parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/307622 | 3 | If $X$ is a Kan complex, then it is an easy consequence of the existence of the Quillen model structure or of the basic theory of anodyne extensions, that $X^{K}$ is also Kan. However, I am interested in a "direct" proof of this fact, appealing to neither of the above (which is apparently how these sorts of things were... | https://mathoverflow.net/users/113828 | $X^K$ a Kan complex, without model structure or anodyne extensions | For me, the 'usual' way is as suggested by მამუკა ჯიბლაძე. This is based on a systematic analysis of the simplices of $\Delta^m\times \Delta^n$ and of those missing in the corresponding subcomplex involving the k-horn of $\delta^n$. This sort of thing is classically well known and is based on collapsing arguments from ... | 4 | https://mathoverflow.net/users/3502 | 307639 | 134,010 |
https://mathoverflow.net/questions/307635 | 11 | Is it consistent in [$\mathsf{ZF}$](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) that there is a set $X$ with more than $1$ point such that every injective map $f:X\to X$ has a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_(mathematics))?
| https://mathoverflow.net/users/8628 | Fixed points of injective self-maps | As Yair suggests, a strongly amorphous set has this property.
Recall that an amorphous set is a set which cannot be split into two infinite sets. A strongly amorphous set is a set such that in addition to being amorphous, every partition has only finitely many non-singletons. While we didn't require that it is an inf... | 12 | https://mathoverflow.net/users/7206 | 307641 | 134,011 |
https://mathoverflow.net/questions/307577 | 1 | Given the following differential equations:
\begin{equation}
\begin{aligned}
\dot{x}\_1 &= f\_1(x\_1,\ldots,x\_n) \\
\vdots \\
\dot{x}\_n &= f\_n(x\_1,\ldots,x\_n)
\end{aligned}
\end{equation}
In a compact way: $$\dot{\hat{x}} = \hat{F}(\hat{x})$$
Let the group $\Psi\subset S\_n$, where $S\_n$ is a symmetric ... | https://mathoverflow.net/users/93600 | Does stability of equilibrium point preserved by permutation matrix (symmetry)? | I think that the question is **more suitable for the [Mathematics StackExchange](https://math.stackexchange.com)**, but I'll try to (reasonably fully) answer it here.
In order not to be burdened with unnecessary technicalities, let us assume that $F$ is so regular that Picard's theorem holds. Also, let $\lVert \cdot ... | 1 | https://mathoverflow.net/users/121784 | 307642 | 134,012 |
https://mathoverflow.net/questions/307640 | 8 | Is there a continuous surjective function $f:[0,1] \to [0,1]^2$ such that
every level set $f^{-1}(y)$ is a finite set? If the answer is no, what about if we replace the finiteness of level sets by "countable level sets"?
| https://mathoverflow.net/users/36688 | Space filling curve whose all level sets are finite (countable) | Recall the definition of the Peano square-filling curve $f:[0,1]\to[0,1]^2$, which is given in terms infinite ternary strings. If $a\in [0,1]$ has a base $3$ representation of the form $0,a\_1a\_2a\_3\dots$, the point $f(a):=(b,c)$ has base $3$ digits resp. $$b\_n:={\bf k}^{a\_2+a\_4+\dots a\_{2n-2}}a\_{2n-1}$$
$$c\_n:... | 14 | https://mathoverflow.net/users/6101 | 307646 | 134,014 |
https://mathoverflow.net/questions/307645 | 6 | Let $X$ be a positive real number. Can someone help me by providing an asymptotic formula for this sum.
$$\sum\_{n \leq X, \; n\, \equiv\, a \mod{b}} \log{n},$$
where $a$ and $b$ are two coprime integers.
Thanks in advance.
| https://mathoverflow.net/users/76102 | An asymptotic formula for this sum | The sum
$$F(X)=\sum\_{n \leq X, \; n\, \equiv\, a \mod{b}} \log{n}=\sum\_{p={\rm Int}\,[-a/b]}^{{\rm Int}\,[(x-a)/b]}\log(a+pb)$$
can be approximated in the large-$X$ limit by
$$F\_\infty(X)=\sum\_{p=1}^{(X-a)/b}\log(pb)=\frac{X-a}{b}\log b+\log\Gamma\left(\frac{X-a}{b}+1\right)$$
Here is a plot of $F(X)$ (gold) and ... | 10 | https://mathoverflow.net/users/11260 | 307648 | 134,016 |
https://mathoverflow.net/questions/307651 | 1 | Let $A$ be a compact DVR in characteristic $0$, uniformizer $\pi$ and residue field $k$. Let $A\subset B$ be a complete DVR with the same uniformizer $\pi$ and algebraicly closed residue field $F$. Let $\varphi:B\rightarrow B$ be a surjective morphism of $A$-Algebras, such that the induced morphism of $k$-Algebras $\ov... | https://mathoverflow.net/users/92310 | Lifting Lang-Steinberg to DVR's in Characteristic 0 | Yes, as long as you are careful about the hypotheses, this is a result of M. J. Greenberg [Schemata over local rings II, Section 3, Proposition 3]. You need to assume that $\mathbb{G}$ is smooth over $A$ (Greenberg says 'simple') and of course that its fibre over $F$ is connected (otherwise the Lang--Steinberg theorem ... | 4 | https://mathoverflow.net/users/2381 | 307657 | 134,018 |
https://mathoverflow.net/questions/307629 | 33 | My [three](https://mathoverflow.net/questions/307526/real-numbers-with-given-complexity) [computability](https://mathoverflow.net/questions/307288/time-functions-of-non-deterministic-turing-machines) [questions](https://mathoverflow.net/questions/307607/time-functions-of-non-deterministic-turing-machines-a-better-quest... | https://mathoverflow.net/users/nan | Is this conjecture strictly weaker than P=NP? | The conjecture is indeed strictly weaker than $\mathrm{P = NP}$, in the sense that it follows from $\mathrm E=\Sigma^E\_2$, which is not known to imply $\mathrm{P = NP}$. Of course, we cannot prove this unconditionally with current technology, as it would establish $\mathrm{P\ne NP}$.
Here, [$\mathrm E$](https://en.... | 24 | https://mathoverflow.net/users/12705 | 307659 | 134,020 |
https://mathoverflow.net/questions/307619 | 22 | Some mathematics journals publish "research announcements", a class of publication that before today I had not heard of. An example is [Electronic Research Announcements in Mathematical Sciences](http://eramath.s3-website-us-east-1.amazonaws.com/).
I presume that after publishing a research announcement in such a jou... | https://mathoverflow.net/users/799 | Properties of a "research announcement" | This is an extended comment on history of research announcements. The principal mathematical journal which did this was Comptes Rendus published by the French Academy. It published (and still publishes) very short notes (1-2 pages), usually without complete proofs or with very short sketches of proofs. The papers were ... | 18 | https://mathoverflow.net/users/25510 | 307662 | 134,022 |
https://mathoverflow.net/questions/307572 | 1 | Let $f\_i$, $i=1,\dotsc,n$, be mutually orthogonal Abelian projections in a von Numann algebra, and let $e\leq\sum f\_i$. Is it true that there exist mutually orthogonal Abelian projections $e\_j$, $j=1,\dotsc,m$ such that $e=\sum e\_j$?
| https://mathoverflow.net/users/84700 | Subprojections of the sum of mutually orthogonal Abelian projections | This is true, more generally, in a C\*-algebra of real rank zero (i.e., such that the selfadjoint elements with finite spectrum are dense in the set of selfadjoints). Zhang proves in "A Riesz decomposition property and ideal structure of multiplier algebras" that the monoid of Murray-von Neumann equivalence classes of ... | 3 | https://mathoverflow.net/users/13381 | 307665 | 134,024 |
https://mathoverflow.net/questions/307669 | 1 | I was trying to understand the basic ideas of the operator-stable distributions. I found the papers by [Hudson](https://ac.els-cdn.com/0047259X80900792/1-s2.0-0047259X80900792-main.pdf?_tid=2953aa8d-c0a2-4a76-91f6-3f41bfdf0994&acdnat=1533563860_f86a125d48c19655037145541942ca02) and [Sato](https://ac.els-cdn.com/0047259... | https://mathoverflow.net/users/109447 | Definition and examples of operator-stable distributions | $\newcommand{\R}{\mathbb{R}}
\newcommand{\D}{\overset{\text{D}}=}
\newcommand{\E}{\operatorname{\mathsf E}}$
The fundamental paper in this area is the one by [Sharpe](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=2ahUKEwiO5KTD2NjcAhVIQ6wKHfyQD3kQFjAAegQIABAB&url=http%3A%2F%2Fwww.... | 2 | https://mathoverflow.net/users/36721 | 307675 | 134,027 |
https://mathoverflow.net/questions/307654 | 5 | Two players, Green and Red, play a zero-sum game. It is parametrized by two integers $n\geq 0, k\geq 0$, and a finite family $F$ of sets of size $n$ (each set may appear multiple times in $F$).
Each turn, Green colors an uncolored element green, and then Red colors an uncolored element red.
A set in $F$ is consider... | https://mathoverflow.net/users/34461 | A set-family game | $G(3,2)=\frac{1}{2}$.
For any set $F$ composed of 3-element sets, let's assume there's a sequence of choices $a\_1,b\_1,\dots a\_k,b\_k$ where B gains the upper hand and has two elements chosen out of more than half the sets in $F$, despite A playing optimally. If B can force such a sequence, then A can force the seq... | 3 | https://mathoverflow.net/users/103753 | 307677 | 134,028 |
https://mathoverflow.net/questions/307656 | 7 | Let $X$ be an integral normal flat finite type scheme over $\mathbb{Z}$.
>
>
> >
> > Does there exist a proper surjective generically finite morphism of schemes $Y\to X$ with $Y$ an integral regular finite type scheme over $\mathbb{Z}$?
> >
> >
> >
>
>
>
I could not find such a result in the literature.
... | https://mathoverflow.net/users/127408 | Do arithmetic schemes have non-singular alterations? | This is Theorem 8.2 in de Jong's original paper [dJ].
[dJ] de Jong, A. J., [*Smoothness, semi-stability and alterations*](http://dx.doi.org/10.1007/BF02698644). Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996). [ZBL0916.14005](https://zbmath.org/?q=an:0916.14005).
| 4 | https://mathoverflow.net/users/82179 | 307680 | 134,029 |
https://mathoverflow.net/questions/307199 | 8 | Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H\_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$
**Question** *Can we say anything about Betti numbers or minimal resolution of $H\_{\mathfrak m}^i(R)?$*
In particular, can we say anything (whether the Betti numbers ... | https://mathoverflow.net/users/9485 | Minimal resolution of local cohomology module | It is hard to give a useful answer. I suspect whatever you want/need would be more specific. In particular, details on how such $R$ arises in your research would make it easier to say something more concrete. But anyhow, there are a couple of remarks and references one can point to.
First, your ring is not Cohen-Mac... | 5 | https://mathoverflow.net/users/2083 | 307685 | 134,031 |
https://mathoverflow.net/questions/307141 | 0 | Consider the affine hyperquadric $Q:=\biggl\{(z\_1,...,z\_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum\_{i=1}^{n+1}z\_i^2=1\biggr\}\cong TS^n$.
>
> What is a reasonable Kähler metric for $Q$ (induced by the pullback of the metric from the ambient space $\mathbb{C}^{n+1})$?
> Furthermore, how do we explicitly calculate the... | https://mathoverflow.net/users/98896 | How do we compute the even cohomology $H^{2i}(Q)$ of the affine hyperquadric? | I'm going to answer questions 1 and 2.
1. You can restrict the Kaehler metric from the ambient affine space to get a Kaehler metric on this hypersurface if you want (there are, of course, many others).
2. The "curvature form" definition of Chern classes is not usually very useful for calculations. Instead, as you hav... | 4 | https://mathoverflow.net/users/10839 | 307686 | 134,032 |
https://mathoverflow.net/questions/307687 | 4 | Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the intersection form (or cup-product) on $H^1(X,\mathbb{Z})$. My question is: Is this the only principal polarization on $J(X)$? I... | https://mathoverflow.net/users/32151 | Naive question on the Jacobian of a curve | It is possible for the Jacobian's of non-isomorphic curves to be isomorphic as abelian varieties, but obviously, not as principally polarized abelian varieties. This paper <https://arxiv.org/pdf/math/0304471.pdf> by Howe gives examples. Also from the same paper - "it has been known since the late 1800s that distinct cu... | 7 | https://mathoverflow.net/users/5100 | 307691 | 134,033 |
https://mathoverflow.net/questions/307676 | 2 | Important note
==============
@MateuszKwaśnicki in the comment section has raised a fundamental issue with the current statement of the problem. I'm trying to bugfix it.
Setup
=====
I wish to show that a Lipschitz function with a "very small" Lipschitz constant can't be a "very good" *classifier* (machine learnin... | https://mathoverflow.net/users/78539 | Lower bound on misclassification rate of Lipschitz functions in terms of Lipschitz constant | If we let $P$ put probability $\frac{1}{2}$ each on $Y \in \{\pm 1\}$ independent of $X$, then for every classifier (from any class!)
$$P(\text{sign}(h(x)) \neq y) = \frac{1}{2}$$
so $\text{err}\_{\mathcal{H}} = \frac{1}{2}$ of course.
On the other hand, most reasonable classes $\mathcal{H}$ can guarantee $\text{err... | 1 | https://mathoverflow.net/users/29697 | 307695 | 134,035 |
https://mathoverflow.net/questions/307700 | 3 | More precisely, if I have a complete bounded finite lattice $C$, can I compute a lattice-operation-preserving map $C \to P(S)$? for some $S$. If not, is there another universal lattice structure that has efficient meet and join implementations on a computer?
| https://mathoverflow.net/users/127441 | Is every complete bounded finite lattice equivalent to a sublattice of a powerset lattice? | This answer addresses the vague question "is there another universal lattice structure that has efficient meet and join implementations on a computer?" The question of whether boolean lattices work has already been addressed in comments.
It is important to be clear what efficient means here. If we just want lattice ... | 1 | https://mathoverflow.net/users/297 | 307705 | 134,036 |
https://mathoverflow.net/questions/307704 | 10 | Let $X$ be a manifold without boundary and let $Y$ and $Z$ be two manifolds with boundary such that $X$ is homeomorphic to their interiors: $X \cong Y^\circ \cong Z^\circ$. Does it follow that $Y \cong Z$ as well? That is, does there exist a homeomorphism that extends to the boundaries as well? If not, can we at least ... | https://mathoverflow.net/users/56938 | Is the boundary of a manifold topologically unique? | Since you say in a comment that you might be satisfied with a homotopy equivalence, let me sketch a proof that the homotopy type of the boundary depends only on the interior.
Let $Y$ be the interior of $X$ and let $DY$ be the space of all proper maps $[0,1)\to Y$, suitably topologized. Then $DY$ must be homotopy equ... | 9 | https://mathoverflow.net/users/6666 | 307710 | 134,038 |
https://mathoverflow.net/questions/307712 | 9 | Oversimplification: Newton & Leibnitz &c build the calculus and other methods that solve a vast number of practical problems. Weierstrass, Dedekind, Cantor &c build a foundation under it dependent on transfinite quantities.
Kronecker, Brouwer, etc, were appalled by this. Later, Bishop &c actually demonstrates approac... | https://mathoverflow.net/users/2937 | What did the Intuitionists want to do with applied mathematics? | What did the logicians of the 20th century think? Perhaps this was best described by Michael Beeson in his book ("Foundations of constructive mathematics: metamathematical studies", 1985, Springer):
>
> The thrust of Bishop's work was that both Hilbert and Brouwer had been wrong about an important point on which th... | 11 | https://mathoverflow.net/users/1176 | 307719 | 134,040 |
https://mathoverflow.net/questions/307714 | 0 | This is related to [Dirac's theorem](http://en.wikipedia.org/wiki/Dirac%27s_theorem_on_Hamiltonian_cycles#Bondy.E2.80.93Chv.C3.A1tal_theorem).
For any finite, simple, undirected graph $G=(V,E)$ let $\delta(G)$ denote the minimal [degree](https://en.wikipedia.org/wiki/Degree_(graph_theory)) of all vertices.
Are ther... | https://mathoverflow.net/users/8628 | A weaker version of Dirac's theorem | No. Suppose that $n$ and $c$ are natural numbers. We consider the complete bipartite graph $G=K\_{n+c+1,n}$. Then $\delta(G)=n$. However, every matching $M$ of $G$ contains at most $n$ edges; hence it leaves at least $c+1$ vertices uncovered.
| 4 | https://mathoverflow.net/users/296 | 307721 | 134,042 |
https://mathoverflow.net/questions/307713 | 10 | $\newcommand{\Ga}{\Gamma}$
I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by very similar proofs by Verblunsky and Potts. The idea in those three papers, which looks quite natural, is to subdivide $R... | https://mathoverflow.net/users/36721 | Proof of Green's formula for rectifiable Jordan curves | One can circumvent the technical difficulties as follows. Consider a large ball $K$ containing $\Gamma$ and any $p>2$. Given a function $f\in L^p(K)$, we can define its Cauchy transform
$$
\left(\mathcal{C}f \right)(z)=\frac{1}{\pi}\int\_K\frac{f(w)}{z-w}.
$$
By Hölder inequality, this is a continuous operator from $L... | 8 | https://mathoverflow.net/users/56624 | 307724 | 134,044 |
https://mathoverflow.net/questions/307728 | 2 | I assume that a result of the following kind is known, and I would really appreciate a reference for it... Or at least, some hints as to where to start looking.
>
> **Theorem(?)**: Let $\varepsilon>0$ and Let $\Gamma=(V,E)$ be a finite regular non-bipartite graph of degree $d$, with $V=\{v\_1,\dots, v\_n\}$ and wit... | https://mathoverflow.net/users/801 | Random walk and isoperimetric constant | Given the proper keywords, a quick search gives
<https://ocw.mit.edu/courses/mathematics/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/lecture-notes/MIT18_409F09_scribe4.pdf>
as one of many many references.
Assume $\mu\_t$ is the measure at the $t^\text{th}$-step of your random... | 4 | https://mathoverflow.net/users/18974 | 307734 | 134,046 |
https://mathoverflow.net/questions/307726 | 3 | I have a basic question on Voevodsky's stable homotopy category of spectra $\mathbf{SH}(S)$, where $S$ is a finite dimensional noetherian scheme.
Let $E$ be an $\Omega$-spectrum and $\varphi \colon F\to E$ be a morphism of spectra satisfying that for any smooth $S$-scheme $X$ the induced map
$$
\mathrm{Hom}\_{\mathb... | https://mathoverflow.net/users/12204 | Basic questions on spectra | Here is a slightly more fleshed out version of the comment above.
First, the claim that the collection $\mathcal{C} = \{ \Sigma^{p,q} U \mid U \in Sm/S, p,q \in \mathbb{Z} \}$ is a collection of compact generators, is Theorem 9.1 of <https://arxiv.org/pdf/math/0310190.pdf> (there may be earlier references). In parti... | 6 | https://mathoverflow.net/users/16785 | 307739 | 134,047 |
https://mathoverflow.net/questions/307698 | 2 | I am reading through a proof in W. Ding and G. Tian's 1992 paper on the generalised Futaki invariant. To provide context, we are looking for obstructions to the existence of Kähler--Einstein metrics with positive scalar curvature. The Futaki invariant provides an example of such an obstruction. My confusion is not in t... | https://mathoverflow.net/users/105103 | Function is $L^p$-integrable for $p >1$ [Kähler Geometry] | In polar coordinates
$ \int\_{|z|<1} |\log(|z|)|^p = 2 \pi \int\_0^1 (-\log r)^p r dr < +\infty .$
Ok..
In a compact space it is enough to check the intergrability condition locally, only the terms $\log \|S\|$ might give trouble at points on $E = \{z=0\}$ (in local coordinates), so $ \log \|S\| = H + \log|z| $ ... | 1 | https://mathoverflow.net/users/127247 | 307748 | 134,052 |
https://mathoverflow.net/questions/307747 | 18 | Let $A$ be a symmetric square matrix with entries in $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ such that all of its diagonal entries are nonzero. Does there exists always a vector $x$ with all coordinates nonzero in the image of $A$? (this is true for $p=2$, but I don't know the answer for other values of $p$)
| https://mathoverflow.net/users/51663 | A linear algebra problem in positive characteristic | This is false. Let $c$ be a quadratic nonresidue modulo $p$. Our matrix will be $(p^2-1) \times (p^2-1)$, with rows and colums indexed by pairs $(x,y) \in \mathbb{F}\_p^2 \setminus \{ (0,0) \}$.
Our matrix is defined by
$$A\_{(x\_1,y\_1) \ (x\_2, y\_2)} = x\_1 x\_2 - c y\_1 y\_2.$$
This is obviously symmetric. Since... | 29 | https://mathoverflow.net/users/297 | 307751 | 134,053 |
https://mathoverflow.net/questions/307612 | 10 | In the paper "[Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$](https://arxiv.org/pdf/math/0301260.pdf)" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning that the NLS
$$i\partial\_t\phi(x,t)+\Delta\phi(x,t)=\vert\phi(x,t)\vert^2\phi... | https://mathoverflow.net/users/nan | Nonlinear Schrödinger equation with discrete Laplacian | My impression is, that in the discrete case the interesting questions and tools are sometimes different from the continuous case.
Studying the discrete Laplacion also means that one is interested in discrete solution. For example, in one dimension people are often investigating the discrete nonlinear Schrödinger equa... | 1 | https://mathoverflow.net/users/50551 | 307752 | 134,054 |
https://mathoverflow.net/questions/307668 | 2 | For $v, w \in \{0,1\}^n$, denote $v w = (v\_1 w\_1, \ldots, v\_n w\_n)$ and $|v|=\sum\_{i} v\_i$.
Let $v\_1, v\_2 \in \{0,1\}^n$ and
\begin{align\*}
f(x\_1, x\_2) = \sum\_{d=0}^{|v\_1 v\_2|} \frac{1}{2^{|v\_1|+|v\_2|-|v\_1 v\_2|}} {|v\_1| - |v\_1 v\_2| \choose x\_1 - d} {|v\_2| - |v\_1 v\_2| \choose x\_2 - d}.
\end... | https://mathoverflow.net/users/11877 | How to estimate a summation? | Put $a=|v\_1|$, $b=|v\_2|$, $c=|v\_1v\_2|$. Then we have
$$
\sum\_{i=0}^a\sum\_{j=0}^b\sum\_{k=0}^c\binom{a-c}{i-k}\binom{b-c}{j-k}= \sum\_{k=0}^c\left(\sum\_{i=0}^a\binom{a-c}{i-k}\right)\left(\sum\_{j=0}^b\binom{b-c}{j-k}\right).
$$
As $i$ runs from $0$ to $a$, $i-k$ runs from $-k$ to $a-k$. Since $0\leq k\leq c$, th... | 3 | https://mathoverflow.net/users/37555 | 307755 | 134,056 |
https://mathoverflow.net/questions/307699 | 2 | I am trying to develop a good geometric intuition and to understand the motivation behind Kawamata's definition of a relative movable Cartier divisor in Section 2 of reference [1]:
[1] Y. Kawamata, Crepant blowing-ups of three-dimensional canonical singularities and its application to degenerations of surfaces; Ann.... | https://mathoverflow.net/users/4046 | Intuition behind Kawamata's definition of a relative movable Cartier divisor | The base locus of a divisor $D$ on $X$ is the same as those points where $\mathscr O\_X(D)$ is **not** generated by global sections, which can be identified with the locus where the natural map
$$
\tag{$\star$}
H^0(X,\mathscr O\_X(D))\otimes \mathscr O\_X \to \mathscr O\_X(D)
$$
is **not** surjective.
The relative... | 2 | https://mathoverflow.net/users/10076 | 307760 | 134,058 |
https://mathoverflow.net/questions/307730 | 3 | Let $A$ be a finite dimensional algebra and assume all modules are also finite dimensional. A module $M$ is said to have dominant dimension at least $n$ in case the term $I\_i$ for $i=0,1,...,n-1$ are projective when $(I\_i)$ denotes a minimal injective coresolution of $M$. The dominant dimension of the algebra is the ... | https://mathoverflow.net/users/61949 | Identity for $Ext^1$ for special algebras | The conjecture does not appear to be true as stated. However, this might just mean that it needs to be clarified or refined.
To see a counterexample, assume $A$ has dominant dimension exactly $n-1$ and take $X=A$. By your definition, $A$ is $n$-torsionfree since $\tau(A) = 0$. Then the conjectured isomorphism becomes... | 2 | https://mathoverflow.net/users/11791 | 307761 | 134,059 |
https://mathoverflow.net/questions/307222 | 8 | Let $E\neq \{0\}$ be a Banach space.
For each $p\in[1,\infty), $ we define
$$E\oplus\_p E = \{(x,y): x\in E, y\in E, \|(x,y)\| = \sqrt[p]{\|x\|^p + \|y\|^p}\}.$$
Let $F$ be another Banach space.
By $E\cong F,$ I mean that $E$ and $F$ are isometrically isomorphic.
>
> Question: Suppose that $p,q\in [1,\infty).$ If ... | https://mathoverflow.net/users/42411 | If $E\oplus_\phi E \cong E\oplus_\psi E,$ does it imply that $\phi= \psi$? | It is a sketch of the proof in the case of $p\ne q$ (a complete
proof on these lines is rather lengthy). Something similar can be
done in more general case, possibly with some exceptions.
Our plan is the following: We assume that $p,q\in[1,\infty)$,
$p\ne q$ and $W=E\_1\oplus\_p E\_2=F\_1\oplus\_q F\_2$ (isometricall... | 5 | https://mathoverflow.net/users/37822 | 307780 | 134,068 |
https://mathoverflow.net/questions/307762 | 10 | I want to develop a basic background in number theory with a goal towards contributing to some part of the Langlands program (which I know is vast, and has many different aspects; I want to do something involving Shimura varieties, Galois representation, modular forms). I have been reading (but not doing the exercises ... | https://mathoverflow.net/users/80739 | Books with exercises to learn Langlands program, Galois representations, modular forms | This is perhaps not the type of answer you were looking for (in which case I apologize), but it isn't clear to me that doing all of the exercises in an enormous number of books is the most efficient way to get to your goal of contributing to some aspect of the Langlands program, Shimura varieties, Galois representation... | 26 | https://mathoverflow.net/users/nan | 307784 | 134,070 |
https://mathoverflow.net/questions/307804 | 0 | Today I have heard that statement but I can't find the reference.
Can somebody know a reference and/or a proof?
| https://mathoverflow.net/users/62218 | Is the numerable product of finite abelian groups a cantor set? | The idea is that a base for the product topology can be formed by sets of the following form:
$$U\_{\alpha\_0,....,\alpha\_{n}} = \{(x\_i : i \in \mathbb{N}) \in \prod\_{i \in \mathbb{N}} G\_i : x\_0 = \alpha\_0, x\_1 = \alpha\_1, x\_2 = \alpha\_2 ..., x\_k = \alpha\_k\}$$
where $\alpha\_0 \in G\_0$, $\alpha\_1 \i... | 2 | https://mathoverflow.net/users/45707 | 307805 | 134,082 |
https://mathoverflow.net/questions/259129 | 3 | In dimension 3 we have that for $T=\int\_{[0,\infty)}1\_{B\_{t}\in B(0,1)}dt$ has the Laplace transform
$$E[e^{-\lambda T}]=sech(\sqrt{2\lambda}).$$
And in dimension 1 we have the same for $\tau=\min\{t: |B(t)|=1\}$:
$$E[e^{-\lambda \tau}]=sech(\sqrt{2\lambda}).$$
Inverting this in Mathematica didn't give a clean... | https://mathoverflow.net/users/99863 | Inverse Laplace transform of $sech(\sqrt{2\lambda})$ and Brownian motion occupation time | **Q2** $\qquad$ *"Is there a closed formula for the inverse Laplace transform of ${\rm sech}(\sqrt{2\lambda})$"*
might still benefit from an explicit answer in terms of a special function.
The inverse Laplace transform of ${\rm sech}(\sqrt{2\lambda})=1/\cosh(\sqrt{2\lambda})$ follows from an entry in Table 2 in [T... | 2 | https://mathoverflow.net/users/11260 | 307826 | 134,089 |
https://mathoverflow.net/questions/307808 | 5 | Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$.
Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ with $\det A > 0$ a.e.
>
> Do there exist $u\_n \in C^{\infty}\big(\Omega,\text{GL}(\mathbb{R}^n)\big)$ such that $u\_n \t... | https://mathoverflow.net/users/46290 | Can we stay invertible while approximating linear maps in Sobolev spaces? | I will recycle the answer I've been writing into some words of explanation on Alex Gavrilov's example, which is more simple and elegant. We can focus on the first column $p\_n$ of an approximating sequence $u\_n:=[p\_n,q\_n]$ of $A(x,y):=\begin{bmatrix}x&-y\\y&x\end{bmatrix}$. If $p\_n\in C^0\cap W^{1,1}\_{loc}(\mathbb... | 4 | https://mathoverflow.net/users/6101 | 307832 | 134,092 |
https://mathoverflow.net/questions/295952 | 9 | We know the answers to some questions like *What is the maximal number of singularities of (reduced) plane curves of degree $d$?* for general $d$ (in this case $\tfrac{1}{2}d(d-1)$, obtained by $d$ lines in general position). On the other hand, more subtle questions like *Given $d$, what is the largest $n$ such that so... | https://mathoverflow.net/users/15782 | Classification of singularities of plane curves of fixed degree (reference request) | Here is what is known for quintics and sextics. Starting from $d=7$, I do not know the answer to your question.
* **Quintics**: The maximal $A\_n$ on a $C\_5$ is $n=12$. For example, the curve given by the zero set of $$
(y^2-xz)^2(\frac{1}{4}x+y+z)-x^2(y^2-xz)(x+2y)+x^5
$$ has an $A\_{12}$-singularity at $(0:0:1)$, ... | 6 | https://mathoverflow.net/users/127497 | 307835 | 134,094 |
https://mathoverflow.net/questions/307824 | 4 | Let $U^{m} \subset \mathbb{R}^{m}$ be an open set. Suppose $\varphi$ is an immersion of $U^{m}$ into $\mathbb{R}^{m+n}$ satisfying the following condition:
For each point $p \in \varphi(U^{m})$, the nullity of the second fundamental form of $\varphi(U^{m})$ is equal to $m-1$.
Then, it is well-known that
1. $\var... | https://mathoverflow.net/users/74033 | Codimension reduction for developable Euclidean submanifold | Fix a generic smooth curve $\gamma$ in $\mathbb{R}^{n+1}$.
The productt $\gamma\times\mathbb{R}^{m-1}\subset\mathbb{R}^{n+m}$ is full, is not it?
| 1 | https://mathoverflow.net/users/1441 | 307837 | 134,096 |
https://mathoverflow.net/questions/99953 | 15 | In HTT, a version of the adjoint functor theorem for (locally) presentable infinity categories is proven (Corollary 5.5.2.9). Is there a more refined version of this somewhere, which more closely resembles Freyd’s original version? I.e., is there a version for infinity categories which are *not* necessarily (locally) p... | https://mathoverflow.net/users/4528 | Adjoint functor theorem for infinity categories | The general adjoint functor theorem didn't exist in the literature when this question was originally asked, but now it does: [Nguyen, Raptis, and Schrade](https://arxiv.org/abs/1803.01664).
| 9 | https://mathoverflow.net/users/2362 | 307839 | 134,097 |
https://mathoverflow.net/questions/307778 | 7 | Let $G$ be a finite group. Assume that $\chi$ is a complex irreducible character of $G$ of degree $n\geq 2$, with the property that for each element $g\in G$ either $\chi(g)=0$ or $|\chi(g)|=n$.
>
> 1. Does it necessarily follow that $\chi$ is imprimitive, i.e., induced
> from a character of a subgroup?
>
>
>
... | https://mathoverflow.net/users/106628 | Finite group with a character having one nonzero absolute value | As Geoff said, a group having such an irreducible character is called a *group of central type*, and Howlett and Isaacs have shown, using the classification of finite simple groups, that such groups are solvable: see
*Howlett, Robert B.; Isaacs, I. Martin*, [**On groups of central type**](http://dx.doi.org/10.1007/B... | 10 | https://mathoverflow.net/users/10266 | 307841 | 134,098 |
https://mathoverflow.net/questions/307844 | 15 | I have recently started to read a bit about geometry and topology. Hopf fibration, Lense spaces, CW complexes, stuff that are discussed in Hatcher's Algebraic Topology and other things that require good visualization. What is apparent to me is that the further I go, the less I understand what is going on. I have search... | https://mathoverflow.net/users/126279 | What kind of computer tools topologists/geometrists use to visualize the objects they deal with? | Here is one case study:
An impressive [animation](https://www.youtube.com/watch?time_continue=33&v=AKotMPGFJYk) of the Hopf fibration created by [Niles Johnson](https://nilesjohnson.net/hopf.html) using only open-source tools, available for all platforms: *The Py... | 10 | https://mathoverflow.net/users/11260 | 307852 | 134,102 |
https://mathoverflow.net/questions/307840 | 7 | I hope this is research level. Suppose $E$ is the direct limit of finite spectra, say $E=\mathrm{colim }\ E\_i$, which itself is not finite. I wonder how much and under which conditions the inverse limit $\mathrm{lim}\ D(E\_i)$ is a good candidate for playing role of $D(E)$? Here, I write $D$ for the $S$-duality functo... | https://mathoverflow.net/users/51223 | $S$-dual of filtered spectra | One key property of duality is that the canonical map from $E$ to its double dual $D(D(E))$ is a homotopy equivalence when $E$ is a finite spectrum. This fails (in general) for infinite spectra. Moreover, the map $E\to D^2(E)$ is not always a monomorphism in the homotopy category, or even injective on homotopy groups. ... | 10 | https://mathoverflow.net/users/6668 | 307853 | 134,103 |
https://mathoverflow.net/questions/307851 | 9 | The sequence [A006318](https://oeis.org/A006318) at OEIS stands for the [Schröder numbers](https://en.wikipedia.org/wiki/Schr%C3%B6der_number).
>
> They describes the number of lattice paths from the southwest corner $(0,0)$ of an $n\times n$ grid to the northeast corner $(n,n)$, using only single steps north, $(0... | https://mathoverflow.net/users/63938 | Why is the number of Perfect Matchings in a triangular grid equivalent to the number of Royal Paths? | I was able to find the result at **Ex. 6.39 s** in Stanley's EC Vol.2. The reference given there is to
>
> Ciucu, M. "[Perfect Matchings of Cellular Graphs](https://doi.org/10.1023/A:1022408900061)" *Journal of Algebraic Combinatorics* **5** (1996), 87-103
>
>
>
In theorem 4.1, Ciucu computes the generating fu... | 10 | https://mathoverflow.net/users/2384 | 307856 | 134,106 |
https://mathoverflow.net/questions/307866 | 12 | The sequence $a\_n$ given by
$$a\_n=\sum\_{k=0}^n\frac{n!}{k!}$$
is found at A000522 on [OEIS](https://oeis.org/A000522) with a description: total number of arrangements of a set with $n$ elements. Let $\nu\_2(x)$ denote the [$2$-adic valuation](https://en.wikipedia.org/wiki/P-adic_order) of the integer $x$. My first ... | https://mathoverflow.net/users/66131 | A curious valuation of this sequence | *To your first question:* No, it is false. For $n = 256 - 13$, the number $a\_n$ has $\nu\_2\left(a\_n\right) = 7 < 8 = \nu\_2 \left(n+13\right)$.
**HOWEVER**, it is **almost** correct: namely, it is correct whenever $128 \nmid n+13$ (so the smallest counterexample is $n = 128 - 13 = 115$). This is the following theo... | 24 | https://mathoverflow.net/users/2530 | 307869 | 134,112 |
https://mathoverflow.net/questions/307867 | 4 | Suppose $G$ is a connected reductive group defined over a field $F$ of characteristic $0$. Does every maximal torus contain a regular semisimple element defined over $F$?
I know that over an algebraically closed field this is true because being regular corresponds to being in the intersection of the complements of t... | https://mathoverflow.net/users/97316 | Existence of Regular Semisimple elements of reductive groups in characteristic 0 | A torus is unirational. The set of regular elements is open, hence also unirational. Over an infinite field, any unirational variety has (a Zariski dense set of) F-points. This answers the question in the affirmative.
| 5 | https://mathoverflow.net/users/425 | 307877 | 134,116 |
https://mathoverflow.net/questions/307868 | 6 | Suppose $G$ is a connected reductive group over an algebraically closed field. Then given a maximal torus $T$, we can define a Weyl group $W$ and consider $T^W$, the Weyl-invariants of $T$. This clearly contains the center $Z(G)$ but can be larger. For instance, in $\mathrm{PGL}\_2$, the element with $\mathrm{GL}\_2$ r... | https://mathoverflow.net/users/97316 | Fixed Points of the Weyl Group action on a Maximal Torus and the Center of a Reductive Group | Notice that nothing in the problem is harmed by base change, so that it doesn't matter that the ground field is algebraically closed.
The identity component of $T^W$ is generated by the images of the $W$-invariant cocharacters of $T$, and the $W$-invariant part of the cocharacter lattice consists precisely of the cen... | 5 | https://mathoverflow.net/users/2383 | 307884 | 134,119 |
https://mathoverflow.net/questions/307880 | 6 | Setup: $\pi: \mathcal X \to C$ is a flat morphism from a germ of a smooth curve $(C, o)$. Suppose the special fiber $\pi^{-1}(o) := \mathcal X\_o$ has a certain class of singularities, one wants to study if $\mathcal X$ preserves such singularities.
It has been shown (see "[Deformations of canonical singularities](ht... | https://mathoverflow.net/users/29730 | A paradox on the deformation of singularities | I don't think it is true that $\mathcal X$ is $\mathbb Q$-Gorenstein. Suppose in fact that $\dim \mathcal X \_t=2$ for all $t\in C$ and $\mathcal X \to Z$ is a flipping contraction with exceptional locus contained in the central fiber $\mathcal X \_0$, then $K\_Z$ is not $\mathbb Q$-Cartier, but if $\mathcal X \_0$ is ... | 9 | https://mathoverflow.net/users/19369 | 307886 | 134,121 |
https://mathoverflow.net/questions/307882 | 3 | If the trace of all positive powers of a $n \times n$ complex matrix is $0$, then the matrix must be nilpotent. <https://math.stackexchange.com/questions/159167/traces-of-all-positive-powers-of-a-matrix-are-zero-implies-it-is-nilpotent>
Is a similar conclusion known to be valid in the setting of finite von Neumann al... | https://mathoverflow.net/users/127523 | Quasinilpotent operator in finite von Neumann algebra | The answer is negative. Consider the algebra of essentially bounded functions on the unit circle with the trace induced by the Lebesgue measure. Then the identity function $z \mapsto z$ is a counterexample. Indeed, it is a unitary element, so its spectrum is contained in the unit circle; it does not even contain zero.
... | 4 | https://mathoverflow.net/users/24953 | 307887 | 134,122 |
https://mathoverflow.net/questions/307891 | 1 | Let $B\_t$ denote a standard Brownian motion, and $0 < l < u$. I am wondering if the law of $\sup\_{l \leq t \leq u} \frac{|B\_t|}{\sqrt{t}}$ is atomless, that is, $\mathbb{P}\left(\sup\_{l \leq t \leq u} \frac{|B\_t|}{\sqrt{t}} = c \right) = 0$ for all constant $c$.
Any help will be greatly appreciated!
| https://mathoverflow.net/users/122927 | Is the law of $\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}}$ atomless? | This seems to follow easily from the strong Markov property. Informally: if this probability were positive for some $c$, then we could stop the process when $t^{-1/2} |B\_t| = c$ for the first time, and get a contradiction with the oscillatory character of $B\_t$ for small times.
---
Write $$M = \sup\_{l\le t\le ... | 3 | https://mathoverflow.net/users/108637 | 307905 | 134,129 |
https://mathoverflow.net/questions/307263 | 4 | Let $N < 2^a$ be a positive integer chosen uniformly at random. Let $\tilde{N}$ be the result of removing from $N$ all its prime factors less than $2^b$. What is the probability that $\tilde{N}$ is composite and $\tilde{N} > 2^c$?
The problem is similar to [Integers with a large smooth divisor](https://pdfs.semantics... | https://mathoverflow.net/users/416 | Density of integers with a large rough divisor | If $a/b$ is not too large, you can compute the probability using arguments as in the computation leading to the asymptotics for smooth numbers. In theory, you can compute for all $\beta, \gamma$ a real number $\rho$, such that
$$
\#\{n\leq x: \tilde{n}>x^\gamma\} \sim \rho x,
$$
where $\tilde{n}$ is $n$ divided by all... | 2 | https://mathoverflow.net/users/37555 | 307909 | 134,130 |
https://mathoverflow.net/questions/307899 | 1 | Let $x$ and $y$ be two positive real numbers.
What is the mean value of the function $\log$ on $y$ friables integers less than $x$ i.e the value of the following sum
$$\sum\_{\substack{n \leq x \\ P(n)\leq y}} \log(n),$$
where $P(n)$ is the greatest prime factor of $n.$
Thanks in advance.
| https://mathoverflow.net/users/76102 | Sum of log over friables | De La Breteche and Tenenbaum established in their paper **'Propriétés statistiques des entiers friables'** the asymptotic formula for the above sum. They find it as a corollary; Uniformly for $2 \leq y \leq x,$ we have
$$\sum\_{\substack{n \leq x \\ P(n)\leq y}} \log(n)=\left\{\log(x)-\frac{\log(y)+O(\log\log(y))}{\lo... | 7 | https://mathoverflow.net/users/76102 | 307911 | 134,131 |
https://mathoverflow.net/questions/307823 | 14 | Let $p=8k+1\equiv 1\pmod 8$ be a prime, thus $2$ is a quadratic residue module $p$. Euler's criterion show that $$2^{\frac{p-1}{2}}\equiv 1 \pmod p.$$
So we must have
$$2^{\frac{p-1}{4}}\equiv \delta(p) \pmod p$$
where $\delta(p)=\pm1$.
**Now my question is how to determine $\delta(p)$.**
I calculated many exam... | https://mathoverflow.net/users/98430 | Determine $2^{\frac{p-1}{4}}\equiv 1\pmod p$ or $2^{\frac{p-1}{4}}\equiv -1\pmod p$ when $p\equiv 1 \pmod 8$ | Barrucand and Cohn (MR0249396, *Note on primes of type $x^2+32y^2$, class number, and residuacity*. J. Reine Angew. Math. 238, 1969, 67--70) have proved that for primes $p \equiv 1 \textrm{ mod } 8$, the condition $h(−4p) \equiv 0 \textrm{ mod } 8$ is equivalent to $−4$ being a $8$-th power mod $p$. This implies your o... | 10 | https://mathoverflow.net/users/6506 | 307913 | 134,132 |
https://mathoverflow.net/questions/307918 | 11 | Assume the first three obstruction classes of a rank 4 vector bundle vanish and look at the fourth obstruction class. This fourth obstruction class can be decomposed as the Euler class and the first Pontryagin class (since $\pi\_3(SO\_4) \simeq \mathbb{Z} \oplus \mathbb{Z}$). Is there a geometric description of a syste... | https://mathoverflow.net/users/18974 | Fourth obstruction, Pontryagin and Euler class | Geometric generators for $\pi\_3(SO(4))$ have been identified in §22 of Steenrod's "Topology of fibre bundles", using the identification of $S^3$ as unit quaternions. Conjugation of quaternions induces an element of $\pi\_3(SO(4))$ denoted by $\alpha\_3$ and left multiplication induces an element denoted by $\beta\_3$.... | 14 | https://mathoverflow.net/users/50846 | 307922 | 134,134 |
https://mathoverflow.net/questions/275177 | 4 | Let $Y$ be a complete intersection in a complete simplicial toric variety $X\_\Sigma$ such that $\DeclareMathOperator{Sing}{Sing}\Sing(Y)\subset\Sing(X\_\Sigma)$. Suppose that $\phi:X\_{\widehat{\Sigma}}\to X\_\Sigma$ is a toric resolution induced by a refinement $\widehat{\Sigma}$ of the fan $\Sigma$.
Is there a si... | https://mathoverflow.net/users/33377 | When does a discrepant toric resolution induce a crepant resolution of a subvariety? | If the subvariety $Y$ is transversal to all the strata, which is the case, for example, for generic complete intersections of base point free linear systems, then it is easy.
The restriction of the resolution to Y is crepant if and only if Y misses all strata that are images of exceptional divisors with nonzero discr... | 1 | https://mathoverflow.net/users/38468 | 307925 | 134,136 |
https://mathoverflow.net/questions/307914 | 4 | There is a result due to Siegel that, for a number field $K$, any totally positive element of $K$ is the sum of four squares of $K$. This is discussed in another question ([sum of squares in ring of integers](https://mathoverflow.net/questions/14456/sum-of-squares-in-ring-of-integers)).
Is the result true in the mor... | https://mathoverflow.net/users/126815 | Sums of squares in global fields (Reference Request) | Corollary 1.5 on page 379 of Lam's *Introduction to quadratic forms over fields* (in the Sums of Squares section) states that if $F$ is any nonreal global field then every element of $F$ is a sum of four squares. Since you say you are interested in finite extensions of $\mathbb F\_q(t)$, and all fields of characteristi... | 4 | https://mathoverflow.net/users/nan | 307941 | 134,142 |
https://mathoverflow.net/questions/307950 | 10 | Today the set-theoretic operations of intersection $\cap$ [German: *Durchschnitt*] and union $\cup$ [German: *Vereinigung*] are standard.
The modern notations are present in the first edition of van der Waerden's *Moderne Algebra* (1930). However, the notation is mostly missing from Steinitz' *Algebraische Theorie de... | https://mathoverflow.net/users/33757 | Whence "Durchschnitt" and "Vereinigung"? | An extensive discussion of the origin of **"Menge"** is given in [Earliest Known Uses of Some of the Words of Mathematics](http://jeff560.tripod.com/s.html) (scroll down to "Set and Set Theory"). Cantor's (1880) [Über unendliche linear Punktmannigfaltigkeiten](https://books.google.nl/books?id=_oaABwAAQBAJ&pg=PA145) is ... | 15 | https://mathoverflow.net/users/11260 | 307955 | 134,146 |
https://mathoverflow.net/questions/307943 | 11 | Let $f:\mathbb{S}^2\to \mathbb{S}^2$ with degree $d$.
It is well known that the induced map $$f\_\ast:\pi\_3(\mathbb{S}^2)=\mathbb{Z}\to \pi\_3(\mathbb{S}^2)=\mathbb{Z}$$ is given by multiplication by $d^2$.
But, $\mathbb{S}^2=\mathbb{C}P^1$ and $[\mathbb{C}P^n,\mathbb{C}P^n]=\mathbb{Z}$ for $n\geq 1$ and for any $... | https://mathoverflow.net/users/35872 | Induced maps on homotopy groups by self maps of $\mathbb{CP}^n$ | I guess it should be enough to prove this statement for one particular example (for each $d,n$), since homotopy classes of self-maps of $\mathbb CP^n$ are classified by their degree (aren't they?).
Let us identify $S^{2n+1}$ with $|z\_1|^2+\ldots+|z\_{n+1}|^2=1$.
Consider the map $\phi(n,d):S^{2n+1}\to S^{2n+1}$ whic... | 12 | https://mathoverflow.net/users/943 | 307965 | 134,151 |
https://mathoverflow.net/questions/307973 | 29 | i.e. does there exist an integer $C > 0$ such that $11, 11 + C, ..., 11 + 10C$ are all prime?
| https://mathoverflow.net/users/126543 | Is there an 11-term arithmetic progression of primes beginning with 11? | Such an integer $C$ exists. The smallest $C$ with this property is $C=1536160080$.
I found this $C$ by computing the analogous number $C$ for a $3$-term prime arithmetic progression beginning with $3$, a $5$-term prime arithmetic progression beginning with $5$ and a $7$-term prime arithmetic progression beginning wit... | 67 | https://mathoverflow.net/users/nan | 307975 | 134,157 |
https://mathoverflow.net/questions/307979 | 3 | Let $\Sigma$ a closed oriented embedded surface in $R^3$. When $\Sigma$ is a round sphere, then for any smooth curve $A(t) \in SL(3)$ through the identity (i.e. $A(t) \in R^{3\times 3}$, $\det(A(t))=1$, $A(0)=I$), we have
$$
\frac{d}{dt}\bigg|\_{t=0} {\rm Area} (A(t) \Sigma) = 0.
$$
This can be proved either by a dire... | https://mathoverflow.net/users/102458 | area variation of a closed surface under ${\rm SL}(3)$ | In fact, for any surface $\Sigma$, there exists a matrix $A\_0 \in SL\_3(\mathbb{R})$ so that $A\_0\cdot\Sigma$ is a critical point for variations by $sl(3)$. The point is that if we look at the orbit $\{A\cdot\Sigma, A\in SL\_3(\mathbb{R})\}$, the area function $\Phi: SL\_3(\mathbb{R})\to \mathbb{R}$ given by $\Phi(A)... | 5 | https://mathoverflow.net/users/1345 | 307982 | 134,160 |
https://mathoverflow.net/questions/307995 | 2 | What is an example of a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph) $H=(V,E)$ with $|e|\geq \aleph\_0$ for all $e\in E$ and the property that $H\cong H^\*$ where $H^\*$ is the [dual hypergraph](https://en.wikipedia.org/wiki/Hypergraph#Terminology) of $H$?
| https://mathoverflow.net/users/8628 | Example of self-dual hypergraph with infinite edges | Let $V = \mathbb R$ and $E = \{(-\infty,r] : r \in \mathbb R\}$.
| 4 | https://mathoverflow.net/users/70618 | 307996 | 134,166 |
https://mathoverflow.net/questions/307287 | 9 | According to [Brouwer's fixed point theorem](https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem), for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point.
However, these fixed points cannot be chosen continuously, even for $K=[0,1]$, in the sense that there is no cont... | https://mathoverflow.net/users/83073 | Is it possible to continuously select a probability distribution over fixed points in Brouwer's fixed point theorem? | The problem here is not restricted to obtaining continuous choice functions, it already fails at the level of continuous multivalued functions.
**Theorem** The multivalued function $\mathrm{BFT}\_k : \mathcal{C}([0,1]^k,[0,1]^k) \rightrightarrows [0,1]^k$ mapping a continuous function to some arbtirary fixed point is... | 1 | https://mathoverflow.net/users/15002 | 308003 | 134,169 |
https://mathoverflow.net/questions/307834 | 2 | Disclaimer
==========
Sorry in advance for vagueness. I'm still trying to get my ideas right on this one.
Setup
=====
So, let $P$ be a distribution on a Euclidean space $X$ with an $\ell\_p$ metric, and let $P\_\epsilon$ be another distribution on $X$ whose Wasserstein distance (or KL diverence, for simplicity an... | https://mathoverflow.net/users/78539 | Draw samples from distribitions in the neighborhood of a fixed distribution | Maybe to add to the point of calculating $\max\_{P\_\varepsilon} \mathbb{E}\_{P\_\varepsilon}[f]$: I will write this a bit more in line with the literature I will refer to. Let $(X, d)$ be some polish space and a probability measure $\bar{\nu} \in \mathcal{P}(X)$ fix. Then the problem is
\begin{equation}
\Phi(f) :=\sup... | 1 | https://mathoverflow.net/users/106046 | 308014 | 134,175 |
https://mathoverflow.net/questions/308012 | 5 | Let $\pi\_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that
$$ \pi\_m \otimes \pi\_n = \bigoplus\_{k=0}^{\min(m,n)}\pi\_{m+n-2k}.$$
But suppose I want to think of this decomposition as matrices. Evaluating at a point $x \in \mathrm{SU}(2)$, on the left I have
$$... | https://mathoverflow.net/users/104213 | Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices | You can find such a formula with indices here:
<https://en.wikipedia.org/wiki/Wigner_D-matrix#Kronecker_product_of_Wigner_D-matrices,_Clebsch-Gordan_series>
| 3 | https://mathoverflow.net/users/7410 | 308015 | 134,176 |
https://mathoverflow.net/questions/307951 | 4 | Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $\pi{:}\ X\to\mathbb{R}^m$ be a linear map, with $m<n$ (for example, a projection). Let $\pi^{-1}(y)=\{x\in X\mid\pi(x)=y\}$ denote the inverse of $\pi$. If $F$ is a face of $\pi(X)$, what is the set $\pi^{-1}(F)$? Is it a face of $X$?
| https://mathoverflow.net/users/114221 | Is the preimage of a face under an affine map a face? | The question in the title is not how I'd phrase your question. I'd say what you're asking is "Is the preimage of a face under an affine map a face?", and the answer to that is yes.
The argument is simple. Let $f : X \rightarrow Y$ be an affine map (in your case, $\pi : X \rightarrow \pi(X)$), and let $F$ be a face o... | 6 | https://mathoverflow.net/users/61785 | 308021 | 134,178 |
https://mathoverflow.net/questions/308020 | 12 | Do we have an exact formula to compute the KL divergence between 2 mixtures of Gaussians (i.e convex combinations of a finite number of Gaussian distributions)?
If not exactly known, are there good upperbounds that are known for this quantity?
| https://mathoverflow.net/users/89451 | KL divergence and mixture of Gaussians | There is no closed form expression, for approximations see:
* [Lower and upper bounds for approximation of the Kullback-Leibler divergence between Gaussian mixture models](https://infoscience.epfl.ch/record/174055/files/durrieuThiranKelly_kldiv_icassp2012_R1.pdf) (2012)
>
> A lower and an upper bound for the Kull... | 8 | https://mathoverflow.net/users/11260 | 308022 | 134,179 |
https://mathoverflow.net/questions/308030 | 9 | The question of existence of sets $x,y$ such that
$$|x|<|y| \wedge |P(x)|=|P(y)|$$
is known to be independent of $\text{ZFC}$!
But are there known examples of sets fulfilling the above condition
that necessitates violation of choice?
| https://mathoverflow.net/users/95347 | Are there known examples of sets whose power set is equal in size to power set of larger sets only in absence of choice? | If there is an infinite Dedekind-finite set, then there are two infinite Dedekind-finite sets which have equipotent power sets, and one is larger than the other.
This is due to the fact that the existence of an infinite Dedekind-finite set implies that there are two sets $X$ and $Y$ such that $|X|<|Y|$ and $|Y|\leq^\... | 16 | https://mathoverflow.net/users/7206 | 308033 | 134,183 |
https://mathoverflow.net/questions/307601 | 10 | Can you provide a proof or a counterexample for the following claim :
>
> Let $P\_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$
>
>
>
>
> Let $N=k\cdot 2^n+1$ such that $n>2$ , $0< k <2^n$ and
>
>
>
>
> $\begin{cases} k \equiv 1,7 \pmod{30} \text{ with ... | https://mathoverflow.net/users/88804 | Primality test for specific class of Proth numbers | Your criterion is equivalent to:
$$(4+\sqrt{15})^{k2^{n-1}}\equiv (4+\sqrt{15})^{\frac{N-1}{2}}\equiv -1 (\bmod N \mathbb{Z}[\sqrt{15}]).$$
The point is that $P\_m(8)=(4+\sqrt{15})^m+(4-\sqrt{15})^m$. Moreover, $S\_i=(4+\sqrt{15})^{k2^i}+(4-\sqrt{15})^{k2^i}$, which one may prove by induction, using the fact that... | 7 | https://mathoverflow.net/users/1345 | 308036 | 134,185 |
https://mathoverflow.net/questions/308006 | 1 | I am looking for a good accessible reference that would summarize properties of zeros of complex analytic functions.
For my purpose, it would be interesting to see a discussion on the following topics:
1. Zeros of single variable analytic function are discrete (isolated)
2. Zeros of multivariable analytic function... | https://mathoverflow.net/users/69661 | Zeros of Multivariate Complex Functions [need reference] | MR1111477 Chirka, E. M. Complex analytic sets. Kluwer Academic Publishers Group, Dordrecht, 1989.
| 2 | https://mathoverflow.net/users/25510 | 308049 | 134,187 |
https://mathoverflow.net/questions/308048 | 7 | Let $x,y$ be positive real numbers then
$$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$
we obtain $1/2$-Hölder continuity for the square-root.
I would like to know if $x,y$ are positive Hilbert-Schmidt operators. Does... | https://mathoverflow.net/users/nan | Hölder continuity for operators | Your proposed inequality doesn't even work on diagonal matrices. Let $x\in\mathbb M\_n$ be diagonal with entries $x\_i\geq 0$. Then $x^{1/2}$ is diagonal with entries $\sqrt{x\_i}$. Thus
$$ \|x^{1/2}\|\_{HS}^2 = \sum\_i x\_i, \qquad
\|x\|\_{HS} = \Big( \sum\_i x\_i^2 \Big)^{1/2}. $$
Taking e.g. $x\_i=1/\sqrt n$ we obta... | 13 | https://mathoverflow.net/users/406 | 308050 | 134,188 |
https://mathoverflow.net/questions/308053 | 2 | Let $R$ be a real closed field. Recall that the *ladder* of $R$ is the divisible, ordered abelian group obtained by quotienting $R$ by a certain equivalence relation.
Note that $R$ has trivial ladder iff $R$ is a subfield of $\mathbb R$. If $R$ has trivial ladder and $L$ is a divisible ordered abelian group, let $R\l... | https://mathoverflow.net/users/2362 | Is any real closed extension of $\mathbb R$ characterized up to isomorphism by its ladder? | First, the initial claims in the question are false: the ladder of $R^L$ is *not* $L$. For example, for any $x\in L$, the field $R^L$ contains the element $\sqrt x$, which is not “comparable” to any element of $L$.
Not every linear order is isomorphic to a ladder of a real-closed field; for example, any such ladder i... | 5 | https://mathoverflow.net/users/12705 | 308058 | 134,192 |
https://mathoverflow.net/questions/307999 | 8 | Let $S\subset \mathbb{R}^3$ be a smooth surface with the Gauss normal map $N:S\to S^2$.
Then for every $x\in S$, the differential $(dN)\_x:T\_xS\to T\_{N(x)}S^2$ can be considered as an endomorphism of the tangent space $T\_xS$ since $T\_xS$ is parallel to $T\_{N(x)}S^2$. So from now on, without any ambiguity and in ... | https://mathoverflow.net/users/36688 | The differential of the Gauss normal map from a Lie algebraic view point | ADDED: I checked my calculation of the first displayed equation, and it appears to be correct. If so, it looks to me that it already implies that $A$ is either the identity or zero and therefore the second paragraph isn't even needed.
The shape operator is an example of a bundle map $A: S\rightarrow S$, where $S$ is ... | 8 | https://mathoverflow.net/users/613 | 308062 | 134,193 |
https://mathoverflow.net/questions/308061 | 0 | Let $X=TS^2\setminus Z$ where $Z$ is the zero section of the tangent bundle of $S^2$.
Is there a $S^2$- fiber bundle structure on $(X,\mathbb{R}^2\setminus\{0\},q)$ for some continuous fibre map $q$?
| https://mathoverflow.net/users/36688 | Is $TS^2\setminus Z$ a $S^2$- fibre bundle on the puntured plane?(Swapping the role of fibre points and base space) | Assuming that your question means: Is $X$ a fiber bundle over $\mathbb{R}^2 -0$, the answer is no. $X$ is homotopy equivalent to the unit tangent bundle of $S^2$, which is in turn homotopy equivalent to $\mathbb{R}P^3$. Also, $R^2 -0 \simeq S^1$. Now look at the exact sequence of the fibration; you'd conclude $\pi\_2(\... | 5 | https://mathoverflow.net/users/3460 | 308064 | 134,195 |
https://mathoverflow.net/questions/308057 | 2 | The ordinary [circle packing problem](https://en.wikipedia.org/wiki/Circle_packing) in the variant with equal radii asks for the largest radius $r\_{max}$ that allows placing $n$ non-overlapping circles with radius $r\_{max}$ e.g. in the unit square, in the unit circle or, on the unit sphere ([Tammes' problem](http://e... | https://mathoverflow.net/users/31310 | Name and Algorithms for a Sparsest Circle Packing | An overview of this "sparse packing problem" is given by Matthew Kahle in [Sparse locally-jammed disk packings](https://pdfs.semanticscholar.org/2827/b52f9626d1f09b918208d793a7ea4975439b.pdf). Quite generally, $r\_{\rm min}=O(1/n)$, so the fraction of the unit square or unit disc covered by the $n$ circles goes to 0 as... | 5 | https://mathoverflow.net/users/11260 | 308066 | 134,197 |
https://mathoverflow.net/questions/308017 | 21 | A neural network can be considered as a function
$$\mathbf{R}^m\to\mathbf{R}^n\quad
\text{by}\quad x\mapsto w\_N\sigma(h\_{N-1}+w\_{N-1}\sigma(\dotso h\_2+w\_2\sigma(h\_1+w\_1 x)\dotso)),$$
where the $w\_i$ are linear functions (matrices) $\mathbf{R}^{d\_{i-1}}\to\mathbf{R}^{d\_i}$, the $h\_i\in \mathbf{R}^{d\_i}$,... | https://mathoverflow.net/users/127583 | Structures of the space of neural networks | In information geometry, people study structures of Riemannian manifolds with dual affine connections on sets of neural networks. (The metric measures how close neural networks are in their input-output behaviour.) Riemannian geometry can then be used to study learning algorithms (like gradient descent methods). Here a... | 9 | https://mathoverflow.net/users/50846 | 308091 | 134,205 |
https://mathoverflow.net/questions/308095 | 9 | Is the following consistent with $\text{ZF}$?
There exists a set $S=\{x\_1,x\_2,x\_3,...\}$ such that:
1. $|x\_{i+1}| < |x\_i|$
2. $\forall m,n \in S (|P(m)|=|P(n)|)$
Where cardinality $``||"$ is defined after Scott's.
| https://mathoverflow.net/users/95347 | Can we have an infinite sequence of decreasing cardinality all terms of which have equal sized power sets? | Yes.
For silly reasons.
Suppose that $X$ is a Dedekind-finite set, then $S(X)$, the set of all injective finite *sequences* from $X$ is also Dedekind-finite. Let $S\_n(X)$ denote the subset of $S(X)$ of sequences whose domain is *at least* $n$. It is easy to see why $S\_n(X)$ surjects onto $S(X)$. Simply erase the ... | 14 | https://mathoverflow.net/users/7206 | 308103 | 134,209 |
https://mathoverflow.net/questions/308099 | 0 | If $(P,\leq)$ is a poset and $S\subseteq P$ we let $$\uparrow S = \{p\in P: p\geq s\text{ for some }s\in S\}.$$
Let $([\omega]^\omega,\subseteq)$ denote the collection of infinite subsets of $\omega$, ordered by set inclusion. If $S\subseteq [\omega]^\omega$ has the property that $\uparrow S = [\omega]^\omega$, does ... | https://mathoverflow.net/users/8628 | Upward generators of $[\omega]^\omega$ | As Wojowu says, the existence of an almost disjoint family of size continuum gives an affirmative answer. For completeness, here's one way to construct one of those:
* Fix a bijection $b$ from $2^{<\omega}$ to $\omega$.
* For each $f\in 2^{\omega}$, let $X\_f=\{b(\sigma): \sigma\prec f\}$.
Since any two distinct el... | 5 | https://mathoverflow.net/users/8133 | 308106 | 134,211 |
https://mathoverflow.net/questions/308109 | 8 | The homology of an $E\_2$-algebra is a Gerstenhaber algebra.
How precisely is the Gerstenhaber structure related to the $E\_2$-structure?
Obviously, the Gerstenhaber product is the commutative product that the $E\_2$-product induces in homology.
But what precisely is the interpretation of the Gerstenhaber bracket... | https://mathoverflow.net/users/119240 | What is the interpretation of the Gerstenhaber bracket? | I think the most transparent interpretation is by identifying $E\_2$ algebras with brace algebras (which was proved by [McClure and Smith](https://arxiv.org/abs/math/9910126)).
Namely, a brace algebra satisfies the relation
$$ab - (-1)^{|a||b|}ba = (-1)^{|a|} d(a\{b\}) - (-1)^{|a|}(da)\{b\} + a\{db\},$$
where $a\{b\}... | 13 | https://mathoverflow.net/users/18512 | 308111 | 134,213 |
https://mathoverflow.net/questions/307697 | 13 | *The motivation for this question is a bit convoluted, so in the interests of conciseness I'm just asking it as a curiosity (and I do find it interesting on its own); if anyone is interested, feel free to email me.*
I'm playing around with various notions of describing topological spaces, and I've found the following... | https://mathoverflow.net/users/8133 | When can I "draw" a topology in Baire space? | I have two things to offer, the first of which could help with getting better characterizations, the latter should give ample of examples.
Since I am not aware of standard terminology, call $(X,\tau')$ a topological weakening of $(X,\tau)$ if $\tau' \subseteq \tau$.
**Theorem**: A space is pictorial iff it is homeo... | 5 | https://mathoverflow.net/users/15002 | 308119 | 134,217 |
https://mathoverflow.net/questions/308121 | 4 | By combining two methods I've stumbled into a rather messy random walk situation. I have the typical random walk setup
$$\theta\_{i+1} = \theta\_{i} + \hat{\theta}\_{i+1}$$
Where $\hat{\theta}\_{i+1} \sim \mathcal{N}(0,1)$. However, $\hat{\theta}\_{i+1}$ depends on the previous $\hat{\theta}\_{i}$:
$$\hat{\theta}... | https://mathoverflow.net/users/127651 | Is this a random walk? Does it have a name? | I presume you mean $\epsilon = \epsilon\_i$, where $\epsilon\_i \sim \mathscr N(0,1)$ is independent of all the previous random variables. The pairs $(\theta\_i, \hat{\theta}\_i)$ form a Gaussian Markov process.
| 4 | https://mathoverflow.net/users/13650 | 308126 | 134,219 |
https://mathoverflow.net/questions/307947 | 25 | I'll state my questions upfront and attempt to motivate/explain them afterwards.
>
> **Q1:** Is there a direct way of expressing the relation "*$y$ is a function of $x$*" inside set theory?
> More precisely: Can you provide a formula of first order logic + $\in$, containing only two free variables $y$ and $x$, w... | https://mathoverflow.net/users/745 | Formalizations of the idea that something is a function of something else? | First of all, it seems to me as though the *real* question here is "what is a variable quantity?" Most of the definitions you quote from pre-20th century mathematicians assume that the notion of "variable quantity" is already understood. But this is already not a standard part of modern formalizations of mathematics; s... | 28 | https://mathoverflow.net/users/49 | 308132 | 134,220 |
https://mathoverflow.net/questions/308136 | 2 | Let $X$ be a set and ${\cal P}(X)$ its powerset. We say that ${\cal F} \subseteq {\cal P}(X)$ has the *splitting property (SP)* if there is $A\in {\cal P}(X)$ such that for all $F\in {\cal F}$ we have $$F \cap A \neq \emptyset \neq F\cap (X\setminus A).$$
Let $\text{SP}(X)$ denote the collection of all subsets of ${\... | https://mathoverflow.net/users/8628 | Maximality with respect to the splitting property | Yes, any family of subsets of $X$ with the splitting property can be extended to a maximal such family. The axiom of choice is not needed for this.
Suppose $\mathcal F\in \operatorname{SP}(X),$ and let $A\in\mathcal P(X)$ be such that for all $F\in\mathcal F$ we have
$$F\cap A\ne\emptyset\ne F\cap(X\setminus A).$$
... | 5 | https://mathoverflow.net/users/43266 | 308139 | 134,224 |
https://mathoverflow.net/questions/308147 | 0 | I am reading this paper by S.H Karin titled [Norm attaining operators and pseudospectrum](https://arxiv.org/pdf/1209.1218.pdf).
In page 2 he gives the definition of $l\_p$ direct sum of a family of Banach spaces as follows:
If $1\leq p< \infty$ and $\{X\_\alpha\}\_{\alpha\in\Lambda}$ is a faily of Banach spaces, then t... | https://mathoverflow.net/users/120523 | Regarding $\ell_p$ direct sums | Yes, $X\_\alpha$ can be nonzero for uncountably many $\alpha$. Of course, each individual vector $x \in X$ is zero except for countably many coodinates, in order for $\sum\_{\alpha\in\Lambda}\|x\_\alpha\|^p<\infty$.
**added**
Suppose (for purposes of contradiction) $x \in X$ is such that there are uncontably many ... | 0 | https://mathoverflow.net/users/454 | 308150 | 134,230 |
https://mathoverflow.net/questions/301257 | 3 | Let $X$ be an operator space such that there is a weak$^\*$-continuous complete isometry $\phi$ from its second dual $X^{\*\*}$ into a $W^\*$-algebra $M$ in which $\phi(X^{\*\*})$ is a (necessarily weak$^\*$-closed) left ideal. Put $J:=\phi(\hat{X})$, where $\hat{X}$ is the canonical copy of $X$ in $X^{\*\*}$.
>
> ... | https://mathoverflow.net/users/25499 | Is the ideal property of $X^{**}$ inheritable to $X$? |
>
> The answer is **no**.
>
>
>
*Proof*. Theorem 9 in the reference [3] below implies that there exist a $C^{\star}$-algebra $\mathcal{A}$ and open projections $p,q\in\mathcal{A}^{\*\*}$ such that $p$ and $q$ are Murray-von Neumann equivalent, but the $C^{\star}$-algebras $\mathcal{A}\_p\,(:=p\mathcal{A}^{\*\*}p... | 0 | https://mathoverflow.net/users/25499 | 308162 | 134,234 |
https://mathoverflow.net/questions/308120 | 4 | Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a *bijective, continuous* map $\phi:K \to K$ which satisfies the condition
$$
\hat u + \hat v \supset \hat w \implies \phi(\hat u) + \phi(\hat v) \supset \phi(\hat w) \tag{1}
$$
for al... | https://mathoverflow.net/users/34894 | Extending a continuous map over projective space | Your condition (1) means: if $\hat{u}$, $\hat{v}$, and $\hat{w}$ are linearly dependent, then so are $\widehat{\varphi(u)}$, $\widehat{\varphi(v)}$, and $\widehat{\varphi(w)}$. So $\varphi$ preserves linear dependence of $3$ vectors. But it doesn't detect linear dependence among $4$ or more vectors.
The other answers... | 1 | https://mathoverflow.net/users/88133 | 308167 | 134,236 |
https://mathoverflow.net/questions/307534 | 29 | Let $P$ be a compact connected set in the plane and $x,y\in P$.
>
> Is it always possible to connect $x$ to $y$ by a path $\gamma$ such that the length of $\gamma\backslash P$ is arbitrary small?
>
>
>
**Comments:**
* Be aware of [pseudoarc](https://en.wikipedia.org/wiki/Pseudo-arc) --- it is a compact conn... | https://mathoverflow.net/users/1441 | Running most of the time in a connected set | The answer to this question is positive. A required path $\gamma$ can be constructed inductively using the following
**Lemma.** For any continuum $P\subset\mathbb R^2$, distinct points $x,y\in P$, and $\varepsilon>0$ there are continua $P\_1,\dots,P\_{k}\subset P$ of diameter $<\varepsilon$, and horizontal or vertica... | 9 | https://mathoverflow.net/users/61536 | 308172 | 134,238 |
https://mathoverflow.net/questions/308205 | 3 | Let $X\_1,\ldots,X\_n$ be iid Rademacher variables, i.e., $P(X\_1=1)=P(X\_1=-1)=1/2$. CLT says that $Y\_n\equiv \sqrt{n}\bar{X}$ converges in distribution to $N(0,1)$ as $n\to\infty$. So $Y\_n^2$ is asymptotically $\chi^2$ which means $E e^{tY\_n^2}=O(1)$ provided that $t<1/2$. Intuitively, $E e^{t Y\_n^3}\to\infty$ si... | https://mathoverflow.net/users/71254 | about an interesting moment generating function | Let us prove
>
> **Theorem 1:** There is some $t\_\*\in[0.672,0.694]$ such that $E\_n:=E e^{t Y\_n^3/\sqrt{n}}$ is bounded for $t\in(0,t\_\*)$ and unbounded for $t\in(t\_\*,\infty)$.
>
>
>
*Proof*. The unboundedness of $E\_n$ for $t\ge\ln2=0.693\ldots$ follows by Christian Remling's comment. It remains to sh... | 4 | https://mathoverflow.net/users/36721 | 308215 | 134,250 |
https://mathoverflow.net/questions/308206 | 6 | Let $X$ be a compact finite dimensional Alexandrov space with curvature bounded below.
>
> Does there exist $\varepsilon\_0>0$ (depending on $X$) such that for any $\varepsilon \in (0,\varepsilon\_0)$ and any point $x\in X$ the open ball $B(x,\varepsilon)$ is contractible? Is similar statement true for closed ball... | https://mathoverflow.net/users/16183 | Contractibility of balls in Alexandrov spaces | Formally speaking the answer is "no".
Take a 2-dimensional cone with small total angle. Then for any $\varepsilon>0$ there is a point $x$ close enuf to the tip of the cone such that $B(x,\varepsilon)$ is an annulus.
| 7 | https://mathoverflow.net/users/1441 | 308230 | 134,257 |
https://mathoverflow.net/questions/308178 | 13 | Let $\mathrm{ODD}(n)$ be the set of permutations in $\mathfrak{S}\_n$ whose cycle lengths are all odd. It is known that
$$ \#\mathrm{ODD}(n) = \begin{cases} ((n-1)!!)^2 &\textrm{ if $n$ is even}; \\ n\cdot((n-2)!!)^2 &\textrm{ if $n$ is odd}. \end{cases}$$
Set $\mathcal{C}\_{\mathrm{ODD}}(n,t) = \sum\_{\pi \in \mathr... | https://mathoverflow.net/users/25028 | Cycle generating function of permutations with only odd cycles | I'll start by addressing conjecture 2. By summing your generating fun over all values of $k$ we obtain
$$F(z,w)=\sum\_{n\geq 0}\sum\_{k\geq 0}C\_{\text{ODD}}(n,2k)\frac{z^n}{n!}w^k=\sum\_{k\geq 0}w^k\left(\frac{1+z}{1-z}\right)^k=\frac{1-z}{1-w-z-wz}$$
From here we see that
$$\sum\_{n\geq 1}\sum\_{k\geq 1}\frac{P\_k(n)... | 5 | https://mathoverflow.net/users/2384 | 308233 | 134,259 |
https://mathoverflow.net/questions/307610 | 13 | From this Temkin's [paper](https://arxiv.org/pdf/1010.2235.pdf) (at the end of section 1.1.3), I know that one may define Berkovich spaces that include both **archimedean** and **non-archimedean** worlds. This looks very interesting.
Temkin mentions an example: the affine line over $\mathbb Z$. But I was wondering if... | https://mathoverflow.net/users/69190 | Berkovich space including both archimedean and non-archimedean worlds | The definition of analytic space over $\mathbf{Z}$ was given by Berkovich in his foundational book "Spectral theory and analytic geometry over non-Archimedean fields" (see the beginning of section 1.4 and section 1.5). It is quite general, and anyway enough so that analytifications of schemes locally of finite type ove... | 20 | https://mathoverflow.net/users/4069 | 308246 | 134,264 |
https://mathoverflow.net/questions/308240 | 3 | Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed.
Let $p \in M$, and let $\alpha\_p \in \bigwedge^k(T\_pM)^\*$.
>
> Does there exist an open neighbourhood $U$ of $p$, and a closed and co-closed $k$-form $\omega \in \Omega^k(U)$ satisfying $\omega\_p=\alpha\_p$?
>
> ... | https://mathoverflow.net/users/46290 | Can we specify the value of harmonic forms at a point? | The answer to your question is 'yes', but it may take me a little while to look up the appropriate references, since I'm traveling now.
The main point is that the (usually overdetermined) system of PDE defined by $\mathrm{d}\alpha = \mathrm{d}^\*\alpha = 0$ for $\alpha\in\Omega^k(M)$ is involutive and hence, if the ... | 5 | https://mathoverflow.net/users/13972 | 308254 | 134,265 |
https://mathoverflow.net/questions/308253 | 4 | In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$.
>
> $H^2(M,\mathbb{Z})$ is the group of isomorphism classes of line bundles over $M$.
>
>
>
I guess they mean there is a natural i... | https://mathoverflow.net/users/118688 | Classification of line bundles by second cohomology of a manifold | I think $H^2(M;\mathbb{Z})$ cannot mean the de Rham cohomology group. The coefficients are wrong.
Anyway: $\mathbb{CP}^\infty$ is an amazing space. It is both a model for $K(\mathbb{Z},2)$ and a model of $BU(1)$. Homotopy classes into $K(\mathbb Z,2)$ is in bijection with $H^2(M;\mathbb{Z})$ (By pulling back the fun... | 10 | https://mathoverflow.net/users/12156 | 308255 | 134,266 |
https://mathoverflow.net/questions/308251 | 19 | I asked this question on Mathematics Stackexchange ([link](https://math.stackexchange.com/q/2863312/660)), but got no answer.
Let $K$ be a field, let $x\_1,x\_2,\dots$ be indeterminates, and form the $K$-algebra $A:=K[[x\_1,x\_2,\dots]]$.
Recall that $A$ can be defined as the set of expressions of the form $\sum\_... | https://mathoverflow.net/users/461 | Is $K[[x_1,x_2,\dots]]$ an $\mathfrak m$-adically complete ring? | [Edit: The lemma was revised and proved, changed the point of view from series to sequences.]
[2nd Edit: The proof of the lemma was improved, and now the argument can show that Cauchy series such as $x\_1+(x\_{1^3+1}^2+\dots x\_{2^3}^2)+(x\_{2^3+1}^{3}+\dots+x\_{3^3}^3)+\dots$, diverge in the $\mathfrak{m}$-adic topo... | 9 | https://mathoverflow.net/users/86006 | 308266 | 134,272 |
https://mathoverflow.net/questions/306673 | 0 | I asked a question here [order of a permutation and lexicographic order](https://mathoverflow.net/questions/306572/order-of-a-permutation-and-lexicographic-order) but it seems\*\*\* that a very powerful and rich generalization can be made!
Let $A$ be a finite ring together with an arbitrary **total order** $<^\*$ and... | https://mathoverflow.net/users/112382 | root of identity matrix and lexicographic order | Iterating $\mathsf L\_Q$ over an initial matrix shall always yield a periodic sequence at some point because there is only finitely many possible matrices, but the period is not necessarily $q$ or a multiple or divisor of $q$.
Here is an example where the period is $4$ while $q=3$. It is a modification of [yours](htt... | 1 | https://mathoverflow.net/users/127616 | 308268 | 134,273 |
https://mathoverflow.net/questions/308211 | 4 | Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $Rep\_V(\mathfrak{g}) \subset Rep(\mathfrak{g})$ to be the smallest symmetric monoidal, idempotent complete, abelian subcateg... | https://mathoverflow.net/users/22810 | Generating Irreducible representations of a simple lie algebra with Schur functors | You can replace "simple Lie algebra over $\mathbb{C}$" with "simply connected, simple compact Lie group" and the category of representations will be equivalent as a tensor category.
It's a standard result that a representation of a compact group with finite center is a tensor generator if and only if it is faithful, ... | 4 | https://mathoverflow.net/users/66 | 308271 | 134,275 |
https://mathoverflow.net/questions/308244 | 0 | I would like to prove such a matrix as a positive definite one,
$$
(\omega^T\Sigma\omega) \Sigma - \Sigma\omega \omega^T\Sigma
$$
where $\Sigma$ is a positive definite symetric covariance matrix while $\omega$ is weight column vector (without constraints of positive elements)
I would apply an arbitrary $x$ belongin... | https://mathoverflow.net/users/127575 | Proof of A Positive Definite Covariance Matrix | To elaborate a bit on Mahdi's comment, recall that a positive definite matrix $\Sigma$ can be used to define a scalar product, i.e. $\langle a,b \rangle := a^\top\Sigma\, b$, and $\langle a,a\rangle = ||a||^2$.
You can multiply out the left side of the inequality you end up with scalar products and norms. That's whe... | 0 | https://mathoverflow.net/users/127682 | 308278 | 134,279 |
https://mathoverflow.net/questions/299712 | 3 | In my research I need to compute the group homology of the dicyclic group Dic3, which is a semi-direct product of $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$, and let's denote it by $G$, we have a short exact sequence,
\begin{equation}
0 \rightarrow \mathbb{Z}/3\mathbb{Z} \rightarrow G \rightarrow \mathbb{Z}/... | https://mathoverflow.net/users/87910 | Computation of group homology $H_2 ((\mathbb{Z}/3\mathbb{Z}) \rtimes (\mathbb{Z}/4\mathbb{Z}),\mathbb{Z})$ | In the spectral sequence, notice that by the remarks of YCor and Derek Holt, it is almost trivial: since the orders of $Z/4$ and $Z/3$ are prime to each other, all homology groups of the form $H\_p(Z/4,H\_q(Z/3,Z))$ are zero when $p\neq 0$ or $q\neq 0$. The only remaining groups are on the ``bounadry'' of the spectral ... | 5 | https://mathoverflow.net/users/41644 | 308288 | 134,282 |
https://mathoverflow.net/questions/308274 | 6 | Let $X$ be a connected compact metric space. Given a positive $\varepsilon$ and two points $x,y\in X$ we write $x\sim\_\varepsilon y$ if there exists a sequence $C\_1,\dots,C\_n$ of connected subsets of diameter $<\varepsilon$ in $X$ such that $x\in C\_1$, $y\in C\_n$ and $C\_i\cap C\_{i+1}\ne\emptyset$ for all $i<n$. ... | https://mathoverflow.net/users/61536 | Are $\varepsilon$-connected components dense? | I believe $[x]\_\varepsilon$ is always $\sigma$-compact. First consider a slightly modified definition where we replace 'diameter $<\varepsilon$' with 'diameter $\leq \varepsilon$' and call the corresponding set $[x]\_{\leq\varepsilon}$.
Note that $[x]\_\varepsilon = \bigcup\_{\delta < \varepsilon}[x]\_{\leq\delta}$... | 3 | https://mathoverflow.net/users/83901 | 308294 | 134,284 |
https://mathoverflow.net/questions/308270 | 14 | Let $a\_1,a\_2,\dots$ be a sequence of positive numbers less than $1$, such that $$\sum\_{n=1}^\infty a\_i= \infty,$$ and $S^1 = \mathbb{R}/\mathbb{Z}$.
Suppose $I\_1,I\_2,\dots$ be random intervals with respective lengths $a\_1,a\_2, \dots$in $S^1$ such that the distribution of the centers of $I\_n$ (for every $n$) ... | https://mathoverflow.net/users/51663 | Union of random intervals with total length equal to infinity | This is a refinement of Iosif Pinelis's answer, so we shall be somewhat brief. For a punchline, jump to the "Added" section below.
We claim that if $a\_n>c/n$ holds some $c>1$ and for all $n\geq n\_0$, then $P(I=S^1)=1$. To see this, fix a large integer $N$ and any interval $J$ of length $1/N$. Then,
$$P(J\not\subset... | 8 | https://mathoverflow.net/users/11919 | 308295 | 134,285 |
https://mathoverflow.net/questions/308156 | 24 | In his 2004 paper [Conformal Field Theory and Torsion Elements of the Bloch Group](https://arxiv.org/pdf/hep-th/0404120.pdf), Nahm explains a physical argument due to Kadem, Klassen, McCoy, and Melzer for the following remarkable identity. Let $C \in \operatorname{Mat}\_8(\mathbb{Z})$ denote the inverse of the Cartan m... | https://mathoverflow.net/users/78 | Has the $E_8$-based generating function for squares numbers been proven? | This identity was actually proven 24 years ago in
>
> S.O. Warnaar and P.A. Pearce, ["Exceptional structure of the dilute A 3 model: E8 and E7 Rogers-Ramanujan identities"](https://people.smp.uq.edu.au/OleWarnaar/pubs/Exceptional.pdf) J.Phys. A27 (1994) L891-L898
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The proof essentially establishes a fini... | 24 | https://mathoverflow.net/users/2384 | 308303 | 134,287 |
https://mathoverflow.net/questions/308217 | 7 | Let $p\in {\mathbb{R}}[x\_1,\ldots, x\_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of squares of $d$ dimensional Fourier harmonics up to degree $n$.
My question is if $p$ is positive on the unit sphere ... | https://mathoverflow.net/users/127701 | Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics | This Wikipedia page has many references: <https://en.wikipedia.org/wiki/Positive_polynomial>, including for example
* Marshall, Murray *Positive polynomials and sums of squares.* Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008.
* B. Reznick, *Uniform denominators in Hilbe... | 6 | https://mathoverflow.net/users/88133 | 308308 | 134,289 |
https://mathoverflow.net/questions/295492 | 4 | Let $S$ be a complex algebraic (smooth) surface and $\widetilde{S}$ be the blowup of $S$ at a point $p\in S$.
I would like to understand the statement:
>
> As a topological manifold, $\widetilde{S}$ is a connected sum of $S$ and the complex projective plane $\Bbb{P}^2$ with reversed orientation.
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Even ... | https://mathoverflow.net/users/40038 | Topology of the blowup of a surface at a point (connected sum) | Here's a proof of the statement in Michael's answer (repeated below) that conceptually has more steps but hopefully more illuminating than the computation found in Huybrechts - we hope to make clear why the $\mathbb{P}^n$ should have the opposite orientation.
Proposition: Let $x\in X$ be a point in a complex manifold... | 8 | https://mathoverflow.net/users/119160 | 308314 | 134,290 |
https://mathoverflow.net/questions/308325 | 10 | There are two "opposite" functors:
$$ op\_\Delta\colon sSet\to sSet$$
and
$$op\_s\colon sCat\to sCat.$$
The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite of a functor $\Delta\to \Delta$ which is the identity on objects and takes a morphism $\langle k\_0,\ldots,k\_n\rangl... | https://mathoverflow.net/users/11546 | Simplicial nerve functor commutes with opposites | It all follows from the following elementary lemma:
$\mathfrak{C}([n]^{op})$ is isomorphic to $\mathfrak{C}([n])^{op}$ as a cosimplicial simplicial category (in fact, they are actually equal, since the components of the natural isomorphism are all identities).
proof: It is an immediate calculation from the defini... | 4 | https://mathoverflow.net/users/1353 | 308327 | 134,292 |
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