 605$.
Let's consider the prime factorisat... | 4 | https://mathoverflow.net/users/2481 | 412494 | 168,309 |
https://mathoverflow.net/questions/412495 | 1 | Is $\sum\_{\rho \text{ irred. }} \deg(\rho) \chi\_{\rho}(g)=0$ for every froup element $1\neq g \in G$ of the finite group $G$?
I have searched for but not found a proof to this. Probably it is not so difficult, but has as application that:
$$\det(T\_G) = 1$$
where $T\_G = (t\_{gh^{-1}})\_{g,h \in G}$ is the grou... | https://mathoverflow.net/users/165920 | Is $\sum_{\rho \text{ irred. }} \deg(\rho) \chi_{\rho}(g)=0$ for every group element $1 \neq g \in G$ of the finite group $G$? | The assertion holds with deg$(\rho)^2$ replaced with deg$(\rho)$.
You also need the extra requirement that the group element $g$ is non-trivial.
Then the Plancherel Theorem implies that the right hand side equals the trace of the left translation $L\_g$ on $\ell^2(G)$. Then you compute this trace as
$$
\mathrm{tr}(L\_g... | 3 | https://mathoverflow.net/users/nan | 412496 | 168,310 |
https://mathoverflow.net/questions/412483 | 1 | Let $A$ be a non atomic measure on $\mathbb R$. Consider the product measure $\mu := A \times \dots \times A$ on $\mathbb R^n$.
**Question:** Let $M$ be a $n-1$ dimensional smooth submanifold of $\mathbb R^n$. Is it true that $M$ has measure $0$ under $\mu$?
| https://mathoverflow.net/users/173490 | Does a submanifold of nonzero codimension have measure zero under the product of non atomic measures? | A submanifold $M$ of ${\mathbb R}^{n}$ (with codimension 1) can always (after permuting the coordinates) be locally represented in the form $G\_f=\{(x,y) : y=f(x)\}$ where $x$ runs over $V \subset {\mathbb R}^{n-1}$ and $f:V \to {\mathbb R}$. That implies that $M$ is contained in a countable union of such graphs. Each ... | 2 | https://mathoverflow.net/users/7691 | 412502 | 168,311 |
https://mathoverflow.net/questions/412504 | 22 | This question is about Joyal and Tierney's famous [*An extension of the Galois theory of Grothendieck*](https://doi.org/10.1090/memo/0309). One of the main results states (see the [MathSciNet review](https://mathscinet.ams.org/mathscinet-getitem?mr=756176) by Peter Johnstone):
**Joyal and Tierney's theorem.** *Each G... | https://mathoverflow.net/users/471475 | An extension of the Galois theory of Grothendieck | The point of view where this title comes from is that Grothendieck's theorem can be seen as a characterization of toposes of the form $BG$ for $G$ a profinite group. It shows that some toposes can be represented as $BG$.
I think before Joyal–Tierney's paper it was also known how to generalize from profinite group to ... | 18 | https://mathoverflow.net/users/22131 | 412510 | 168,314 |
https://mathoverflow.net/questions/412381 | 10 | Let $p, q$ be two distinct prime number. I'm trying to provide a non-trivial upper bound for the sum
$$S(p, q) = \sum\_{1 \leq x < p} \sum\_{1 \leq y < q} \frac{1}{\|x / p\| \, \|y / q\| \, \|x/p + y/q\|},$$
where $\|t\|$ denotes the distance of $t \in \mathbb{R}$ from the nearest integer.
Precisely, I'm interested i... | https://mathoverflow.net/users/357523 | Non-trivial upper bound for a sum related to $p^{-1}z \pmod q$ and $q^{-1}z \pmod p$ | If you just want $o()$, the story is rather simple. Let $a\_{xy}$ be the remainder of $qx+py\mod pq$ where all remainders modulo $P$ are assumed to be between $-P/2$ and $P/2$. Note that all $a\_{xy}$ are distinct, so if we have any set $Z$ of pairs $(x,y)$, then $\sum\_{(x,y)\in Z}\frac 1{a\_{xy}}\le 2(1+\log|Z|)$. Wh... | 8 | https://mathoverflow.net/users/1131 | 412520 | 168,319 |
https://mathoverflow.net/questions/407352 | 3 | It is known that an algebraic function with non-solvable monodromy group can not be represented by radicals. Where can we find a detailed proof about the nonrepresentability by radicals and entire (or meromorphic) functions of algebraic functions in some English references?
| https://mathoverflow.net/users/11966 | Nonrepresentability by radicals and entire (or meromorphic) functions of algebraic functions | See e.g.:
* Vladimir Arnold: Collected Works. Springer
* [Khovanskii, A.: Topological Galois Theory - Solvability and Unsolvability of Equations in Finite Terms. Springer 2014](https://doi.org/10.1007/978-3-642-38871-2)
* [Khovanskii, A.: One dimensional topological Galois theory. 2019](https://arxiv.org/abs/1904.033... | 3 | https://mathoverflow.net/users/94085 | 412521 | 168,320 |
https://mathoverflow.net/questions/412527 | 2 | Easton's theorem can give a very weak nontrivial constraint on continuum function, but it does not hold for singular cardinals. So:
1. What are the non-trivial constraints on continuum function in singular cardinals?
2. Is it possible to well-define a concept of "**Continuum function maximum of singular cardinals**" ... | https://mathoverflow.net/users/143814 | Continuum function maximum | The intricacies of arithmetic at singular cardinals notwithstanding, I think you're looking for something which doesn't exist.
Given any cardinals $\kappa,\lambda$, singular or regular, there is a (set) forcing extension preserving cardinals and cofinalities in which $2^\kappa>\lambda$. Upper bounds only appear in a ... | 2 | https://mathoverflow.net/users/8133 | 412528 | 168,324 |
https://mathoverflow.net/questions/412534 | 1 | $Define: X \text { is cardinal definable} \iff \\\exists \text { cardinal } \kappa \, \exists \text { cardinals } \lambda\_1,.., \lambda\_n <^\rho \kappa \ \exists \phi : \\ X=\{ y \in V\_{\rho(\kappa)} \mid \phi^{V\_{\rho (\kappa)}} (y,\lambda\_1,..,\lambda\_n)\}$
Where: $\lambda\_i <^\rho \kappa \iff \rho(\lambda\_... | https://mathoverflow.net/users/95347 | Is every set being cardinal definable consistent with ZF? | Yes, assuming ZF is consistent it is, because it follows from ZFC + V=HOD. This is because the ordertype of the class of (ordinal-)cardinals is just that of the ordinals. That is, if $X$ is definable over $V\_\alpha$ from ordinal parameter $\beta<\alpha$, then $X$ is definable over $V\_{\aleph\_\alpha}$ from ordinal pa... | 5 | https://mathoverflow.net/users/160347 | 412536 | 168,327 |
https://mathoverflow.net/questions/412551 | 0 | I would like to know if some widely believed conjecture, be it GRH, Hardy-Littlewood conjecture, or any other would imply the following statement for some $l>1$:
$l$-th power radioprimal growth conjecture ($l$-PRG conjecture for short)
Let $r\_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $r\_{i+1}(n):=\inf\{r>r... | https://mathoverflow.net/users/13625 | $l$-th power radioprimal conjecture | Assuming the strengthened version of Hardy-Littlewood conjecture I discuss [here](https://mathoverflow.net/a/395878/30186) (which follows from Dickson's conjecture), the following much stronger result holds: let $a\_0,\dots,a\_m$ be any sequence of natural numbers with $a\_0\geq 1$ and $a\_{i+1}\geq a\_i+2$ for $i<m$. ... | 2 | https://mathoverflow.net/users/30186 | 412554 | 168,333 |
https://mathoverflow.net/questions/412201 | 5 | I am reading the paper "Mori Dream Spaces and GIT" by Hu and Keel.
<https://arxiv.org/abs/math/0004017>
I cannot understand the proof of Lemma 1.6 in it.
Let $X$ be a normal projective variety.
Assume that $D$ is a divisor on $X$ such that $R(X,D) = \bigoplus\_{m \in \mathbb{Z}\_{\ge0} } H^0(X,\mathcal{O}\_X(mD))... | https://mathoverflow.net/users/472874 | Rational contraction and Proj of section ring | Let's assume that $|kD|=|kM|+kF$ where $|M|$ is base point free and moreover $Sym ^kH^0(M)\to H^0(kM)$ is surjective for any $k>0$. This can be achieved replacing $X$ by an appropriate resolution and $k$ by a multiple. Let $f:X\to Y$ be the morphism induced by $|M|$ so that $f^\*A=M$.
For [1], we wish to show that $f... | 4 | https://mathoverflow.net/users/19369 | 412560 | 168,334 |
https://mathoverflow.net/questions/412532 | 3 | I am looking for a paper "**Linear spaces with disjoint elements and their conversion into vector lattices**" by A. I. **Veksler**.
It was published in 1967 in Research Notes of Leningrad State Pedagogical University.
The paper is in Russian, and is called "**Линейные пространства с дизъюнктными элементами и преврщ... | https://mathoverflow.net/users/53155 | Looking for a paper on axiomatic orthogonality in a vector space | This journal published by the [Herzen University](https://en.wikipedia.org/wiki/Herzen_University) is not yet available in electronic form.
A paper version can be found in multiple libraries, including the [National Library of Russia](https://webservices.nlr.ru/util/?method=recordFormat&vid=07NLR_VU1&sysid=002516234&... | 2 | https://mathoverflow.net/users/402 | 412562 | 168,335 |
https://mathoverflow.net/questions/412566 | 5 | Let $G$ be a locally compact group, $C\_0(G)$ the $C^\*$-algebra of continuous functions on $G$ that vanish at infinity, $C\_b(G)$ the $C^\*$-algebra of bounded continuous functions on $G$. We know that $C\_b(G)$ is the multiplier algebra of $C\_0(G)$, and we denote the strict topology on $C\_b(G) = \mathcal{M}\bigl( C... | https://mathoverflow.net/users/128540 | Density of matrix coefficients of unitary representations of a locally compact group | First, some remarks that may help with literature-searching.$\newcommand{\fsnorm}[1]{{\Vert#1\Vert}\_{\rm B}}$
$\newcommand{\supnorm}[1]{{\Vert#1\Vert}\_\infty}$
The algebra you have denoted by $A\_0(G)$ is known as the *Fourier--Stieltjes algebra* of $G$, and is usually denoted by $B(G)$, so I will do that from now ... | 8 | https://mathoverflow.net/users/763 | 412574 | 168,338 |
https://mathoverflow.net/questions/412575 | 1 | I would like example of measures which shows that the following propositions are false:
>
> **Proposition 1:** Let $\mathfrak{B}$ be the Borel $\sigma$-algebra of a topological space $X$ and $\mu:\mathfrak{B}\to\overline{\mathbb{R}}$ be a measure. Then for all
> $B\in\mathfrak{B}$ with $\mu (B)<\infty$ and $\vareps... | https://mathoverflow.net/users/143671 | Find a Borel measure such that the closed sets aren't arbitrarily close to the Borel sets with finite measure | Let $X$ be the Sierpinski space $\{∅,\{b\},\{a,b\}\}$. Then the point $b$ is Borel but not closed. Let $\mu(a)=\mu(b)=1$. Take $B=\{b\}$, and it's a counterexample to proposition 1. Take $B=\{a\}$, and it's a counterexample to proposition 2.
| 3 | https://mathoverflow.net/users/125498 | 412579 | 168,339 |
https://mathoverflow.net/questions/412587 | 2 | Where can I find any more or less explicit semi-orthogonal decompositions of derived categories of perfect complexes or of bounded derived categories for singular schemes that are proper over a ring R?
I guess that one can take a "Beilinson-type" decomposition of $D^{perf}(\mathbb{P}^n(\operatorname{Spec}\,R))$ where... | https://mathoverflow.net/users/2191 | Semi-orthogonal decompositions over singular schemes | There is a notion of base change for semiorthogonal decompositions, applying which you can induce semiorthogonal decompositions over rings from those over fields, see Kuznetsov, Alexander. Base change for semiorthogonal decompositions. Compos. Math. 147 (2011), no. 3, 852--876. You can also find nontrivial examples of ... | 3 | https://mathoverflow.net/users/4428 | 412589 | 168,341 |
https://mathoverflow.net/questions/412585 | 2 | Let $P$ be the polytope obtained as the convex hull of $n$ points in $\mathbb R^d$. This is the $V$-representation of $P$. Note that $P$ can also be represented as an intersection of closed half-spaces in $\mathbb R^d$. This is the $H$-representation of $P$.
>
> **Question.** *In terms of $n$ and $d$, what is a goo... | https://mathoverflow.net/users/78539 | Optimal number of half-spaces in the $H$-representation of the convex hull of $n$ points in $\mathbb R^d$ | By the [Upper Bound Theorem](https://en.wikipedia.org/wiki/Upper_bound_theorem), the maximum number of $(d-1)$-dimensional faces of an $n$-vertex polytope is achieved by the cyclic polytopes. This number can be explicitly written via the [Dehn-Sommerville equations](https://en.wikipedia.org/wiki/Dehn%E2%80%93Sommervill... | 3 | https://mathoverflow.net/users/2233 | 412590 | 168,342 |
https://mathoverflow.net/questions/412541 | 11 | Recall the definition of *[cardinal definable](https://mathoverflow.net/questions/412534/is-every-set-being-cardinal-definable-consistent-with-zf)*, where every set being cardinal definable is [proved](https://mathoverflow.net/questions/412534/is-every-set-being-cardinal-definable-consistent-with-zf/412536#412536) cons... | https://mathoverflow.net/users/95347 | Is every set being cardinal definable consistent with ZF + negation of Choice? | This is consistent. Kanovei constructed a model $M$ with an infinite Dedekind finite set of reals which is lightface projectively definable. By descending to $L(R),$ we can further assume it satisfies $V=L(R).$
Clearly choice fails in this model. Since it satisfies $V=L(R),$ every set is definable from an ordinal and... | 12 | https://mathoverflow.net/users/109573 | 412591 | 168,343 |
https://mathoverflow.net/questions/412594 | 1 | (I've tried [Math SE](https://math.stackexchange.com/questions/4342665/intuition-behind-using-random-variables-in-monte-carlo-methods-localization), but have so far come up empty handed, so I'm trying my luck here.)
I would like to get a better intuitive understanding of why [Monte Carlo](https://en.wikipedia.org/wik... | https://mathoverflow.net/users/102097 | Intuition of the "work" done by random variables in Monte Carlo methods (incl. MCL) | **Q:** *Are random variables acting as a tool for us to be able to create multiple iterations (to measure and average) of an otherwise deterministic experiment?*
**A:** An answer has two ingredients:
* firstly, many deterministic questions can be formulated as asking for the expectation of a random variable. For ex... | 4 | https://mathoverflow.net/users/11260 | 412596 | 168,345 |
https://mathoverflow.net/questions/412581 | 6 | $\DeclareMathOperator\GL{GL}$Let $K$ be a number field and $R$ its ring of integers. Let $G$ be a connected reductive closed subgroup of $\GL\_{n,K}$. On p55 of Brian Conrad's notes [Reductive group schemes](http://math.stanford.edu/%7Econrad/papers/luminysga3.pdf), he claims that the schematic closure $\overline{G}$ o... | https://mathoverflow.net/users/88840 | Flatness of the closure of a closed subgroup of the generic fiber of an algebraic group inside an integral model of the ambient group | $\DeclareMathOperator\GL{GL}$A colleague asked me this question some time ago and here is the answer I sent him.
Let $R$ be a domain with fraction field $K$. Let $A$ be the $R$-algebra underlying the group scheme $\GL\_{n,R}$ over $R$.
Suppose given a closed subgroup scheme $H$ of $\GL\_{n,K}$. Let $B$ be the under... | 9 | https://mathoverflow.net/users/17308 | 412597 | 168,346 |
https://mathoverflow.net/questions/412441 | 1 | Let $G$ be a discrete group. Let $(A,\alpha)$ and $(B,\beta)$ be $G$-$C^\*$-algebras and $\varphi: A \to B$ be $G$-equivariant and completely positive. All crossed products in this post are full (= universal).
I want to prove that there is a completely positive map
$$\varphi \rtimes G: A \rtimes\_\alpha G \to B \rtim... | https://mathoverflow.net/users/216007 | A completely positive equivariant map $\varphi: A \to B$ induces a map on the full crossed products | (Too long for a comment). Your notation is confusing you, I think.
A *covariant* representation is not really of $B\rtimes G$, but is of the pair $(B,\beta)$. In particular, $\pi:B\rightarrow B(H)$ (notice the domain!) and not a representation of $B\rtimes G$. The covariant condition, that $\pi(\beta\_t(b)) = u\_g \p... | 2 | https://mathoverflow.net/users/406 | 412598 | 168,347 |
https://mathoverflow.net/questions/412599 | 3 | Let $\phi: S^{k-1}\to E\setminus Q$ be a continuous map, here
* $E$ is an infinite dimensional Hilbert space
* $Q$ is a compact set in $E$
I need extend $\phi$ to $B^{k}$ such that
$\phi: B^{k}\to E\setminus Q$ is still continuous.
If $Q := \{u\in E:\|u\|\_E=1\}\setminus E\_{k+1}~ \text{where}~ E\_{k+1} ~\text{is... | https://mathoverflow.net/users/166368 | extend a continuous map on sphere to ball such that the image is out of a compact set | Extending $\phi$ to the ball is equivalent to proving that $\phi:S^{k-1}\to H\setminus Q$ is nulhomotopic.
To prove this, you can consider the map $F:\phi(S^{k-1})\times Q\to E;(x,y)\to\frac{y-x}{|y-x|}$, which has compact image, so there is some vector $v$ with $|v|=1$ outside the image of $F$. So you can homotope $... | 2 | https://mathoverflow.net/users/172802 | 412602 | 168,348 |
https://mathoverflow.net/questions/412605 | 7 | The OEIS sequence [A000085](http://oeis.org/A000085) is defined by
$$ a\_n \!=\! (n-1)a\_{n-2} + a\_{n-1} \;\text{with }\; a\_0\!=\!1, a\_1\!=\!1.$$
If $n$ of the form $b^2-b+1, b \in \mathbb{N}, b > 2, \;\text{then: }\;$ $$ \left\lfloor \frac{a\_n}{a\_{n-1}} \right\rfloor > \left\lfloor \frac{a\_{n-1}}{a\_{n-2}} \... | https://mathoverflow.net/users/33646 | Question on OEIS A000085 | Your statement is (almost) proved in the paper [On solutions of $x^d=1$ in symmetric groups](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/on-solutions-of-xd-1-in-symmetric-groups/8F90642D9472FA7326164E54BE3BE57B). The $a\_n$ in your post corresponds to $T\_n$ in the paper, and the aut... | 18 | https://mathoverflow.net/users/125498 | 412612 | 168,351 |
https://mathoverflow.net/questions/412617 | 0 | Let $G$ be the heat kernal, i.e. for $0\le t<s$ and $x,y\in\mathbb R$
$$G(t,x;s,y):=\frac{1}{\sqrt{4\pi(s-t)}}\exp\left(-\frac{(y-x)^2}{4(s-t)}\right).$$
For $T>0$, let $\mathcal H\_T:=\{h:[0,T]\to [0,1]:~ h,~h' \mbox{ are both continuous on } [0,T]\}$ be endowed with the norm
$$\|h\|\_T := \max\_{0\le t\le T}|h(... | https://mathoverflow.net/users/261243 | Is this a contraction mapping for small $T$? | $\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}\newcommand{\De}{\Delta}\newcommand\R{\mathbb R}$**Edit:** This answer is insufficient, even though (almost) all the reasoning appears relevant to the problem. I will try to come back and fix it.
---
Suppose that $T\in(0,1]$.
Integrating in $y$, simplify the... | 1 | https://mathoverflow.net/users/36721 | 412632 | 168,355 |
https://mathoverflow.net/questions/412636 | 0 | How can I visualise PDF of distribution defined by quantiles, that I predict with my neural network? Now I'm passing quantiles to the histogram, but I don't think it is the correct way for visualising. I don't know it's simple/effectively to find the derivation of quantile function defined by the neural network. Exist ... | https://mathoverflow.net/users/473380 | Visualization PDF of distribution defined by quantiles | Smoothed histograms should be a way to go. See e.g. [this paper](https://www.cs.cmu.edu/%7Ejgc/publication/PublicationPDF/Analysis_Of_Uncertain_Data_Smoothing_Of_Histograms.pdf) and, more generally, these [search results](https://www.google.com/search?q=smoothed%20histogram&oq=smoothed%20histogram&aqs=chrome..69i57.741... | 0 | https://mathoverflow.net/users/36721 | 412638 | 168,356 |
https://mathoverflow.net/questions/412630 | 6 | I have recently encountered a triangular array $(a\_{i,j})\_{0\le i\le j}$, each line of which might (should?) have a combinatorial interpretation in terms of $S\_{2n+1}$. Here it is (the first entry of each line is the label $2n+1$, and the first $1$ on the right is $a\_{0,0}$):
```
1 | 1
3 | 2 ... | https://mathoverflow.net/users/29783 | Is there a combinatorial interpretation of this array in terms of $S_{2n+1}$? | You can use <https://www.findstat.org> to obtain a candidate for a conjectural solution, see <https://www.findstat.org/StatisticsDatabase/St000389oMp00093oMp00127oMp00066oMp00090> for details:
```
after adding 1 to every value
and applying
Mp00090: cycle-as-one-line notation: Permutations -> Permutations
Mp0... | 9 | https://mathoverflow.net/users/3032 | 412639 | 168,357 |
https://mathoverflow.net/questions/412648 | 6 | Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$.
Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$.
For $\chi\in$ Irr$(G)$ define $e\_{\chi} := \frac{\chi(1)}{|G|}\sum\_{g\in G} {\chi(g^{-1})\cdot g}$.
Let $B$ denote a $p$-block of $G$. Let Irr$(B)$ be the set of th... | https://mathoverflow.net/users/12826 | Question concerning the coefficients of block idempotents | Yes, this is true. By block orthogonality relations due to R. Brauer, it is true that $a\_{g}=0$ whenever the element $g$ has order divisible by $p$. But when $g$ has order prime to $p$, it is clear that $a\_{g}$ lies in $\mathbb{Q}[\omega]$ for some $t$-th root of unity $\omega$, where $t$ is not divisible by $p$.
Fur... | 9 | https://mathoverflow.net/users/14450 | 412659 | 168,361 |
https://mathoverflow.net/questions/412670 | 1 | I accidently found a class of rank-deficient square matrices that are their own Moore-Penrose pseudo inverse:
$$\boldsymbol{A}\in\mathbb{R}^{n\times n}: a\_{(i,i)}=n-1,\, a\_{\lbrace i,j\rbrace}=-1\implies\Big(\frac{1}{n}\boldsymbol{A}\Big)^+ =\Big(\frac{1}{n}\boldsymbol{A}\Big)$$
as an exemplary $4\times 4$ calcul... | https://mathoverflow.net/users/31310 | Characterization of pseudo unit-matrices | All symmetric idempotent matrices are their own Moore-Penrose pseudo-inverse; these are matrices with eigenvalues equal to either 0 or 1.
Your matrices are in this class.
| 1 | https://mathoverflow.net/users/11260 | 412674 | 168,367 |
https://mathoverflow.net/questions/412663 | 1 | Let $\mu$ be a probability measure on $\mathbb R^n$ and let $P$ be a compact polytope in $\mathbb R^n$. For any $x \in \mathbb R^n \setminus P$, let $p(x) \in P$ be (unique!) point in $P$ which is closest to $x$ and let $d(x) := \|x-p(x)\|$ be the distance from $x$ to $p$ and $u(x) := (x-p(x))/d(x) \in S\_{n-1}$, where... | https://mathoverflow.net/users/78539 | On the Lipschitz continuity of the unit-normal vector field of a polytope | **EDIT:** The OP changed the question while I was writing this answer. In this answer, we take the domain of $u$ to be $\{x\in\mathbb{R}^n:d(x)\geq\epsilon\}$ for some fixed $\epsilon>0$. This is a different use of $\epsilon$ than in OP's current problem formulation.
Let $B\_\epsilon$ denote the open ball centered at... | 1 | https://mathoverflow.net/users/29873 | 412676 | 168,368 |
https://mathoverflow.net/questions/412592 | 2 | Let $K$ be a convex body in $\mathbb R^d$ which contains the origin and let $\theta \in (0,1)$.
>
> **Question.** *Is it always possible to find $n$ points $x\_1,\dotsc,x\_n \in \mathbb R^d$ such that
> $$
> \theta K \subseteq \operatorname{conv}\{x\_1,\dotsc,x\_n\} \subseteq K
> \tag{1}\label{1}
> $$
> and $n \le ... | https://mathoverflow.net/users/78539 | Existence of fine approximate of a convex body in $\mathbb R^d$ with convex hull of $\mathcal O(d)$ points | This is false for any fixed $\theta\in(\frac{1}{2},1)$, we will use the balls of radius $1$ in $\mathbb{R}^d$ as a counterexample. Given $\theta$, if we want $
\theta K \subseteq \operatorname{conv}\{x\_1,\dotsc,x\_n\} \subseteq K
$, we will need at least one of the $x\_i$ to be in the convex closure of each [hypersphe... | 2 | https://mathoverflow.net/users/172802 | 412685 | 168,371 |
https://mathoverflow.net/questions/412683 | 1 | Reading a proof of the Schwarz lemma for the Kähler-Ricci flow from p22 of [these lecture notes](https://arxiv.org/pdf/1212.3653.pdf). I am confused as to what they mean by taking $$\inf \_{x \in M} \{\hat{R}\_{i \bar i j \bar j}(x) \mid \{\partial\_{z^{1}}, \ldots, \partial\_{z^{n}} \} \text{ is orthornormal w.r.t. } ... | https://mathoverflow.net/users/473447 | Schwarz lemma and bisectional curvature lower bound | Their idea is correct, but its formulation (and the chosen notation) is indeed a bit sloppy. What they seem to do, in reality, is to define
$$ - \hat C = \inf \_{x \in M} \inf \{ \hat R (e\_i, \bar {e\_i}, e\_j, \bar {e\_j}) \mid \{e\_1, \bar {e\_1}, \ldots, e\_n, \bar {e\_n} \} \text{ is orthornormal w.r.t. } \hat{g... | 2 | https://mathoverflow.net/users/54780 | 412688 | 168,372 |
https://mathoverflow.net/questions/412323 | 1 | The well-known theorem of [Abhyankar–Moh–Suzuki](https://en.wikipedia.org/wiki/Abhyankar%E2%80%93Moh_theorem) says the following:
Let $f=f(t), g=g(t) \in k[t]$, $k$ is a field of characteristic zero.
If $k[f,g]=k[t]$, then $\deg(f) \mid \deg(g)$ or $\deg(g) \mid \deg (f)$.
Let us concentrate on the case $k=\mathbb{C}... | https://mathoverflow.net/users/72288 | A variation on Abhyankar–Moh–Suzuki theorem | The answer is no. You can simply choose $$f=t(t^2-t+1)$$
$$g=t(t^2+1)$$
These satisfy all your conditions, but $\mathbb{C}[f,g]\subsetneq \mathbb{C}[t]$.
Let us check this:
$(1)-(2)$: $f=tf\_1$ and $g=tg\_1$ where $f\_1=t^2-t+1$, $g\_1= t^2+1$ are both of degree $2$ and without common factor.
$(3)$: $\mathbb{C}(f... | 3 | https://mathoverflow.net/users/23758 | 412696 | 168,377 |
https://mathoverflow.net/questions/412656 | 1 | Let $C$ be a nonempty closed subset of $\mathbb R^n$. It is known that any such set satisfies the following condition
>
> **(Unique CPP a.e).** *For **almost every** $x \in \mathbb R^n$, there exists a unique point in $c(x) \in C$ such that $\|x-c(x)\| = \mbox{dist}(x,C) := \inf\_{c \in C}\|x-c\|$.*
>
>
>
For ... | https://mathoverflow.net/users/78539 | Conditions for Lipschitzness of boundary normal vector, almost everywhere | Given a nonempty closed set $C\subseteq\mathbb{R}^n$, let $S\_C\subseteq\mathbb{R}^n$ denote the set of points for which the closest point in $C$ is not unique. Suppose $C$ satisfies each of the following:
* $S\_C$ is nonempty (i.e., $C$ is not convex),
* there exists $\rho>0$ such that $C^\rho\cap S\_C=\emptyset$, a... | 1 | https://mathoverflow.net/users/29873 | 412698 | 168,378 |
https://mathoverflow.net/questions/412697 | 5 | Let $M$ be a topological manifold. We know that $M$ is stably smoothable if and only its tangent microbundle, up to stabilization, admits a reduction to vector bundle.
Now I wonder if there is a relative version of this fact. Suppose $M$ and $N$ are topological manifolds and $f: M \to N$ is a locally flat topological... | https://mathoverflow.net/users/46433 | Stable smoothing of topological manifolds relative to an embedding | I think the answer is yes, it follows from this: if $M$ is a triangulable manifold of dimension $n$ greater than 4, and if the total space $E$ of a vector bundle over $M$ is smoothable, then the smooth structure is concordant to one where the zero section is a smooth submanifold.
To see why it follows, stabilize $M$ ... | 4 | https://mathoverflow.net/users/134512 | 412707 | 168,381 |
https://mathoverflow.net/questions/412039 | 12 | Let $u\in\mathrm{GL}\_n$ be a unipotent element, let $\mathcal{B}\_u$ be the variety of Borel subgroups containing $u$, and let $d=\dim \mathcal{B}\_u$. Then Springer theory tells us that $H^{2d}(\mathcal{B}\_u,\overline{\mathbb{Q}}\_{\ell})$ is an irreducible representation of $S\_n$ in a natural way, and that all irr... | https://mathoverflow.net/users/56217 | Branching rule of $S_n$ and Springer theory | This is a nice question. I have never seen this before.
Let us write
$$\mathcal B\_u = \{ V\_0 \subset V\_1 \subset \cdots \subset V\_{n-1} \subset V\_n = \mathbb C^n : u V\_i \subset V\_i \}. $$
Let $ \lambda $ be the Jordan type of $ u $. Then we can partition $ B\_u $ into locally closed subsets depending on the J... | 5 | https://mathoverflow.net/users/438 | 412709 | 168,382 |
https://mathoverflow.net/questions/412733 | 2 | Given a graph $G$ and $k\le \delta(G)$, the $k$-out model, $\mathcal{G}\_{k-out}(G)$ is a distribution on subgraphs $H$ of $G$ determined by the following: for $v\in G$, we choose a uniformly random subset $S\_v\subset N(v)$ of size $k$, we conclude by taking $H$ to have the edges $\{(v,u):v \in G, u \in S\_v\}$.
Boh... | https://mathoverflow.net/users/130484 | $k$-out model containing the square of a hamiltonian cycle | Your statement is correct, because there are too few triangles in $\mathcal G\_{k-out}(K\_n)$.
For three vertices $a,b,c \in \mathcal G\_{k-out}(K\_n)$ to form a triangle, the edges $ab$, $bc$ and $ca$ should all be present, and the probability is $O(n^{-3})$ after $k$ is fixed. Thus the expected number of triangles ... | 3 | https://mathoverflow.net/users/125498 | 412735 | 168,392 |
https://mathoverflow.net/questions/412684 | 1 | I'm looking for a solution to the following integral. However, it seems it doesn't have a solution.
$$\int\limits\_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{\displaystyle1}{\displaystyle \epsilon+3}} y^{-\frac{\displaystyle\epsilon+4}{\displaystyle\epsilon+3}}dy,$$
where $\theta \ge 1$, $\epsil... | https://mathoverflow.net/users/103291 | Is there a solution to $\int_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{1}{\epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy$? | With some effort (the lower integration limit requires care) I found this answer for the definite integral:
$$I(x)=\int\limits\_{\delta}^{x} \left(y-\delta\right)^{-\frac{1}{ \epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy=\frac{ (-2/\delta)^{\frac{2}{{\varepsilon}+3}} \pi ^{3/2} }{\sin \left(\pi\frac{{\varepsilon}... | 5 | https://mathoverflow.net/users/11260 | 412737 | 168,393 |
https://mathoverflow.net/questions/412722 | 3 | I am looking for some metric for distribution with support on the interval $[0,1-\epsilon]$, that will be based on the ratio of their moments.
That is, if $X\sim f(x)$, $Y\sim g(y)$, I'm looking for a metric $d(f,g)$ such that
$\frac{\lvert\mathbb{E}X^k-\mathbb{E}Y^k\rvert}{\mathbb{E}X^k}$ is small for all $0<k$ $\iff$... | https://mathoverflow.net/users/103133 | Comparing two distributions based of the ratio of their moments | $\newcommand\ep\epsilon$There is no such metric, because for any $\ep\in(0,1)$ and any real $k>0$ there are random variables $X$ and $Y$ with different pdf's $f$ and $g$ supported on $[0,1-\ep]$ such that $EX^k=EY^k$.
However, the [pseudometric](https://en.wikipedia.org/wiki/Pseudometric_space) $d\_k$ defined by the ... | 1 | https://mathoverflow.net/users/36721 | 412740 | 168,394 |
https://mathoverflow.net/questions/412743 | -1 | For sets $A,B$ we write $A\approx B$ if there is a bijection between $S$ sand $B$.
If $\kappa$ is a cardinal, let $\kappa^{<\kappa}$ denote the collection of subsets of $\kappa$ having cardinality $<\kappa$.
If $\kappa,\lambda$ are cardinals, does $\kappa^{<\kappa} \approx \lambda^{<\lambda}$ imply $\kappa=\lambda$... | https://mathoverflow.net/users/8628 | Does $|\kappa^{<\kappa}|=|\lambda^{<\lambda}|$ imply $\kappa=\lambda$? | No.
Consider when $2^{\aleph\_0}=2^{\aleph\_1}=2^{\aleph\_2}=\aleph\_3$, with $\kappa=\aleph\_1$ and $\lambda=\aleph\_2$.
---
If you allow for one of these to be singular, then consider $\kappa=\beth\_\omega$ and $\lambda=\kappa^+$. Then $\lambda^{<\lambda}=\lambda^\kappa=2^\kappa\cdot\lambda$, and on the other... | 7 | https://mathoverflow.net/users/7206 | 412744 | 168,395 |
https://mathoverflow.net/questions/412742 | 1 | This is followup to the following question: [On the Lipschitz continuity of the unit-normal vector field of a polytope](https://mathoverflow.net/q/412663/78539)
---
Let $C$ be a (nonempty) closed convex subset of $\mathbb R^n$. Note that to every $x \in \mathbb R^n$ corresponds a point $c(x) \in C$ which is close... | https://mathoverflow.net/users/78539 | Sufficient conditions for the boundary unit-normal vector field of a closed convex set to be Lipschitz continuous | (1) A necessary condition: For every $x\in\partial C$, it holds that the [polar cone](https://en.wikipedia.org/wiki/Dual_cone_and_polar_cone) of $C-x$ is one-dimensional.
(The polar cone is always at least one-dimensional by part 4 of Prop 4 of [these lecture notes](https://people.orie.cornell.edu/dsd95/teaching/orie... | 1 | https://mathoverflow.net/users/29873 | 412746 | 168,396 |
https://mathoverflow.net/questions/412729 | 1 | Perhaps stupid question.
>
> **Question**: Can "almost supermartingale" theorem be equally applicable to prove the convergence of some algorithms solving non-random optimization problems?
>
>
>
Attempt for a non-random version of "almost supermartingale" theorem (without any proof):
>
> Let the non-negativ... | https://mathoverflow.net/users/123235 | Can we invoke "almost supermartingale" Theorem for deterministic sequences? | For $k=0,1,\dots$, let $v\_k:=V^k$, $s\_k:=S^k$, $u\_k:=U^k$, and $b\_k:=\beta\_k$, so that the $v\_k$'s, $s\_k$'s, $u\_k$'s, and $b\_k$'s are nonnegative real numbers such that $\sum\_{k=0}^\infty b\_k<\infty$,
\begin{equation\*}
\sum\_{k=0}^\infty u\_k<\infty, \tag{1}
\end{equation\*}
and
\begin{equation\*}
v\_{k+1... | 1 | https://mathoverflow.net/users/36721 | 412747 | 168,397 |
https://mathoverflow.net/questions/412409 | 5 | $\newcommand{\C}{\mathbb C}$A question [asked recently](https://mathoverflow.net/questions/412398/how-to-construct-non-abelian-functions) was as follows:
>
> For the symmetric group $G:=S\_3$, is it possible to construct functions $t\_g\colon\C\to\C$ that satisfy the convolution identity
> \begin{equation}
> t\_g(... | https://mathoverflow.net/users/36721 | A functional equation for a family of functions indexed by the symmetric group $S_3$ | I think there is another "degenerate" way to deal with the question as posed, for a general finite group $G$ (later note: which may be developed into more interesting solutions, as we have done in later edits). The key is the connection with idempotents (not necessarily central) of the group algebra $\mathbb{C}G.$
Wh... | 4 | https://mathoverflow.net/users/14450 | 412750 | 168,398 |
https://mathoverflow.net/questions/412736 | 6 | Let $\mathfrak{g}$ be a (finite dimensional) semi-simple Lie algebra over a field $k$ and let $x \in \mathfrak{g}$. By definition, we have the equivalence:
$$ \mathrm{rk}(\mathrm{ad}\_x) = 0 \iff x = 0,$$
where $\mathrm{rk}(\mathrm{ad}\_x)$ is the rank of $\mathrm{ad}\_x$ seen as an element of $\mathrm{End}(\mathfrak{g... | https://mathoverflow.net/users/37214 | Rank one adjoint operators on a Lie algebra | Another approach.
To show it's impossible (the rank can't be 1), it is enough to show this when the field (assumed of char 0) is algebraically closed, and in turn it's enough to show the result in case $\mathfrak{g}$ is simple. If $x$ has $\mathrm{ad}(x)$ of rank 1, $x$ has centralizer of codimension 1. It is known (... | 6 | https://mathoverflow.net/users/14094 | 412759 | 168,400 |
https://mathoverflow.net/questions/412758 | 4 | Let $G$ be a finite group. Does there exist a finite group $\Gamma$ and a surjection $f\colon \Gamma \rightarrow G$ such that the center $Z(\Gamma)$ lies in the kernel of $f$?
Of course, this is only an interesting question if $Z(G) \neq 1$. It is also pretty trivial if we don't insist that $\Gamma$ is finite since w... | https://mathoverflow.net/users/473591 | Extending a finite group to avoid the center | Every finite group $G$ is quotient of a finite group with trivial center. Namely choose $p$ odd prime not dividing the order of $G$. Let $V\_p$ be the abelian group of functions $G\to\mathbf{Z}/p\mathbf{Z}$ with sum zero. The action of $G$ on $V\_p$ by translation $g\cdot f(h)=f(g^{-1}h)$ is faithful and fixes no nonze... | 12 | https://mathoverflow.net/users/14094 | 412760 | 168,401 |
https://mathoverflow.net/questions/412751 | 2 | Let $G\subseteq \mathbb{C}^n$ be a bounded domain. Consider the Carathéodory metric $C\_G$ on $G$. If $G=\mathbb{D}^n$ (unit polydisc), then $C\_G(a,z)=\max\_{1\leq j\leq n}p(a\_j,z\_j)$, where $p$ denotes the Poincaré metric on $\mathbb{D}$.
My question: Is there a similar formula in case $D=\mathbb{B}^2\times \math... | https://mathoverflow.net/users/159066 | Carathéodory metric on product domain | For a general formula, you should check out Kobayashi's book *Hyperbolic Complex Spaces*. In Proposition 3.1.11 of that text, he proves the following:
>
> Let $X$ and $Y$ be complex spaces. For $(x,y), (x',y') \in X\times Y$, we have
> $$
> \text{max}\{ c\_X(x,x'),c\_Y(y,y')\} \leq c\_{X\times Y}((x,y),(x',y')) \le... | 2 | https://mathoverflow.net/users/56667 | 412765 | 168,402 |
https://mathoverflow.net/questions/412767 | 3 | Recall the Coifman–Meyer Theorem as stated in [Grafakos and Oh - The Kato-Ponce Inequality](https://arxiv.org/abs/1303.5144).
**Theorem**: Let $m \in L^{\infty}\left(\mathbf{R}^{2 n}\right)$ be smooth away from the origin and satisfy
$$
\left|\partial\_{\xi}^{\alpha} \partial\_{\eta}^{\beta} m\right|(\xi, \eta) \leq ... | https://mathoverflow.net/users/68232 | Does the following version of the Coifman–Meyer Theorem exist? | No. For instance, if $f,g$ have disjoint supports, the right-hand side vanishes, but there is no reason to expect the left-hand side to vanish; and one can easily construct examples where it does not, for instance by first selecting a pair $f,g$ of bump functions with disjoint support and then localising $m$ to a small... | 11 | https://mathoverflow.net/users/766 | 412770 | 168,404 |
https://mathoverflow.net/questions/412773 | 1 | Let $A$ be a $C^\*$-algebra. Endow $A$ with the strict topology for which a net $\{a\_i\}\_{i \in I}$ converges to $a \in A$ if $$\|a\_i b-ab\| + \|ba\_i-ba\| \to 0$$
for all $b \in A$. Is it true that if $\{a\_i\}\_{i \in I}$ is a bounded net of positive elements that converges strictly to $a \in A$, then $a$ is also ... | https://mathoverflow.net/users/216007 | Convergent bounded net of positive operators converges to a positive operator | Yes, $a$ is positive, too.
*Proof.*
For every $b \in A$ we have $b^\* a\_i b \to b^\* a b$, so $b^\*a b$ is positive.
Now, let $(e\_j)$ be an approximate identity in $A$. Then it follows that $(e\_j a e\_j)$ is a net of positive elements that converges in norm to $a$; hence, $a$ is positive.
| 4 | https://mathoverflow.net/users/102946 | 412779 | 168,409 |
https://mathoverflow.net/questions/412784 | 1 | Sorry in advance if this is not sufficiently research-level, it is really more of a reference request since the proof is not difficult. Let $\mathcal{Y}$ be a compact set, let $\{X\_n\}$ denote a sequence of random variables, and let $f(x,y)$ and $g(y)$ be "nice" functions. Suppose that for each fixed $y\in\mathcal{Y}$... | https://mathoverflow.net/users/70190 | Pointwise almost sure convergence implies global convergence | First here, without loss of generality (wlog) $g=0$ (otherwise, replace $f(x,y)$ and $g(y)$ by $f(x,y)-g(y)$ and $0$, respectively). Second, in view of the almost-sure (a.s.) condition and the a.s. desired conclusion, wlog each random variable $X\_n$ takes only one value, say $x\_n$ -- which let us assume for simplicit... | 3 | https://mathoverflow.net/users/36721 | 412788 | 168,413 |
https://mathoverflow.net/questions/396004 | 1 | I'm currently reading this [paper](https://www.ams.org/journals/proc/1996-124-02/S0002-9939-96-03154-1/S0002-9939-96-03154-1.pdf) (and working on a similar one). The main goal is to study the Hammerstein integral equation (in $\mathcal{C}(I,E))$:
$$x(t) = \int\_{0}^{t} K(t,s)f\big(s,x(s)\big)ds,\quad t\in I;$$
wher... | https://mathoverflow.net/users/102228 | Simple example of Hammerstein integral equation | It is a question what is reasonable and whether you can conclude something interesting from it: If, for instance, the integral operator with kernel $\lvert K\rvert$ acts in $L\_\infty$ and $f$ grows sublinear in its second argument (with a bound independent of its first argument), the hypothesis will be satisfied with ... | 1 | https://mathoverflow.net/users/165275 | 412799 | 168,418 |
https://mathoverflow.net/questions/412525 | 21 | We have two well known definitions of the [semidirect product](https://en.wikipedia.org/wiki/Semidirect_product) $N \rtimes H$ of groups:
1. (Internal semidirect product) We write $G = N \rtimes H$ if $N$ is a normal subgroup of $G$, $H$ is another subgroup of $G$, $N \cap H = \{1\}$, and $G = NH$.
2. (External semid... | https://mathoverflow.net/users/766 | What is the name of this relative semidirect product of groups? | This construction appears in the literature as Remark 1.4.5 in "Pseudo-reductive groups" by Conrad, Gabber and Prasad.
There, it is called a 'non-commutative pushout'. It is a generalization of their 'standard construction' which produces most of the pseudo-reductive groups.
| 7 | https://mathoverflow.net/users/44668 | 412800 | 168,419 |
https://mathoverflow.net/questions/412798 | 0 | Let $A$ be a $C^\*$-algebra acting on a Hilbert space that admits a cyclic unit vector $\xi \in H$. Pose $S\_\xi = \{\eta \in A \xi: \| \eta \| = 1\}$, and for each $\eta \in S\_\xi$, pose $A\_\eta = \{a \in A : a \xi = \eta\}$, so $A\_\eta \ne \emptyset$ by definition; and let $c\_\eta = \inf\{\| a \| : a \in A\_\eta\... | https://mathoverflow.net/users/128540 | Controlling norm of operators sending a fixed vector to another | Set $A = C([0,1])$ and $H = L^2([0,1])$ (with respect to Lebesgue measure) with $\xi=1$ the constant function. Then $A\xi$ is the image of $C([0,1])$ in $L^2([0,1])$, which is a norm-decreasing injective but not bounded below inclusion. Then $S\_\xi$ is the intersection of the unit ball of $L^2$ with the continuous fun... | 2 | https://mathoverflow.net/users/406 | 412807 | 168,422 |
https://mathoverflow.net/questions/412713 | 4 | Consider a smooth projective threefold $\overline W$, constructed in section 4 of [this paper](https://arxiv.org/pdf/0810.0957.pdf).
This threefold is a resolution of singularities of the quotient of a product of a K3 surface and $\mathbb CP^1$ by $(\rho, \psi)$, where $\rho$ is a non-symplectic, non-fixed-point-free i... | https://mathoverflow.net/users/69559 | Irrationality of some threefolds | I am just posting my comment as an answer. All such threefolds are rational.
By the hypotheses on the involution of the K3 surface, the quotient surface is a rational surface. The projection from the threefold to the rational surface has geometric generic fiber isomorphic to $\mathbb{P}^1$, the other factor in the pr... | 3 | https://mathoverflow.net/users/13265 | 412810 | 168,425 |
https://mathoverflow.net/questions/412815 | 2 | Let $X,Y$ be complex manifolds of $\dim X=n$, $\dim Y=m>1$, $U\subset X$ open and $g\colon U\to Y$ holomorphic embedding. Then $g(U)$ is a submanifold of codimension $m-n\ge1$. It seems clear that $Y\setminus g(U)$ is connected, think for example at $\Bbb C\setminus\{0\}$. I'm searching for a reference, as it seems a w... | https://mathoverflow.net/users/70148 | Complement of complex submanifolds of codimension $\ge1$ is connected | Every (real) codimension $2^+$ embedding has path connected complement.
The proof is by using the tubular neighborhood theorem to reduce to showing that the total space of the normal bundle of $g(U)$ in $Y$ minus the zero section is connected, and noticing that the normal bundle is a rank $2^+$ vector bundle.
See [... | 3 | https://mathoverflow.net/users/125498 | 412819 | 168,429 |
https://mathoverflow.net/questions/412657 | 4 | Let $F$ be a field which has a positive characteristic $p \ge 2$ and $(\mathfrak{g},[p])$ be a restricted Lie algebras over a field $F$ where $[p]$ is a $p$-th power map on $\mathfrak{g}$. $(\mathfrak{g},[p])$ is called **simple restricted** if $(\mathfrak{g},[p])$ has no non-trivial $p$-ideals (i.e. ideals $\mathfrak{... | https://mathoverflow.net/users/167704 | Simple restricted but not restricted simple Lie algebras | The simple restricted Lie algebras are exactly the minimal $p$-envelopes of the simple Lie algebras. In fact, if $(\mathfrak{g}, [p])$ is a simple restricted Lie algebra over a field $\mathbb{F}$ of characteristic $p>0$, then $[\mathfrak{g}, \mathfrak{g}]$ is simple as an ordinary Lie algebra and its minimal $p$-envelo... | 6 | https://mathoverflow.net/users/14653 | 412830 | 168,433 |
https://mathoverflow.net/questions/412705 | 2 | I am aware that the implicit and inverse function theorems can be generalized to infinite dimensional cases, but I am having difficulty in applying it to a specific calculation.
Let $C^1\_{\mathbb{R}}[0,1]$ be the space of real-valued $C^1$ functions on the interval $[0,1]$. If we impose the following norm:
$$
\begin... | https://mathoverflow.net/users/56524 | Showing that a nonlinear operator over function spaces is differentiable and locally invertible? | The derivative of $F$ at $f \in C^1[0,1])$ is by definition the linear operator $F'(f): C^1[0,1] \rightarrow C[0,1]$ given by
$$ F'(f)\dot{f} = \lim\_{t\rightarrow 0} \frac{F(f+t\dot{f}) - F(f)}{t}. $$
Here, you get
$$
F'(f) = (\cosh f)\dot{f} + 2f'\dot{f}'.
$$
To apply the inverse function theorem, you need a linear o... | 2 | https://mathoverflow.net/users/613 | 412837 | 168,436 |
https://mathoverflow.net/questions/412701 | 2 | Let $C$ be a (nonempty) compact subset of euclidean $\mathbb R^n$, and consider the set-valued map $p\_C:\mathbb R^n \to 2^C$ defined by
$$
p\_C(x) = \{c \in C \mid \|x-c\| = \mbox{dist}(x,C)\},
$$
where $\mbox{dist}(x,C) := \inf\_{c \in C}\|x-c\|$ is the distance of $x$ from $C$.
>
> **Question.** *Under what mini... | https://mathoverflow.net/users/78539 | On the Lipschitz continuity of $x \mapsto \arg\min_{c \in C}d(x,c)$ w.r.t Hausdorff distance | Seems like in fact convex sets are the only ones for which $p\_C$ is continuous. To prove this, we can begin by noticing that for a set $C$ with the property, $p\_C$ can only take one point sets as values. If not, let $x,a\_1,a\_2$ be different points with $a\_1,a\_2\in p\_C(x)$. Then any point $y$ in the open segment ... | 1 | https://mathoverflow.net/users/172802 | 412839 | 168,438 |
https://mathoverflow.net/questions/412764 | 3 | In their classical paper on fluctuations in coin tossing [On Fluctuations in Coin-Tossing](https://www.pnas.org/content/35/10/605), Chung and Feller give a precise formula for the conditional probability of the number of positive “sides” of a random walk with an even number of steps, given a particular outcome for the ... | https://mathoverflow.net/users/471081 | What is the generalization of the formula for Chung and Feller's Theorem 2 to odd numbers of steps? | Speaking only for myself, the reason that I don't publish all my results is that I write very slowly. (It has nothing to do with the nature of combinatorics.) It takes a long time for me to arrange my work into something publishable; it's much quicker to put some results into a slide presentation for a talk.
But I'd ... | 5 | https://mathoverflow.net/users/10744 | 412841 | 168,439 |
https://mathoverflow.net/questions/412818 | 1 | I'm looking for a solution to the following integral.
$$\int\_{\lambda}^{y}(x-a)^{-b}x^{-c}\exp\left( -d x^{-e} \right)dx,$$
where $b,c,d,e> 0$ and $0< a < \lambda < y$.
This equation appears in the context of Physical Layer Security, which is an area of study in digital communications (telecom).
| https://mathoverflow.net/users/103291 | Is there a solution to $\int_{\lambda}^{y}(x-a)^{-b}x^{-c}\exp\left( -d x^{-e} \right)dx$? | There is no closed-form expression in terms of known functions, but if $a$ is small, you could use the power series in terms of the [exponential integral](https://en.wikipedia.org/wiki/Exponential_integral#Generalization),
$$\int\_{\lambda}^{y}(x-a)^{-b}x^{-c}\exp\left( -d x^{-e} \right)\,dx=\sum\_{p=0}^\infty \frac{b\... | 3 | https://mathoverflow.net/users/11260 | 412846 | 168,442 |
https://mathoverflow.net/questions/412826 | 2 | Let $X,Y$ be real vector spaces, $T: X\to Y$ be a linear map, and fix a nonempty $S\subseteq X$ (we do not assume that $S$ is neither convex nor compact (indeed, right now we do not assume any topological structure on $X$ and $Y$)). Everything below is just basic linear algebra, hence you may prefer to go directly to t... | https://mathoverflow.net/users/32898 | Commutation of linear maps and extreme points | The answer to the question is **yes**.
*Proof.* Let $y$ be an extreme point of $T[S]$. Then
$$
F :=T^{-1}(\{y\}) \cap S
$$
is non-empty, compact and a face of $S$. By the Krein-Milman theorem, $F$ has an extreme point $x$, and since $F$ is a face of $S$, it follows that $x$ is also an etreme point of $S$. Since $Tx ... | 4 | https://mathoverflow.net/users/102946 | 412849 | 168,443 |
https://mathoverflow.net/questions/410447 | 2 | It is known that Brownian motion is almost surely locally Holder continuous, on a range that is random, i.e. depends on the particular path. This question explores the maximal range on which Brownian motion is Holder continuous.
Let $W$ be a standard Brownian motion, and let $C > 0$ and $0 < \alpha < \frac{1}{2}$ be ... | https://mathoverflow.net/users/173490 | On the range of Holder continuity of Brownian motion | **Claim:** Let $0<\alpha<1/2$ and $0<C<\infty$. The random variable $H\_{C,\alpha}$ has an absolutely continuous distribution.
**Proof:** Let $\{{\cal F}\_t\}\_{t \ge 0}$ be the standard Brownian Filtration. Given a non-negative rational $q$, define the ${\cal F}\_q$-measurable random boundary functions $\psi\_q^+:[q... | 2 | https://mathoverflow.net/users/7691 | 412850 | 168,444 |
https://mathoverflow.net/questions/412854 | 1 | In [this paper](https://www.researchgate.net/publication/254211850_Non-archimedean_function_spaces_and_the_Lebesgue_dominated_convergence_theorem), the authors explain that the full generality of the Lebesgue dominated convergence theorem holds for functions on a compact zero-dimensional space $X$ taking values in a me... | https://mathoverflow.net/users/120369 | Non-Archimedean Lebesgue dominated convergence theorem | Here's the correct DOI link for the paper referenced: <https://doi.org/10.1016/j.jmaa.2009.10.059>
Paper is in Open Archive so I am pretty sure you can view it yourself.
AFAICT, the reference is correct (Theorem 4.13 in the above-linked paper is a Dominated Convergence Theorem). Here is the full bibliographic refer... | 2 | https://mathoverflow.net/users/3948 | 412856 | 168,446 |
https://mathoverflow.net/questions/412844 | 1 | Consider
$$X\_t=X\_0 + \int\_0^t b(s)ds+ \int\_0^t \sigma(s)dW\_s,\quad \forall t\ge 0,$$
where $X\_0\ge 0$ is a random variable of density $\rho$, $(W\_t)\_{t\ge 0}$ is an independent Brownian motion and $b,\sigma$ are measurable function "as nice as possible". Set $\tau:=\inf\{t\ge 0: X\_t\le 0\}$ and $Y\_t:=X\_{... | https://mathoverflow.net/users/261243 | PDE interpretation of some properties of the solution to Fokker–Planck equations | Differentiating $m(t)$ and using the equation leads to
$$
m'(t):=\int\_0^{\infty}\left(
\frac{\sigma^2(t)}{2}\partial^2\_{xx} p(x,t) - b(t)\partial\_x p(x,t)\right)\,dx=
-\frac{\sigma(t)^2}{2}\partial\_{x} p(0,t)<0.
$$
The last inequality follows from the Zaremba-Giraud theorem for the sign of the solution's normal der... | 2 | https://mathoverflow.net/users/14551 | 412869 | 168,450 |
https://mathoverflow.net/questions/412855 | 4 | As the title suggests, I have the following question:
>
> Is there a compact complex manifold with $b\_1(X)=b\_2(X)=b\_3(X)=b\_4(X)=0$?
>
>
>
Clarification:
Denote by $b\_k$ the $k$th Betti number of a compact complex manifold of positive dimension.
| https://mathoverflow.net/users/105103 | Is there a compact complex manifold with $b_1(X)=b_2(X)=b_3(X)=b_4(X)=0$? | There are complex manifolds with reduced cohomology vanishing in arbitrarily high degrees. Namely, product of two odd-dimensional spheres admits complex structure coming from representing it as a quotient of $(\Bbb C^n \setminus {0})\times (\Bbb C^m \setminus {0})$ by diagonal action of $\Bbb C$. They are known as Cala... | 10 | https://mathoverflow.net/users/81055 | 412874 | 168,452 |
https://mathoverflow.net/questions/412892 | 8 | Let $\mathbb{S}$ be the Sierpiński space, the two pointed space $\{ 0, 1 \}$ with open sets $\{0 \}$, $\emptyset$, $\{ 0, 1 \}$. We give $\{ 0, 1 \}$ a partial order where $0 < 1$.
Let $X$ be a topological space. Consider the space $Y = \prod\_{f : X \rightarrow \mathbb{S}} \mathbb{S}$. This space is compact by Tycho... | https://mathoverflow.net/users/30211 | Is this space the Stone–Čech compactification? | No, the closure of the image of $f$ in $Y$ is never the Stone-Čech compactification of $X$ unless $X$ is empty. In particular, consider the element $a\in Y$ which is $1$ on every coordinate. Note that the only open subset of $Y$ that contains $a$ is $Y$ itself. So, if $X$ is nonempty, then $a$ will be in the closure $X... | 16 | https://mathoverflow.net/users/75 | 412894 | 168,463 |
https://mathoverflow.net/questions/412891 | 2 | Is the maximum nearest neighbor distance between points of the process, over all the infinitely many points of a stationary Poisson point process of intensity $\lambda$ in $\mathbb{R}^d$, almost surely finite? What is its distribution?
I am interested in the case $d=1$ and $d=2$. Also, if it is any easier, you can re... | https://mathoverflow.net/users/140356 | Maximum nearest neighbor distance for a Poisson point process | $\newcommand\la\lambda\newcommand\de\delta\newcommand\R{\mathbb R}$This maximum (or, rather, supremum) distance, say $M$, is $\infty$ almost surely (a.s.).
Indeed, recall that a (simple) Poisson point process of intensity $\la\in(0,\infty)$ on $\R^d$ is a random Borel measure $m$ over $\R^d$ such that, for any pairwi... | 4 | https://mathoverflow.net/users/36721 | 412895 | 168,464 |
https://mathoverflow.net/questions/412899 | 5 | I am investigating whether or not there exist $\epsilon > 0$ such that $\zeta(s) \neq 0$ on the strip $1-\epsilon < \Re(s) \leq 1$.
Suppose not. Then given $\delta > 0$ there exists a zero of zeta $\rho$ such that $1 -\delta < \Re(\rho) < 1 $. Hence, there exists a sequence of zeros $\{ z\_n \}\_{n=1}^\infty$ with in... | https://mathoverflow.net/users/8435 | Density of fake zeros of Zeta | There are, provably, very few zeros with real part close to $1$ (or bigger than $0.51$ for that matter). These theorems go under the name of "zero density estimates", and they have a vast literature. See Chapter 11 in Ivić: The Riemann zeta function (1985). The book was reprinted in 2013 by Dover Publications.
| 19 | https://mathoverflow.net/users/11919 | 412901 | 168,467 |
https://mathoverflow.net/questions/326116 | 22 | I noticed that the only official reason given for awarding Edward Witten the Fields medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully rigorous.
However, I came across this paper <https://projecteuclid.org/download/pdf_1/euclid.cmp/1103921154> by Taubes and ... | https://mathoverflow.net/users/119114 | Is Witten's proof of the positive mass theorem rigorous? | The positive mass theorem is more or less to do with the geometry of a type of positive scalar curvature condition.
Witten's work considers harmonic spinors, which are solutions to a certain linear elliptic system of partial differential equations. In his paper he presents a calculation which proves a rigidity theore... | 9 | https://mathoverflow.net/users/156492 | 412909 | 168,471 |
https://mathoverflow.net/questions/412917 | 2 | **I. General Question**
Consider a one-parameter family of vector bundles $E\_t$ on a smooth projective variety $X$ with fixed Chern character $v$. Suppose $E\_t$ is Gieseker stable when $t\neq 0$ and $E\_0$ is *not* Gieseker semi-stable. **Is there a way to find the limit $\lim\_{t\to 0}E\_t$ in the Gieseker semi-st... | https://mathoverflow.net/users/74322 | Is there a stable reduction for a family of vector bundles? | Consider the Harder-Narasimhan filtration of $E\_0$. Assume for simplicity it has length 2:
$$
0 \to F\_0 \to E\_0 \to E\_0/F\_0 \to 0,
$$
where $F\_0$ and $E\_0/F\_0$ are semistable and the slope of $F\_0$ is bigger than the slope of $E\_0/F\_0$. If $B$ is the parameter space of your family and $i \colon X \to X \time... | 5 | https://mathoverflow.net/users/4428 | 412919 | 168,472 |
https://mathoverflow.net/questions/412924 | 5 | In the book R. M. Young, An introduction to non-harmonic Fourier series, I came across the following problem (page 18):
**Problem.** Show that the sequence $\left \{ \frac{1}{x+1},\frac{1}{x+2},\frac{1}{x+3}, \dots \right \}$ is complete in $L^2[0,1]$.
I tried to apply the Müntz-Szasz theorem but it didn't work out... | https://mathoverflow.net/users/106425 | Completeness of the sequence $\left \{ \frac{1}{x+1},\frac{1}{x+2},\frac{1}{x+3}, \dots \right \}$ in $L^2[0,1]$ | Let $S$ be the Hilbert subspace of $L^2[0,1]$ spanned by the sequence. For every positive integer $n$, the function $\frac{n}{x+n}=\frac{1}{1+x/n}$ lies in $S$. For $n$ large, this function is very close to the constant $1$ function in sup-norm, hence $1\in S$. Now we prove by induction that, for every nonnegative inte... | 18 | https://mathoverflow.net/users/11919 | 412929 | 168,475 |
https://mathoverflow.net/questions/412923 | 33 | I recently came across this problem from [USAMO 2005](https://artofproblemsolving.com/wiki/index.php/1995_USAMO_Problems):
"A calculator is broken so that the only keys that still
work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The display initially shows $0$. Given any positive ratio... | https://mathoverflow.net/users/7113 | What is the smallest set of real continuous functions generating all rational numbers by iteration? | It is enough with one continuous function. First, I'll give a simple example with one function which is discontinuous at one point. To do it, consider the function $$f:(0,\pi+1)\to(0,\pi+1)$$ with
$$
f(x) = \begin{cases}
x+1 &\text{if $x<\pi$,} \\
x-\pi &\text{if $x>\pi$,} \\
1 &\text{if $x=\pi$.}\\
\end{cases}
$$
**Cl... | 33 | https://mathoverflow.net/users/172802 | 412931 | 168,476 |
https://mathoverflow.net/questions/412933 | 5 | Consider the following statements in $\sf ZF$:
>
> (S) If $A, B$ are nonempty sets, then there is a surjection $s:A \to B$, or there is a surjection $t:B\to A$.
>
>
>
>
> (I) If $A, B$ are sets, then there is an injection $i:A\to B$, or there is an injection $j:B\to A$.
>
>
>
Note that (I) implies (S). ... | https://mathoverflow.net/users/8628 | Existence of surjection vs injection over $\sf ZF$ | The answer is no. Both (I) and (S) are equivalent to AC over ZF. Indeed, for any set $S$ the class of ordinals $\alpha$ such that $\alpha$ injects into $S$ (resp. $S$ surjects onto $\alpha$) is a set, so there is some ordinal $\beta$ outside this set. Assuming (I) (resp. (S)), there is an injection $S\to\beta$ (resp. s... | 16 | https://mathoverflow.net/users/30186 | 412934 | 168,478 |
https://mathoverflow.net/questions/412893 | 4 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$Let $ \Sigma\_g $ be a compact orientable surface of genus $ g $. Let the subgroup $ \pi\_1(\Sigma) $ of $ \SL\_2(\mathbb{R}) $ be a Fuchsian representation of the fundamental group. Then $ \Sigma\_g $ admits a hyperbolic structure
$$
\p... | https://mathoverflow.net/users/387190 | Generalizing a result about hyperbolic 2-folds to hyperbolic 3-folds | The (orientation-preserving) isometry group $G=PSL(2,{\bf C})$ acts on the bundle of (positively oriented) orthonormal frames on ${\bf H}^3$.
An ad hoc argument using the Iwasawa decomposition $G=KAN$ shows that this action is transitive. Indeed, considering the upper half space model and fixing the standard frame in... | 5 | https://mathoverflow.net/users/39082 | 412937 | 168,481 |
https://mathoverflow.net/questions/412927 | 4 | Let $G(t)$ be a probability generating function of some integer and non-negative r. v. $X$. Suppose that
$$\lim\_{t\to1}G'(t)=+\infty.$$
That is
$$
\mathbb{E}X=+\infty.
$$
Can you show that
$$
\lim\_{t\to1}\frac{(G'(t))^2}{G''(t)}=0\,?
$$
There is an example satisfying the conjecture:
$$
G(t)=\frac{6}{\pi^2}\text{P... | https://mathoverflow.net/users/43017 | Ratio of the first squared and the second moment | This is correct.
Denote $G(t)=\sum\_{k=0}^\infty p\_k t^k$, where $p\_i\geqslant 0$ and $\sum p\_i=1$. Then we are given that $\sum kp\_k=\infty$ and should prove that
$$
\lim\_{t\to 1-0} \frac{(\sum\_{k=1}^\infty kp\_k t^{k-1})^2}{\sum\_{k=2}^\infty k(k-1)p\_kt^{k-2}}=0.
$$
Since both numerator and denominator tend ... | 4 | https://mathoverflow.net/users/4312 | 412941 | 168,484 |
https://mathoverflow.net/questions/412896 | 3 | Suppose I have a measurable space $(\Omega, \Sigma)$ and a function $f: \mathbb{R} \times \mathbb{\Sigma} \rightarrow [0,1]$ such that for any $x \in \mathbb{R}$ the tuple $(\Omega, \Sigma, f(x, \\_))$ is a measure space.
Is it the case that
$$\int\_{\mathbb{R}} \left[ \int\_{\Omega} df(x, \\_) \right] dx = \int\_{\O... | https://mathoverflow.net/users/265390 | Does the Radon-Nikodym derivative commute with integration? | Provided $x\mapsto f(x,B)$ is Borel measurable for each $B\in\Sigma$, you can define
$$
\mu(B):=\int\_{\Bbb R} f(x,B) dx,
$$ and check (using Tonelli's theorem) that $\mu$ is a measure.
If $H\ge 0$ is a measurable function on $\Omega$,
then
$$
\int\_\Omega H d\mu=\int\_{\Bbb R}\left[\int\_\Omega H(\omega) d\_\omega f(x... | 3 | https://mathoverflow.net/users/42851 | 412942 | 168,485 |
https://mathoverflow.net/questions/412906 | 3 | I was reading this question [The connected component of the idele class group](https://mathoverflow.net/questions/350676/the-kernel-of-the-global-class-field-theory-homomorphism) but I am very confused about the structure of the *solenoids* $(\widehat{\mathbb{Z}}\times\mathbb{R})/\mathbb{Z}$, (where $\mathbb{Z}$ acts d... | https://mathoverflow.net/users/177957 | Is $(\mathbb{Z}_p\times \mathbb{R})/\mathbb{Z}$ connected? | No originality here, but I would tell the story as follows. Consider the subgroup $(\mathbb{Z}\times\mathbb{R})/\mathbb{Z}$ of $(\mathbb{Z}\_p\times\mathbb{R})/\mathbb{Z}$. It is dense, because $\mathbb{Z}$ is dense in $\mathbb{Z}\_p$. It is also connected, because it is isomorphic to $\mathbb{R}$ (with a coarser topol... | 6 | https://mathoverflow.net/users/11919 | 412943 | 168,486 |
https://mathoverflow.net/questions/412951 | 8 | It looks like there is some literature out there on what might be called 'non-Archimedean complex analysis' e.g. [Benedetto - An Ahlfors Islands Theorem for non-archimedean meromorphic functions](https://arxiv.org/abs/math/0407142) and [Cherry - Lectures on Non-Archimedean Function Theory](https://arxiv.org/abs/0909.45... | https://mathoverflow.net/users/197447 | Literature on non-Archimedean analogues of basic complex analysis results | Benedetto has a textbook that discusses basic $p$-adic analysis, although his aim is to study $p$-adic dynamics. And it's for a single variable. But might be a good place to get some information.
[Dynamics in One Non-Archimedean Variable, Robert L. Benedetto, Graduate Studies in Mathematics, Volume 198, 2019, America... | 7 | https://mathoverflow.net/users/11926 | 412954 | 168,490 |
https://mathoverflow.net/questions/412965 | 8 | Let $f$ be a weight $2$ cusp form for the group $\Gamma\_0(N)$. I was experimenting with integrals of the form
$$ \int\_r^s f(z) \, dz$$
where $r, s \in \mathbf{P}^1(\mathbf{Q})$ and the integral above is over the geodesic in the upper-half plane connecting $r$ and $s$. When I plotted these period integrals, I noticed ... | https://mathoverflow.net/users/394740 | What is the rank of the period lattice of modular forms? | For Q1: I'm assuming your $f$ is a normalized Hecke *newform* of level $N$ with coefficients in $\mathbf{Z}$. Then the choice of $f$ determines a splitting of the homology as
$$H\_1(X\_0(N), \{cusps\}, \mathbf{Q}) = (f\text{-generalised eigenspace}) \oplus \text{(other stuff)}.$$
By the Eichler--Shimura isomorphism... | 14 | https://mathoverflow.net/users/2481 | 412968 | 168,493 |
https://mathoverflow.net/questions/412969 | 4 | This is essentially a reference request. According to the classification of the finite simple groups, there are 26 (or arguably 27) sporadic groups. Let us denote these by G\_1 , ... , G\_{26} (resp. G\_{27}).
>
> **Question**: *Which G\_i can be embedded into which G\_j?*
>
>
>
I was unable to find the answer... | https://mathoverflow.net/users/203598 | Embeddings of the sporadic simple groups | See Table 2, p. 362 in:
* Wilson, Robert A.
Is the Suzuki group Sz(8) a subgroup of the Monster? Bull. Lond. Math. Soc. 48 (2016), no. 2, 355-364. <https://doi.org/10.1112/blms/bdw012>
Which lists all simple groups that are contained in a sporadic group.
It seems the question you ask about (which sporadics can be... | 10 | https://mathoverflow.net/users/38068 | 412974 | 168,494 |
https://mathoverflow.net/questions/412960 | 14 | In Quantum theory, groups and representations, Peter Woit writes:
>
> A fundamental principle of modern mathematics is that the way to
> understand a space $M$, given as some set of points, is to look at $F(M)$,
> the set of functions on this space.
>
>
>
I was wondering what some examples of this "fundamental... | https://mathoverflow.net/users/473920 | What are some examples of understanding a space by studying the functions on this space? | The idea of studying the relationship between structured spaces and appropriate spaces of functions thereon could be described as one of the basic principles of functional analysis, perhaps even the defining one.
Examples:
* completely regular spaces and continuous functions—general, bounded or of compact support (... | 9 | https://mathoverflow.net/users/159073 | 412976 | 168,495 |
https://mathoverflow.net/questions/412944 | 6 | Let $F$ be a number field. For an irreducible cuspidal automorphic representation $\pi$ of $\operatorname{GL}\_n(\mathbb{A}\_F)$, we say that $\pi$ is symplectic (or orthogonal) if $L(s,\pi,\bigwedge^{2})$ (or $L(s,\pi,\operatorname{Sym}^2)$) has a pole at $s=1$.
I am wondering whether if $\pi=\bigotimes \pi\_v$ is s... | https://mathoverflow.net/users/35898 | Global symplectic (orthogonal) type of automorphic representation compels its type to all its local components? | Here is a proof of the claim using results from Arthur's monograph [The Endoscopic Classification of
Representations: Orthogonal and
Symplectic Groups](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.294.6966&rep=rep1&type=pdf).
Let $N = 2n$ be an even integer, and $\pi$ a cuspidal automorphic representatio... | 4 | https://mathoverflow.net/users/2481 | 412977 | 168,496 |
https://mathoverflow.net/questions/412958 | 3 | I use the convention of the Weinberg QFT textbooks, that is, $(-,+,+,+)$.
According to Weinberg QFT vol 2 p. 369, he says the Euclidean Dirac operator
\begin{equation}
{D}:=[i\partial\_i +t\_\alpha A\_{i \alpha}]\gamma\_i
\end{equation}
is Hermitian. Here, $\partial\_4:=-i\partial\_0$, $A\_{4\alpha}:= iA^0\_\alpha$ a... | https://mathoverflow.net/users/56524 | The exact domain on which the Euclidean Dirac operator is self-adjoint | Recall that a densely defined operator $T$ on a Hilbert space $H$ is *essentially self-adjoint* if and only if its minimal closure $\overline{T}$ is self-adjoint, if and only if $T$ is symmetric and has a unique extension to a self-adjoint operator (i.e., the minimal closure $\overline{T}$).
In general, you can consi... | 4 | https://mathoverflow.net/users/6999 | 412979 | 168,498 |
https://mathoverflow.net/questions/410524 | 4 | Let $S\_{g,b}$ denote the orientable connected compact surface of genus $g$ with $b$ boundary components. A group homomorphism $\varphi\colon G\to \text{Homeo}^+(S\_{g,b})$ is said to be free $G$-action if $\varphi(a)$ has no fixed point for all non-trivial $a\in G$. Two free group actions $\varphi\_1,\varphi\_2\colon ... | https://mathoverflow.net/users/363264 | Inequivalent free $\Bbb Z/n\Bbb Z$-actions on orientable compact bordered surface | Did you try to use the double $T$ of the surface $S$? Any fixed point-free action of $\mathbf Z/n$ on $S$ induces a fixed point-free action on the closed orientable surface $T$. Moreover, the induced action commutes with the natural orientation-reversing involution on $T$. By Nielsen's Theorem that you mentioned, you'v... | 5 | https://mathoverflow.net/users/85592 | 412992 | 168,502 |
https://mathoverflow.net/questions/412989 | 10 | Consider the below $q$-series identity. One of the things I like about this expansion is how nicely the difference on the left hand side factors to the right hand side of the equation.
$$\prod\_{k\geq1}(1+q^k)^3-\prod\_{k\geq1}(1+q^{3k})
=3q\prod\_{n\geq1}(1+q^n)(1+q^{9n})^2(1+q^n+q^{2n}+\cdots+q^{8n}).$$
I have a ... | https://mathoverflow.net/users/66131 | Looking for a "clever" argument for a $q$-series identity | Here's a proof that indicates a systematic method for proving such identities. Let $\eta(z) = q^{1/24} \prod\_{n=1}^{\infty} (1-q^{n})$, with $q = e^{2 \pi i z}$. The identity you state in the equation is the same as
$$
\frac{\eta^{3}(2z)}{\eta^{3}(z)} - \frac{\eta(6z)}{\eta(3z)} = 3 \frac{\eta(2z) \eta^{2}(18z)}{\eta... | 16 | https://mathoverflow.net/users/48142 | 412994 | 168,503 |
https://mathoverflow.net/questions/412988 | 64 | I am an enthusiastic but ever-so-slightly naive PhD student and have been 'following my nose' a lot recently, seeing whether topics that I have studied can be generalised or translated in various ways into unfamiliar settings; exploring where the theory breaks down etc.
When doing this, I have found it very difficult... | https://mathoverflow.net/users/197447 | How do pure mathematicians assess whether their research ambitions can be realistically achieved? | Over decades, and across multiple research fields, I've noticed a way to predict I'm on track to make progress. **I discover something interesting, only to learn it is already known**.
As a student, this was incredibly discouraging, and in fact I stopped some lines of research for this very reason. But by now I'm use... | 99 | https://mathoverflow.net/users/1227 | 413001 | 168,506 |
https://mathoverflow.net/questions/413011 | 14 | I preface my question by admitting I know no algebraic geometry nor algebraic number theory. I do know some algebraic topology. I'm a student.
Recently I learned about sheaf cohomology. Then a little bit of etale cohomology, as much as I could stomach having never studied algebraic geometry. Then I came across [Artin... | https://mathoverflow.net/users/472967 | Is Mazur's analogy between arithmetic and topology formal, in any sense? | For this analogy, like most analogies in mathematics, and indeed like most philosophical principles in mathematics, one can certainly make *a part of it* formal and rigorous, but I don't think any true formal statement could ever capture *all* of what we mean by the analogy.
In particular, by well-chosen definitions,... | 14 | https://mathoverflow.net/users/18060 | 413014 | 168,514 |
https://mathoverflow.net/questions/336755 | 13 | Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it.
[Volterra series](https://en.wikipedia.org/wiki/Volterra_series) are a generalization of Taylor series that can also model "memor... | https://mathoverflow.net/users/24611 | “Taylor series” is to “Volterra series” as “Padé approximant” is to _________? |
>
> "I would be at least happy to know the answer for the discrete Volterra series, which (I think) would be equivalent to something like a multivariate Padé approximant."
>
>
>
Multi-variate versions of Padé approximants got attention in the 1970s, for example a generalization to functions of two variables was ... | 11 | https://mathoverflow.net/users/66334 | 413017 | 168,515 |
https://mathoverflow.net/questions/413015 | 3 | This is a continuation of [my previous question](https://mathoverflow.net/questions/412137/a-baire-subset-of-reals-that-is-not-suslin-measurable). Recall that a subset $A \subseteq {}^\omega\omega$ is *analytic* if it is the continuous image of the Baire space. I would like to know if there exist two models $N \subsete... | https://mathoverflow.net/users/146831 | A submodel of set theory with all reals which every set is analytic | In fact, the principle "Every set is analytic" is not consistent with $\mathsf{ZF}$ in the first place. We don't need choice to get a surjection $h$ from Baire space to the set of continuous maps on Baire space. But once we have such an $h$, the "diagonalizing" set $\{x: x\not\in h(x)\}$ can't be analytic.
---
Le... | 8 | https://mathoverflow.net/users/8133 | 413018 | 168,516 |
https://mathoverflow.net/questions/413002 | 1 | We know that various [Markov categories have deterministic morphisms](https://scholar.google.com/citations?view_op=view_citation&hl=en&user=ofP7CgYAAAAJ&alert_preview_top_rm=2&citation_for_view=ofP7CgYAAAAJ:W7OEmFMy1HYC). This suggests that they support faithful functors into them from categories of categories. One int... | https://mathoverflow.net/users/417530 | What category of categories have faithful functors into Markov categories? | Following N. Virgo's point (but even more simply), any concrete category $\mathcal C$ admits (by definition) a faithful functor $F: \mathcal C \to Set$. The free functor $G$ from $Set$ to the Kleisli category of the distribution monad is faithful as well. So $GF: \mathcal C \to Kl(Dist)$ is a faithful functor. Converse... | 2 | https://mathoverflow.net/users/2362 | 413021 | 168,517 |
https://mathoverflow.net/questions/413007 | 5 | For integer-valued sequences $(x\_n)\_{n=0}^\infty$, consider recurrences of the form
$$x\_n=ax\_{n-1}+(bn+c)x\_{n-2} \tag{$\*$}\label{star}$$
for $n\ge2$, where $a,b,c$ are integers.
There seem to be many such sequences in the OEIS — see e.g. [A000085](http://oeis.org/A000085), [A001475](https://oeis.org/A001475), [A0... | https://mathoverflow.net/users/36721 | A common combinatorial description for a certain type of recurrences | So, the Fibonacci numbers can be constructed from this recursion, so it is natural to look for generalizations of those (and combinatorial models for the Fibonacci numbers).
The Fibonacci number $f\_n$ (up to some index shift), can be seen as the number of integer compositions of $n$ (list of numbers summing to $n$), u... | 5 | https://mathoverflow.net/users/1056 | 413023 | 168,519 |
https://mathoverflow.net/questions/412940 | 54 | Define the function $$S(N, n) = \sum\_{k=0}^n \binom{N}{k}.$$
For what values of $N$ and $n$ does this function equal a power of 2?
There are three classes of solutions:
* $n = 0$ or $n = N$,
* $N$ is odd and $n = (N-1)/2$, or
* $n = 1$ and $N$ is one less than a power of two.
There are only two solutions $(N, ... | https://mathoverflow.net/users/136 | When do binomial coefficients sum to a power of 2? | The case $n=2$ was settled by Nagell in 1948 and suspected (?) by Ramanujan in 1913, but in an equivalent form.
As John points out in his growing [blog post](https://www.johndcook.com/blog/2022/01/01/turning-the-golay-problem-sideways/), the $n = 2$ case is a quadratic equation which, via the quadratic formula, requi... | 24 | https://mathoverflow.net/users/14807 | 413024 | 168,520 |
https://mathoverflow.net/questions/413020 | 5 | 1. Consider the co-monad $M:=\Sigma^\infty \Omega^\infty$ on the category of spectra.
It is clear that given a pointed space $X$, $M\Sigma^\infty X=\Sigma^\infty E(X)$,where $E(X)$ is the free unital $E\_\infty$-algebra on $X$.
As $\Sigma^\infty$ commutes with colimit, $M\Sigma^\infty X$ is equivalent to
the free $E\_\... | https://mathoverflow.net/users/170573 | On the endofunctor $\Sigma^\infty\Omega^\infty$ | For connected spectra $A$, there is an equivalence
$$\Sigma^\infty \Omega^\infty A \simeq \bigvee\_{n=1}^\infty A^{\wedge n}\_{h\Sigma\_n}$$
if and only if $A$ is a wedge summand of a suspension spectrum.
This is Theorem 1.2 in N. Kuhn's paper *Suspension Spectra and Homology Equivalences* (TAMS, 1983). Kuhn calls sp... | 4 | https://mathoverflow.net/users/6668 | 413030 | 168,524 |
https://mathoverflow.net/questions/412984 | 2 | A uniformly random $r$-regular bipartite graph on $n$ vertices is known to be $r$-edge connected. That is, with high probability as $n$ grows large, the minimum size of a cut in a random $r$-regular bipartite graph is $r$.
I was wondering if there is a similar statement for the following small extension: Is a uniform... | https://mathoverflow.net/users/256325 | Is a random $(r+1,r)$-biregular bipartite graph $r$-edge connected w.h.p? | The problem can be solved for $r\geq7$ by the following 2nd-eigenvalue results.
The first is from the paper [Edge-Disjoint Spanning Trees, Edge Connectivity, and Eigenvalues in Graphs](https://onlinelibrary.wiley.com/doi/abs/10.1002/jgt.21857).
>
> Theorem 1.6. Let $k$ be an integer with $k \geq 2$ and $G$ be a g... | 1 | https://mathoverflow.net/users/125498 | 413035 | 168,526 |
https://mathoverflow.net/questions/413040 | 5 | Does the 2-category of Grothendieck topoi have exponential objects?
There are size issues: Since Grothendieck topoi are supposed to have a small set of generators, the collection of objects of a Grothendieck topos has to be a *class* and cannot be a [conglomerate](https://en.wikipedia.org/wiki/Conglomerate_(mathemati... | https://mathoverflow.net/users/473904 | Does the 2-category of topoi have exponential objects? | No. [Some, but not all topoi are exponentiable](https://ncatlab.org/nlab/show/exponentiable+topos).
| 11 | https://mathoverflow.net/users/2362 | 413041 | 168,528 |
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