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https://mathoverflow.net/questions/441352 | 6 | Dale Peterson famously gave a series of lectures on the quantum cohomology of flag varieties $G/P$ at MIT in 1997. These lectures are often cited in subsequent papers by other authors on the subject (where many of the proofs appear for the first time in the literature), as Peterson himself never published any of this w... | https://mathoverflow.net/users/138296 | Peterson's quantum cohomology of G/P lectures | I put it as an answer to keep the page from jumping to the front from time to time:
There is this typed version, see the last two lines of the last page:
[http://math.soimeme.org/~arunram/Resources/QuantumCohomologyOfGPL16-18.html](http://math.soimeme.org/%7Earunram/Resources/QuantumCohomologyOfGPL16-18.html)
Fol... | 2 | https://mathoverflow.net/users/3903 | 441466 | 178,179 |
https://mathoverflow.net/questions/441329 | 2 | Let a non trivial $\alpha\in \pi\_m(S^n)$ with a finite order $|\alpha|$. Write $G\_\alpha(S^m, S^n)$ for the path-component determined by $\alpha$ of the mapping space $G(S^m, S^n)$ of free maps. Next, take a natural number $k>1$ relatively prime to $|\alpha|$. Then, for a self-map $f\colon S^{n-1} \to S^{n-1}$ with d... | https://mathoverflow.net/users/35872 | Path component of the mapping spaces $G(S^m, S^n)$ | This is not true. The short version is that even though $\alpha$ has order relatively prime to $k$, it's still a whole path component of $G(S^m,S^n)$. The homotopy groups of $G\_\alpha(S^m,S^n)$ then still contain a lot of $k$-torsion that tends to be killed by this degree-$k$ map.
$\require{AMScd}$
---
Pick base... | 2 | https://mathoverflow.net/users/360 | 441470 | 178,181 |
https://mathoverflow.net/questions/441465 | 3 | I have the following problem. It is well known that $H^\ast(D\_8,\mathbb{Z}/2)\cong \mathbb{F}\_2[x,y,w]/(xy=0)$ with $|x|=|y|=1$ and $|w|=2$ (see Adem,Milgram "Cohomology of finite groups"). So we get that.
$$
H^2(D\_8,\mathbb{Z}/2)=\mathbb{F}\_2\langle x^2, y^2,w\rangle.
$$
I wanted to ask if there is any method/re... | https://mathoverflow.net/users/123432 | Identifying group extension from cohomology class of $D_8$ | Here is a start for your specific questions. It is easy to see that under the inclusion $C\_4 < D\_8$, $x$ and $y$ both map to the 1 dimensional class and $w$ maps to the nonzero 2 dimensional class. So one concludes that a group $P$ of order 16 fitting into a central extension
$$ C\_2 \rightarrow P \rightarrow D\_8$$
... | 6 | https://mathoverflow.net/users/102519 | 441481 | 178,187 |
https://mathoverflow.net/questions/441467 | 3 | $G$ is a group of odd order, $\sigma$ is an automorphism of $G$, and $\sigma^2=\mathrm{id}$.
I want to find an example to show that
$G\_s= \{ g \in G \mid \sigma \left( g \right)= g^{-1} \} $ might not be a subgroup of $G$.
$G$ has to be a non-Abelian group.
If $G\_s$ is a group, it should be an Abelian group.
I tr... | https://mathoverflow.net/users/151339 | Find an example where a subset of “inverse fixed points“ is not a subgroup | For an odd prime $p$, let $$G = \langle a,b \mid a^p=b^p=[[a,b],a]=[[a,b],b]=1 \rangle$$ be an extraspecial group of order $p^3$ and exponent $p$.
Then there is an automorphism $\sigma$ of $G$ with $\sigma^2=1$, $\sigma(a)=a^{-1}$, $\sigma(b)=b^{-1}$, and $\sigma([a,b])=[a,b]$.
Since $a,b \in G\_s$ but $G\_s \ne G$... | 12 | https://mathoverflow.net/users/35840 | 441483 | 178,188 |
https://mathoverflow.net/questions/441460 | 0 | I am trying to proof the uniqueness of a maximum for a two-dimensional function (well behaved, twice differentiable, domain $R^2$, etc.), yet cannot compute the exact derivatives or the Hessian.
I have $f(x,y) = g(x,y) - bx - cy$ and know that $g\_{x}>0, \ g\_{y}>0, \ g\_{xx}<0$ and $g\_{yy}<0$, but do not know $g\_{... | https://mathoverflow.net/users/476037 | Identify maxima for 2-Dimensional Function without knowing cross-derivative | Of course, nothing definite can be said here. E.g., let $b=c=1$ and $g(x,y)=x+y-x^2-y^2+axy$ for some real $a$ and all real $x,y$, so that $f(x,y)=-x^2-y^2+axy$.
Then, if $|a|<2$, then $(0,0)$ is the only point of (local and global) maximum of $f$. If $|a|>2$, then $(0,0)$ is a saddle point of $f$ and there is no poi... | 2 | https://mathoverflow.net/users/36721 | 441489 | 178,190 |
https://mathoverflow.net/questions/441504 | 1 | Let $ X , S $ be proper varieties over $ \mathbb{C} $ and $ \pi : X \rightarrow S $ be a smooth, proper morphism with relative canonical line bundle $ K\_{X/S} $. If $ L $ is a $ \pi $-relatively ample line bundle on $ X $, is there a class of examples where it is true that $ H^q(X, K\_{X/S} \otimes L) = 0 $ for all $ ... | https://mathoverflow.net/users/152391 | Relative Kodaira Vanishing? | The natural analogue of Kodaira vanishing is $R^q\pi\_\*(K\_{X/S} \otimes \mathscr L) = 0$ for $q > 0$, which follows from Kodaira vanishing plus 'cohomology and base change' [Hartshorne, Thm. III.12.11(a)].
Thus the Leray spectral sequence for $\pi$ collapses on the $E\_2$ page, giving
$$H^q(X,K\_{X/S} \otimes \math... | 4 | https://mathoverflow.net/users/82179 | 441507 | 178,196 |
https://mathoverflow.net/questions/441486 | 2 | Suppose $X$ is a smooth projective complex variety, connected of dimension $n$.
Let $a$ be an algebraic correspondence in $A^n(X\times X)$, the group of cycles modulo homological equivalence in $H^{2n}((X\times X)(\mathbf{C}),\mathbf{Q})$. This is a ring with the usual composition of correspondences.
$a$ acts on th... | https://mathoverflow.net/users/nan | Cup products and correspondences | It is not true that $a^\*(\alpha \smile \beta) = a^\*\alpha \smile a^\*\beta$:
**Example.** Let $X = \mathbf P^2$, and let $a$ be the Künneth projector onto $H^2(\mathbf P^2) \subseteq H^\*(\mathbf P^2)$. Explicitly, $a$ can be represented by the algebraic cycle $\Delta\_{\mathbf P^2} - \mathbf P^2 \times x - x \time... | 1 | https://mathoverflow.net/users/82179 | 441510 | 178,197 |
https://mathoverflow.net/questions/441503 | 2 | Suppose that $X,Y$ are nonnegative random variables satisfying, for each $t>0$,
$$ P(X>t,Y>t)\le P(X>t)P(Y>t).$$
Is there a standard term for this type of negative dependence?
(Edit: I've seen
the condition $E[XY] \le E[X]E[Y]$ called *negative covariance*.
Edit 2: the nonnegativity condition really isn't essential. ... | https://mathoverflow.net/users/12518 | Name of type of negative dependence | It is called negative quadrant dependence. See Chapter 5 of "An Introduction to Copulas" by R. B. Nelson.
| 3 | https://mathoverflow.net/users/25622 | 441511 | 178,198 |
https://mathoverflow.net/questions/441519 | 0 | Following this question: [Can we get that $ P(N^{2/3}(\lambda\_N-\lambda\_{N-1})\le c)\ge 1-\epsilon$?](https://mathoverflow.net/questions/436173/can-we-get-that-pn2-3-lambda-n-lambda-n-1-le-c-ge-1-epsilon).
We know that for $\lambda\_N\le \lambda\_{N\_1}\le \dots le\lambda\_1$ (eigenvalues of GOE matrix)
>
> $$
... | https://mathoverflow.net/users/168083 | Does there exist a constant $c>0$ such that $$ P(N^{2/3}(\lambda_N-\lambda_{N-1})\ge c)\ge 1-\epsilon? $$ | The probability density function of the spacing $\delta\_N=\lambda\_N-\lambda\_{N-1}$ of the eigenvalues $\lambda\_N$ and $\lambda\_{N-1}$ at the edge of the spectrum decays linearly for $\delta\_N\ll N^{-2/3}$, with a slope that is independent of $N$. So no matter how small $\epsilon$, you can always find a $c>0$ such... | 1 | https://mathoverflow.net/users/11260 | 441526 | 178,200 |
https://mathoverflow.net/questions/441532 | 4 | In the paper "Convex Sets of Cardinals", Truss mentioned a result of Jech:
>
> If $M$ is a countable transitive model of ZFC, and $(P,<)∈M$ is a poset, then there exists a Cohen extension of $M$ such that $(P,<)$ is isomorphic to a set of cardinalities of that model.
>
>
>
The result is referenced to to "On or... | https://mathoverflow.net/users/113405 | Exactly how much (and how little) can partial ordered sets (classes) embed to the cardinalities | Yes. This was essentially proved by Honsel and Forti in the 1980s by analysing a model that generalises the Cohen model (essentially, the one Monro used to show it can be consistent for Dedekind finite sets to have large Lindenbaum numbers).
In my preprint Iterated Failures of Choice I provide a different constructio... | 6 | https://mathoverflow.net/users/7206 | 441533 | 178,203 |
https://mathoverflow.net/questions/441536 | 7 | Are there any formulas for the irreducible off-diagonal elements $E^{\lambda}\_{ij}$ in the Gelfand-Tsetlin basis of the symmetric group algebra $\mathbb{C}[S\_n]$?
Here is the context for my question. There exists well-known formula for *minimal idempotents* (sometimes also called *primitive idempotents*) in the Gel... | https://mathoverflow.net/users/146845 | Formula for the off-diagonal "elementary matrices" in the Gelfand-Tsetlin basis of the symmetric group algebra? | Yes, these exist and are known as Young's orthogonal matrix units.
To construct $E\_{T,T'}$ note that $E\_T \,\mathbb{C}[S\_n] \, E\_{T'}$ is one dimensional. Thus it suffices to pick any $x\in \mathbb{C}[S\_n]$ such that $E\_T \, x\, E\_{T'}$ is nonzero, for example $x \in S\_n$ the unique permutation that sends $T'... | 9 | https://mathoverflow.net/users/45956 | 441543 | 178,205 |
https://mathoverflow.net/questions/441540 | 2 | (In following we are working in "classical" complex setting: i.e. all involved schemes are considered to be varieties over $k=\mathbb{C}$)
Let $X \subset \mathbb{P}^n$ be irreducible *surface* and $L $ some general $(n-3)$-plane disjoint from $X$.
We consider now the projection map $f = \pi\_L: X \to \mathbb{P}^2$ f... | https://mathoverflow.net/users/108274 | Generically finite projection $\pi_L: X \to \mathbb{P}^2$ from plane $L$ and critical points | The morphism $f$ is not just generically finite but finite. I think this resolves all the difficulties.
To see this, since $f$ is proper it just suffices to check it is quasi-finite, and thus to check the fiber over a point can't contain a curve. But the fiber over a point is a $(n-2)$-plane containing $L$, and any c... | 1 | https://mathoverflow.net/users/18060 | 441546 | 178,207 |
https://mathoverflow.net/questions/441542 | 4 | A *symmetric function* is a function $f:\mathbb R^n\to \mathbb R$ such that $f(x\_1,\ldots,x\_n)=f(\sigma(x\_1,\ldots,x\_n))$ for every permutation $\sigma\in S\_n.$
The most commonly encountered symmetric functions are polynomial functions.
**Question 1:** Is it true that, given a symmetric function $f:\mathbb R^n... | https://mathoverflow.net/users/165036 | Can every symmetric function be factorized through symmetric polynomials? | $\newcommand\R{\mathbb R}$For any $(x\_1,\dots,x\_n)\in\R^n$,
$$P(x):=\prod \_{i=1}^{n}(x-x\_{i})=\sum \_{k=0}^{n}(-1)^{k}e\_k(x\_1,\dots,x\_n)x^{n-k},$$
where the $e\_k$'s are the [elementary symmetric polynomials](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial) in $x\_1,\dots,x\_n$. Therefore and be... | 2 | https://mathoverflow.net/users/36721 | 441548 | 178,208 |
https://mathoverflow.net/questions/441550 | 2 | Let $\Theta \subset \mathbb{R}^n, \mathcal{X} \subset \mathbb{R^m}$, and suppose that $C: \Theta \rightrightarrows \mathcal{X}$ is a correspondence defined by $f: \Theta \times \mathcal{X}\to \mathbb{R}^d$ as follows:
$$
C(\theta) = \{ x\in \mathcal{X} \mid f\_1(\theta, x) \geq 0, \dots, f\_d(\theta, x) \geq 0 \}
$$
... | https://mathoverflow.net/users/160399 | Can continuous correspondence be represented via continuous functions? | Neither implication holds.
Let $\Theta=\mathcal{X}=[-1,1]$.
First, define $f$ by $f(x,y)=xy$. Then $C$ is not lower hemicontinuous. Indeed, $$\big\{\theta\mid C(\theta)\cap (-1,0)\neq\emptyset\big\}=[-1,0]$$
is not open.
Next, let $f$ be any discontinuous function with nonnegative values. Then $C$ is constant wit... | 2 | https://mathoverflow.net/users/35357 | 441557 | 178,212 |
https://mathoverflow.net/questions/441438 | 5 | Let $\epsilon\_1, \dots, \epsilon\_n$ be random signs, equiprobably in $\{-1, 1\}$, independently.
Let $S\_k = \sum\_{j=1}^k \epsilon\_j$. I am wondering what is known about the expectation
$$
\mathbb{E}\Big[\max\_{k \leq n} |S\_k| \Big].
$$
It can be seen as the maximum distance from the origin over $n$ steps of a ... | https://mathoverflow.net/users/121486 | Maximum distance from origin of simple random walk | If you just care about the asymptotics, it is indeed just $(1+o(1))\sqrt{\pi n/2}$, where the $o(1)$ term decays like $\tilde{O}(1/n^{1/4})$; this can be done using the approach that I suggested of using the natural embedding to compare to Brownian motion at the obvious stopping times, the fact that the Brownian motion... | 3 | https://mathoverflow.net/users/170770 | 441560 | 178,214 |
https://mathoverflow.net/questions/441552 | 1 | I wonder if we can replace the binomial terms in Bernstein polynomial with multinomial terms. To be more specific, given a vector $f=(f\_1, f\_2,\dotsc, f\_M)\in \mathbb{Z}^M$ such that $\forall m, f\_m\ge0, \sum\_mf\_m=N\in\mathbb{Z}$, define
$F\_{N,f}(\sigma)=\frac{N!}{\prod\_{m} f\_{m}!}\prod\_{m:\sigma\_m>0}(\sigma... | https://mathoverflow.net/users/136078 | Is there a name for an extension of Bernstein basis polynomial based on multinomial distribution? | $\newcommand{\de}{\delta}\newcommand{\De}{\Delta}\newcommand\R{\mathbb R}\newcommand{\om}{\omega}\newcommand{\si}{\sigma}$Such polynomials are called [multivariate Bernstein polynomials](https://www.google.com/search?q=multivariate%20Bernstein%20polynomials&oq=multivariate%20Bernstein%20polynomials&aqs=chrome..69i57.44... | 1 | https://mathoverflow.net/users/36721 | 441564 | 178,216 |
https://mathoverflow.net/questions/441562 | 0 | Consider two $n-$dimensional random vectors $u$ and $v$ uniformly distributed on the sphere. Define $X\_n :=u\cdot v$. Note that as $n\to \infty$, $\sqrt{n}X\_n \to N(0,1)$ as $n\to \infty$. Fix $\epsilon>0$ (very small).
Can we show that for every $\delta>0$, there exist constants $\alpha>0, \beta>0$ so that
$$
\lim... | https://mathoverflow.net/users/168083 | Can we show that for every $\delta>0$, there exist constants $\alpha>0, \beta>0$ so that the following inequality holds with high probability? | $\newcommand\al\alpha\newcommand\be\beta\newcommand\ep\epsilon\newcommand\de\delta$Let $\al=\beta=1$. Let $Z\_n:=\sqrt n\,X\_n$, so that $Z\_n\to Z\sim N(0,1)$ in distribution. Then for real $\ep>0$
$$
P\Big(\frac{X\_n^{-2}-1}{\ep^{-2}-1}<\al n^\be\Big)
=P\Big(|Z\_n|>\sqrt{\frac n{1+(\ep^{-2}-1)n}}\,\Big) \\
\to P\Big(... | 1 | https://mathoverflow.net/users/36721 | 441567 | 178,218 |
https://mathoverflow.net/questions/441559 | 2 | A rational normal curve $C \subset \mathbb{P}\_k^d$ (assume $k= \mathbb{C}$) can be defined usually up to projective equivalence in two equivalent ways:
* smooth irreducible nondegenerate curve $C \subset \mathbb{P}^d$ of minimal degree $\text{deg}(C)=d$
* projectively equivalent to the image of the Veronese map
$$... | https://mathoverflow.net/users/108274 | Equivalent characterizations of rational normal curve | Let $\mathbb{P}^d = \mathbb{P}(V)$. A hyperplane in $\mathbb{P}(V)$ corresponds to an epimorphism $V \to k$, a pencil of hyperplanes to an epimorphism
$$
V \otimes \mathcal{O}\_{\mathbb{P}^1} \to \mathcal{O}\_{\mathbb{P}^1}(1),
$$
and a collection of $d$ pencils to a morphism
$$
f \colon V \otimes \mathcal{O}\_{\mathbb... | 4 | https://mathoverflow.net/users/4428 | 441593 | 178,229 |
https://mathoverflow.net/questions/441595 | 7 | I have difficulty even in finding a Russian version of the next paper:
"Aleksandrov, A. D., Almost everywhere existence of the second differential of a convex
function and some properties of convex surfaces connected with it (in Russian).
Uchenye Zapiski Leningrad. Gos. Univ., Math. Ser. 6 (1939), 3–35"
This is a q... | https://mathoverflow.net/users/113353 | Asking for an English version of Aleksandrov's famous 1939 paper in Convex Geometry | This paper is contained in the 1-st volume of Aleksandrov's Selected works. This has an English translation:
Alexandrov, A. D.
Reshetnyak, Yu. G. (ed.)
Selected works. Part 1: Selected scientific papers. Ed. by Yu. G. Reshetnyak and S. S. Kutateladze, transl. from the Russian by P. S. V. Naidu. (English) Zbl 0960.010... | 13 | https://mathoverflow.net/users/25510 | 441597 | 178,230 |
https://mathoverflow.net/questions/441599 | 3 | $\newcommand\tetra[2]{{^{#1}{#2}}}$In a recent discussion on the Tetration Forum (see <https://math.eretrandre.org/tetrationforum/showthread.php?tid=1703&page=2>), it has been pointed out how my results on the constancy of the "congruence speed" of the integer tetration $\tetra b a$ (a peculiar property of hyper-$4$ de... | https://mathoverflow.net/users/481829 | $\lim_{b \rightarrow \infty} {^{b}a} \in \mathbb{Q}_p$ for any $a \in \mathbb{Z}^+$? | A sequence given by $x\_1=a$, $x\_{n+1}=a^{x\_n}$, where $a$ is a positive integer, is eventually constant modulo every positive integer $T$. This is widely known.
A short proof. We induct in $T$, so assume that $T>1$ and the claim is proven for smaller numbers. Denote $T=T\_1T\_2$, where $T\_1$ has prime divisors wh... | 9 | https://mathoverflow.net/users/4312 | 441602 | 178,232 |
https://mathoverflow.net/questions/441608 | 5 | McCoy's theorem (one of them) says that for any commutative ring $A$, $f\in A[x]$ is a zero-divisor iff it's annihilated by a scalar in $A$.
There's a widespread [proof by contradiction](https://math.stackexchange.com/a/83171/223002). There's also a constructive proof using the Dedekind-Mertens lemma about polynomial... | https://mathoverflow.net/users/69037 | Constructive proof of univariate McCoy theorem without Dedekind-Mertens? | The question you're asking might not be the question you want to ask. "What is the scalar" should be "what is the nonzero scalar", because $0$ wouldn't be interesting, right? But we don't know how to find any nonzero elements of $A$ in the first place, and they might not even exist ($A$ can be trivial). So the most nat... | 5 | https://mathoverflow.net/users/2530 | 441611 | 178,235 |
https://mathoverflow.net/questions/441612 | 3 | I am trying to understand something which is probably basic for experts so I am sorry if this is not suited for this forum.
Let $\mathcal{O}\_n$ denote the ring of germs at $0 \in \mathbb{R}^n$ of real-analytic functions. It is known that $\mathcal{O}\_n$ is a [Noetherian ring](https://en.wikipedia.org/wiki/Noetheria... | https://mathoverflow.net/users/50777 | Does Noetherianity imply division theorem? | This is a consequence of the noetherianity of $\mathcal{O}\_n$, in some way.
Let me write $A\_n$ for the ring of real formal formal series around the origin, i.e., $A\_n=\mathbb{R}[[t\_1,\dotsc,t\_n]]$. Let $I$ be the ideal of $\mathcal{O}\_n$ generated by $\varphi\_1,\dotsc,\varphi\_q$. Then your question is equival... | 3 | https://mathoverflow.net/users/86006 | 441621 | 178,239 |
https://mathoverflow.net/questions/441634 | 7 | A classic result, of Murray and Von Neumann I believe, is that if $\mathcal M\subseteq B(H)$ is a factor then the $\*$-homomorphism $\pi : \mathcal M \odot \mathcal M' \rightarrow B(H)$ given by $\pi(x\_1\otimes x\_2) = x\_1x\_2$ on simple tensors is injective.
On the other hand, if $A \subseteq B(H)$ is a commutative ... | https://mathoverflow.net/users/76593 | When is the multiplication map of the algebraic tensor product of C*-algebras injective? | For a commuting pair of C\*-algebras $A,B\subset B(H)$, the multiplication map is injective if and only if there are no nonzero elements $a\in A$ and $b\in B$ with $ab=0$. In particular, if $A$ is simple (and non-degenerate on $H$), then the multiplication map is injective.
This is because the pure states on C\*-alge... | 10 | https://mathoverflow.net/users/7591 | 441638 | 178,244 |
https://mathoverflow.net/questions/440731 | 20 | Let $V\_0, V\_1, \dots, V\_n$ denote a series of finite-dimensional vector spaces. We write $v\_i : = \dim V\_i$ for $i=0, 1, \dots, n$. I am thinking of these as real vector spaces, but I think the answer to my question ultimately doesn't depend on that.
Consider the affine space $A^N := \oplus\_{i=0}^{n-1} \mbox{Ho... | https://mathoverflow.net/users/4558 | What is the dimension of the variety of chain complexes? | There is a general special case where the problem has a nice answer. The situation is when the dimensions of the vector spaces are equal in pairs, or in other words $\dim V\_{2i}=\dim V\_{2i+1}$ for $0\le i\le \lfloor (n-1)/2 \rfloor$.
For convenience, we keep the auxiliary $r\_{0}=r\_{n+1}=0$, and in the case of eve... | 5 | https://mathoverflow.net/users/2384 | 441643 | 178,247 |
https://mathoverflow.net/questions/436099 | 7 | Are there peer-reviewed journals that focus on "fun mathematics"?
By this I mean fun things that do involve nontrivial mathematics and which I think other mathematicians would enjoy reading in their leisure time but that are not interesting from the point of view of contemporary mathematical research.
For instance ... | https://mathoverflow.net/users/36563 | Are there journals for "fun mathematics"? | I agree this usually goes under the name "recreational mathematics." I'd suggest looking at The Mathematical Gazette as well, which has a range of articles including short notes and problems.
| 2 | https://mathoverflow.net/users/87779 | 441647 | 178,249 |
https://mathoverflow.net/questions/441636 | 3 | Given a quadratic program,
$$\begin{array}{ll} \text{minimize} & \displaystyle \frac12 x^TAx + b^Tx \\ \text{subject to} & Cx \le d \end{array}$$
Suppose $A \succ 0$, so the program strongly convex. The question is, is the solution $x^\*$ continuous with respect to the weights $A$ and $b$ ?
If we only have equality c... | https://mathoverflow.net/users/500057 | Sensitivity of the solution of QP with respect to parameters | $\newcommand\R{\mathbb R}\newcommand\tz{\tilde z}\newcommand{\de}{\delta}$Yes, the minimizer $x\_{A,b}$ of $\frac12 x^TAx + b^Tx$ subject to $Cx\le d$ is
continuous with respect to $A$ and $b$ -- provided that the set $X:=\{x\in\R^n\colon Cx\le d\}$ is nonempty (otherwise, such a minimizer does not exist).
Indeed,... | 3 | https://mathoverflow.net/users/36721 | 441648 | 178,250 |
https://mathoverflow.net/questions/441127 | 4 | Let $G$ be an abelian group and let $q:G \to \mathbb{Q/Z}$ be a quadratic form, i.e. $q(a)=q(-a)$ and $b(x,y)=q(x+y)-q(x)-q(y)$ is a bihomomorphism. On vector spaces, when people speak about the kernel of a quadratic form they mean the radical of $\frac{b}{2}$ which is obviously not possible here, at least not straight... | https://mathoverflow.net/users/58211 | Is there a good notion of kernels of quadratic forms on abelian groups? | As requested, I post my comment (with mild changes) as answer:
In the theory of quadratic forms over a field of characteristic $2$, the radical of a quadratic form $q$ is sometimes defined as $R(q)=\{x\in \mathrm{rad}(b):q(x)=0\}$, where $b$ is the polar form of $q$. This should work in your situation as well: $R(q)$... | 2 | https://mathoverflow.net/users/86006 | 441655 | 178,251 |
https://mathoverflow.net/questions/441637 | 8 | $\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Out{Out}$Here I mean $\vcd(G)$ to be the virtual cohomological dimension of $G$. Some possible examples of what I mean by $H$ are $H=\Aut(G)$, $\Inn(G)$ or $\Out(G)$. I could extend this to other natural, more t... | https://mathoverflow.net/users/495429 | If G is a finitely generated group with vcd(G) finite, is vcd(H) finite for H, where H is an automorphism group of G? | Sorry, it is Rips' and not Mikhailova's construction: I should not comment when I am half-asleep.
Let me start with the Rips construction.
Let $Q$ be a finitely presented group. Rips in [1] constructed $C'(1/\lambda)$-[small cancellation](https://en.wikipedia.org/wiki/Small_cancellation_theory) groups $G$ (with arb... | 4 | https://mathoverflow.net/users/39654 | 441675 | 178,256 |
https://mathoverflow.net/questions/441576 | 5 | Suppose that $f \colon [0, 1] \to \mathbb{R}$ is a $C^\infty$ function satisfying the constraints
$$
f(0) = f'(0) = f(1) = f'(1) = 0, \quad \mbox{and} \quad \int\_0^1 (f''(y))^2 \, dy \leq 1.
$$
Denote this class of functions $\mathcal{F}.$
I want to know what is the best approximation one can give of the $L^2$ norm... | https://mathoverflow.net/users/121486 | Optimal constant to compare $L^2$ norm of smooth function on $[0, 1]$ to a grid | By the [Euler–Maclaurin formula](https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula#The_formula) (with $p=4$, $m=0$, and $g(x):=\frac1n\,f^2(\frac xn)$ in place of $f(x)$ there in the formula),
$$d\_n(f):=\int\_0^1 f^2(y) \, dy - \frac{1}{n}\sum\_{i=1}^n f^2(i/n) \\
=-\frac1{2n}\,f^2(1)-\frac1{6n^3}\,f(1)f'(... | 2 | https://mathoverflow.net/users/36721 | 441682 | 178,259 |
https://mathoverflow.net/questions/441663 | 1 | It comes from estimates for wave equations.
>
> For any $ u=u(t,x)\in C\_{0}^{\infty}(\mathbb{R}^{1+2}) $, which is a smooth compactly supported function, prove that
> $$
> \|u\|\_{L^{\infty}(\mathbb{R}^{1+2}\,\,)}\leq C\|\partial\_{x\_1}\square u\|\_{L^1(\mathbb{R}^{1+2}\,\,)},
> $$
> where $ \square $ is the wav... | https://mathoverflow.net/users/241460 | How to prove $ \|u\|_{L^{\infty}}\leq C\|\partial_1\square u\|_{L^1} $ for any $ u\in C_0^{\infty}(\mathbb{R}^{1+2}) $? | Let $E$ be a fondamental solution of $\partial\_{x\_1}\square$. Then you have for $u$ compactly supported
$$
u=u\ast \delta=u\ast (\partial\_{x\_1}\square E)= (\partial\_{x\_1}\square u)\ast E,
$$
so that
$
\Vert u\Vert\_{L^\infty}\le \Vert \partial\_{x\_1}\square u\Vert\_{L^1}
\Vert E\Vert\_{L^\infty}
$
and
it is now ... | 2 | https://mathoverflow.net/users/21907 | 441683 | 178,260 |
https://mathoverflow.net/questions/441661 | 2 | In the literature, there are a number of results called *Rosenlicht's lemma*, but I am talking about the following one:
Let $T$ be a torus over a, in my case, number field $k$. Denote by $\bar{k}[T]$ the ring of regular functions on $T$, and by $\widehat{T} := \mathrm{Hom}\_{\bar{k}}(T\_{\bar{k}},\mathbb{G}\_m)$ the ... | https://mathoverflow.net/users/172132 | Is there a Rosenlicht's lemma for semi-abelian varieties? | I am not sure which Lemma of Rosenlicht you mean. In "Toroidal algebraic groups. Proc. Amer. Math. Soc. 12 (1961), 984–988" he proves:
**Theorem 3:**
Let $\phi:\Gamma\to G$ be an everywhere defined rational map from
a connected algebraic group $\Gamma$ into a toroidal algebraic group $G$, with
$\phi(e)=0$. Then $\phi... | 3 | https://mathoverflow.net/users/89948 | 441685 | 178,261 |
https://mathoverflow.net/questions/441087 | 19 | **TL;DR.** Given a topological space $X$, is there a natural way to "induce" a topology on $\mathcal{P}(X)$ from the topology of $X$ in such a way that 1) all the basic operations of set theory (intersections, unions, direct images, etc.) become continuous and 2) the topologies on $X$ and $\mathcal{P}(X)$ are compatibl... | https://mathoverflow.net/users/130058 | Existence of a *really* nice topology on the powerset of a topological space | 1-7 together are pretty strong. You aren't going to have a procedure for doing this without most of the powerset topologies being indiscrete.
First note that if $X$ is a set and $\tau$ is at topology on $\mathcal{P}(X)$ satisfying 2-4, then the function $A \mathbin{\Delta} B = (A \cup B) \setminus (A \cap B)$ is cont... | 4 | https://mathoverflow.net/users/83901 | 441689 | 178,264 |
https://mathoverflow.net/questions/441610 | 10 | In Peter Johnstone's 1979 paper [On a topological topos](http://plms.oxfordjournals.org/content/s3-38/2/237.full.pdf), he proposed the topos of sheaves on the full subcategory of topological spaces spanned by the single object $\mathbb{N}\_\infty$, the one-point compactification of the natural numbers, as a convenient ... | https://mathoverflow.net/users/49 | Properties of pyknotic sets | Let me recall a little bit of the background. The question is about the relation between topological spaces and pyknotic sets, and properties of the topos of pyknotic sets. Recall that pyknotic sets are sheaves on the site of compact Hausdorff spaces, for the Grothendieck topology generated by finite families of jointl... | 16 | https://mathoverflow.net/users/6074 | 441690 | 178,265 |
https://mathoverflow.net/questions/441681 | 4 | Consider the 1-dimensional Burger's equation on a finite interval $I=(0,1)$,
$$u\_t+uu\_x=u\_{xx}$$
with initial condition
$$u(x,0)=f(x)$$
and boundary conditions
$$u(0,t)=A(t) \qquad u(1,t)=B(t).$$
>
> **QUESTION.** Suppose that $A(t)$ and $B(t)$ are periodic functions, of integer period $m$ and $n$, respectively.... | https://mathoverflow.net/users/66131 | Periodicity and Burger's equation | Of course **not**. For instance if $A=B\equiv0$ (these are periodic), then the solution decays to $0$ as $t\to+\infty$.
Instead, and this is classical in dynamical system theory, if $A$ and $B$ are periodic as in your assumption, then you might be able to prove that there exists an initial data $f$ for which the solu... | 5 | https://mathoverflow.net/users/8799 | 441692 | 178,266 |
https://mathoverflow.net/questions/441633 | 4 | I found a statement involving a homeomorphism $f:X\to X$ of a compact metric space $X$, with Lipshitz coefficient 1, i.e., a non-expansive map, and cannot think of an example where $f$ is not an isometry. Must it be?
| https://mathoverflow.net/users/172802 | Is every 1-Lipschitz homeomorphism $f:X\to X$ from a compact metric space to itself an isometry? | It follows from 1.6.15(1) in "A Course in Metric Geometry" by Burago, Burago, and Ivanov and 1.6.15(2) is a more general statement:
>
> Any distance-noncontracting map from a compact metric space to itself is an isometry.
>
>
>
This statement is needed in the proof that Gromov--Hausdorff metric is a metric.
Al... | 5 | https://mathoverflow.net/users/1441 | 441693 | 178,267 |
https://mathoverflow.net/questions/441707 | 13 | It's probably a really naive question, but I didn't find any references. Given a category $V$ and we know that it is equivalent to the category $\mathbf{Vect}(X)$ of smooth vector bundles over a smooth manifold $X$, under what assumptions could we determine $X$? Or is this generally hopeless?
| https://mathoverflow.net/users/104710 | Reconstructing base manifold from its category of smooth vector bundles | Yes, it is possible to recover the manifold through the following steps:
* [Smooth Serre-Swan theorem](https://ncatlab.org/nlab/show/smooth+Serre-Swan+theorem): $\mathbf{Vect}(X)$ is equivalent to the category of finitely generated projective modules over $C^\infty(X)$.
* If $R$ is any commutative ring, then $R$ is c... | 23 | https://mathoverflow.net/users/27013 | 441709 | 178,270 |
https://mathoverflow.net/questions/441713 | 2 | In the context of generic embeddings, we fix a set $Z$, an ideal $I$ on $Z$, and $G$ a generic ultrafilter on $\mathcal{P}(Z) / I$. In $V[G]$, we can define the generic ultrapower $Ult(V, G)$ and an elementary embedding from $V$ into $Ult(V, G)$. If $I$ is precipitous then $Ult(V, G) \cong M$ for some transitive class ... | https://mathoverflow.net/users/29231 | Closure properties of elementary embeddings resulting from generic iterations | It is impossible that $M^{\omega}\subseteq M$ in $V[G]$. The map $j$ is continuous at $\omega\_2^V$, i.e. $\omega\_2^M=\sup\_{\alpha<\omega\_2^V} j(\alpha)$ (this can be seen by induction along the length of the iteration). As $\omega\_2^V$ is countable in $V[G]$, this means that $\omega\_2^M$ has countable cofinality ... | 3 | https://mathoverflow.net/users/125703 | 441726 | 178,273 |
https://mathoverflow.net/questions/441714 | 8 | Does $x\_0=1/3$ lead to periodicity in the logistic map $x\_{k+1}=4x\_k(1-x\_k)$?
I believe it does not, but this is equivalent to proving that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. I am wondering if there are any known results. After all, $1/3$ is the most rudimentary seed that (I suspect) leads to non-per... | https://mathoverflow.net/users/140356 | Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$? | The orbit of $1/3$ is infinite. You can show this via the [$3$-adic valuation $\nu\_3$](https://en.wikipedia.org/wiki/P-adic_valuation).
Let us show by induction that $\nu\_3(x\_n) = -2^{n}$:
We have $\nu\_3(x\_0) = \nu\_3(1/3) = -1 = -2^0$.
Now $\nu\_3(x\_{n+1}) = \nu\_3(4 x\_{n} (1 - x\_{n})) =
\nu\_3(4)+ \nu\_3(... | 20 | https://mathoverflow.net/users/500150 | 441730 | 178,276 |
https://mathoverflow.net/questions/441735 | 1 | We are considering the following problem:
Given an integer $n$ and a sequence of integers $r\_i,\ 1\le i\le n$, with $0\le r\_i\le n-1$ does there exists a symmetric matrix $A$ such that the diagonal elements of $A$ are $0$'s, the off-diagonal elements are only $-1$ and $1$ and for each $i$ there are exactly $r\_i$ $... | https://mathoverflow.net/users/500153 | On the existence of symmetric matrices with prescribed number of 1's on each row | Replace each $-1$ with $0$. Then the matrix you look for is the adjacency matrix of a graph with a given degree sequence $r\_1, r\_2, \dots$. Reconstruction of such a graph (and its adjacency matrix) is known as the *graph realization* problem. E.g., see [Wikipedia](https://en.wikipedia.org/wiki/Graph_realization_probl... | 3 | https://mathoverflow.net/users/7076 | 441737 | 178,279 |
https://mathoverflow.net/questions/441739 | 3 | Recently, a problem in my research has appeared and now I need to construct some algebraic numbers with special properties (related to its degree and some other fields extensions).
Now, in order to help me, I would like to prove the following result:
**Proposition.** *Let $\alpha$ be a real algebraic number and let... | https://mathoverflow.net/users/120084 | Irreducibility of polynomials over some number fields | **Lemma.** *Let $K$ be any number field, and $p$ a prime unramified in $K$. Then $X^n-p$ is irreducible over $K$.*
*Proof.* It suffices to show that the field $L = K(\sqrt[n\ \ ]{p})$ has degree $n$ over $K$. Let $\mathfrak q \subseteq \mathcal O\_L$ be a prime above $p$, and let $\mathfrak p = \mathcal O\_K \cap \ma... | 11 | https://mathoverflow.net/users/82179 | 441741 | 178,280 |
https://mathoverflow.net/questions/441740 | -2 | Recall the definition of [cardinal definable](https://mathoverflow.net/questions/412541/is-every-set-being-cardinal-definable-consistent-with-zf-negation-of-choice) sets, to re-iterate:
$Define: X \text { is cardinal definable} \iff \\\exists \text { cardinal } \kappa \, \exists \text { cardinals } \lambda\_1,.., \la... | https://mathoverflow.net/users/95347 | Does cardinal definable choice imply AC? | Over ZF yes, it does.
Let $X$ be any family of nonempty sets, and let $κ$ be cardinal such that $ρ(κ)>ρ(X)$, then $V\_{ρ(κ)}\setminus\{∅\}$ is cardinal definable, hence has a choice function that induce a choice function on $X$.
In ZF-Regularity Scott trick doesn't work anymore, in fact Lévy has shown that it is co... | 4 | https://mathoverflow.net/users/113405 | 441750 | 178,282 |
https://mathoverflow.net/questions/441751 | 0 | Suppose we are given a scheme $S$ and two vector bundles $V$ and $W$ over $S$. Is it always true that $\mathbb{P}(V)\cong \mathbb{P}(W)$ implies that $V\cong W$ as $S$-schemes?
If the statement is false then what is the most general condition on $S$ for which it becomes true?
| https://mathoverflow.net/users/99988 | Are projective bundles corresponding to non-isomorphic vector bundles always non-isomorphic? | This is not true as stated, because for any line bundle $L$ on $S$ one has
$$
\mathbb{P}(V \otimes L) \cong \mathbb{P}(V),
$$
but this is the only issue. Indeed, if $X = \mathbb{P}(V) \stackrel{p}\to S$ and $S$ is connected, the relative Picard group $\mathrm{Pic}(X/S)$ is cyclic, and if $H$ is a lift to $\mathrm{Pic}(... | 7 | https://mathoverflow.net/users/4428 | 441755 | 178,283 |
https://mathoverflow.net/questions/441295 | 3 | Are there any results that allow us to characterize affine schemes via morphisms to/from other schemes?
Suppose we have a category $\mathcal{C}$ which is equivalent to $\mathbf{Sch}$, the category of schemes. We do not know anything about the equivalence. We want to find the objects $X \in \mathcal{C}$ such that thes... | https://mathoverflow.net/users/104710 | Characterizing affine schemes via morphisms | As pointed out in the comments, [this answer](https://mathoverflow.net/a/349015/82179) implies much more: for a category $\mathscr C$ equivalent to $\mathbf{Sch}\_{/X}$ for any $X \in \mathbf{Sch}$, there is an explicit formula producing a functor $F\_{\mathscr C} \colon \mathscr C \to \mathbf{Sch}$ that only depends o... | 4 | https://mathoverflow.net/users/82179 | 441763 | 178,286 |
https://mathoverflow.net/questions/441715 | 1 | $\newcommand{\Ex}{\mathbb E}$ I'm reading an argument in the proof of *Proposition 3.8.* in the paper [Nonlinear self-stabilizing processes - I Existence, invariant probability, propagation of chaos](https://www.sciencedirect.com/science/article/pii/S0304414998000180).
---
Let $X\_0$ and $X\_0^{\prime}$ be two in... | https://mathoverflow.net/users/99469 | How to obtain this differential relation about moments of a stochastic process? | I believe you meant lemma 3.8. Here at the final step of your calculation, you just apply [Lebesgue-differentiation theorem](https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem) since those quantities are continuous and the integral is Lebesgue. That will give you the $\mu'(0)$.
So to get the $\mu'(t\_{0})... | 2 | https://mathoverflow.net/users/99863 | 441764 | 178,287 |
https://mathoverflow.net/questions/441762 | 3 | Let $P$ be a positive, self-adjoint (unbounded) operator in a Hilbert space $H$ with $0\notin \sigma(P)$. Consider its spectral decomposition
$$P = \int\_{\sigma(P)} t dE(t).$$
Since $0 \notin \sigma(P)$, we can define (the functional calculus)
$$P^{-}= \int\_{\sigma(P)} t^{-1}dE(t).$$
This is an unbounded operator wit... | https://mathoverflow.net/users/216007 | Unbounded positive self-adjoint without $0$ in its spectrum: can we construct its inverse using functional calculus? | $\newcommand\si\sigma\newcommand\D{\mathscr D}$Yes, $P^-=P^{-1}$.
Indeed, the resolvent set of $P$ is open and $0$ is in this set. So, $\si(P)$ is bounded away from $0$, and hence the domain $\D(P^-)$ of $P^-$ is the entire space $H$ -- because, by Theorem 13.24 (a) in [1],
$$\D(P^-)=\D\_g:=\Big\{x\in H\colon \int\_{... | 4 | https://mathoverflow.net/users/36721 | 441773 | 178,292 |
https://mathoverflow.net/questions/441777 | 3 | Let $X$ be a space and $\mathcal U$ a cover of $X$. Instead of Čech cohomology, I would like to take the following construction:
let
$$I= \{ \text{finite nonempty intersections of elements of }\,\mathcal U\},$$
which is a poset, and now I can take the nerve $N(I)$ in the sense of category theory,
whose $n$-simplices ar... | https://mathoverflow.net/users/125523 | Čech-like cohomology with the “other nerve” | This construction as stated gives exactly the barycentric subdivision of the Čech nerve of $\mathcal{V}$, as any simplicial complex yields a face poset and the categorical nerve of that is the barycentric subdivision of the original complex. There are several alternative nerve constructions that can be used; see H. Abe... | 5 | https://mathoverflow.net/users/3502 | 441786 | 178,297 |
https://mathoverflow.net/questions/441789 | 3 | It is well know that the convergence in distributions does not necessarily imply convergence in expectation, but implies convergence in expectation of bounded continuous functions.
Let $\{X\_n\}$ be a sequence of random variables that converge in distribution to $X$. I would like to ask two examples as follows.
1. ... | https://mathoverflow.net/users/170508 | Examples of convergence in distribution not implying convergence in moments | Let $P(X\_{n} = n) = \frac{1}{n}$ and $P(X\_{n} = 0) = 1 - \frac{1}{n}$. Then $X\_{n}$ converges in distribution to $X=0$, but $\mathbb{E}(X\_{n}^{k}) = \frac{1}{n} n^{k} + 0 = n^{k-1} \not\xrightarrow{n\to\infty} 0$ for each $k \in \mathbb{N}$.
**UPDATE:**
If you prefer continuous random variables on $\mathbb{R}$, y... | 6 | https://mathoverflow.net/users/143828 | 441791 | 178,300 |
https://mathoverflow.net/questions/441711 | 4 | Assume that $\sigma\in S\_n$ has the cycle type $(p,.,p,1,..,1)$ where $p>2$ is a prime and the numbers of $1$ maybe $0$. If $\sigma\_1$ and $\sigma\_2$ are chosen uniformly in the conjugacy class of $\sigma$. Assume the cycle type of $\sigma\_1 \sigma\_2=(k\_1,k\_2,.,k\_l)$ . Is there a lower bound $B$ such that $$\ma... | https://mathoverflow.net/users/482299 | A probability problem in the conjugacy classes of symmetric group | Let $kp$ be the size of the support of $\sigma$. Let $1,2,3,4$ be four points of the ground set. The probability that $\sigma\_1$ maps $1 \mapsto 2$ is $kp/n(n-1)$, because there is a $kp/n$ chance that $1$ is in the support and conditionally a $1/(n-1)$ chance that $1 \mapsto 2$. Conditional on that, the probability t... | 5 | https://mathoverflow.net/users/20598 | 441798 | 178,303 |
https://mathoverflow.net/questions/441794 | 6 | Let $\mu$ be a Borel measure on the real line $\mathbb{R}$ taking values in a separable Banach algebra $A$. Assume that $\mu$ is such that the total variation measure $|\mu|$ is finite. Let $f$ be a bounded measurable function on the real line $\mathbb{R}$ taking values in a Banach algebra $A$. Is there a general theor... | https://mathoverflow.net/users/47256 | Integration in Banach algebra | I believe OP's setting is an instance of the more general so called *bilinear integral*, defined for the following data:
* a measurable space $(S,\mathfrak{S})$;
* a function $f: S \to X$, where $X$ is a normed linear space;
* an *additive* measure $\mu: \mathfrak{S} \to Y$, where $Y$ is another normed linear space;
... | 12 | https://mathoverflow.net/users/1849 | 441803 | 178,304 |
https://mathoverflow.net/questions/77845 | 9 | This question is not directly related to, but was inspired by, [this question](https://mathoverflow.net/questions/77583/is-the-free-product-of-arbitrarily-many-copies-of-mathbbz-and-mathbb). We know that a finitely generated residually nilpotent group is residually of prime-power order. However, we may need to use diff... | https://mathoverflow.net/users/1392 | Is there a residually nilpotent one-relator group that is not residually a finite p-group for any prime p? | The answer is yes, such a group exists: this is the main result of [1]. In fact, one can take the Baumslag-Solitar group $\operatorname{BS}(p^r, -p^r) = \langle a, b \mid ba^{p^r}b^{-1} = a^{-p^r} \rangle$ where $p$ is an odd prime and $r \geq 1$. These are residually nilpotent, but are not residually a finite $q$-grou... | 6 | https://mathoverflow.net/users/120914 | 441807 | 178,306 |
https://mathoverflow.net/questions/441800 | 0 | $z\_i=f+a\_i+\epsilon\_i$ ，where $f\sim N(\bar{f},\sigma\_{f}^2)$ ; $a\_i\sim N(\bar{a\_{i}},\sigma\_{a}^2)$; $\epsilon\_i\sim N(0,\sigma\_{\epsilon}^2)$. We can see the signals $\{z\_i\}$ where $i\subseteq {1,2,……,M}$, and note that $f$ is a common term for all signals. The question is we want to know the **Posterior ... | https://mathoverflow.net/users/500213 | How does this Bayesian updating work $z_i=f+a_i+\epsilon_i$ | $\newcommand{\bR}{\mathbb{R}}
\newcommand{\one}{\mathbf{1}}
\newcommand{\diag}{\textrm{diag}}
\newcommand{\Id}{\textrm{Id}}$
I guess you assume independence of the random variables $f,a,\varepsilon$, which I am going to use.
**General strategy** for Gaussian conditioning in linear models:
Let $a\sim N(\bar{a},Q\_{a... | 1 | https://mathoverflow.net/users/143828 | 441810 | 178,307 |
https://mathoverflow.net/questions/441591 | 6 | Recall that for $P$ a forcing order and $\dot{Q}$ a $P$-name for a forcing order, the termspace forcing $T(P,\dot{Q})$ consists of minimal-rank names for elements of $\dot{Q}$, ordered by $\dot{q}'\leq\dot{q}$ iff $1\_{P}\Vdash\dot{q}'\leq\_{\dot{Q}}\dot{q}$. In his chapter in the Handbook of Set Theory, James Cummings... | https://mathoverflow.net/users/138274 | Proof (or reference) about the cc-ness of termspace forcing | The answer is consistently negative, and it seems likely that it is actually always negative.
Let us look at the property: for every $|\mathbb P| < \kappa$, if $\Vdash\_{\mathbb{P}} \dot{\mathbb{Q}}$ is $\kappa$-c.c. then $T(\mathbb P, \dot{\mathbb Q})$ is $\kappa$-c.c.
It is known that for $\kappa$ which is weakly... | 7 | https://mathoverflow.net/users/41953 | 441814 | 178,310 |
https://mathoverflow.net/questions/441820 | 9 | Is it known if there are any examples of a finitely generated group $G$ such that:
1. $G$ has a finite index subgroup $H$ which is free-by-cyclic
2. $G$ itself is **not** free-by-cyclic
3. $G$ is torsion-free
Since subgroups of free-by-cyclic groups are free-by-cyclic, one may strengthen (1) and ask that $H$ is nor... | https://mathoverflow.net/users/117514 | Torsion-free virtually free-by-cyclic groups | The group $$G=\langle a, b, x, y\mid [a, b]^2=[x, y]^2\rangle$$
is a torsion-free group which is not free by cyclic. However, $G$ is free-by-$D\_{\infty}$ and so virtually free-by-cyclic (containing an index-two subgroup which is free-by-cyclic).
This example is from the paper Baumslag, Fine, Miller and Troeger, *Vir... | 12 | https://mathoverflow.net/users/6503 | 441826 | 178,315 |
https://mathoverflow.net/questions/441677 | 1 | Let $X$ be a smooth projective complex variety and $t\in H^{2p}(X,\mathbf{Q})(p)$ a rational cohomology class.
Assume that there exist
$$t\_1,\ldots,t\_N\in H^{2\*}(X,\mathbf{Q})(\*)$$
algebraic classes (i.e. in the image of the cycle map $cl : A^\*(X)\to H^{2\*}(X,\mathbf{Q})(\*)$) and an algebraic subgroup $$G\subs... | https://mathoverflow.net/users/nan | Cohomology classes fixed by algebraic automorphism subgroups | Essentially you are asking whether the category of homological motives generated by $X$ is tannakian, which is true if and only if homological equivalence coincides with numerical equivalence (Jannsen, Deligne). This coincidence is known for abelian varieties (Lieberman) but not much else, so I think the answer to your... | 1 | https://mathoverflow.net/users/500240 | 441831 | 178,316 |
https://mathoverflow.net/questions/441813 | 1 | Let $k$ be a number field and $\mathbb{G}\_m$ be the multiplicative group sheaf. For an algebraic group $G$, we define the character group $\widehat{G}:= \mathrm{Hom}\_{\bar{k}}(\bar{G},\mathbb{G}\_{m,\bar{k}})$, where $\bar{G}$ is the base change to an algebraic closure of $k$. This is a contravariant functor and I be... | https://mathoverflow.net/users/172132 | Character group functor of an exact sequence of algebraic groups | The functor isn't right exact. For example,
$\mathrm{Ext}^1(A,\mathbb{G}\_m)\simeq$
the set of primitive elements in $H^1(A,\mathcal{O}\_A^{\times})$.
| 4 | https://mathoverflow.net/users/500240 | 441843 | 178,321 |
https://mathoverflow.net/questions/441838 | 20 | As is probably clear from my previous questions, I am coming to "condensed mathematics" from the naive perspective of a category theorist, without much knowledge of the intended applications in algebraic geometry and functional analysis. I hope therefore that this question is not *too* naive, but I haven't found a plac... | https://mathoverflow.net/users/49 | Condensed vs pyknotic vs consequential | Some comments:
Regarding 1): They are quite different. Johnstone actually uses a very general notion of "cover" in his sequential topos -- his site is a full subcategory of metrizable profinite sets (=$\aleph\_1$-small profinite sets=countable limits of finite sets=sequential Pro-category of finite sets), but not all... | 19 | https://mathoverflow.net/users/6074 | 441847 | 178,322 |
https://mathoverflow.net/questions/441835 | 5 | This question is a question about nomenclature more than anything. I have shown all the math, but I don't know what to search for for similar results. In such a sense, it is more so a reference request, or a request for someone to point me in the right direction.
We start with the function:
$$
f\_1(z) = \sum\_{n=0}... | https://mathoverflow.net/users/133882 | Analytic continuation for disjoint domains | As I already hinted at in my comment, your formula is based on a miscalculation (the only alternative is that $f\_1$ can actually be continued past $|z|=1$, which even without closer analysis looks highly suspicious since the summands have poles at $n$th roots of $-1$, as you pointed out). Here's a direct quick test th... | 4 | https://mathoverflow.net/users/48839 | 441853 | 178,323 |
https://mathoverflow.net/questions/441844 | 4 | I heard that the Langlands functoriality conjecture implies the Ramanujan conjecture for $GL(2)$. especially for the Maass form.
There are various versions of the Langlands functoriality concerning to which groups are associated.
I am wondering which version of the Langlands functorial conjecture could prove the Ra... | https://mathoverflow.net/users/29422 | Which Langlands functoriality conjecture implies the original Ramanujan conjecture? | Let $F$ be a number field, let $\mathbb{A}\_F$ be the ring of adeles of $F$, and let $\mathcal{A}(n)$ be the set of cuspidal automorphic representation of $\mathrm{GL}\_n(\mathbb{A}\_F)$ with unitary central character. For $\pi\in\mathcal{A}(n)$, I will express the generalized Ramanujan conjecture (GRC) for $\pi$ as th... | 9 | https://mathoverflow.net/users/111215 | 441856 | 178,324 |
https://mathoverflow.net/questions/441848 | 1 | I happen to encounter the following inequality which I need to prove:
$$\left(k+(1+k)\left(1-\frac{1}{e}\left(1+\frac{1}{k}\right)^k\right)\right)\log\left(1+\frac{1}{k}\right)>1,$$ for $k\in\mathbb{Z}^{+}$.
My current idea is to expand $$k+(1+k)\left(1-\frac{1}{e}\left(1+\frac{1}{k}\right)^k\right)=k+\frac{1}{2}+\fr... | https://mathoverflow.net/users/136078 | An inequality about e | We can actually prove this directly for $k \ge 7$ and it should be easy to check it for $k<7$
We note that for $x < 1$ we have $x/2-x^2/3+x^3/4-x^4/5... \ge x/2-x^2/3$ (series converges absolutely and grouping in pairs the remainder is positive).
For $y<1$ we also have $1-e^{-y} \ge y-y^2/2$ again by using the Tayl... | 2 | https://mathoverflow.net/users/133811 | 441858 | 178,325 |
https://mathoverflow.net/questions/441871 | 0 | Consider a $N\times N$ normalized matrix sample from GOE (the definition see [https://www.lpthe.jussieu.fr/~leticia/TEACHING/Master2019/GOE-cuentas.pdf](https://www.lpthe.jussieu.fr/%7Eleticia/TEACHING/Master2019/GOE-cuentas.pdf)). If we apply the following result of the edge of the spectrum,
>
> If we denote the $... | https://mathoverflow.net/users/168083 | The probability upper bound on the ratio of the eigenvalues | The Tracy-Widom distribution says that the level spacing near the edge of the spectrum at $\pm 2$ is of order $N^{-2/3}$, hence we may define
$$|\sigma\_N|\equiv 2+N^{-2/3}\delta,\;\;|\sigma\_{N-k+1}|\equiv 2+N^{-2/3}\delta',$$
with $\delta,\delta'$ of order $N^0$. Now consider
$$\Delta=N^{2/3}\left(\frac{|\sigma\_N|}{... | 1 | https://mathoverflow.net/users/11260 | 441873 | 178,327 |
https://mathoverflow.net/questions/441861 | 0 | Suppose that for all $n$ natural numbers, $d\_{n}$ is a pseudometric on set $X
$. Define $d=\sum\_{n=1}^{\infty }a\_{n}\frac{d\_{n}}{1+d\_{n}}$, where $\left(
a\_{n}\right) $ is a sequence of positive numbers such that $\sum\_{n\in
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}a\_{n}<\infty $.
It i... | https://mathoverflow.net/users/86099 | A question about uniformities generated by pseudometrics | One direction:
let $\varepsilon>0$ and taken $N$ such that $\sum\_{n>N}a\_n<\frac12\varepsilon$. Take $\delta>0$ such that $\delta\cdot\sum\_{n\le N}a\_n<\frac12\varepsilon$. Now if $d\_n(x,y)<\delta$ for all $n\le N$ then
$$
d(x,y)< \delta\cdot\sum\_{n\le N}a\_n+\sum\_{n>N}a\_n<\varepsilon
$$
(because $d\_n(x,y)/(1+d\... | 1 | https://mathoverflow.net/users/5903 | 441875 | 178,328 |
https://mathoverflow.net/questions/441877 | 2 | [Here](https://mathoverflow.net/questions/371010/how-to-calculate-inverse-of-sum-of-two-kronecker-products-with-specific-form-eff) and [here](https://www.sciencedirect.com/science/article/pii/S0024379587903144?via%3Dihub), specific ways to address the equation in $x$, for $N=2$, are given:
$$\sum\_{i=1}^N (A\_i\otime... | https://mathoverflow.net/users/155442 | Linear system with sum of Kronecker products | The recent state of the art is described in section 7.2 of Simoncini, V. "Computational methods for linear matrix equations." *SIAM Rev.* **58**, 377 (2016), <https://doi.org/10.1137/130912839>. Your equation is equivalent to equation (2) in that reference. A lightly reformatted quote from there:
>
> Equation (2) i... | 1 | https://mathoverflow.net/users/1847 | 441893 | 178,332 |
https://mathoverflow.net/questions/441870 | 0 | (**Preliminaries:**) 1.) Let $S\subset\mathbb{R}^n$ and define $\mathcal{M}(S) = \{\text{$\mu$ a Borel measure}: \text{$0 < \mu(S) < \infty$ and $\mathrm{support}(\mu)\subset S$}\}$.
2.) Define the Fourier transform of a measure as $\hat{\mu}(x):= \int\_{\mathbb{R}^n}e^{-2\pi i\xi \cdot x}d\mu(\xi)$.
3.) Define the... | https://mathoverflow.net/users/172696 | Calculating the Fourier dimension of a real interval $\left[a, b\right]$ | If $-\infty\le a<b\le\infty$, then $\dim\_F[a,b]=\infty$.
Indeed, without loss of generality $[a,b]=[-1,1]$. Take any natural $n$ and let
$$\nu\_n:=\frac1{c\_n}\,\mu\_{1/2}\*\cdots\*\mu\_{1/2^n},$$
where $\mu\_L$ is the uniform distribution on $[-L,L]$ and $c\_n:=\prod\_{k=1}^n 2^k$. Then for all real $t\ne0$
$$\hat\... | 2 | https://mathoverflow.net/users/36721 | 441899 | 178,335 |
https://mathoverflow.net/questions/441834 | 3 | Usually, Berkovich analytic spaces are derived from some Banach rings (or chains of Banach rings) over a completely normed field $k$ through Berkovich spectrum. But when the base field is the complex number $\Bbb{C}$, there is another way of getting analytic spaces, namely gluing open sets in $\Bbb{C}^n$ via biholomorp... | https://mathoverflow.net/users/166298 | "Non-algebraic" Berkovich spaces | Indeed, every complex manifold is locally isomorphic to an open in $\mathbf{C}^n$. More generally, we define complex analytic spaces as those locally isomorphic to a locally ringed space of the form $(Z, i^{-1}\mathcal{O}\_U/I)$ where $U$ is an open in $\mathbf{C}^n$, $I$ is the ideal in the sheaf $\mathcal{O}\_U$ of h... | 5 | https://mathoverflow.net/users/3847 | 441901 | 178,336 |
https://mathoverflow.net/questions/441894 | 3 | Let $G$ be a discrete group, and let $\mathcal{F}$ be a family of subgroups of $G$ (closed under conjugation and taking subgroups). Then we may define the geometric and cohomological dimensions of $G$ with respect to $\mathcal{F}$, denoted $\operatorname{gd}\_\mathcal{F}(G)$ and $\operatorname{cd}\_\mathcal{F}(G)$ resp... | https://mathoverflow.net/users/8103 | Geometric vs cohomological dimension with families - on a proof of Lueck and Meintrup | **Note** This is based on misreading the question as being about projective resolutions, not free ones. I had deleted it but the OP found it helpful so I have undeleted.
I have managed to cobble this out of the Brown references I gave in now deleted comments. In this answer all chain complexes are assumed to be bound... | 1 | https://mathoverflow.net/users/15934 | 441908 | 178,340 |
https://mathoverflow.net/questions/441829 | 3 | Let $V$ be a finite-dimensional vector space over $\mathbb{R}$ equipped with an inner product $\omega(-,-)$. One standard fact is that there is an induced inner product on $\wedge^k V$. For instance, this shows up when you're setting up Hodge theory.
All the constructions I've seen in books construct the inner produc... | https://mathoverflow.net/users/500239 | Coordinate free way to construct inner product on exterior powers | I'm going to answer my own question, summarizing the comments and adding a little more from my own reflections (marked community wiki, though it doesn't matter since this is not a registered account and I can't earn reputation from it).
The first observation (from Tom Goodwillie) is that it is actually very easy to s... | 3 | https://mathoverflow.net/users/500239 | 441911 | 178,342 |
https://mathoverflow.net/questions/441752 | 2 | I am working with subsets of $[n]$ of the form $(A+B)\cap A$, where $A+B$ is a sumset. Namely, I am interested if there are nonempty sets $B$ such that whenever $A$ covers a positive proportion of $[n]$, the set $(A+B)\cap A$ also covers a positive proportion of $[n]$. In fact it would be nice if the proportion is the ... | https://mathoverflow.net/users/168142 | Sets with certain property concerning density of sumsets | The set of factorials $B$ is not intersective, hence there is a set $A$ of positive density for which $A$ and $A+B$ are disjoint.
Indeed, every natural number $n$ has a unique "factorial base" representation
$$ n = \sum\_{k=1}^\infty a\_k k!$$
with $0 \leq a\_k \leq k$ and all but finitely many of the $a\_k$ non-zero... | 4 | https://mathoverflow.net/users/766 | 441912 | 178,343 |
https://mathoverflow.net/questions/441907 | 0 | Let $G$ be a group and let $\mathbb{G}$ be the associated one object category. Is there an explicit presentation of representable functors from $\mathbb{G} \to $**Set**? If so how does the Yoneda lemma look like explicitly in this setting?
| https://mathoverflow.net/users/160055 | Yoneda lemma for one object categories | Let me write $\def\B{\mathbf{B}}\B G$ for what you call $\mathbb{G}$.
You can check that presheaves on $\B G$ are precisely the right $G$-sets: the unique point of $\B G$ is sent to some set $X$, and functoriality defines a group homomorphism $\def\op{\mathrm{op}}G^\op\to\operatorname{Aut}(X)$.
In particular, the (... | 2 | https://mathoverflow.net/users/160838 | 441913 | 178,344 |
https://mathoverflow.net/questions/441321 | 6 | Where can I find the construction for a **skew Hadamard matrix** of order 756?
According to multiple papers (e.g. [Koukouvinos and Stylianou - On skew-Hadamard matrices](https://www.sciencedirect.com/science/article/pii/S0012365X07004013?via%3Dihub) and [Seberry - On skew Hadamard matrices](https://ro.uow.edu.au/cgi/... | https://mathoverflow.net/users/498306 | How to construct a skew Hadamard matrix of order 756? | The matrix can be constructed using Theorem 7 from [Seberry - On skew Hadamard matrices](https://ro.uow.edu.au/cgi/viewcontent.cgi?referer=&httpsredir=1&article=2003&context=infopapers), by setting $m = 1$ and $n = 28$.
The amicable Orthogonal Design of type $((1, 27); (28))$ are constructed from amicable hadamard ma... | 4 | https://mathoverflow.net/users/498306 | 441915 | 178,345 |
https://mathoverflow.net/questions/441393 | 4 | In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which:
1. there are vector spaces without a basis;
2. the field of complex numbers $\mathbb{C}$ only has two automorphisms (identity and complex conjugation).
Before I even ask my main question, I want ... | https://mathoverflow.net/users/12884 | Automorphisms of vector spaces and the complex numbers without choice | This is not a full answer, but it is too long to be a comment.
Let $B(F)$ for field $F$ be the statement "every vector space over $F$ has a basis" and let $AL19(F)$ be the statement "for every vector space $V$ over $F$, every generating subset of $V$ contains a basis", $AL20(F)$ means "for every vector space $V$ over... | 4 | https://mathoverflow.net/users/113405 | 441918 | 178,346 |
https://mathoverflow.net/questions/440661 | 2 | Let $f:[0, \infty)\to [0, \infty)$ be non-decreasing and satisfy for all $t>t\_{0}$, $$f(t)+C\int\_{t\_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t\_{0}}\int\_{t\_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ and $C>0$.
Of course, if $f$ is differentiable and the somewhat similar differential inequality $$f'+Cf^{\gamma}\leq 0$$ h... | https://mathoverflow.net/users/163368 | Property so that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$? | As $f$ is non-decreasing, the integral inequality implies that $f$ is $0$.
**Proof.** The integral inequality implies, in particular, that the same inequality is satisfied for $C=0$, so
$$
\int\_{t\_0}^t f(t) \, ds = (t-t\_0)f(t) \le \int\_{t\_0}^t f(s) \, ds
$$
for all $t \ge t\_0$. Now fix $t > t\_0$. Then it foll... | 1 | https://mathoverflow.net/users/102946 | 441954 | 178,357 |
https://mathoverflow.net/questions/441956 | 1 | Let $f:[0, \infty)\to [0, \infty)$ be **non-increasing** and satisfy for all $t>t\_{0}$, $$f(t)+C\int\_{t\_{0}}^{t}f^{\gamma}(s)ds\leq \frac{1}{t-t\_{0}}\int\_{t\_{0}}^{t}f(s)ds,$$ where $0<\gamma<1$ and $C>0$.
**My question**: Is it true that $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$?
**Remark**: If $... | https://mathoverflow.net/users/163368 | Integral inequality implies $f(t)\equiv 0$ for all $t\geq T$ for some finite $T>0$? | Since $f$ is nonnegative and nonincreasing, we have $f(t)\to L$ as $t\to\infty$ for some $L\in[0,\infty)$. Then from your first display we get
$$L+C\int\_{t\_{0}}^\infty f^{\gamma}(s)\,ds\le L.$$
So, $f=0$ on $(t\_0,\infty)$, as desired.
| 3 | https://mathoverflow.net/users/36721 | 441959 | 178,358 |
https://mathoverflow.net/questions/441898 | 3 | Let $X$ be a compact metric space, and let $K\_X$ be the set of non-empty closed subsets of $X$, equipped with the $\sigma$-algebra
$$ \mathcal{B}(K\_X) \ := \ \sigma(\{C \in K\_X : C \cap U = \emptyset\} \, : \, \text{open } U \subset X ) . $$
(This is precisely the Borel $\sigma$-algebra of the Hausdorff metric.) Def... | https://mathoverflow.net/users/15570 | Can the set of compact metrisable topologies naturally be equipped with the structure of a standard Borel space? | In most cases the answer is no:
A much stronger result was proved in
J. Zielinski: The complexity of the homeomorphism relation between compact metric spaces, Adv. Math. 291 (2016), 635–645.
It is shown that the homeomorphism equivalence relation on compact subspaces of the Hilbert cube X=Q is Borel bireducible to the ... | 4 | https://mathoverflow.net/users/128723 | 441962 | 178,359 |
https://mathoverflow.net/questions/441931 | 0 | Here is the definition of the frog model we are interested in:
"... consider the homogeneous tree $\mathbb{T}\_{d}$, that is, the rooted tree in which each
vertex has (is connected by edges to) $d + 1$ neighbours. One frog is put at each vertex and
all but the one of the root start inactive. Active frogs perform simp... | https://mathoverflow.net/users/476677 | What is the probability space corresponding to the probability measure $\mathbb{P}_{p}$ in the context of this paper? | (1) As you read the probability literature, you'll soon discover that people tend not to be very specific about the sample space that they are working with. The details of that space are not important; what matters is the collection of observables (events and random variables) and their joint distribution. Any probabil... | 3 | https://mathoverflow.net/users/5784 | 441977 | 178,364 |
https://mathoverflow.net/questions/441942 | 2 | I have a block matrix
$$M=\begin{bmatrix}
I\_0& I\_1& \cdots& I\_1\\
I\_2& I\_0& \ddots& \vdots\\
\vdots& \ddots& \ddots& I\_1\\
I\_2& \cdots& I\_2& I\_0\\
\end{bmatrix}\_{n \times n}$$
with
$$I\_0=\begin{bmatrix}
0& 1\\
1& 0\\
\end{bmatrix}, \qquad
I\_1=\begin{bmatrix}
0& 1\\
-1& 0\\
\end{bmatrix},\qquad... | https://mathoverflow.net/users/495317 | Eigenvalues of a specific matrix | For the signed circulant matrix
$$U:=\left[\begin{matrix}
& 1 & & & \\
& & \ddots & & \\
& & & 1 &\\
-1 & & & &
\end{matrix}\right]
\mbox{ in }
M\_n(\mathbb{C}),$$
one has
$$M= 1 \otimes I\_0 + (U+U^2+ \cdots + U^{n-1}) \otimes I\_1
\mbox{ in }
M\_n(\mathbb{C})\otimes M\_2(\mathbb{C}).$$
For $\omega:=\exp\frac{i\... | 4 | https://mathoverflow.net/users/7591 | 441988 | 178,369 |
https://mathoverflow.net/questions/441980 | 14 | It is not too hard to find examples of finite groups which have fewer automorphisms than one of their subgroups. For example $\mathcal D\_4 \times \mathbb Z/2\mathbb Z$ (where $\mathcal D\_4$ is the dihedral group of order $8$) has at most (in fact exactly) $64$ automorphisms but contains a subgroup isomorphic to $\lef... | https://mathoverflow.net/users/133679 | Finite abelian groups with fewer automorphisms than a subgroup | From the Hiller-Rhea formula
$$|\operatorname{Aut} H\_p| = \prod\_k (p^{d\_k} - p^{k-1}) \prod\_j (p^{e\_j})^{n-d\_j} \prod\_i (p^{e\_i-1})^{n-c\_i+1},$$
given an abelian $p$-group of type $p^{e\_1}\cdots p^{e\_n}$ where $1 \leq e\_1 \leq \cdots \leq e\_n$, you can try to increment one of the exponents $e\_l$ by 1, and... | 19 | https://mathoverflow.net/users/121 | 441999 | 178,372 |
https://mathoverflow.net/questions/441994 | 3 | In thinking about my question [here](https://mathoverflow.net/questions/441982) for the Linial arrangement, the following limit arose:
$$ \lim\_{n\to\infty}\frac{(n-1)\sum\_{k=0}^n {n\choose k}(k+1)^{n-2}}
{\sum\_{k=0}^n {n\choose k}(k+1)^{n-1}}. $$
Is this limit finite? What is its value? It might be around $2.27\cdo... | https://mathoverflow.net/users/2807 | A limit involving the quotient of two sums | First of all, it seems like the value of the limit is more like $1.27...$, and not $2.27...$.
Using some heuristics outlined below it is possible to find the limit:
$$
a=1.278464542761...,
$$
where $a$ is the root of $(x-1)e^x=1$ (can be expressed in terms of Lambert W-function).
The approach is standard, based on... | 12 | https://mathoverflow.net/users/82588 | 442008 | 178,374 |
https://mathoverflow.net/questions/441797 | 8 | I am a PhD student in the represention theory of finite groups. One of my friends and I solved all exercises in the book [I M Isaacs - Algebra A Graduate Course](https://bookstore.ams.org/view?ProductCode=GSM/100) except for the following exercise in Chapter 9(transfer theory):
**(Exercise 9.4)**
Let $G$ be a finit... | https://mathoverflow.net/users/138348 | I M Isaacs Algebra Exercise 9.4 | I think the exercise is incorrect (and so is the hint).
There was a question before about this same exercise at math.stackexchange; the answer there gives a counterexample. Link: [Math.SE](https://math.stackexchange.com/questions/4577477/fusion-in-the-normaliser-of-a-sylow-subgroup/).
| 7 | https://mathoverflow.net/users/38068 | 442010 | 178,375 |
https://mathoverflow.net/questions/441990 | 1 | Let $N(h)$ be the number of solutions of the following linear diophantine equation:
\begin{equation}
x\_1 + 2x\_2 + 3x\_3 + \dots + (h-1)x\_{h-1} = 6h-6;
\end{equation}
where $h\geq 2$ and solution means a vector $(z\_1,\dots,z\_{h-1})$ of non-negative integers satisfying the equation.
Does there exist a formula for ... | https://mathoverflow.net/users/14514 | Solutions of a linear diophantine equation | As Max Alekseyev observed, $N(h)$ is the number of partitions of $6h-6$ into at most $h-1$ parts. A complicated asymptotic formula exists for this quantity, as a special case of a result of Szekeres (1953). See [this paper by Canfield](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v4i2r6/pdf) for more d... | 3 | https://mathoverflow.net/users/11919 | 442012 | 178,376 |
https://mathoverflow.net/questions/441792 | 9 | There are many objects in mathematics that have the term "chiral" in their name, for instance, chiral algebra by Beilinson and Drinfeld, chiral de Rham complex, chiral Koszul duality etc. Some people told me that chiral algebras are $2$-dimensional analogue of associative algebras, which are considered to be $1$-dimens... | https://mathoverflow.net/users/466793 | What is the meaning of chiral in the context of vertex algebras? | A vertex operator algebra describes the algebra of local operators in the chiral part of a 2d CFT. Typically one sees a VOA depending on a complex coordinate $z$. To describe a full 2d CFT, you would need to also include an "anti-chiral" VOA depending on a conjugate coordinate $\bar{z}$. So by considering only a single... | 2 | https://mathoverflow.net/users/88421 | 442020 | 178,378 |
https://mathoverflow.net/questions/442017 | 5 | Given a homeomorphism between complex manifolds, $f : X → Y$, is it then true that the rational Pontrjagin class $p\_1(X) \in H^4(X,\mathbb Q)$ equals the pull-back $f^\* p\_1(Y)$?
If $X$ and $Y$ are compact, then I understand that this is the famous Novikov result. I am, however, unsure if the result holds in the no... | https://mathoverflow.net/users/33531 | Topological invariance of rational Pontrjagin classes for non-compact spaces | I am not sure about the reference, but here is an argument: the map $BO\to BTOP$ induces an isomorphism on rational cohomology, as mentioned, e.g., on p.2 of [Dalian notes on rational Pontryagin classes](https://arxiv.org/pdf/1507.00153.pdf) by Weiss. The topological rational Pontryagin class $p\_i$ is an element in $H... | 8 | https://mathoverflow.net/users/1573 | 442025 | 178,380 |
https://mathoverflow.net/questions/316226 | 11 | Let $G$ be a finite $p$- group and let $\varphi$ be an automorphism of $\mathbb{F}\_pG$ as $\mathbb{F}\_p$-algebras and let $n = \sum\_{g\in G} g$ be the norm element. Does it follow that $\varphi(n)=n$?
Since the $\mathbb{F}\_p$-vector space spanned by $n$ is exactly the annihilator of the maximal ideal, it follows ... | https://mathoverflow.net/users/3969 | Is the norm element characteristic in modular group rings? | I know this is an old question, but I think the answer is yes. The idea is to pass to the associated graded for the Jennings filtration, and look at the $p$-restricted Lie algebra $L(G)$ you get that way. The element $n$ corresponds to the socle element in the restricted universal enveloping algebra, which is one dimen... | 4 | https://mathoverflow.net/users/460592 | 442037 | 178,384 |
https://mathoverflow.net/questions/442049 | 2 | We call the natural number $n$ a partition number $\iff$
$$
\exists d | n: \gcd\left(d,\frac{n}{d}\right)=1 \text{ and } \Omega(d) = \Omega\left(\frac{n}{d}\right)\;,
$$
where $\Omega$ counts the prime divisors with multiplicities.
Then we have:
$n= p\_1^{a\_1} \cdots p\_r^{a\_r}$ is a partition number $\iff S = (a\_... | https://mathoverflow.net/users/165920 | Using Kolmogorov complexity to measure difficulty of problems? | It is worth mentioning that *variants of* Kolmogorov complexity have been investigated in the context of the study of pseudorandomness.
See [On One-Way Functions from NP-Complete Problems by Yanyi Liu and Rafael Pass](https://drops.dagstuhl.de/opus/volltexte/2022/16598/) ([arXiv link](https://arxiv.org/abs/2009.11514))... | 5 | https://mathoverflow.net/users/101207 | 442054 | 178,388 |
https://mathoverflow.net/questions/442058 | -4 | I have been working on a problem concerning the "line of sight" from a fixed integer co-ordinate — let's say $(0,0)$ — to a variable co-ordinate $(a,b)$. Having a line of sight means that there are no other integer co-ordinates on a straight line connecting the two points.
This is a a well documented problem and ther... | https://mathoverflow.net/users/497178 | Proving that $P($$\{\text{$a$ and $b$ are co-prime}$ }$)=0$ for $a,b$ following the Uniform distribution over $[n, 2n]$ as $n \rightarrow \infty$ | I seem to get the probability is still the usual $1/\zeta(2)$ by the usual inclusion/exclusion argument, but possibly there's a mistake in the following calculation:
$$
\begin{aligned}
\frac{1}{n^2}\sum\_{a,b\in[n,2n]} \;\;\sum\_{d\mid(a,b)} \mu(d)
&= \frac{1}{n^2}
\sum\_{1\le d\le 2n} \mu(d)\sum\_{\substack{A,B\\ A\in... | 6 | https://mathoverflow.net/users/11926 | 442059 | 178,389 |
https://mathoverflow.net/questions/442063 | 2 | Let $(E, d)$ be a metric space. For $\varepsilon>0$, we define two notions of $\varepsilon$-covering number as follows, i.e.,
* $N\_\varepsilon^o (E)$ is the smallest number of *open* balls whose radii are $\varepsilon$ that cover $E$.
* $N\_\varepsilon^c (E)$ is the smallest number of *closed* balls whose radii are ... | https://mathoverflow.net/users/99469 | Are two metric spaces isometric if they have the same $\varepsilon$-covering numbers for all $\varepsilon>0$? | No. For $0 < \delta \leq 2$ let $E\_\delta$ be the metric space
consisting of three points $A,B,C$ with
$d(A,B) = d(A,C) = 1$ and $d(B,C) = \delta$.
I claim that the $E\_\delta$ for $1 \leq \delta \leq 2$ all have
the same covering numbers for all radii $\varepsilon$.
Indeed in every such $E\_\delta$
all open $\varepsi... | 6 | https://mathoverflow.net/users/14830 | 442064 | 178,391 |
https://mathoverflow.net/questions/441724 | 15 | The ordinary generating function $T\_n=T\_n(x)$ for the $n$-ary trees satisfies the functional equation
$$
T\_n=1+xT\_n^n.
$$
This is usually defined for $n\ge 0$, but the functional equation can be extended to negative $n$. Writing
$$
T\_{-n}=1+xT\_{-n}^{-n}
$$
and dividing through by $T\_{-n}$, we obtain that
$$
T\_{... | https://mathoverflow.net/users/113161 | A combinatorial interpretation for $n$-ary trees for negative $n$ | Here's an explanation of the combinatorial meaning of $T\_{-n}(x)$.
The combinatorial interpretation $T\_n(x)$ is that it counts $n$-ary trees. More precisely, it counts ordered trees in which every vertex has 0 or $n$ children, and each internal vertex (with $n$ children) is weighted $x$ and each leaf is weighted 1.... | 12 | https://mathoverflow.net/users/10744 | 442068 | 178,394 |
https://mathoverflow.net/questions/442070 | 2 | Let $(X, d)$ be a compact metric space.
* We say that $\{x\_1, \cdots, x\_n\} \subseteq X$ is an $\varepsilon$-**covering** of $X$ if for any $x \in X$, there exists $i \in \{1, \ldots, n\}$ such that $d(x, x\_i) \leq \varepsilon$. Let
$$
\operatorname{Cov} (X, \varepsilon) := \min \{n: \exists \varepsilon \text {-co... | https://mathoverflow.net/users/99469 | Are two metric spaces isometric if they have the same $\varepsilon$-covering and $\varepsilon$-packing numbers for all $\varepsilon>0$? | Certainly no. Consider metric spaces on $n$ points and all distance 1 and 2. There are $2^{n^2/2+o(n^2)}$ such spaces. But only polynomially many different covering and packing functions.
| 10 | https://mathoverflow.net/users/4312 | 442072 | 178,395 |
https://mathoverflow.net/questions/442035 | 6 | Let $ \mathbb{P}$ be any directed, well-founded poset of cofinality $ \aleph\_{n+1}$, where $n$ is a natural.
Can we write it as an increasing union $ \mathbb{P} = \bigcup\_{\alpha < \omega\_{n+1} } \mathbb{P}\_\alpha $
where each $\mathbb{P}\_\alpha $ is a directed poset of cofinality $\aleph\_n$ ?
| https://mathoverflow.net/users/495743 | Poset as union of posets of lower cofinality | Yes, and well-foundedness is irrelevant (this is working in ZFC). Let $\mathbb{P}$ be the poset, with ordering $\leq$. Let $f:[\mathbb{P}]^2\to\mathbb{P}$ be a function such that for all $x,y\in\mathbb{P}$, we have $x\leq f(\{x,y\})$ and $y\leq f(\{x,y\})$.
Let $c:\aleph\_{n+1}\to\mathbb{P}$ be cofinal in $\mathbb{P}$,... | 6 | https://mathoverflow.net/users/160347 | 442080 | 178,398 |
https://mathoverflow.net/questions/442081 | 0 | For a one-dimensional $f(x)$, the dilation operator $f(ax)$ can be expressed as $\exp(g(D))f(x)$, where $g$ is a closed-form function. This is easily checked by e.g. formal Taylor series expansion.
However, it is not clear what to do in the multidimensional case- even if one restricts $U$ to be orthogonal with determ... | https://mathoverflow.net/users/113020 | Differential form of the multidimensional "orthogonal dilation" operator | In 3 dimensions, rotations, i.e., transformations corresponding to orthogonal $U$ with determinant 1, are generated by the (orbital) angular momentum operator $\vec{L} $ with components $L\_i =-i \epsilon\_{ijk} x\_j \,\partial / \partial x\_k $. By Euler's rotation theorem, any given such transformation can be effecte... | 1 | https://mathoverflow.net/users/134299 | 442084 | 178,399 |
https://mathoverflow.net/questions/441609 | 2 | In Jacob Rasmussen's paper [Khovanov homology and the slice genus](https://arxiv.org/pdf/math/0402131.pdf), he states as Corollary 3.6 that $s(\mathfrak s\_o)=s(\mathfrak s\_{\bar o})=s\_{min}(K)$, where $s$ is the $s$-grading and $\mathfrak s\_o,\mathfrak s\_{\bar o}$ are Lee's canonical generators for her homology th... | https://mathoverflow.net/users/146012 | Corollary in Rasmussen's paper about $s$-grading of Lee's canonical generators | Let $ C ( D) $ denote the Lee chain complex and $ Kh ' ( K ) $ its homology, $ q $ denote the grading on $ C ( D) $ (associated to the filtration) and $ s $ the induced grading on $ Kh ' ( K ) $.
Recall that $ \mathfrak{s}\_\overline{o} $ is obtained from $ \mathfrak{s}\_o $ by interchanging $ r = v\_+ + v\_- $ and $... | 3 | https://mathoverflow.net/users/61064 | 442085 | 178,400 |
https://mathoverflow.net/questions/442078 | 1 | Let
$$\ell(n)=\left\lfloor\log\_2 n\right\rfloor$$
Let
$$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$
Here $T(n,k)$ is the $(k+1)$-th bit from the right side in the binary expansion of $n$.
Let $a(n)$ be the sequence of positive integers such that we start from $A:=0$ and then for $k=0..\ell(n)$... | https://mathoverflow.net/users/231922 | Coefficients of number of the same terms which are arising from iterations based on binary expansion of $n$ | In other words, if $(b\_\ell b\_{\ell-1}\dots b\_0)\_2$ is the binary representation of $n$, then
$$a(n) = g(g(\dots g(g(0,b\_0),b\_1)\dots ),b\_{\ell-1}), b\_\ell),$$
where
$$g(A,b) = \begin{cases} A+2, &\text{if } b=0;\\
\left\lfloor \frac{A+2}2\right\rfloor, &\text{if } b=1.
\end{cases}$$
Consider a number triangl... | 2 | https://mathoverflow.net/users/7076 | 442095 | 178,404 |
https://mathoverflow.net/questions/441910 | 4 | Let $\Omega$ be a Polish locally compact space and $(\Omega, \mathscr{F}, \mathbb{P})$ be a probability space. Consider a measurable map
\begin{align\*}
\theta\colon T\times \Omega &\to \Omega\\
(t,\omega)&\mapsto \theta\_{t}\omega
\end{align\*}
such that $\theta\_{t+s}=\theta\_{t}\circ \theta\_{s}$ for
every $s, t\in ... | https://mathoverflow.net/users/98969 | Finite number of ergodic random Dirac measures | Let $\theta$ be a discrete time dynamical system as in your question. And for each $\omega$, let $A(\cdot)$ be a measurable map from $\Omega$ into $\text{GL}(d,\mathbb R)$ such that $\log \|A(\cdot)\|$ and $\log\|A(\omega)^{-1}\|$ are integrable.
Let me also assume $\mathbb P$ is ergodic. Then the multiplicative etgo... | 1 | https://mathoverflow.net/users/11054 | 442096 | 178,405 |
https://mathoverflow.net/questions/431840 | 0 | We have a inhomogeneous continous $K$-State Markov chain $X(t)$ with transition intensity matrix $Q(t)$. Therefore its entries are:
$$q\_{ij}(t)= \lim\_{\delta \to 0} \frac{1}{\delta} \mathbb{P}(X(t+\delta) = j | X(t) = i).$$
The transistion probabilities are in a matrix $P(s,t)$ with entries
$$P\_{ij}(s,t) = \mathbb{P... | https://mathoverflow.net/users/476504 | Inhomogeneous Markov chains and the product-integral as a solution to the Kolmogorov forward equation | I see it as follows: $$\frac{\partial}{\partial t}P(s,t)=P(s,t)Q(t) \rightarrow P(x,y+\Delta y)=P(x,y)+P(x,y)Q(y)\Delta y+o(\Delta y)\ ,$$ so we also have $$P(x,y+\Delta y)=P(x,y)[I+Q(y)\Delta y]+o(\Delta y) \rightarrow P(y,y+\Delta y)=I+Q(y)\Delta y+o(\Delta y)\ .$$Now, as $P(a,c)=P(a,b)P(b,c)$ whenever $a\le b \le c$... | 0 | https://mathoverflow.net/users/470349 | 442098 | 178,406 |
https://mathoverflow.net/questions/442104 | 8 | I'm trying to come up with a good explanation for my students of why the [finite lattice representation problem](https://en.wikipedia.org/wiki/Finite_lattice_representation_problem) is difficult. I've already shown that the "greedy approach" to representing the lattice drawn in [this post](https://mathoverflow.net/ques... | https://mathoverflow.net/users/8133 | Example of trickiness of finite lattice representation problem? | $M\_4$, the modular lattice of height two with four atoms is an example.
$M\_4$ arises as the subgroup lattice of the symmetric group on $3$ letters, hence it arises as the congruence lattice of a regular $S\_3$-set. This is an algebra with $6$ elements. There is no representation of $M\_4$ as the congruence lattice of... | 8 | https://mathoverflow.net/users/75735 | 442105 | 178,408 |
https://mathoverflow.net/questions/442099 | 5 | Let $\mathscr{C}$, $\mathscr{D}$, and $\mathscr{E}$ be (infinity) categories, and assume we're given a Cartesian diagram: $\require{AMScd}$
\begin{CD}
\mathscr{C} \times\_{\mathscr{E}} \mathscr{D} @>\operatorname{pr}\_1>> \mathscr{D}\\
@V \operatorname{pr}\_2 V V @VV \varphi V\\
\mathscr{C} @>>\psi> \mathscr{E}
\end{CD... | https://mathoverflow.net/users/101861 | Base change isomorphism for left Kan extensions | I believe one set of conditions is for either $\varphi$ to be proper or $\psi$ to be smooth. The dual of this (using right Kan extensions rather than left Kan extensions) is proven by Cisinski in "Higher Categories and Homotopical Algebra", Theorem 6.4.13.
For completeness, a map between simplicial sets $p : X \to Y$... | 6 | https://mathoverflow.net/users/76636 | 442113 | 178,411 |
https://mathoverflow.net/questions/442107 | 5 | The unknot and the Hopf link are (as far as I know) the only links whose complements have abelian fundamental groups. Are there more examples whose complement have amenable fundamental group?
| https://mathoverflow.net/users/89741 | Amenable link groups | No. Suppose that $K$ is a non-trivial knot. Then its knot genus is at least one. Thus $X = S^3 - K$ contains a $\pi\_1$-essential surface (with boundary) of genus at least one. Thus $\pi\_1(X)$ contains a free group of rank at least two.
The story for links is similar.
---
A more subtle proof uses the geometris... | 7 | https://mathoverflow.net/users/1650 | 442120 | 178,414 |
https://mathoverflow.net/questions/442114 | 7 | I apologise in advance for what must be a naive question. Let $\mathcal O\_K$ be the ring of integers of the algebraic number field $K.$ Let $p$ be a rational prime, and factorize $$(p)=\mathfrak p\_1^{e\_1}\cdots\mathfrak p\_r^{e\_r}$$ where the $\mathfrak p\_i$ are primes in $\mathcal O\_K.$ Let $k\_i=\mathcal O\_K/\... | https://mathoverflow.net/users/313687 | Quotients of number fields by certain prime powers | I remember working this out 25 years ago. The main idea is to view both rings as quotient rings of completions: $\mathcal O\_K/\mathfrak p^e \cong \widehat{\mathcal O\_{\mathfrak p}}/\widehat{\mathfrak p}^e$ and $k[t]/(t^e) \cong k[[t]]/(t^e)$. Then the structure of the ring of integers of local fields, in characterist... | 15 | https://mathoverflow.net/users/3272 | 442125 | 178,415 |
https://mathoverflow.net/questions/442124 | 15 | **INTRODUCTION**. I am teaching a course in Harmonic Analysis. In class, very often I find myself stressing out the fundamental property that the functions
$$
e\_n(x)=\exp(2\pi i n x), \quad \text{where }\quad x\in \mathbb R / \mathbb Z$$
are simultaneous eigenfunctions of all translations. By this I mean that, defi... | https://mathoverflow.net/users/13042 | Is there a non-constant function on the sphere that diagonalizes all rotations simultaneously? | For $\mathrm{C}^1$-functions, the argument can be reduced to the circle case: Write the sphere as a union of circles meeting at the poles. On each circle the considered function is equivariant under all rotations, therefore its restriction to the circle must be of the form $t\mapsto C\exp(i\lambda t)$ for some real num... | 27 | https://mathoverflow.net/users/58125 | 442129 | 178,416 |
https://mathoverflow.net/questions/442071 | 1 | [Crossposted](https://math.stackexchange.com/q/4651807/339790) on Mathematics SE
---
I've seen in many optimisation papers the statement that general non-convex optimisation problem is NP-hard. If we assume that non-convex optimisation is in NP class, it can be shown, as I remember, that some NP-complete problems... | https://mathoverflow.net/users/479332 | Is non-convex optimisation really in NP class? | As noted in a [comment by Emil Jeřábek](https://mathoverflow.net/questions/442071/is-non-convex-optimisation-really-in-np-class#comment1140811_442071), $\mathsf{NP}$ is a class of decision problems, so on the face of it, an optimization problem cannot be in $\mathsf{NP}$ for the rather trivial reason that it is the wro... | 4 | https://mathoverflow.net/users/3106 | 442139 | 178,420 |
https://mathoverflow.net/questions/442138 | -7 | Is the group of symmetries of the rectangle-not-square related to the Klein bottle mathematically?
The reason I am asking is because I want to put a Klein bottle coffee cup in a joke about V\_4 and wonder if this would be a good idea.
| https://mathoverflow.net/users/498715 | Is the Klein group related to the Klein bottle? | No, according to wikipedia ([bottle](https://en.wikipedia.org/wiki/Klein_bottle), [group](https://en.wikipedia.org/wiki/Klein_four-group)) they're both named after Felix Klein, but appeared in different papers on completely different topics. The bottle comes from his [notes](https://www.kleinbottle.com/The%20First%20Kl... | 3 | https://mathoverflow.net/users/22 | 442140 | 178,421 |
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