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https://mathoverflow.net/questions/419155 | 2 | Let $k[x,y]$ be the polynomial ring in two variables over a field $k$ of characteristic zero. Every $k$-algebra automorphism of $k[x,y]$ is tame (e.g. the paper of McKay and Wang). It was pointed out in an answer to [this question](https://mathoverflow.net/questions/228657/is-a-wild-automorphism-of-kx-1-ldots-x-n-n-geq... | https://mathoverflow.net/users/361094 | Is a finite order automorphism of k[x,y] necessarily linear? | The answer seems to be no. The automorphism group of $k[x,y]$ can be described as the amalgamated product of the affine transformations with the group of maps of type (i) as in the abstract of McKay and Wang. The torsion elements in an amalgamated product are those elements conjugate to a torsion element of one of the ... | 1 | https://mathoverflow.net/users/361094 | 419201 | 170,646 |
https://mathoverflow.net/questions/419172 | 11 | Classical Galois theory gives necessary and sufficient conditions for the roots of a polynomial in $k[x]$ to be expressible in terms of nested radicals of the coefficients.
Suppose instead that a single root $\alpha$ of $p(x)\in \mathbb{Q}[x]$ is known. Are there known necessary and sufficient conditions on $p(x)$ su... | https://mathoverflow.net/users/27513 | Polynomials for which roots can be expressed as polynomials in a single root | Let $\alpha=\alpha\_1$, $\alpha\_2$, ..., $\alpha\_n$ be the roots of $p(x)$. You want $\mathbb{Q}(\alpha\_1,\alpha\_2, \ldots, \alpha\_n) = \mathbb{Q}(\alpha)$. If the Galois group is $G \subseteq S\_n$, then $\mathbb{Q}(\alpha\_1)$ corresponds to the stabilizer of $1$ in $G$, and $\mathbb{Q}(\alpha\_1,\alpha\_2, \ldo... | 9 | https://mathoverflow.net/users/297 | 419203 | 170,647 |
https://mathoverflow.net/questions/419182 | -1 | As we all know, the complex number field $\mathbb{C}$ be a finite Galois extension field of the real number field that contains all algebraic numbers.
I want to know the proof of the following proposition:
Any p-adic number field $\mathbb{Q}\_p$ has a finite Galois extension field $E$ such that $E$ contains $\overline{... | https://mathoverflow.net/users/278738 | $p$-adic number field $\mathbb{Q}_p $and algebraic numbers | I am trying to answer what it seems you ask.
About your last question: "Is there an extension field of $\mathbb{Q}$, such that it can be embedded into $\mathbb{Q}\_p$ and it can be embedded into the real number field? If the answer is yes, is there the largest such field?"
If you consider **algebraic** extensions, ... | 1 | https://mathoverflow.net/users/158462 | 419204 | 170,648 |
https://mathoverflow.net/questions/419096 | 6 | Let $A,B$ be a pair of quasi-hereditary algebras and assume that $A$ and $B$ are both standard Koszul. Further assume that the graded decomposition matrices of $A$ and $B$ coincide (that is, the multiplicities of standard modules in projective modules coincide, as do their grading shifts). Under what circumstances coul... | https://mathoverflow.net/users/19113 | Are standard Koszul algebras with the same Kazhdan-Lusztig polynomials Morita equivalent? | Let $k$ be an infinite field. I will descibe an infinite family of standard Koszul
algebras with the same graded decomposition matrix.
They will be algebras given by a quiver with relations. The quiver
will be the same in all cases: nine vertices $a\_{1}, a\_{2}, a\_{3},
b\_{1}, b\_{2}, b\_{3}, c\_{1}, c\_{2}, c\_{3}... | 3 | https://mathoverflow.net/users/22989 | 419207 | 170,651 |
https://mathoverflow.net/questions/418832 | 1 | Recall that the extension of function from $u:\mathbb{R}^n\to \mathbb{R}$ can be defined using the Poisson Kernel as follows:
$$u^{\mathrm{e}}(\mathbf{x}):=\gamma\_{n} \int\_{\mathbb{R}^{n}} \frac{x\_{n+1} u(y)}{\left(|x-y|^{2}+x\_{n+1}^{2}\right)^{\frac{n+1}{2}}} \mathrm{~d} y \quad \text { for } \mathbf{x}=\left(x, x... | https://mathoverflow.net/users/68232 | Fourier transform of the fractional Poisson kernel | Yes indeed, for fractional $s$ it is related to Modified Bessel Functions of 2nd Kind (Macdonald Function). For $r=|\mathbf{x}|,\ a=|x\_{n+1}|$ $$P(r,a)=\frac{a^{2s}}{(r^2+a^2)^{n/2+s}}$$ The Fourier Transform of an n-dimensional radial function $f(r)$ is a radial function in the trasformed space $\mathcal{F}(|\xi|)$. ... | 2 | https://mathoverflow.net/users/141375 | 419223 | 170,655 |
https://mathoverflow.net/questions/419220 | 2 | Let $X \sim \operatorname{Bin}(n,p)$. Suppose we estimate $p$ by $\hat{p}=\frac{X}{n}$. By Hoeffding’s inequality
it holds for all $\delta \in (0,1)$ with probability at least $1-\delta$ that, $$\lvert\hat{p}-p\rvert\le \sqrt{\frac{\log\frac{2}{\delta}}{2n}}.
$$
I am interested in a matching non-asymptotic high probabi... | https://mathoverflow.net/users/134624 | Lower bound on the error of proportion estimation | $\newcommand{\de}{\delta}$You want a huge deal more than what there is in reality.
Indeed, you want
\begin{equation\*}
p\_n:=P\Big(|\hat p-p|\ge c\sqrt{\frac{\ln(1/\de)}n}\,\Big)\ge1-\de \tag{1}\label{1}
\end{equation\*}
for some $c\in(0,\infty)$, all large enough $n$, and all small enough $\de>0$.
By the central ... | 6 | https://mathoverflow.net/users/36721 | 419225 | 170,656 |
https://mathoverflow.net/questions/418997 | 3 | Does there exist some method for finding an analytic expression for the coefficient of $z\_1^kz\_2^kz\_3^k$ in:
$$[(1+z\_1)(1+z\_2)(1+z\_3)(1+z\_1z\_2)(1+z\_1z\_3)(1+z\_2z\_3)(1+z\_1z\_2z\_3)]^{k}$$
or is it hopeless?
I can't think of any other method than trying to expand each factor.
Background: the above pol... | https://mathoverflow.net/users/136218 | Analytic expression for the coefficient of a multivariate polynomial | I was coming to the same conclusion that Brendan McKay posted in the comments at about the same time: the efficient way to calculate this is the direct approach $$\sum\_{r,s,t,u} \binom{k}{r} \binom{k}{s} \binom{k}{t} \binom{k}{u} \binom{k}{k-r-s-u} \binom{k}{k-r-t-u} \binom{k}{k-s-t-u}$$ where the sum is over the supp... | 2 | https://mathoverflow.net/users/46140 | 419226 | 170,657 |
https://mathoverflow.net/questions/419232 | 7 | Let $G$ be a semisimple Lie group and let $G = KAK$ be a Cartan decomposition.
For $\mathrm{SL}\_2(\mathbb{R})$ it holds for every $g \in G$ that $KgK = Kg^{-1}K$.
Does the same hold for every semisimple Lie group?
| https://mathoverflow.net/users/122635 | Question on KAK decomposition | No, this fails already in $\mathrm{SL}\_3(\mathbb{R})$. If $Kg\_1K=Kg\_2K$, then $g\_1^Tg\_1$ is conjugate (by an element of $K$) to $g\_2^Tg\_2$. So $g\_1^Tg\_1$ and $g\_2^Tg\_2$ have the same (positive eigenvalues). In particular, if $g\_1$ and $g\_2$ are positive diagonal matrices, then they have the same diagonal e... | 8 | https://mathoverflow.net/users/11919 | 419235 | 170,661 |
https://mathoverflow.net/questions/419238 | 3 | Let $(a\_n)\_{n\in\mathbb{N}}$ be a sequence of positive number such that $\sum\_n a\_n < +\infty$ (i.e. $a\_n \in \ell^1$) but $\sum\_n r^n a\_n = +\infty$ for every $r > 1$.
Given $\sigma \in (0,1)$, I would like to prove a lower bound on the function which maps $t \geq 1$ to
$$
H\_\sigma(t) := \sum\_n \frac{4^{n \... | https://mathoverflow.net/users/50777 | Tauberian lower bound for a series | $\newcommand{\si}{\sigma}\newcommand{\ep}{\varepsilon}$The answer is no. If the sequence $(a\_n)$ is lacunary enough, then $H\_\si(t)$ may behave for some large $t$ roughly as just one of the summands $\frac{4^{n \si}}{4^n+t^2} a\_n$, so that for such $t$ we have $H\_\si(t)\ll t^{-2+\ep}$ for every real $\ep>0$, and he... | 2 | https://mathoverflow.net/users/36721 | 419249 | 170,666 |
https://mathoverflow.net/questions/419208 | 1 | Perhaps there is a simple answer, but I'm very puzzled by the following question:
**Question**: Does there exist a (smooth, connected) algebraic group $G$ such that the general centralizer (i.e. the centralizer in a Zariski open set) is finite?
I'm pretty sure the answer is no (maybe some extra assumptions such as ... | https://mathoverflow.net/users/86596 | General centralizer of algebraic group | Every element $g\in G$ is contained in a Borel subgroup $B\subseteq G$. The quotient $B^{ab}:=B/(B,B)$ has positive dimension since $B$ is solvable. Moreover, the $B$-conjugacy class of $g$ maps to a point in $B^{ab}$. Hence $\dim B/C\_B(g)\le\dim (B,B)$ and therefore $\dim C\_G(g)\ge\dim C\_B(g)\ge \dim B^{ab}>0$.
| 4 | https://mathoverflow.net/users/89948 | 419250 | 170,667 |
https://mathoverflow.net/questions/419212 | 12 | ### Motivation
The question "[Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?](https://mathoverflow.net/q/418554/78525)" asks for an explanation of the sequence 2, 8, 8, 18, 18, 32, … of row lengths in the periodic table. The question currently has t... | https://mathoverflow.net/users/78525 | Mathematical explanation of orbital shell sizes: why is it sufficient to consider single-electron wave functions? | Let me expand a bit on my comment, focusing on points 1 and 2. (I have no meaningful response to 3.) It may be instructive to consider the simplest multi-electron atom, Helium, with two electrons. In the shell model one says these occupy two hydrogenic 1s states with opposite spin. This is a qualitative, approximate st... | 8 | https://mathoverflow.net/users/11260 | 419252 | 170,668 |
https://mathoverflow.net/questions/419241 | 11 | There are many closed manifolds with universal cover homotopy equivalent to $\mathbb{R}^n$, they are precisely the closed aspherical manifolds. There are also many closed smooth manifolds with universal cover diffeomorphic to $\mathbb{R}^n$, e.g. those which admit a metric of non-positive curvature. If one weakens diff... | https://mathoverflow.net/users/21564 | Is there a closed manifold whose universal cover is $\mathbb{R}^n\setminus\{x_1, \dots, x_k\}$ for some $k > 1$? | If we demand that the universal cover is homeomorphic / diffeomorphic to $\mathbb{R}^n \setminus \{x\_1,\ldots,x\_k\}$ with $k>1$ the answer is no, there are no such closed manifolds. Each missing point (together with the "infinity" of the one-point compactification of $\mathbb{R}^n$) is an end of the covering space, a... | 18 | https://mathoverflow.net/users/75344 | 419258 | 170,670 |
https://mathoverflow.net/questions/419270 | 2 | [This article](https://link.springer.com/article/10.1007/BF03022866) states that in 1930 Skolem independently (of Presburger) published quantifier-eliminations for linear algebra over the integers.
I have checked the Ω-Bibliography of Mathematical Logic and was not able to find such a result.
According to Hao Wang'... | https://mathoverflow.net/users/82839 | Skolem's version of quantifier elimination for linear algebra over the integers | I don’t have access to Skolem’s treatise, but according to its reviews by [Gödel](https://zbmath.org/?q=an:0002.00302) and [Ackermann](https://zbmath.org/?q=an:57.1320.03), he proves quantifier elimination for a somewhat unusual system with variables ranging over integers, function symbols for addition, multiplication ... | 5 | https://mathoverflow.net/users/12705 | 419278 | 170,675 |
https://mathoverflow.net/questions/419253 | 3 | It is well known that a complex harmonic function $f$ on a simply connected domain $D$ has a canonical decomposition of the form $$f=g+\bar{h},$$ where $g$ and $h$ are analytic functions on $D.$ In fact, this decomposition is unique if we assume $g(z\_0)=0,$ for some $z\_0\in D.$
Let $\{f\_n\}$ be a sequence of non c... | https://mathoverflow.net/users/143655 | Component wise convergence of a sequence of complex harmonic functions | The answer is positive. Consider a small disk $D(a,r)$ in the domain of
uniform convergence (the center is at $a$, radius $r$), and expand $f\_n(a+te^{i\theta})$ into a harmonic Fourier series:
$$f\_k(a+te^{i\theta})=\sum\_{n=-\infty}^\infty c\_{k,n}t^ne^{in\theta},\quad t<r.$$
Then we have $c\_{k,n}\to c\_n$, as $k\to... | 2 | https://mathoverflow.net/users/25510 | 419280 | 170,677 |
https://mathoverflow.net/questions/419284 | 3 | I am looking for examples of invertible sheaves in smooth, projective families such that the associated base locus (i.e., the intersection of all the effective divisors in the complete linear system) jumps. More precisely, take a discrete valuation ring $R$ and $\pi: X \to \mathrm{Spec}(R)$ be a smooth, projective morp... | https://mathoverflow.net/users/43198 | Examples of jumping base locus of complete linear systems | Take $X$ to be $\mathbb P^2$ blown up at three $R$-points which are colinear on the special fiber and not on the generic, and take $L$ to be $\mathcal O(2)$ minus the three exceptional divisors. The line containing the three points will be the base locus of the special divisor, because its intersection number with $L$ ... | 5 | https://mathoverflow.net/users/18060 | 419287 | 170,678 |
https://mathoverflow.net/questions/419251 | 5 | Assume that $g$ and $g'$ are metric tensors with one dimensional kernel and the same signature.
Does the classical results of Weyl (dim >3) or of Cotton (dim=3) generalise to that case, i.e. $g$ and $g'$ are conformally equivalent iff their Weyl (resp. Cotton) tensors are equal?
| https://mathoverflow.net/users/153070 | Conformal equivalence degenerate metric tensors | In order to define an actual Weyl or Cotton tensor, one has to have a non-degenerate conformal structure. In the OP's case, we aren't given such a structure, so the only way to get an actual Weyl or Cotton tensor would be to construct a conformal structure out of the given data, and that, as Ben McKay points out, is no... | 6 | https://mathoverflow.net/users/13972 | 419295 | 170,679 |
https://mathoverflow.net/questions/419298 | 0 | Let $S=K[x\_1,\ldots, x\_n]$ polynomial ring. Let $I \subseteq S$ an ideal and $<$ be a (global) monomial order in $S$. If in$\_<(I)$ a radical ideal, then in$\_<(I)=$ in$\_<(P\_1) \;\cap$ in$\_<(P\_2)\cap \ldots \cap$ in$\_<(P\_l)$, where $P\_1,\ldots, P\_l$ are the minimal prime ideals of $I$.
Question
Is this tr... | https://mathoverflow.net/users/479175 | Monomial order and initial ideals | Since $in(I)$ is a radical ideal, $I$ is a radical ideal. Thus, $I=\bigcap\_{i=1}^{l} P\_i$, and so, $in(I)=in\left( \bigcap\_{i=1}^{l} P\_i \right) \subseteq \bigcap\_{i=1}^{l} in(P\_i)$.
Now, we show that $\bigcap\_{i=1}^{l} in(P\_i) \subseteq in(I)$. As each $in(P\_i)$ is a monomial ideal, $\bigcap\_{i=1}^{l} in(P... | 0 | https://mathoverflow.net/users/479175 | 419303 | 170,681 |
https://mathoverflow.net/questions/419297 | 8 | I like to use MathSciNet's BibTeX feature. However, I recently found out that sometimes their BibTeX entry removes the initial zeros on a MR number, and other times it includes the initial zeros. Is there a preferred choice between these two options? Alternatively, is there a reason I'm missing that explains why MathSc... | https://mathoverflow.net/users/3199 | Math Review #'s — Include the initial zeros? | In most instances, when displaying items with MR numbers that are less than 1000000, we pad the number by prepending with zeros to obtain a 7-digit number. The numbers, however, are stored just as numbers in the database. When we export the record to either BibTeX or AMSRefs, we use the number, no padding. So, "short" ... | 16 | https://mathoverflow.net/users/49409 | 419307 | 170,682 |
https://mathoverflow.net/questions/412795 | 3 | Let $V$ be an affine real algebraic set. That is, $V$ is the zero set of some polynomials in $\mathbb{R}^n$. I would like to show that there is not a proper algebraic subset $W\subset V$ which admits a surjective polynomial map $W\twoheadrightarrow V$. The plan to do this is to take the ordered list of dimensions of ir... | https://mathoverflow.net/users/1345 | Reference request: ordered list of dimensions of components of a variety? | You can prove this without needing to keep track of dimensions.
Suppose that $W$ and $V$ are real algebraic sets with $W \subsetneq V$, and that there exists a surjective polynomial map
$\phi: W \twoheadrightarrow V$. Then define a sequence of algebraic sets $W\_i$
with $W\_0 = W$ and $W\_{i+1} = \phi^{-1}(W\_i)$. Si... | 5 | https://mathoverflow.net/users/479686 | 419312 | 170,683 |
https://mathoverflow.net/questions/419305 | 4 | We write $[N]$ to denote $\{1,\dots,N\}$. We say a set $S$ is $k$-AP-free if it lacks non-trivial arithmetic progressions of length $k$.
We define the 2-color van der Waerden number, $w(2;k)$, to be the largest integer $N$ where $[N]$ can be partitioned into two $k$-AP-free sets. For general $k$, the best lower bound... | https://mathoverflow.net/users/130484 | Can we do better than random when constructing dense $k$-AP-free sets | We can take the set of all numbers in base $k$ that don't contain the digit $0$, for $k$ prime.
This is $k$-term-progression-free since every $k$-term progression in $\mathbb F\_k$ is either constant or contains $0$, thus any $k$-term progression in $\mathbb Z$ takes all possible values in the last nonconstant digit.... | 10 | https://mathoverflow.net/users/18060 | 419315 | 170,684 |
https://mathoverflow.net/questions/419308 | 5 | A *rooted, labeled tree on $n$ vertices* is a tree with vertex set $[n] := \{1,2,\ldots,n\}$ in which one vertex has been designated the root. A *leaf* of a rooted tree is a vertex $v$ for which either: $v$ is not the root and has degree $1$; or $v$ is the root and has degree $0$.
Using standard generating function t... | https://mathoverflow.net/users/25028 | Bijectively counting labeled trees by number of leaves | If you identify a function $f: [n-1] \to [n]$ with the Prüfer code $(f(1), f(2), \ldots, f(n-1))$ then it corresponds to an unrooted labelled tree on $n+1$ vertices in which the label $n+1$ is a leaf. Designate the neighbour of that leaf as the root and delete the leaf and you have the desired bijection.
| 5 | https://mathoverflow.net/users/46140 | 419316 | 170,685 |
https://mathoverflow.net/questions/419321 | 4 | Is it true
$$ \frac{1}{\Gamma(1-\nu)}\frac{1}{\Gamma(\nu)} \int\_{0}^{x}(x-y)^{-\nu}dy\int\_0^y (y-t)^{\nu-1}f(t)dt = \int\_0^x f(u)du$$
for any continuous function $f(x)$ such that $f(0)=0$ and $0<\nu<1$?
| https://mathoverflow.net/users/152618 | How to validate the exponentiality of fractional calculus? | Without loss of generality we may assume $x>0$. Then also $0<t<x$ and I can use the identity
$$\int\_t^x(y-t)^{\nu-1}(x-y)^{-\nu}\,dy=\frac{\pi}{\sin\pi \nu}=\Gamma(1-\nu)\Gamma(\nu),$$
valid for $0<\nu<1$, $0<x<t$, to conclude that
$$\frac{1}{\Gamma(1-\nu)}\frac{1}{\Gamma(\nu)} \int\_{0}^{x}(x-y)^{-\nu}\,dy\int\_0^y (... | 2 | https://mathoverflow.net/users/11260 | 419335 | 170,693 |
https://mathoverflow.net/questions/297622 | 4 | Let us first recall [Specht's Theorem](https://en.wikipedia.org/wiki/Specht%27s_theorem). Denote by $\text{Mat}\_{\mathbb{C}}(n)$ the set of all $n\times n$ matrices over the complex field $\mathbb{C}$. Let $A$ be a matrix in $\text{Mat}\_{\mathbb{C}}(n)$ and denote by $A^\*$ its conjugate transpose. A word $w(A,A^\*)$... | https://mathoverflow.net/users/9839 | A variant of Specht's Theorem using sum of elements (rather than trace) of complex matrices? | A sum-of-entries version of Specht's theorem was proven in [Theorem 20 of Grohe et al. (2022): Homomorphism Tensors and Linear Equations](https://drops.dagstuhl.de/opus/volltexte/2022/16411). The condition you're describing is also sufficient. More precisely, for matrices $A$ and $B$ it holds that $\sigma(w(A,A^\*)) = ... | 4 | https://mathoverflow.net/users/479711 | 419340 | 170,694 |
https://mathoverflow.net/questions/419349 | 2 | I am reading Lurie's Elliptic Cohomology II and it claims (Section 4.1.3) that for an $\mathbb{E}\_\infty$-ring $A$ "there is an essentially unique symmetric monoidal functor $\mathcal{S} \to \operatorname{Mod}\_A$ which preserves small colimits", where $\mathcal{S}$ is the category of spaces.
At first I thought this... | https://mathoverflow.net/users/170467 | Monoidal colimit-preserving functor from spaces to $A$-modules | Your description of the functor is correct. The suspension spectrum functor is homotopy colimit preserving, as you say. Smashing with A is left adjoint to the forgetful functor from A-modules to spectra, and this means that it also preserves homotopy colimits.
| 3 | https://mathoverflow.net/users/360 | 419356 | 170,698 |
https://mathoverflow.net/questions/419343 | 2 | Let $X$ be a smooth projective variety and $E$ be a line bundle on $X$. Let $F$ be a line bundle on $X\times \mathbb{P}^1$. It is known that, if $p\_1^\*E|\_{X\_t}\cong F|\_{X\_t}$ for every $t\in \mathbb{P}^1$, then we have $F\cong E\boxtimes \mathcal{O}\_{\mathbb{P}^1}(k)$ for some $k$.
>
> **Question.** Can this... | https://mathoverflow.net/users/153842 | Coherent sheaves that are isomorphic on every fibre | It seems to me that the statement is false already for rank $2$ vector bundles.
In fact, take $X=\mathbb{P}^1$. Then, it is known **[1]** that there exist indecomposable, uniform rank $2$ vector bundles $F$ on $\mathbb{P}^1 \times \mathbb{P}^1$, namely, indecomposable bundles such that $$F|\_{X\_t} = \mathcal{O}(a) \... | 2 | https://mathoverflow.net/users/7460 | 419361 | 170,699 |
https://mathoverflow.net/questions/419319 | 25 | For each positive integer $n$, split the integers $1$ to $2n$ into two sets of $n$ elements each, and such that the products of the elements in each of these sets are as close as possible, say they differ by $a(n)$.
It can be checked that $a(1)=1$, $a(2)=2$, $a(3)=6$, $a(4)=18$, and $a(5)=30$.
Is this sequence stri... | https://mathoverflow.net/users/60732 | Splitting the integers from $1$ to $2n$ into two sets with products as close as possible | Original question
-----------------
>
> Is this sequence strictly increasing?
>
>
>
No.
```
n difference smaller half
16 16753029012720 [3, 5, 6, 7, 9, 10, 11, 13, 15, 18, 19, 21, 25, 27, 29, 30]
17 10176199188480 [4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 18, 19, 21, 22, 27, 28, 33]
```
>
> What about if... | 19 | https://mathoverflow.net/users/46140 | 419370 | 170,701 |
https://mathoverflow.net/questions/419362 | 21 | After reading some recent questions on mathoverflow about universal coverings, I am curious about the following:
Is it possible to construct a closed $6$-manifold $M$, with universal cover homeomorphic to $\mathbb{CP}^3 \setminus \{p\_1,p\_2\}$?
| https://mathoverflow.net/users/99732 | Is $\mathbb{CP}^3$ minus two points the universal cover of a compact manifold? | More is true. Let $M^n$ be a closed connected simply connected manifold of dimension $\ge 3$. Let $p,q\in M$ be two distinct points. Suppose $M\setminus\{p,q\}$ is the universal cover of a closed manifold.
Then $M$ is homeomorphic to $\mathbb S^n$.
Suppose it is and $M=\tilde N$ is the universal cover of a closed m... | 35 | https://mathoverflow.net/users/18050 | 419376 | 170,705 |
https://mathoverflow.net/questions/419371 | 3 | Let's consider the $1$-variable rational function
$$F(z):=\frac{1-z}{(z^3 - z^2 + 2z - 1)\,(z^3 + z^2 + z - 1)}.$$
Numerical evidence convinces me of the truth of the following.
>
> **QUESTION.** Can you prove that $F(z)$ is positive, in the sense that its Taylor series at $z=0$ has positive coefficients?
>
>
>... | https://mathoverflow.net/users/66131 | Positivity of a one-variable rational function | Let $1/(1-2z+z^2-z^3)=\sum\_{n\geq 0} f(n)z^n$. Then $f(0)=1$,
$f(1)=2$, $f(2)=3$, and
$f(n+1)=2f(n)-f(n-1)+f(n-2)=f(n)+(f(n)-f(n-1))+f(n-2)$, $n>2$. It
follows that $f(n)$ is strictly increasing, so $(1-z)/(1-2z+z^2-z^3)$
has positive coefficients. Similarly, or because
$$ \frac{1}{1-z-z^2-z^3} = \sum\_{m\geq 0}(z+z^2... | 20 | https://mathoverflow.net/users/2807 | 419377 | 170,706 |
https://mathoverflow.net/questions/417748 | 1 | I recently looked through the proof of the Gagliardo–Nirenberg Interpolation Inequality, see [proof](https://arxiv.org/pdf/1812.04281.pdf) and it says that for real line $R$, there exists a sequence of open intervals $\{I\_k\}$, which covers the compact support domain with
$$
\sum\_k \chi\_{I\_k}\le 4
$$
I have read th... | https://mathoverflow.net/users/295572 | Constant bound for the 1 dimensional Besicovitch covering theorem on real line | Given a covering by open intervals of a compact set $K \subset {\mathbb R}$, there is a finite subcover $S$, and we will use intervals from this subcover. Let $S\_1$ be the set of intervals in $S$ that cover $x\_1=\min K$. Choose $I\_1$ as the interval in $S\_1$ with the largest right endpoint, (breaking ties arbitrari... | 2 | https://mathoverflow.net/users/7691 | 419392 | 170,713 |
https://mathoverflow.net/questions/419393 | 0 | I'm sure i have read that the following (or something that implies this) is true
>
> Let $X$ be a $\Pi\_1^0$ class with top Medvedev degree. Then for every
> $x\in X$, there is $y\in X$ with $y<\_T x$.
>
>
>
But i don't remember where. If this is true, do you know where can I find this or a similar statement?
... | https://mathoverflow.net/users/282044 | Turing degrees inside the $\Pi_1^0$ class with top Medvedev degree | We can modify the proof of the Kreisel/Shoenfield basis theorem (see Theorem 3.7 in [Diamondstone/Dzhafarov/Soare](https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-51/issue-1/%CE%A0-1-0-Classes-Peano-Arithmetic-Randomness-and-Computable-Domination/10.1215/00294527-2010-009.full)):
Let $T\s... | 4 | https://mathoverflow.net/users/8133 | 419401 | 170,717 |
https://mathoverflow.net/questions/419406 | 2 | How can you prove that
$$
\sum\_{r=0}^{2k-2} (-1)^r \binom{2k-2}{r} (5k-2-r)^{2k-2} =(2k-2)!
$$
This result I have obtained by comparing results of two different approaches for the partitioning of the set of vertices of a convex n-gon into nonintersecting polygons.
| https://mathoverflow.net/users/479476 | Partitioning the set of vertices of a convex n-gon into nonintersecting polygons | Note that more generally
$$
\nabla^{2k-2}[x^{2k-2}](x) = \sum\_{r=0}^{2k-2}(-1)^r\binom{2k-2}{r}(x-r)^{2k-2}
$$
is the order-$(2k-2)$ [backwards finite difference operator](https://en.wikipedia.org/wiki/Finite_difference#Higher-order_differences) acting on the monomial $x^{2k-2}$, which is the constant $(2k-2)!$.
| 8 | https://mathoverflow.net/users/47484 | 419411 | 170,719 |
https://mathoverflow.net/questions/419374 | 3 | While thinking about convex functions, I managed to put together the following proof which I find a bit too good to be true. $X$ is a topological vector space that is also a Baire space.
**Lemma:** Let $f : X \to \mathbb{R}$ be convex and locally bounded. Then $f$ is continuous.
*proof:* Let $x \in X$ and $U \subse... | https://mathoverflow.net/users/18936 | Is a convex, lower semicontinuous function that is bounded from below, actually continuous? | Though not entirely in the same setting, as can be seen from [these lecture notes](https://people.math.ethz.ch/%7Epatrickc/CA2013.pdf) my reasoning seems to hold. In the lecture notes, one considers barrelled spaces but the local boundedness at some point can easily be obtained in either the barrelled or Baire setting,... | 2 | https://mathoverflow.net/users/18936 | 419412 | 170,720 |
https://mathoverflow.net/questions/419084 | 5 | Let G be a finite simple graph. Consider the independent number $\alpha$, the chromatic number $\chi$ and the path cover number (also called the path partition number) $\rho$.
Then we have $\alpha\chi \ge n$ and $\alpha \ge \rho$.
Is there any relationships between the path cover number $\rho$ and the chromatic numbe... | https://mathoverflow.net/users/160959 | Is there any relationships between path cover number and chromatic number? | @vidyarthi: The answer to your question is too long for a comment.
Let $G$ be a simple graph and let $\rho$ be the smallest integer such that there exists a system of pairwise disjoint paths $P\_1,\ldots,P\_\rho$ containing all vertices of $G$. Then we assume $\rho(G)=\rho$. By the way, sometimes $\rho(G)=-1$ is assu... | 1 | https://mathoverflow.net/users/173068 | 419422 | 170,724 |
https://mathoverflow.net/questions/418943 | 1 | I have a question about Siegel's Lemma (<https://en.wikipedia.org/wiki/Siegel%27s_lemma>) and bounding of underdetermined linear systems (n > m) of the form Ax = b. While the proof provides for an integer solution, is there a way to generalize for a real valued solution with the additional constraint that xi >= 0?
| https://mathoverflow.net/users/479388 | Bounded underdetermined linear system | With the little details you include in your question. I think you can use [Farkas Lemma](https://en.wikipedia.org/wiki/Farkas%27_lemma). It tell you that exactly one of the following two assertions is true: 1. There exists an ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}$ such that ${\displaystyle \mathbf {Ax} =\ma... | 0 | https://mathoverflow.net/users/478686 | 419423 | 170,725 |
https://mathoverflow.net/questions/419410 | 3 | Let $\tau$ be a random variable, which is defined on the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F})\_{t\in T}, P)$ with values in $T$. In most cases, $T=[0,\infty]$. Then $\tau$ is called a stopping time (with respect to the filtration $(\mathcal{F})\_{t\in T}$), if the following condition holds:
$... | https://mathoverflow.net/users/147009 | What is the difference $\{\tau\leq t\}\in (\mathcal{F})_t $ and $\{\tau<t\}\in (\mathcal{F})_t $ in the definition of stopping time? | Two definitions:
(ST1) for all $t \in [0,\infty]$, $\{\tau < t\} \in \mathcal{F}\_t$
(ST2) for all $t \in [0,\infty]$, $\{\tau \le t\} \in \mathcal{F}\_t$
It is true that (ST2) $\Longrightarrow$ (ST1). Indeed, assume (ST2). Given $t$, we have
$$
\{\tau < t\} = \bigcup\_{s < t, s \in \mathbb Q}\{\tau \le s\} \i... | 3 | https://mathoverflow.net/users/454 | 419428 | 170,727 |
https://mathoverflow.net/questions/419429 | 2 | Most examples of ample line bundles that are not globally generated have less number of global sections than the dimension of the variety. Assuming ampleness, is the existence of "enough" global sections sufficient to guarantee globally generated-ness? More precisely:
>
> Let $X$ be a smooth, projective $\mathbb{C}... | https://mathoverflow.net/users/43198 | How far is ample from globally-generated | The answer is no for curves, and then a product construction show that the answer is no in every dimension.
Take a hyperelliptic curve $X$ of genus $g \geq 3$ and let $P$ be a 2-torsion line bundle on $X$, such that $P = \mathcal{O}\_C(p-q)$, where $p$, $q$ are distinct Weierstrass points. Now set $L = \omega\_C \oti... | 5 | https://mathoverflow.net/users/7460 | 419430 | 170,728 |
https://mathoverflow.net/questions/419431 | 3 | Let us consider the linear transport equation
$$
\partial\_t u + \mathrm{div}(a(t,x)u)=0
$$
with initial data $u(0,\cdot) = u\_0$ in $\mathbb R^N$.
Here we consider a smooth Lipschitz vector field $a$.
What happens to the maximum of $u$ along the evolution? Is it true that $\|u(t,\cdot)\|\_{L^\infty} = \|u\_0\|\_{L... | https://mathoverflow.net/users/122620 | Maximum principle and linear transport | This is not a transport equation. It is a conservation law. The difference between these class is that a TE is of the form $\partial\_tu+a(t,x)\cdot\nabla\_xu=0$, for which the essential supremum/infimum in the space variable remains constant as time varies. On the contrary, the space integrals of the positive/negative... | 8 | https://mathoverflow.net/users/8799 | 419433 | 170,729 |
https://mathoverflow.net/questions/419299 | 2 | Let $I=[0,1]$ be the unit segment, and let $(I\_n)\_{1\leq n \leq N}$ be $N$ almost disjoint sub-intervals $I\_n=[t\_n-\delta\_n,t\_n+\delta\_n]$ of $I$ (that is, their interior are disjoint). Let $\chi(x)=\frac{1}{1+x^2}$ (which plays the role as an approximation of $1\_{[-1,1]}$).
>
> How can one prove that there... | https://mathoverflow.net/users/94414 | Exponential integrability of a sum of approximations of disjoint intervals characteristic functions | The simplest argument I can currently come up with is the following:
Let $f\_j$ be non-negative functions on $[0,1]$ with $\sum\_j f\_j\le 1$ (characteristic functions of disjoint intervals in your case) and let $\Phi\_j$ be some even decreasing on $[0,+\infty)$ averaging kernels (Poisson kernels in your case). Let $... | 7 | https://mathoverflow.net/users/1131 | 419436 | 170,731 |
https://mathoverflow.net/questions/419425 | 4 | Let $k$ be a non-archimedean field of characteristic zero. Then let $$f:X \rightarrow Y$$
be a (proper) morphism of smooth projective varieties over $k$. The GAGA functor (for rigid analytic spaces) induces a commutative square of locally G-ringed spaces
$\require{AMScd}$
\begin{CD}
X^\mathrm{rig} @>f^\mathrm{rig}>> ... | https://mathoverflow.net/users/135674 | Higher direct image of coherent sheaf and rigid analytification | Brian Conrad give a positive answer in <http://www.numdam.org/item/10.5802/aif.2207.pdf> Remark A.1.1 p. 1113/1114.
| 3 | https://mathoverflow.net/users/135674 | 419445 | 170,734 |
https://mathoverflow.net/questions/419458 | 4 | I came across the following assertion: if $f\in PW\_\infty([-a,a])$, i.e. the Bernstein space of functions in $L^\infty(\mathbb{R})$ which are the Fourier transform of a distribution supported on $[-a,a]$, then $$\int\_0^{+\infty}\frac{\ln\frac{1}{|f(x)|}}{1+x^2}\mathrm{dx}<+\infty.$$ The authors say it's a classical r... | https://mathoverflow.net/users/39180 | Integral of $\ln(1/|f|)$ for $f$ bandlimited | This is due to Wiener and Paley, (but can be also derived from the Jensen inequality and description of positive harmonic functions in
a half-plane) and a more general formulation is this:
If $f$ is entire, of exponential type, and
$$\int\frac{\log^+|f(x)|}{1+x^2}dx<\infty,$$
then
$$\int\frac{\log^-|f(x)|}{1+x^2}dx<\in... | 5 | https://mathoverflow.net/users/25510 | 419475 | 170,747 |
https://mathoverflow.net/questions/419462 | 6 | The notion of a knot concordance is a rich subject in low-dimensional topology, see Livingston's [survey](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/living.pdf). More precisely:
For $i=0,1$, let $K\_i$ be knots in $S^3$. A *knot concordance* from $K\_0$ to $K\_1$ is a smooth annulus $A=S^1 \times [0,1]$ in $S^3 \t... | https://mathoverflow.net/users/475366 | A (possible) generalization of the unknot, inverses and the knot concordance | One can certainly define concordance by cylinders in $M \times I$ as you suggest. Making the set of concordance classes into a group is problematic, however. Most of the troubles come from the observation that for two knots to be concordant, they must be freely homotopic. This was explored in the Indiana U. PhD thesis ... | 4 | https://mathoverflow.net/users/3460 | 419488 | 170,750 |
https://mathoverflow.net/questions/419493 | 1 | Let $B\_{n}$ for $n\ge0$ denote the Bernoulli number generated by
\begin{equation\*}
\frac{z}{\textrm{e}^z-1}=\sum\_{n=0}^\infty B\_n\frac{z^n}{n!}=1-\frac{z}2+\sum\_{n=1}^\infty B\_{2n}\frac{z^{2n}}{(2n)!}, \quad \vert z\vert<2\pi.
\end{equation\*}
Could you please help find a reference or give a proof of the identity... | https://mathoverflow.net/users/147732 | Ask for a reference or a proof of an identity involving a finite sum and the Bernoulli numbers | Let, $f(x)=\frac{x}{e^x-1}+\frac{x}{2}-1=\frac{x}{2}\coth(\frac{x}{2})-1$
Now the summation in question can be broken in 4 parts. For example, the first part
$$2^{4k}\sum\_{j=1}^{2k}(-1)^j\binom{4k+2}{2j}B\_{2j}B\_{4k-2j+2}$$ is $(4k+2)!$ times the coefficient of $x^{4k+2}$ in $\frac{1}{4}f(2ix)f(2x)$.
Similarly ... | 3 | https://mathoverflow.net/users/156029 | 419498 | 170,753 |
https://mathoverflow.net/questions/419388 | 7 | Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B\_{\mathbb{H}^n}(x,r)$. Are there known estimates on the minimum distortion of a bi-Lipschitz embedding of $B\_{\mathbb{H}^n}(x,r)$ into $\mathcal{W}\_2(\m... | https://mathoverflow.net/users/469470 | Hyperbolic space embeds into Wasserstein space | For global distortion.
Choose a 4 point set --- vertices of an equilateral triangle in $B(x,r)\_{\mathbb{H}^n}$ and its center and observe that it cannot be emebdded isometrically in nonnegatively curved space in the sense of Alexandrov, which includes $\mathcal{W}\_2(\mathbb{R})$.
In is straightforward to get some... | 2 | https://mathoverflow.net/users/1441 | 419500 | 170,754 |
https://mathoverflow.net/questions/419499 | 2 | Let $n\in\mathbb N^\*$, $P(x)=a\_0+\dotsb+a\_{n-1}x^{n-1}+x^n$ and $r\_1,\dotsc,r\_n\in\mathbb C$ the roots of $P$.
>
> Is it true $\lim\limits\_{\max(\lvert a\_i\rvert,i=0\dotsc n-1)\rightarrow 0} \max(\lvert r\_i\rvert,i=1\dotsc n)=0$?
>
>
> If there is an inequality between the two maxes, can you give it?
>
>... | https://mathoverflow.net/users/110301 | About roots of polynomials | See [an application](https://en.wikipedia.org/wiki/Geometrical_properties_of_polynomial_roots#From_Rouch%C3%A9_theorem) of Rouché's theorem to polynomials. As I remember, it is used in one of the proofs of The Fundamental Theorem of Algebra.
| 6 | https://mathoverflow.net/users/157261 | 419501 | 170,755 |
https://mathoverflow.net/questions/419426 | 0 | How can you partition n number of distinguishable objects into m number of indistinguishable blocks given that each of the blocks consists of not less than k number of objects.
(k =1 case can be explained by Stirling numbers of second kind and
k= 3 case can be used to obtain number of different ways to partition the se... | https://mathoverflow.net/users/479476 | Partitioning distinguishable objects into indistinguishable blocks | These are called "$k$-associated Stirling numbers of the 2nd kind": see <https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind#Associated_Stirling_numbers_of_the_second_kind>.
| 4 | https://mathoverflow.net/users/25028 | 419510 | 170,757 |
https://mathoverflow.net/questions/419519 | 6 | I'm currently reading Talenti's paper "Best constant in Sobolev inequality" and am rather stuck on an argument on pg 363 (or pg 11 if you're reading the pdf). In this section of the paper, Talenti is proving the Polya-Szego inequality. More specifically, I don't see how (23a) implies the Lipschitzness of $u^\*$. For so... | https://mathoverflow.net/users/479534 | Lipschitz property of the symmetric rearrangement | $\newcommand{\R}{\mathbb R}$By the continuity of measure, the nonincreasing function $\mu$ is right-continuous. The function $u^\*$ is radial, that is,
\begin{equation\*}
u^\*(x)=U(|x|)
\end{equation\*}
for some nonincreasing function $U\colon[0,\infty)\to[0,\infty)$ and all $x\in\R^n$. So,
\begin{equation\*}
U(a)... | 6 | https://mathoverflow.net/users/36721 | 419522 | 170,758 |
https://mathoverflow.net/questions/419505 | 5 | A cool construction of Hochschild homology (that I saw on B. Antieau's website [here](https://antieau.github.io/2021/01/15/filtered2020.html) ) is the following:
Let $k$ be a commutative ring, then denote by $\mathfrak{a}\text{CAlg}\_k$ the category of animated commutative rings. For any $x\in BS^1$, we have a natura... | https://mathoverflow.net/users/152554 | Defining Hochschild homology of non-commutative DG-algebras with animated rings with a circle action | No, this does not work in the non-commutative case. In general we have $HH(A)=A\otimes\_{A\otimes A^{\mathrm{op}}} A$, and this is only a $k$-module, not an algebra. If $A$ is commutative, the tensor product happens to compute coproducts/pushouts of commutative $k$-algebras, and we have $HH(A)=\operatorname{colim}\_{S^... | 6 | https://mathoverflow.net/users/20233 | 419524 | 170,759 |
https://mathoverflow.net/questions/419529 | 3 | Let $K$ be a field of characteristic $0$ (number fields is a sufficient generality), $A/K$ an abelian variety, and $X\subseteq A$ a closed reduced subscheme.
>
> I am looking for a reference for the statement that the stablizer of $X$ in $A$ is a $K$-abelian subvariety of $A$.
>
>
>
I hope that the reference w... | https://mathoverflow.net/users/2042 | Stabilizers in abelian varieties are also abelian? reference request | This is not true: it does not need to be connected even if $X$ is smooth connected over an algebraically closed field $K$ of characteristic zero. Indeed, if there is an isogeny $\pi \colon A \to B$ and a (necessarily smooth and connected) subscheme $Y \subseteq B$ such that $X = \pi^\*Y$ (the scheme-theoretic inverse i... | 5 | https://mathoverflow.net/users/82179 | 419537 | 170,764 |
https://mathoverflow.net/questions/419496 | 2 | $\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\PMod{PMod}\DeclareMathOperator\Homeo{Homeo}$I am very confused about the definition of mapping class group and pure mapping class group (and their generating sets).
In *Primer on Mapping class groups*:
>
> $\Mod(S)$ is the group of isotopy classes of elements of ... | https://mathoverflow.net/users/nan | Mapping class group and pure mapping class group (and their generating sets) | This really is very similar to your previous [question](https://mathoverflow.net/questions/417191).
**Answer 1:** If $S$ is compact and without boundary then the two groups are isomorphic. (However, be aware that there are yet other definitions of the pure mapping class group where the latter is finite index in the f... | 1 | https://mathoverflow.net/users/1650 | 419540 | 170,765 |
https://mathoverflow.net/questions/419538 | 5 | Let $f:X \rightarrow Y$ be a finite covering map between compact oriented surfaces and let $K$ be the kernel of the induced map $f\_\ast: H\_1(X) \rightarrow H\_1(Y)$. Here homology has rational coefficients.
Question: must $K$ be nondegenerate with respect to the symplectic algebraic intersection pairing $\omega$ on... | https://mathoverflow.net/users/479892 | Nondegeneracy of kernel of map on homology induced by covering of surfaces | The answer is yes. Let me work with $H^1$, which is canonically isomorphic to $H\_1$ by Poincaré duality. I claim that $K=\operatorname{Ker} f\_\*$ is the orthogonal of $\ \operatorname{Im}f^\* $: this is because $(\beta \cdot f^\*\alpha )=(f\_\*\beta \cdot \alpha )$ for $\alpha $ in $H^1(Y)$ and $\beta $ in $H^1(X)$.
... | 7 | https://mathoverflow.net/users/40297 | 419544 | 170,766 |
https://mathoverflow.net/questions/419503 | 3 | Let $\mathbb{G}$ be a compact quantum group in the sense of Woronowicz. We can look at its associated dense Hopf$^\*$-subalgebra $\mathbb{C}[\mathbb{G}]$. Hence, in the framework of multiplier Hopf $\*$-algebras (as introduced by Van Daele) it has a dual
$$\widehat{\mathbb{C}[\mathbb{G}]}.$$
On the other hand, in the... | https://mathoverflow.net/users/216007 | Relating different definitions of dual of a compact quantum group | This is simply a matter of differing conventions. The definition of a discrete quantum group arising as the dual of a compact quantum group, using the multiplicative unitary, uses the conventions of Locally Compact Quantum Groups. A really useful paper bridging between algebraic quantum groups and LCQGs is [Kustermans,... | 3 | https://mathoverflow.net/users/406 | 419549 | 170,768 |
https://mathoverflow.net/questions/419528 | 2 | It is well known that the Bernoulli numbers $B\_{k}$ for $k\in\{0,1,2,\dotsc\}$ can be generated by
\begin{equation\*}
\frac{z}{\textrm{e}^z-1}=\sum\_{k=0}^\infty B\_k\frac{z^k}{k!}=1-\frac{z}2+\sum\_{k=1}^\infty B\_{2k}\frac{z^{2k}}{(2k)!}, \quad \vert z\vert<2\pi.
\end{equation\*}
Could you please find or give a proo... | https://mathoverflow.net/users/147732 | Ask for a proof of an identity involving the product of two Bernoulli numbers | More generally, let $B\_k(x)$, $k\ge0$, be the Bernoulli polynomials defined by the exponential generating function
\begin{equation\*}
\frac{ze^{xz}}{e^z-1}=\sum\_{k=0}^\infty B\_k(x)\frac{z^k}{k!},
\end{equation\*}
thus $B\_k=B\_k(0)$.
These satisfy the general identity (which can be proven using the definition above,... | 4 | https://mathoverflow.net/users/109085 | 419561 | 170,771 |
https://mathoverflow.net/questions/419506 | 3 | I would like to ask for some clarification on the following argument which I can not quite understand.
There is a variety $X$ of dimension $n$ over a number field with a degree two map $f:X\dashrightarrow \mathbb{P}^{\frac{n}{2}}\times\mathbb{P}^{\frac{n}{2}}$ and a rational map $g:\mathbb{P}^n\dashrightarrow X$. The... | https://mathoverflow.net/users/14514 | Rational points of bounded height on a variety | Given the situation, it's understandable that some stuff is missing from the argument.
One part that's missing is the *definition* of the height.
In modern language (Weil's height machine), we usually take a height function to be defined from an ordered pair of a variety *and a line bundle*, well-defined up to a co... | 2 | https://mathoverflow.net/users/18060 | 419569 | 170,773 |
https://mathoverflow.net/questions/419578 | 5 | Suppose $(a\_1,\ldots, a\_k)$ is an integer partition of $n$, and $(b\_1,\ldots,b\_k)$ is a rearrangement of the $a$-sequence. Prove the following identity (preferably combinatorially):
$$
\sum\_{j\_2+\cdots+j\_k=l,\atop l\geq 0, \, a\_t>j\_t\geq 0} \quad\frac{(-1)^l l!}{(a\_1+l+1)\_{l+1}} {a\_2\choose j\_2}\cdots {a\_... | https://mathoverflow.net/users/479910 | direct proof of an identity regarding certain symmetry of integer partitions? | I assume you meant $j\_t\leq a\_t$ (not $j\_t<a\_t$).
It's sufficient to prove that for any analytic function $f(x)$, the function
$$F(a\_1,a\_2) := \sum\_{l\geq 0} \sum\_{j=0}^{a\_2} \frac{(-1)^ll!}{(a\_1+l+1)\_{l+1}} \binom{a\_2}{j} [x^{l-j}]\ f(x)$$
is symmetric, i.e. $F(a\_1,a\_2) = F(a\_2,a\_1)$.
Using the pro... | 4 | https://mathoverflow.net/users/7076 | 419580 | 170,778 |
https://mathoverflow.net/questions/419566 | 0 | Let $a(n)$ be the number of positive integers $k$ such that there exists a nonnegative integer $m$ with $k + k^m = n$.
The sequence begins
$$0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3$$
Let
$$b(n)=\sum\limits\_{i=1}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\left\lfloor\log\_{i+1}(n-i)\right\rfloor$$
... | https://mathoverflow.net/users/231922 | Number of positive integers $k$ such that there exists a nonnegative integer $m$ with $k + k^m = n$ | The formula $c(n)=a(n+1)$ is pretty much straightforward, noticing that
$$\lfloor \log\_{i+1}(n-i)\rfloor - \lfloor\log\_{i+1}(n-1-i)\rfloor=1\quad\text{iff}\quad n-i=(i+1)^m\text{ for some }m.$$
The latter condition means that $n+1=k+k^m$ with $k:=i+1$.
| 5 | https://mathoverflow.net/users/7076 | 419582 | 170,779 |
https://mathoverflow.net/questions/418955 | 3 | $\DeclareMathOperator\null{null}$Let $H$ be a hyperplane of the paving matroid $M$ with $r(M)=n$. How large can $\null(H)$ be?
We know that $\null(H)=|H|-r(H)=|H|-(n-1)$. So everything boils down to finding the size of the largest hyperplane in $M$.
| https://mathoverflow.net/users/165074 | Nontrivial upper bounds for the nullity of hyperplanes in paving matroids | There is no such bound. Take many vectors in a real hyperplane $H$ in $\mathbb{R}^n$ in general position, and one vector outside $H$. They form a paving matroid, since all circuits have size $n-1$ or $n$.
| 1 | https://mathoverflow.net/users/4312 | 419583 | 170,780 |
https://mathoverflow.net/questions/419202 | 1 | [This question arises from a look at the paper
* Shing-Tung Yau, "[On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I](https://jasonpayne.webs.com/Math5339/On%20the%20Ricci%20Curvature%20of%20a%20Compact%20Kahler%20Manifold%20and%20the%20Complex%20Monge-Ampere%20Equation%20I,... | https://mathoverflow.net/users/469129 | A problem of the volume form of Kähler manifold in the paper of Yau's proof of Calabi conjecture | Just to close this off: note that $d=\partial+\bar\partial$ and that $\partial^2=0$ so $\partial\bar\partial=d\bar\partial$, and therefore $\Omega+\partial\bar\partial\varphi=\Omega+d\bar\partial\varphi$ is in the same cohomology class as $\Omega$. Since wedge product of forms descends to the usual product in cohomolog... | 3 | https://mathoverflow.net/users/13268 | 419593 | 170,782 |
https://mathoverflow.net/questions/418861 | 3 | In interpolation theory, given a compatible couple of Banach spaces $(X\_0, X\_1)$ one considers the $J$ and $K$-functionals, defined as follows:
If $x \in X\_0 + X\_1$ and $t > 0$ then
$$K(t, x) = \inf\{\|x\_0\|\_{X\_0} + t\|x\_1\|\_{X\_1} : x = x\_0 + x\_1, x\_0 \in X\_0, x\_1 \in X\_1\}.$$
If $x \in X\_0 \cap X\... | https://mathoverflow.net/users/78173 | Motivation for considering the J and K-functionals of real interpolation | I found the motivation in Luc Tartar's book *An Introduction to Sobolev Spaces and Interpolation Spaces*. It is in the first page of Chapter 24.
I quote:
>
> The K-method is the natural result of investigations which originated in questions of traces: if $u \in L^{p\_0} (R\_+; E\_0)$ and $u′ \in L^{p\_1} (R+; E1)... | 2 | https://mathoverflow.net/users/78173 | 419599 | 170,784 |
https://mathoverflow.net/questions/419597 | 2 | Let $Q$ be the set of squarefree numbers. I'd like to know estimates of following sums:
$$ \sum\_{\substack{n\leq x\\ n\in Q\\}}\log n \qquad\text{and}\qquad \sum\_{\substack{n\leq x\\ n\in Q\\}} n. $$
These estimates may be known in the literature but I don't find an appropriate reference.
| https://mathoverflow.net/users/159935 | Estimate $ \sum_{\substack{n\leq x\\ n\in Q\\}}\log n$ and $\sum_{\substack{n\leq x\\ n\in Q\\}} n$ where $Q$ is the square-free numbers | We can use the asymptotic formula
$$\displaystyle \sum\_{\substack{n \leq x \\ n \in Q}} 1 = \frac{6x}{\pi^2} + O(x^{1/2}).$$
This asymptotic formula is very standard and is easy to prove. For a fixed square-free integer $m$, put $N\_m(x)$ for the number of positive integers $1 \leq n \leq x$ such that $m^2 | n$. T... | 10 | https://mathoverflow.net/users/10898 | 419603 | 170,786 |
https://mathoverflow.net/questions/419607 | 6 | Here $\lambda'$ is the conjugate partition of $\lambda=(\lambda\_1,\lambda\_2,\dots)$ and cells are in the Young diagram.
The **symplectic content** of cell $(i,j)$ of $\lambda$ is defined by
$$c\_{sp}(i,j)=\begin{cases} \lambda\_i+\lambda\_j-i-j+2 \qquad \text{if $i>j$} \\
i+j-\lambda\_i'-\lambda\_j' \qquad \qquad \... | https://mathoverflow.net/users/66131 | What is the motivation behind symplectic/orthogonal content? | A hook-content formula, using the contents $c\_{sp}(i,j)$ and $c\_O(i,j)$, for the dimensions of the irreducible polynomial representations of the symplectic and orthogonal groups, goes back to Ron King. I believe the relevant paper is <https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00... | 12 | https://mathoverflow.net/users/2807 | 419613 | 170,790 |
https://mathoverflow.net/questions/419600 | 5 | Let $m\in\mathbb{N}$ and $p\in(0,1)$ be arbitrary. Is there a sequence $X\_1,\dots,X\_m$ of random variables with the following specs on their distribution:
* Each $X\_i$ is unbiased Bernoulli: $X\_i\sim {\rm Ber}(1/2)$ for $1\le i\le m$.
* $\mathbb{P}\bigl[X\_i=X\_j\bigr] = p$ for all $1\le i<j\le m$.
| https://mathoverflow.net/users/127150 | Existence of a joint distribution on Bernoulli variables with same probability of being pairwise different | $\newcommand{\R}{\mathbb R}\renewcommand{\le}{\leqslant}\renewcommand{\ge}{\geqslant}$Let $n:=m\ge2$.
We will show that the desired condition holds if and only if
$$p\ge\frac{\lceil n/2\rceil-1}{2 \lceil n/2\rceil-1}.$$
Suppose for a moment that there exist random variables $X\_1,\dots,X\_n$ such that $X\_i\sim {\rm ... | 6 | https://mathoverflow.net/users/36721 | 419618 | 170,791 |
https://mathoverflow.net/questions/419615 | 4 | Are there such a complete metric space $X$ of weight $k<\mathfrak{c}$ ($w(X)=k$) and a family $\{F\_{\alpha}: \alpha<k\}$ of closed subsets of $X$ that $k<|X\setminus \bigcup F\_{\alpha}|<\mathfrak{c}$ holds ?
| https://mathoverflow.net/users/112417 | Are there such a complete metric space X of weight k (w(X)=k) and ....? | As K.P. and Ramiro both point out in the comments, it follows from $\mathsf{CH}$ that the answer is no. I claim that it is also consistent that the answer is yes.
It is consistent that $\mathfrak{c} > \aleph\_2$ and that there is a partition $\mathcal P$ of the Cantor space $2^\omega$ into $\aleph\_2$ closed sets, su... | 4 | https://mathoverflow.net/users/70618 | 419625 | 170,793 |
https://mathoverflow.net/questions/419623 | 1 | Are there 10-dimensional irreducible representations of the Lie algebra $so(10,\mathbb{C})$ which are not isomorphic to the standard representation?
| https://mathoverflow.net/users/16183 | 10-dimensional irreducible representations of $so(10,\mathbb{C})$ | A simple application of the Weyl degree formula shows that the degrees of the fundamental representations of $D\_5$ are $10,45,120,16,16$ (the $10$-dimensional fundamental representation being the standard one, the $45$-dimensional being the adjoint one, and the two $16$-dimensional ones being the two half-spin represe... | 10 | https://mathoverflow.net/users/17064 | 419626 | 170,794 |
https://mathoverflow.net/questions/419634 | 11 | Let $\Gamma$ be a finitely presented hyperbolic group with boundary homeomorphic to $S^{n-1}$. Are there any examples of such $\Gamma$ which are known to not be the fundamental group of any $n$-dimensional compact, negatively curved, manifold? When $n=2$, such groups are known not to exist, and when $n=3$ this is an op... | https://mathoverflow.net/users/479962 | Counterexamples to an analog of Cannon's Conjecture which do not arise from manifolds? | The requirement that the manifolds "do not arise at all from negatively curved compact manifolds" is somewhat vague, but here is one known construction.
If one applies Charney-Davis strict hyperbolization to a closed oriented PL manifold with a non-integral Pontryagin number, the result is an aspherical manifold with... | 10 | https://mathoverflow.net/users/1573 | 419635 | 170,797 |
https://mathoverflow.net/questions/419550 | 5 | Given a partition $\lambda=(\lambda\_1\geq\lambda\_2\geq\dots)$, denote the **conjugate** partition by $\lambda'=(\lambda\_1'\geq\lambda\_2'\geq\dots)$. For example, if $\lambda=(4,2,2)$ then $\lambda'=(3,3,1,1)$.
The hook length of a cell $\square=(i,j)$ in the Young diagram of $\lambda$ is given by $h(i,j)=\lambda\... | https://mathoverflow.net/users/66131 | In search of a combinatorial proof on particular set of partitions | **Proposition.** A partition $\lambda \in \mathcal{syP}\_0$ iff it is empty, or both of the following hold:
* $\lambda'\_1 - \lambda\_1 = 1$,
* the partition $\mu$ obtained by removing first row and column of $\lambda$ is also in $\mathcal{syP}\_0$.
Informally, all partitions in $\mathcal{syP}\_0$ (and only those) ... | 6 | https://mathoverflow.net/users/106512 | 419640 | 170,799 |
https://mathoverflow.net/questions/419539 | 2 | Suppose $f$ is a continuous function on $\mathbb{R}$. $0<a<1$. $B(x,r)$ is open ball centered at $x$ with radius $r$. Is it true that
$$ \varlimsup\_{r\rightarrow 0} \frac{|f(x+r)-f(x)|}{|r|^\alpha} \leq C \varliminf\_{r \rightarrow 0^+}\frac{\sup\_{x\_1,x\_2\in B(x,r)} |f(x\_1)-f(x\_2)|}{r^a} $$
for some positive cons... | https://mathoverflow.net/users/152618 | Compare two limits related to Hölder condition | The claim does not hold in general.
I shall give a counterexample.
I interprete the ball $B(x,r)$ as the interval $(x-r,x+r)$.
My example will not be continuous, but one can replace the jumps with linear pieces of fast growing slopes. You will get the drift.
Define
$$
f(t)=\begin{cases} 0&t\le 0,\\
e^{-a(n-1)^2}&e^{-... | 2 | https://mathoverflow.net/users/473423 | 419654 | 170,806 |
https://mathoverflow.net/questions/419651 | 4 | In my research I have come across a divergent asymptotic series $\sum\_{n =0}^\infty a\_n f\_n(x)$ that formally solves a certain fairly simple nonlinear second-order ODE but does not seem to correspond to any standard special functions.
Here is my question: Given such an asymptotic series, what are the standard meth... | https://mathoverflow.net/users/41827 | Reference request: Rigorously solving ODEs using divergent asymptotic series | As I recall, this book deals with it:
*Costin, Ovidiu*, Asymptotics and Borel summability, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 141. Boca Raton, FL: Chapman & Hall/CRC (ISBN 978-1-4200-7031-6/hbk). xiii, 250 p. (2009). [ZBL1169.34001](https://zbmath.org/?q=an:1169.34001).
Assort... | 4 | https://mathoverflow.net/users/454 | 419679 | 170,811 |
https://mathoverflow.net/questions/419695 | 5 | Suppose I have an elliptic curve $E$ defined over a number field $K$.
I know that if it has
* a $2$ $K$-torsion, it has a model of the form:
$E: Y^2=X^3+aX^2+bX$
* a $3$ $K$-torsion, it has a model of the form:
$E: Y^2 +cXY +dY=X^3$
My question is, do we have a nice description for elliptic curves with a $p$ ... | https://mathoverflow.net/users/478525 | Model of an elliptic curve with p-torsion | A standard method to find such an equation (in principle) is the following, shown to me by Tate, but maybe due to Mordell(?). If $P\_0$ is a point of order at least $4$ on $E$ (infinite order is allowed), then after a change of coordinates, we can find an equation for $E$ of the form
$$ E : y^2 + u x y + v y = x^3 + v ... | 6 | https://mathoverflow.net/users/11926 | 419704 | 170,818 |
https://mathoverflow.net/questions/419702 | 2 | Let $A$ and $B$ be abelian schemes over a base scheme $S$. There is the $\underline{\mathrm{Hom}}(A,B)$ functor $T \mapsto \mathrm{Hom}(A \times T, B \times T)$, where $\mathrm{Hom}$ means homomorphisms of group schemes.
>
> Is it true that $\underline{\mathrm{Hom}}(A,B)$ is representable by a scheme of locally fin... | https://mathoverflow.net/users/217216 | Representability of Hom of two abelian schemes | Thanks to Will Sawin for the suggestion. That works, here are the details:
Write $\underline{\mathrm{Mor}}(A,B)$ for the functor describing morphisms of schemes (as opposed to $\underline{\mathrm{Hom}}(A,B)$ for morphisms of group schemes).
By [[Tag 0D1C]](https://stacks.math.columbia.edu/tag/0D1C) we know that $\u... | 2 | https://mathoverflow.net/users/217216 | 419710 | 170,819 |
https://mathoverflow.net/questions/419665 | 3 | As shown in [Strauss: Existence of solitary waves in higher dimensions](https://Existence%20of%20solitary%20waves%20in%20higher%20dimensions), Strauss introduces the Stauss lemma. Precisely speaking, we have the following theorem:
**Theorem** Let $N \ge 2$, every radial function $u \in H^1(\mathbb{R}^N)$ is almost ev... | https://mathoverflow.net/users/137915 | About radial Sobolev inequality (Strauss Lemma) | First, you got the scaling wrong. The correct scaling for
$$ |x|^\alpha u(x) \lesssim \|u\|\_{L^2}^{1-\beta} \|\nabla u\|\_{L^2}^\beta $$
would be $\alpha = \frac{N}{2} - \beta$ where $N$ is the spatial dimension. So for smaller $\alpha$ you need *more* $\beta$, not less.
For $\beta \in [\frac12, 1]$ ($\beta = 1$... | 3 | https://mathoverflow.net/users/3948 | 419713 | 170,820 |
https://mathoverflow.net/questions/419709 | 30 | Let $G$ be a finite group. Let $V\_1, V\_2$ be two finite-dimensional real representations. Suppose $f: V\_1 \to V\_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V\_1$ and $V\_2$ are isomorphic representations?
| https://mathoverflow.net/users/46433 | Are homeomorphic representations isomorphic? | This is a famous problem, originating in work of de Rham, and the answer turns out to be No. The lowest-dimensional examples of non-linear similarity, as it is called, are in dimension 6, and examples only exist if the group has order divisible by (but not equal to) 4. This article contains a summary of the subject:
... | 43 | https://mathoverflow.net/users/4042 | 419719 | 170,821 |
https://mathoverflow.net/questions/419673 | 3 | Let $a, b: \mathbb R\_+ \to [0,1]$ be continuous functions. Let $k: \mathbb R\_+\times\mathbb R \to [1,2]$ be $1-$Lipschitz. Set, for $0<s<t$ and $y>0$,
$$A(s,t,y):=\int\_s^t\frac{k(u,y)}{1+a(u)}du \quad\mbox{and} \quad B(s,t,y):=\int\_s^t\frac{k(u,y)}{1+b(u)}du.$$
Define further
$$f(s,x,t,y):=\frac{1}{\sqrt{2\pi... | https://mathoverflow.net/users/nan | How does the integral of pseudo Gaussian kernel on $(0,\infty)$ depend on its variance? | $\newcommand{\si}{\sigma}\newcommand{\vpi}{\varphi}\newcommand{\R}{\mathbb R}\newcommand{\De}{\Delta}$Let us show that the desired bound holds if
\begin{equation\*}
\int\_0^\infty dx\,|p'(x)|<\infty, \tag{1}\label{1}
\end{equation\*}
which in particular implies that there exists the limit
\begin{equation\*}
p\_0:=p(0... | 2 | https://mathoverflow.net/users/36721 | 419721 | 170,822 |
https://mathoverflow.net/questions/419720 | 1 | The setting is: Let $A, B$ be commutative, Noetherian, local rings, $\phi:A \rightarrow B$ a surjective homomorphism. Both rings also come with surjections $\lambda\_A, \lambda\_B$ to a DVR $\mathcal{O}$ which factor as $\lambda\_A = \lambda\_B \circ \phi$ (if this is helpful at all). Let $e(A)$ denote the Hilbert - Sa... | https://mathoverflow.net/users/478907 | Hilbert - Samuel multiplicity of $B$ when there is a surjection $A \rightarrow B$ | There is a short exact sequence $$0 \to \ker \phi \to A \xrightarrow{\phi} B \to 0.$$ Hilbert-Samuel multiplicity is additive across short exact sequences (see Corollary 4.7.7 in Bruns and Herzog), so $e(A)=e\_A(B)+e\_A(\ker \phi)$, which proves the claim.
| 4 | https://mathoverflow.net/users/155965 | 419723 | 170,823 |
https://mathoverflow.net/questions/418606 | 4 | For an ordinal number $\alpha$, the epsilon number $\varepsilon\_\alpha$ is defined as the "$\alpha$-th" fixed point of the map $n \mapsto \omega^n$, i.e. $\omega^{\varepsilon\_\alpha} = \varepsilon\_\alpha$.
Question: What about fixed points of $n \mapsto \varepsilon\_n$ or $n \mapsto \omega\_n$? Examples for the fo... | https://mathoverflow.net/users/470978 | Set theory: fixed points of $n \mapsto \varepsilon_n$ and $n \mapsto \omega_n$ | The least fixed point of $\mu \mapsto \varepsilon\_\mu$ is called $\zeta\_0$ and is written as $\varphi\_2(0)$ using the Veblen $\varphi$ function. It is also equal to $\psi(\Omega)$ in Madore's psi function and $\psi\_0(\psi\_1(\psi\_1(0)))$ in Buchholz's psi function. You can read more about it here: [https://googolo... | 5 | https://mathoverflow.net/users/473200 | 419730 | 170,828 |
https://mathoverflow.net/questions/419708 | 5 | Let $ \phi : Y \to X $ and $ \psi : Z \to X $ be finite morphisms of integral algebraic curves over a field $ k $.
Let $ \phi^\* : K( X ) \to K( Y ) $ and $ \psi^\* : K( X ) \to K( Z ) $ be the pullbacks of $ \phi $ and $ \psi $.
Is the following true?
The fiber product $ Y \times\_X Z $ is an integral curve over $... | https://mathoverflow.net/users/132492 | Fiber product of algebraic curves | This is ok when $X$ is smooth and $Y$ and $Z$ are integral. The point is that over a *smooth* curve, any finite morphism from an integral curve is automatically flat: over a Dedekind scheme, flat is the same as torsion-free [Tag [0AUW](https://stacks.math.columbia.edu/tag/0AUW)], and a finite integral extension is inde... | 4 | https://mathoverflow.net/users/82179 | 419764 | 170,837 |
https://mathoverflow.net/questions/419485 | 4 | Posted this to MSE several weeks ago and it got 3 upvotes but no answers or even comments so I'm cross-posting to MO
Aschbacher's theorem says that every maximal subgroup of a finite simple classical group falls into at least one of the 9 Aschbacher classes.
Is there a similar result for compact simple classical gr... | https://mathoverflow.net/users/387190 | Aschbacher classes for compact simple group | Every representation of a finite group in characteristic $0$ is equivalent to one over a finite extension of ${\mathbb Q}$ (i.e. a number field), so I guess if your original field for the compact group is ${\mathbb R}$ or ${\mathbb C}$ then the representation is equivalent to one over a proper subfield, which is one of... | 5 | https://mathoverflow.net/users/35840 | 419772 | 170,840 |
https://mathoverflow.net/questions/419701 | 4 | My question is originally related to coding theory, but fairly easy to state in pure combinatorial way.
Fix $k\in\mathbb{N}$, $\beta\in(0,1)$ and consider the binary cube $\Sigma\_n = \{0,1\}^n$ equipped with the Hamming distance. Is it true that there exists nearly equidistant $x\_1,\dots,x\_k\in\Sigma\_n$ with pair... | https://mathoverflow.net/users/127150 | Existence of (near) equidistant codewords | $\beta$ cannot be too much larger than $1/2$;
namely we must have $\beta \leq k/(2k-2)$.
To prove this, identify the $x\_i$ with vectors $v\_i \in {\bf R}^n$
each of whose coordinates is $1$ or $-1$,
and consider these vectors' dot products.
Clearly $v\_i \cdot v\_i = n$, and more generally
$v\_i \cdot v\_j = n - 2 d... | 4 | https://mathoverflow.net/users/14830 | 419780 | 170,842 |
https://mathoverflow.net/questions/419782 | 8 | Suppose that $M$ is a non-compact manifold of finite topological type with one end which is the universal cover of some closed manifold $N$.
Is $M $ necessarily homeomorphic to the total space of some vector bundle over a compact manifold?
In fact the only examples I can think up are much more limited, just of the ... | https://mathoverflow.net/users/99732 | Universal cover with one end | I think not. In any dimension $n\geq 4$ there are examples, constructed by Mike Davis, of contractible manifolds $M^n$ that are not homeomorphic to $\mathbb{R}^n$, and yet are the universal cover of a compact manifold $N$. The only way that $M$ could be a vector bundle over a compact closed manifold $X$ is if $X$ were ... | 11 | https://mathoverflow.net/users/3460 | 419785 | 170,846 |
https://mathoverflow.net/questions/419786 | 3 | Given a prime $p=3m+1$, $(p-1)/3$ of the residues mod $p$ are cubic residues. So heuristically, for any given integer $k>1$ not a perfect cube, we would expect that about 1/3 of the primes $\equiv1\pmod3$ up to $x$ would have $k$ as a cubic residue. Is this known, and what kind of error term has been proved?
| https://mathoverflow.net/users/6043 | Counting cubic residues mod p | This is true. Such primes are exactly the primes that split in the field $\mathbb Q(\mu\_3, \sqrt[3]{k})$, and they split into exactly $6$ prime ideals of norm $p$ since the extension is Galois of degree $6$, and thus their density among the primes is $1/6$ by the [Landau prime ideal theorem](https://en.wikipedia.org/w... | 10 | https://mathoverflow.net/users/18060 | 419787 | 170,847 |
https://mathoverflow.net/questions/419760 | 1 | I recently stumbled upon a formula for the left adjoint of the nerve functor. Let $X$ and $Y$ be simplicial sets, then:
\begin{equation}
\mathbf{sSet}(X,Y)
\cong\mathbf{sSet}(\varinjlim\_{\Delta^n\rightarrow X}\Delta^n,Y)
\cong\varprojlim\_{\Delta^n\rightarrow X}\mathbf{sSet}(\Delta^n,Y)
\cong\varprojlim\_{\Delta^n\rig... | https://mathoverflow.net/users/479945 | Formula for the left adjoint of the nerve functor? | Before anything else, let me point out that you have some typos with some $\varinjlim$'s that should be $\varprojlim$'s.
Otherwise, the first and second formula are correct, but hardly usable in practice. In fact, you typically deduce the second one from a more usable version of the first one - there are probably oth... | 5 | https://mathoverflow.net/users/102343 | 419788 | 170,848 |
https://mathoverflow.net/questions/419736 | 9 | Let $A$ be a (possibly non-unital) algebra over $\mathbb C$. We say that $A$ is *self-induced* if the product map $m:A \otimes\_A A \rightarrow A$ is an isomorphism. Here $A \otimes\_A A$ is the balanced tensor product, the quotient of $A\otimes A$ by the linear span of elements of the form $ab\otimes c - a\otimes bc$.... | https://mathoverflow.net/users/406 | Examples of non-self-induced algebras | In [this paper](http://real.mtak.hu/90053/1/laan_marki_reimaa.pdf) the authors consider the analogous question in the context of semigroups and I think basically the contracted semigroup algebra of their semigroup on page 5 works. This argument should be ok over any base commutative ring with unit but to keep things ea... | 3 | https://mathoverflow.net/users/15934 | 419791 | 170,849 |
https://mathoverflow.net/questions/419767 | 8 | **Short version**: is there a canonical way to adelize a classical Hecke eigenform automorphic form when the adelic quotient has many components? If not, what are the different "choices", how many, etc.?
**Some sketched details**: Let $A$ be a central simple algebra over a number field $F$, e.g. $A$ is the matrix alg... | https://mathoverflow.net/users/168129 | Adelization of automorphic forms for higher class number | $\newcommand{\p}{\mathfrak{p}}$Let $C$ be the class group parametrising the components, say $X = \bigcup\_{c\in C}X\_c$. Then the Hecke operator $T\_\p$ sends component $X\_c$ to $X\_{c\p}$.
In particular, the Hecke operators preserving the components are the $T\_\p$ where the class of $\p$ in $C$ is trivial. If $f$ is... | 8 | https://mathoverflow.net/users/40821 | 419797 | 170,851 |
https://mathoverflow.net/questions/419803 | 8 | We say a measurable subset $S$ of $\mathbb R^n$ is *measure dense* if for every open set $U \subset \mathbb R^n$, $U \cap S$ is of positive Lebesgue measure.
Let $n \geq 2$, and let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz continuous function with *strict* Lipschitz constant $L > 0$.
That is, $|f(x) - f(y)| < ... | https://mathoverflow.net/users/173490 | On functions with strict Lipschitz constant | I guess it suffices to give an example for $n = 1$. If $f: \mathbb{R} \to \mathbb{R}$ is an example then $g(x\_1, \ldots, x\_n) = f(x\_1)$ will be an example for any $n \geq 1$.
All we need is a measurable set $A \subseteq \mathbb{R}$ such that both $A$ and its complement have positive measure in every interval. See ... | 12 | https://mathoverflow.net/users/23141 | 419809 | 170,855 |
https://mathoverflow.net/questions/414687 | 7 | $\DeclareMathOperator\Gr{Gr}$Let $R$ be a local ring, let $A$ be a finite abelian group, and let $I$ be a Hopf ideal of the ring $R[A]$. The quotient $R[A]\twoheadrightarrow R[A]/I$ induces a map on group-like elements $f\colon \Gr(R[A])\to \Gr(R[A]/I)$.
**Is the map $f$ surjective?**
The answer is yes if the order... | https://mathoverflow.net/users/102060 | Group-like elements in quotients of group rings | I found a counter-example, so the answer is no.
Let $B=\mathbb{F}\_2[a,b,c]/J$ where $J=(a,b,c)^3+(ab+ac-bc)$ and put $s=ab$ and $t=ac$. Let $A=(\mathbb{Z}/2\mathbb{Z})^2$ and define $R = B[x\_{10},x\_{01},x\_{11}]/J'$ where
$$
J'=(x\_{10}^2-s, x\_{01}^2-t, x\_{11}^2-(s+t), x\_{10}x\_{01}-ax\_{11}, x\_{10}x\_{11}-bx... | 2 | https://mathoverflow.net/users/102060 | 419814 | 170,857 |
https://mathoverflow.net/questions/419777 | 2 | Let $K$ be an algebraically closed field of characteristic zero, and $X$ be an affine $K$-variety (identify $X$ with its set of $K$-points). Let $G$ be group acting "abstractly" on $X$, by which I mean there is simply a group homomorphism $\rho:G \to \operatorname{Aut}(X)$. Then $G$ also acts on the coordinate ring $K[... | https://mathoverflow.net/users/361094 | Algebraic groups acting on affine varieties with finite-dim orbits in the coordinate ring | A locally finite action on an affine variety is basically algebraic. More precisely, it factors through an algebraic group action. Proof: By assumption there is a $G$-stable finite dimensional subspace $V\subseteq K[X]$ containing a set of generators of $K[X]$ (just take $V=\sum\_i\langle Gf\_i\rangle\_K$ with generato... | 4 | https://mathoverflow.net/users/89948 | 419820 | 170,860 |
https://mathoverflow.net/questions/419802 | 9 | Let $A$ be a C$^\*$-algebra with closed two-sided ideal $I$. Set $B=A/I$ and let $\pi:A\to B$ be the quotient map. Suppose that $b\in B$ is quasi-nilpotent. Does there exist quasi-nilpotent $a\in A$ such that $\pi(a)=b$?
| https://mathoverflow.net/users/142780 | Lifting quasi-nilpotent elements in C$^*$-algebras | If I is the compact elements of A and B is the corresponding Calkin Algebra, the answer is yes. Have you looked at BARNES, B. A., MURPHY, G. J., SMYTH, M. R. F. and WEST, T. T., "Riesz and Fredholm theory in Banach algebras" (Research Notes in Mathematics 67, Pitman, 1982?
| 3 | https://mathoverflow.net/users/356618 | 419821 | 170,861 |
https://mathoverflow.net/questions/419813 | 4 | I have two related questions. Let $\mu$ and $\nu$ be two **distinct** probability measures on $\mathbb{R}^n$ with finite second moments, and $W\_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question is:
(1) As $\mu$ and $\nu$ are distinct, hence $W\_2(\mu,\nu)\neq 0$. Is it possible that there is a sequence of di... | https://mathoverflow.net/users/480106 | 2-Wasserstein metric on convolution of probability distributions | The answer is yes to the second question, and hence yes to the first question as well.
Indeed, it is easy to check that the functions $f$ and $g$ given by
$$f(t):=\max(0,1-|t|)$$
and
$$g(t):=\sum\_{k=-\infty}^\infty f(t-2k)$$
for real $t$ are characteristic functions:
$$f(t)=\int\_{-\infty}^\infty e^{itx}\mu(dx)$$
an... | 4 | https://mathoverflow.net/users/36721 | 419838 | 170,868 |
https://mathoverflow.net/questions/419816 | 3 | Let $k$ be a algebraically closed field and suppose that $A$ and $B$ are finite dimensional $k$-algebras. If we assume that $A$ is a symmetric $k$-algebra and $A\otimes\_k I$ is a projective $A\otimes\_k B$-module for some $B$-module $I$, is it true that $I$ must be a projective $B$-module? Or could someone provide me ... | https://mathoverflow.net/users/134942 | Projectivity of some module | We have in that case (algebraically closed is important) that $\operatorname{pdim} M \otimes\_K N= \operatorname{pdim} M + \operatorname{pdim} N$ and thus if $I$ is not projective then $A \otimes\_K I$ is not projective.
| 4 | https://mathoverflow.net/users/61949 | 419839 | 170,869 |
https://mathoverflow.net/questions/419834 | 3 | I have a series of $n$ independent random variables $X\_1,\ldots, X\_n$, each with the support $[0,1]$, and a monotone convex function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ that is 1-Lipshitz in L1 norm, i.e., for every $x,y \in \mathbb{R}^n$, it holds that $f(x)-f(y) \leq \sum\_{i=1}^{n} |x\_i-y\_i|$.
I want to have... | https://mathoverflow.net/users/480123 | Talagrand's inequality for L1 norm | Yes, you do need more conditions. For instance, if $f(x\_1,\dots,x\_n)\equiv x\_1+\dots+x\_n$ and the $X\_i$'s are (say) iid Bernoulli with parameter $1/2$, then $f$ is $1$-Lipschitz in the $L^1$-norm but, by the central limit theorem, your inequality will hold for all real $t>0$ only if $c\_2\gtrsim n/2$ as $n\to\inft... | 3 | https://mathoverflow.net/users/36721 | 419840 | 170,870 |
https://mathoverflow.net/questions/419801 | 8 | Let $C$ be a $\mathbb{Z}$-linear category, such that $C(x,y)$ is a free abelian group with finite rank, for every $x,y\in\mathrm
{Ob}(C)$. Given a commutative ring with identity $R$, let $RC$ denote the category with the same objects of $C$, and morphisms $RC(x,y):=R\otimes\_{\mathbb{Z}} C(x,y)$.
Does any isomorphism... | https://mathoverflow.net/users/480085 | Lifting isomorphisms between linear categories | It suffices to show that you can lift isomorphisms along $AC\to BC$ whenever $A\to B$ is a square zero extension. (EDIT : here I'm using finite generation of $C(y,x), C(x,y)$ to obtain that $\mathbb Z\_p\otimes C(x,y)$ is $p$-adically complete)
So let $\sum\_i b\_i\otimes f\_i \in BC(x,y)$ be an isomorphism, with inv... | 4 | https://mathoverflow.net/users/102343 | 419841 | 170,871 |
https://mathoverflow.net/questions/419461 | 4 | Let $X$ be a reflexive strictly convex Banach space and $C \subset X$ be a nonempty closed convex subset. Then the metric projection $P\_X : X \rightarrow C$ is well-defined: $P\_C(x)$ is the element satisfying
$$\|x - P\_C(x)\| = \inf\_{c \in C} \|x - c\|.$$
That is, $P\_C(x)$ is the element of $C$ which best approxim... | https://mathoverflow.net/users/78173 | Is the metric projection on a strictly convex space continuous with respect to the set? | Let $X$ be a uniformly convex space and $x\in X$.Let me start by showing the continuity with respect to bounded, closed and convex sets.
We denote by $h$ the Hausdorff distance on the space $\mathcal{CB}(X)$ of bounded closed and convex subsets of $X$ and recall that for all sets $A,B\in\mathcal{CB}(X)$ the inequalit... | 1 | https://mathoverflow.net/users/83700 | 419848 | 170,874 |
https://mathoverflow.net/questions/419694 | 8 | A common action in set theory is making a large cardinal axiom "recursive", i.e. turning it from a large uncountable cardinal to a large countable ordinal. For example:
* Recursively regular = $\alpha$ is admissible, i.e. $L\_\alpha \vDash \text{KP}$.
* Recursively inaccessible = $\alpha$ is admissible and the admiss... | https://mathoverflow.net/users/473200 | How could we define "recursively greatly Mahlos"? | Let me preface this by stating the obvious in that there is no injection from a given uncountable cardinal to any countable ordinal, thus any system of describing "analogous properties" of large countable ordinals relative to an uncountable set of ordinals will be flawed in infinitely many ways. The connection between ... | 3 | https://mathoverflow.net/users/120848 | 419849 | 170,875 |
https://mathoverflow.net/questions/419768 | 1 | While there is abundant literature available on value distribution of meromorphic functions, I am interested to know whether the value distribution theory for bicomplex meromorphic functions has been studied or not. I couldn't find any references for the same.
| https://mathoverflow.net/users/143655 | Reference request for value distribution theory of bicomplex meromorphic functions | Since the bicomplex numbers are isomorphic to $\mathbb{C} \oplus \mathbb{C}$ as a $\mathbb{C}$-algebra, it is not too difficult to see that (after a linear change of coordinates) any bicomplex-holomorphic function is of the form $(f(z),g(w))$ with $f$ and $g$ being ordinary holomorphic functions. So, the behavior of bi... | 2 | https://mathoverflow.net/users/1849 | 419853 | 170,878 |
https://mathoverflow.net/questions/419831 | 5 | Given a measurable set $E \subset \mathbb{R}^d$, with $\mathcal{H}^{d-1} (\partial E) < +\infty$, is it true in general that $E$ is a set of locally finite perimeter? that is, is it true that $\int\_B |D \chi\_E| dx$ is finite, for every bounded ball $B \subset \mathbb{R}^d$?
It is well-known in geometric measure the... | https://mathoverflow.net/users/62739 | Is every set with finite $\mathcal{H}^{n-1}$ measure a set of locally finite perimeter? | The reduced boundary can be defined for just about any (measurable) subset $E \subset \mathbf{R}^n$, whether it is a Caccioppoli set or not.
The precise result you seem to be after should be Theorem 4.5.11 in Federer's book; in my edition this is on page 506. Let me just quickly restate it here.
Define two sets $Q$... | 5 | https://mathoverflow.net/users/103792 | 419862 | 170,882 |
https://mathoverflow.net/questions/417437 | 9 | Let $A$ be a (non-unital) $C^\*$-algebra with multiplier $C^\*$-algebra $M(A)$. Let $\phi: M(A) \to M(A)$ be a $\*$-automorphism. Is it true that $\phi$ is automatically strictly continuous (on bounded subsets)?
Some remarks/observations:
(1) If $A = B\_0(H)$, then this is true because $\*$-automorphisms of $B(H) =... | https://mathoverflow.net/users/470427 | Is a $*$-automorphism $M(A) \to M(A)$ automatically strictly continuous? | I think that the answer is no.
Let $\mu$ be a non-trivial homeomorphism of $\beta \bf N$ with distinct points $y,z\in \beta\bf N\setminus \bf N$ such that $\mu(y)=z$ and $\mu(z)=y$. Set $A=\{f\in C(\beta{\bf N} ): f(y)=0\}$. Then $M(A)=C(\beta {\bf N})$. Let $\phi: M(A)\to M(A)$ be given by $\phi(f)(x)=f(\mu(x))$ $(f... | 5 | https://mathoverflow.net/users/142780 | 419876 | 170,884 |
https://mathoverflow.net/questions/419870 | 1 | I'm considering integrals of the (Hilbert transform) type
$$p.v.\int\_{-\infty}^\infty\frac{f(r)}{r}\,dr$$
where **$f(r)$ is periodic**, say, with period $2\pi$. I'm assuming very little regularity on $f$. To be concrete, let's say that $f(r)$ is **$\alpha$-Holder continuous with $\alpha<1$**. Now I'm wondering if the ... | https://mathoverflow.net/users/166785 | Convergence of oscillatory integrals | For the desired convergence it is enough that the $2\pi$-periodic function $f$ be just locally integrable.
Indeed, let
\begin{equation\*}
g:=f-A,\quad A:=\frac1{2\pi}\int\_\pi^{3\pi}f,
\end{equation\*}
\begin{equation\*}
\int\_\pi^{3\pi}g=0. \tag{1}\label{1}
\end{equation\*}
We want to show that
\begin{equation\*}... | 2 | https://mathoverflow.net/users/36721 | 419877 | 170,885 |
https://mathoverflow.net/questions/419866 | 1 | From [A248667](https://oeis.org/A248667):
>
> The polynomial $p(n,x)$ is defined as the numerator when the sum
> $$1 + \frac{1}{nx + 1} + \frac{1}{(nx + 1)(nx + 2)} + \cdots + \frac{1}{(nx + 1)(nx + 2)\cdots(nx + n - 1)}$$
> is written as a fraction with denominator
> $$(nx + 1)(nx + 2)\cdots(nx + n - 1)$$
>
>
> ... | https://mathoverflow.net/users/231922 | Numbers $m$ for which coefficients of the polynomial $p(m,x)$ are relatively prime | Counterexample: $463 \in b(n)$ (it's a prime and $464 = 2^4 \cdot 29$ is not squarefree), but $463 \not \in a(n)$ because it's a factor of the GCD of the coefficients of $p(463, x)$.
| 7 | https://mathoverflow.net/users/46140 | 419878 | 170,886 |
https://mathoverflow.net/questions/419884 | 3 | This question is inspired but not directly related to this recent [Stanley's MO post](https://mathoverflow.net/questions/419698/number-of-sets-s-for-which-number-of-permutations-in-s-n-with-descent-set-s).
The *descent set* $D(w)$ of a permutation $w=a\_1 a\_2\cdots a\_n\in\frak{S}\_n$ (the symmetric group on $\{1,\d... | https://mathoverflow.net/users/66131 | Generating function for "descents" and "cycle-types", in tandem | Counting permutations by cycle type and descents was first accomplished in I. Gessel and C. Reutenauer, *Counting permutations with given cycle structure and
descent set*, J. Combin. Theory Ser. A 64, No. 2 (1993), 189–215.
Using the results of this paper, an explicit (though somewhat complicated) formula for the gen... | 9 | https://mathoverflow.net/users/10744 | 419889 | 170,891 |
https://mathoverflow.net/questions/419739 | 9 | This question was originally asked at [MSE](https://math.stackexchange.com/questions/4420627/transfinitely-iterating-the-puiseux-levi-civita-or-hahn-series-constructions) but seems too advanced, so I'm reposting it here.
In short, the idea is that many constructions for non-Archimedean fields can naturally be iterate... | https://mathoverflow.net/users/24611 | Transfinitely iterating the Levi-Civita, Hahn or Puiseux constructions | Let us work in NBG set theory with global choice. There is, up to non unique isomorphism, a unique real-closed field that is $\kappa$-saturated for all infinite cardinals $\kappa$. Let's denote it by $\mathbf{K}$. For real-closed fields, being $\kappa$-saturated is the same as having no cut of size $<\kappa$, by which ... | 8 | https://mathoverflow.net/users/45005 | 419895 | 170,893 |
https://mathoverflow.net/questions/419902 | 13 | In his book *Topological Function Spaces* Arhangel'skii says that "it is well known that every nontrivial locally convex linear topological space $X$ is homeomorphic to a space of the form $Y \times \mathbb{R}$, for some space $Y$".
I've been looking for a proof of this result but haven't found anything, which leads ... | https://mathoverflow.net/users/146942 | Topological vector spaces (reference request) | I guess that *non-trivial* means that the locally convex space $X$ is not endowed with trivial topology $\{\emptyset,X\}$.
This implies that $X\neq \overline{\{0\}}$ (since this closure does have the trivial topology). For $y\notin \overline{\{0\}}$,
the Hahn-Banach theorem (applied to $L=\{ty: t\in\mathbb K\}$ and t... | 17 | https://mathoverflow.net/users/21051 | 419907 | 170,895 |
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