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https://mathoverflow.net/questions/233432 | 0 | I wonder if the ODE
$y''+e^{y}=a$
can be solved explicitly. For $a=0$, it is well-known that there is a two-parameter family of explicit solutions
$y=\ln(2)-2\ln(\cosh(cx+d))+2\ln(c)$, $c,d \in R$. Are there explicit solutions for $a>0$?
| https://mathoverflow.net/users/42326 | Explicit solution for one-dimensional Gelfand problem | Multiply by $y'$, we get $((y')^2/2+e^y-ay)'=0$, so $(y')^2/2+e^y-ay=c$, $y'=f(y)$, where $f(y)=\pm \sqrt{2c+2ay-2e^y}$, so $dx/dy=1/f(y)$, $x$ is antiderivative of $1/f(y)$. I doubt that this antiderivative is expressed in elementary functions for general $a,c$.
| 2 | https://mathoverflow.net/users/4312 | 233441 | 108,322 |
https://mathoverflow.net/questions/233438 | 5 | Let $\kappa$ be a strong cardinal. Then for each $\lambda\geq\kappa$ does there exist a $\mu>\lambda$ such that if $U$ is a $\kappa$-complete ultrafilter on $\lambda$ and $j:V\rightarrow M,V\_{\mu}\subseteq M,j(crit(j))>\mu$ then there is some $\alpha$ and $x\in V\_{\alpha}$ so that the ultrafilter $\{R\subseteq V\_{\a... | https://mathoverflow.net/users/22277 | Do strong embeddings always provide all the ultrafilters that exist? | If all we want is that for every measure $U$ there is a strongness
embedding with a seed for $U$, then the answer is yes. To see
this, suppose $\kappa$ is a strong cardinal, and let $U$ be any
$\kappa$-complete ultrafilter on some $\lambda$. Let $j\_0:V\to
M\_0$ be any $\mu$-strong embedding, so that the critical point... | 7 | https://mathoverflow.net/users/1946 | 233451 | 108,326 |
https://mathoverflow.net/questions/233443 | 6 | This is related to my [previous question](https://mathoverflow.net/questions/233152/lifting-of-frobenius-on-semi-abelian-varieties) Assume that $A$ is an abelian variety over a field $k$ of characteristic $p$, $\mathcal{L}$ is a line bundle on $A$. Assume that $A$ is ordinary and $\mathcal{L}$ lifts on the canonical li... | https://mathoverflow.net/users/39304 | Lifting of Frobenius on torsors over abelian varieties | The canonical lifting $\mathcal{A}$ of $A$ has a canonical lift of the relative Frobenius $F\_{\mathcal{A}/W}:\mathcal{A}\to \mathcal{A}'$, where $\mathcal{A}'=F\_W^\* \mathcal{A}$ (=the canonical lift of the Frobenius twist $A'=F\_k^\*(A)$). Moreover, any line bundle $L$ on $A$ has a canonical `Teichmueller' lift $\ma... | 4 | https://mathoverflow.net/users/3847 | 233455 | 108,327 |
https://mathoverflow.net/questions/233152 | 4 | Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume that $A$ lifts to $W(k)$.
When it is possible to lift the Frobenius endomorphism of $A$ to $W(k)$?
Probably, $B$ has t... | https://mathoverflow.net/users/39304 | Lifting of Frobenius on semi-abelian varieties | [My answer to your other question](https://mathoverflow.net/a/233455/3847) seems to show that if $B$ is ordinary, $A$ has a natural lift over $W$ as a scheme, together with a lift of the Frobenius. I don't see however why the group structure of $A$ should lift, and why it should be compatible with the lift of Frobenius... | 4 | https://mathoverflow.net/users/3847 | 233456 | 108,328 |
https://mathoverflow.net/questions/229359 | 22 | I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (*Lie Groups, Lie Algebras, and Representations: An Elementary Introduction*, Springer), which I've enjoyed using. I'm confused about a technical hitch though that I'm not sure how to avoid.
The approach taken in this book has two notable simplifyin... | https://mathoverflow.net/users/22 | Technical issue in the approach to Lie groups taken in a book | Thanks for the question. It is true that I never proved that the simple summands in Theorem 7.8 are themselves semisimple in the sense that I define "semisimple." I would not characterize this omission as a problem, however, since I don't claim they are semisimple. Furthermore, I do not prove the classification of semi... | 17 | https://mathoverflow.net/users/88856 | 233464 | 108,329 |
https://mathoverflow.net/questions/232898 | 9 | Up to topology, the 5D homogeneous space
$$
G\_2/P
$$
of the (real form of the) 14D exceptional Lie group $G\_2$ is the 5D jet space
$$
M:=J^1(2,1)=\{(x,y,u,p,q)\}
$$
of scalar functions in two independent variables.
>
> WHY? Because $G\_2/P=S^2\times S^3\approx\mathbb{P}T^\*S^3\approx\mathbb{P}T^\*\mathbb{R}^3=\ma... | https://mathoverflow.net/users/22606 | Contact distributions on $(G_2,P)$-type Cartan geometries in dimension 5 | As Ben wrote, the question appears to conflate two different parabolic geometries of type $\newcommand{bfD}{{\bf D}}\newcommand{bfE}{{\bf E}}\newcommand{bfH}{{\bf H}}G\_2$:
Let $\Bbb V$ be the standard (i.e., $7$-dimensional irreducible) representation of $\mathfrak{g}\_2$ (either the split real or the complex form);... | 6 | https://mathoverflow.net/users/26266 | 233466 | 108,330 |
https://mathoverflow.net/questions/233430 | 7 | $PSL(2,7)$ acts on the projective plane over $\mathbb{F}\_2$ (the Fano plane) through its identification with $GL(3,2)$. It also acts on the projective plane over $\mathbb{C}$ through either of its pair of 3-dimensional complex representations. Does the Fano plane embed in $\mathbb{P}\_\mathbb{C}^2$ so that the action ... | https://mathoverflow.net/users/12419 | Does the Fano plane "embed" in the complex projective plane? | (As suggested, comment turned into answer) : The answer is "no". ${\rm PSL}(2,7)$ has two non-conjugate subgroups of index $7$, each isomorphic to the symmetric group $S\_{4}$. Either of the three dimensional irreducible representations of ${\rm PSL}(2,7)$ restrict irreducibly to each subgroup of ${\rm PSL}(2,7)$ isomo... | 8 | https://mathoverflow.net/users/14450 | 233467 | 108,331 |
https://mathoverflow.net/questions/233366 | 7 | Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$. Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$ (and on $D^2$ by restriction). Let $\phi$ be a diffeomorphism of $D^2$ which is equal to the identity in a neighborhood of $\partial D^2$, and which preserves area; i.e. $\phi^\* \omega ... | https://mathoverflow.net/users/nan | Example where Calabi invariant is nontrivial? | There are equivalent definitions of the Calabi invariant that allow to build easily such examples.
1) Given a Hamiltonian $H:[0,1]\times \mathbb{R^2}\to \mathbb{R}$, the Calabi invariant of its time-one map is given by
$$\int\_0^1\int H(t,x)\,\omega dt.$$
(See the book "Introduction to symplectic topology" by McDuff ... | 3 | https://mathoverflow.net/users/58620 | 233473 | 108,333 |
https://mathoverflow.net/questions/233475 | 3 | Let $p$ be a prime number, $\mathbb C\_p$ be the completion of an algebraic closure of $\mathbb F\_q\left(\left(\frac1T\right)\right)$ for the valution $v(x)=-\deg(x)$. Let $\sum\_{n\ge0}a\_nz^n$ be a power series that converge for all $z\in\mathbb C\_p$. Schnirelmann's famous result assert that for every $r\in\mathbb ... | https://mathoverflow.net/users/33128 | Growth comparision between an entire function and a related function | No, over the complex field the result is much weaker. Even this is not true:
$$\max\_{|z|\leq r}\left|\sum\_{n=0}^\infty a\_nb\_nz^n\right|\leq K\max\_{|z|\leq r}\left|\sum\_{n=0}^\infty a\_nz^n\right|,$$
with $K$ independent of $z$. I suppose that the simplest counter-example is $a\_n=e^{i\alpha\_n}/n!$ and $b\_n=e^{-... | 5 | https://mathoverflow.net/users/25510 | 233478 | 108,334 |
https://mathoverflow.net/questions/233497 | 4 | Alain Connes being a leading French mathematician today one could ask whether he is a member of the Bourbaki group. Is there a published reference that would either refute or confirm this?
| https://mathoverflow.net/users/28128 | Reference for Connes Bourbaki membership or otherwise | Connes was a member, according to M. Mashaal "Bourbaki: A secret society of mathematicians", AMS 2006 (translated from the French by A. Pierrehumbert). It says so on page 18; see the [link](https://books.google.com/books?id=-CXn6y_1nJ8C&pg=PA18#v=onepage&q&f=false). But, as was also mentioned there, there is a rule tha... | 13 | https://mathoverflow.net/users/2926 | 233509 | 108,345 |
https://mathoverflow.net/questions/233512 | 5 | Let $X$ be a Hausdorff space. Suppose that $C(X)$ (or $C\_0(X)$) is a finitely generated $C^\*$-algebra. What we can say about $X$ ? For example can we characterize its inductive dimension, axioms of countability etc. ?
| https://mathoverflow.net/users/75934 | $C(X)$ as finitely generated $C^*$-algebra | The maximal ideal space $\Delta$ of a finitely generated Banach algebra is homeomorphic to a compact subset of $\mathbb{C}^n$. On the other hand, evaluation at each point of $X$ is clearly a complex homomorphism of $C(X)$. We conclude that $X$ is homeomorphic to a subset of a compact subset of $\mathbb{C}^n$.
Edit: ... | 7 | https://mathoverflow.net/users/88291 | 233517 | 108,348 |
https://mathoverflow.net/questions/233526 | 2 | Let $Z$ be a projective variety embedded into $\mathbb P^n$. Then we can define an affine cone over $Z$ as the inverse image of $Z$ under canonical map $\mathbb A^{n+1}\setminus0 \to \mathbb P^n$. I have to questions about this construction:
1. Does a cone over the given projective variety $Z$ depend on an embedding ... | https://mathoverflow.net/users/88385 | Uniqueness of a (weighted) affine cone | Abotu the first question: yes, it does. Already for $\mathbb{P}^n$ embedded with $\mathcal{O}(d)$, the cone changes with $d$.
About the second, you might view a cone as a variety with a $\mathbb{G}\_m$ and just one fixed point. On a fixed variety, you might have two different actions giving different cone structures.... | 5 | https://mathoverflow.net/users/48866 | 233527 | 108,352 |
https://mathoverflow.net/questions/233525 | 6 | Let $\mathbb A = (A, +\_A)$ be a cancellative, but possibly non-commutative, monoid with identity $0$, and fix an element $x \in A$. Does there always exist a cancellative monoid $\mathbb B = (B, +)$ such that $\mathbb A$ is a submonoid of $\mathbb B$ and $x$ is left-invertible invertible in $\mathbb B$ (see Benjamin S... | https://mathoverflow.net/users/16537 | Embedding a cancellative monoid into another in such a way that a prescribed element becomes left-invertible | The answer is no.
In a cancellative monoid left invertible elements are right invertible. If yx =1 then xyx=x and so xy =1 by cancellation. So you are asking to invert a given element in some cancellative monoid.
Take a fg cancellative monoid M
not embeddable in a group. Invert the generators one by one to embed... | 6 | https://mathoverflow.net/users/15934 | 233529 | 108,353 |
https://mathoverflow.net/questions/233297 | 1 | (Trying to clarify the question; the answer given below is wrong.)
If ${\cal C}$ is a collection of subsets of a set $X$, we associate to ${\cal C}$ a graph $G\_{\cal C} = (V,E)$ where $V = {\cal C}$ and $$E = \big\{\{A,B\}: A\neq B\in {\cal C} \land A\cap B \neq \emptyset\big\}.$$
If $G$ is a simple, undirected gr... | https://mathoverflow.net/users/8628 | Intersection number of the tensor product of graphs | So first, let's get a few problems out of the way, using that the intersection number is the smallest number of cliques needed to cover all edges of the graph.
For all $n$, $i(K\_n) = 1$.
Next, we have $(\*)$: $\exists n$ such that $G \times H = K\_n$ is equivalent to $\exists n\_g n\_h$ such that $G = K\_{n\_g}$ and $... | 1 | https://mathoverflow.net/users/85130 | 233531 | 108,355 |
https://mathoverflow.net/questions/233538 | 7 | (*This is a follow-up to a [previous post](https://mathoverflow.net/questions/233367/generalizing-a-pattern-for-the-diophantine-m-tuples-problem).*) A [*rational Diophantine $m$-tuple*](https://web.math.pmf.unizg.hr/~duje/intro.html) is a set of rationals {$a\_1,a\_2,\dots a\_m$} such that (with $i\neq j$), all $a\_i a... | https://mathoverflow.net/users/12905 | Extending rational Diophantine triples to sextuples | Similar formulas can be obtained by taking multiples $mR$ for the point $R=[0,(t^2+1)^3]$ on the elliptic curve
$$ y^2=x^3+(3t^4-21t^2+3)x^2+(3t^8+12t^6+18t^4+12t^2+3)x+(t^2+1)^6. $$
The above formulas (5) and (6) correspond to $m=2$ and $m=3$.
Other possible formulas appear if the rank of this curve is $\geq 2$. E.g... | 7 | https://mathoverflow.net/users/21337 | 233541 | 108,357 |
https://mathoverflow.net/questions/233545 | 4 | I am interested in when the trace field of a knot complement has the form $F(\sqrt{-d})$ for $F\subset\mathbb{R}$ and $d\in F^+$ (squarefree). Does this occur for infinitely many choices of pairs $(F,d)$?
For knot complements, the trace field and invariant trace field are the same, and the trace field is equal to the... | https://mathoverflow.net/users/14835 | How many quadratic fields occur as trace fields of hyperbolic knot complements? | I think this question is open and even in this narrowly framed context the question still has a number of interesting seemingly weaker questions.
First, as a reference Long and Reid addressed a similar question in section 6 of this paper:
>
> D. D. Long and A. W. Reid, Fields of definition of canonical curves,
... | 4 | https://mathoverflow.net/users/27453 | 233562 | 108,365 |
https://mathoverflow.net/questions/233066 | 2 | Let $I = (0, 1)$ and $1 \le q < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|D^{(m - 1)}u\|\_{L^q(I)} + \sum\_{j = 0}^{m - 2} \|D^ju\|\_{L^\infty(I)} \le \epsilon\|D^mu\|\_{L^1(I)} + C\|u\|\_{L^1(I)}$$for all $u \in W^{m, 1}(I)$?
| https://mathoverflow.net/users/nan | Sobolev inequality involving summing from $j = 0$ to $m - 2$, exists constant | We argue by contradiction. After scaling, we have a sequence $(u\_n)\_{n=1}^\infty \subset W^{m,1}(I)$ with
$$
\| D^{(m-1)} u\_n \|\_q + \sum\_{j=0}^{m-2} \| D^j u\_n \|\_\infty = 1 > \varepsilon \| D^m u\_n \|\_1 + n \| u\_n \|\_1.
$$
We have $(D^{m-1}u\_n)$ is a bounded subset of $W^{1,1}(I)$ from
$$
\| D^{m-1} u\_n ... | 0 | https://mathoverflow.net/users/46135 | 233565 | 108,367 |
https://mathoverflow.net/questions/233564 | 5 | Let $X$ be a curve over an algebraically closed field $k$ (even over $k = \mathbb{C}$ if you want), let $J = Pic^0\_{X/k}$ be its Jacobian, let $P \in X(k)$ be a point, and let $i \colon X \hookrightarrow J$ be the closed immersion that sends $Q$ to the divisor class of $[Q] - [P]$. How does the map
$$Pic^0(i): Pic^0\... | https://mathoverflow.net/users/63877 | Identifying the canonical principal polarization of a Jacobian | The answer depends very much on how you define the homomorphism $p$ associated to the polarization. My personal choice is $p(a)=\mathcal{O}(\Theta \_a-\Theta )$, where $\Theta $ is a theta divisor and $\Theta \_a:=\Theta +a$. Realizing $\Theta $ as $i(\mathrm{Sym}^{g-1}C)$, you find that $i^\*\Theta \_a$ is the divisor... | 9 | https://mathoverflow.net/users/40297 | 233569 | 108,371 |
https://mathoverflow.net/questions/233576 | 6 | Let $(G,+)$ be a finite Abelian group. We say $q\colon G\to \mathbb{T}$ is a non-degenerated quadratic form, if $q(-a)=q(a)$ and the symmetric function
$$
b(g,h) =q(g+h)q(g)^{-1}q(h)^{-1}
$$
is a non-degenenerate bicharacter on $G$, i.e. $b(g+h,k)=b(g,k)b(h,k)$ and
$b(k,g)=1$ for all $k\in G$ implies $g=0$.
Let $(\G... | https://mathoverflow.net/users/10718 | Do all non-degenerate quadratic forms come from positive even lattices? | **Edited:** I have missed your "positive". The signature of the lattice modulo 8 depends on the form only (some people call this Brown invariant and van der Blij theorem; Nikulin below calls this just the signature of the form).
Otherwise (given the right signature), I would suggest that the map is surjective, and th... | 4 | https://mathoverflow.net/users/44953 | 233580 | 108,376 |
https://mathoverflow.net/questions/233589 | 0 | Let $G, H$ be two finite, simple, undirected graphs. We call them "reduce-by-1"-isomorphic, or $r\_1$-isomorphic for short, if there is a bijection $\psi: V(G) \to V(H)$ such that for all $v \in V(G)$ we have that $G \setminus \{v\}$ is isomorphic to $H \setminus \{\psi(v)\}$.
(Ulam's reconstruction conjecture, or so... | https://mathoverflow.net/users/8628 | "Reduce-by-1"-isomorphic graphs | The chromatic polynomial, and therefore the chromatic number, was proved reconstructible by Tutte in his famous paper "All the king's horses". I don't know about the Hadwiger number.
| 6 | https://mathoverflow.net/users/9025 | 233593 | 108,381 |
https://mathoverflow.net/questions/233075 | 1 | I would like to produce an easily-interpretable explicit upper bound (i.e. no unspecified constants) for the function
$$
f(n) := L\_n^{\left(-n-\frac{d}{2}\right)}\left(-\frac{1}{2}\right), \quad n,d \in \mathbb{N}
$$
with $d$ a fixed parameter. This bound needn't be especially sharp (I only need it to characterize ... | https://mathoverflow.net/users/88654 | Upper-bounding the value of a generalized Laguerre polynomial (using recurrence relation?) | You can obtain good bounds by applying the saddle-point method to the generating function. A good introduction to these techniques is in the book *Analytic Combinatorics* by Flajolet & Sedgewick. In your case, you can avoid going to the complex plane and have simple bounds as follows:
1. The generating series of your... | 2 | https://mathoverflow.net/users/80257 | 233603 | 108,383 |
https://mathoverflow.net/questions/233607 | 3 | Sorry for this possibly trivial or stupid question, but I'm very far from being an expert in Algebra.
Let G be the fundamental group of the nonorientable surface of even rank n=2k (n generators, 2n if counting their inverses). Does G contain free subgroups? In the affirmative, how many of them have a maximum number ... | https://mathoverflow.net/users/88933 | free subgroups of the fundamental group of nonorientable surfaces | A double cover (thus a subgroup of index $2$) of your fundamental group is the fundamental group of an orientable surface, which is either closed or not. In the second case, the fundamental group is free. In the first case, it is not free, but any subgroup of infinite index is free. You can have free subgroups of ANY r... | 2 | https://mathoverflow.net/users/11142 | 233610 | 108,385 |
https://mathoverflow.net/questions/233602 | 0 | (I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty)
How can I prove that the following statements are equivalent?
1. $\lambda$ is an eigenvalue of $A+\delta A$, where $\|\delta A\|\_{2}\leq \epsilon$
2.... | https://mathoverflow.net/users/88930 | Eigenvalue-related statements | This is rather classical. The set of such $\lambda$'s is called the $\epsilon$-pseudo-spectrum. It is presented in many books on numerical linear algebra. I suggest S.K. Godunov *Modern aspects of linear algebra*, AMS (1998).
| 1 | https://mathoverflow.net/users/8799 | 233611 | 108,386 |
https://mathoverflow.net/questions/233579 | 7 | Let me call a pair of integers $a, b$ *acceptable* if the equation $ax^2 + by^2 = z^2$ has a non-trivial rational solution. Theorem 4.5.4 of Cojocaru-Murty's book on Sieves says that the number of acceptable pairs of integers $a, b$ with $1 \leq a, b \leq H$ is $\ll H^2 / \log \log H$. This is proved using the Turan si... | https://mathoverflow.net/users/10458 | Hilbert symbol averages | This problem has been considered by a few authors. Versions for $a$ and $b$ *rational* were considered by Hooley and Guo (independently). See
Hooley - On ternary quadratic forms that represent zero.
Guo - On solvability of ternary quadratic forms.
Hooley obtained the (sharp) lower bound of the form $H^3/(\log H)... | 3 | https://mathoverflow.net/users/5101 | 233613 | 108,387 |
https://mathoverflow.net/questions/233540 | 0 | While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over something.
I am currently thinking of
**(1)** Linear independence in linear algebra, for elements of a $R$-module,
**(2)**... | https://mathoverflow.net/users/18583 | Independence in mathematics | The notions of algebraic independence (as in linear independence) and probabilistic independence are both captured in the notions of an independent subset of a Boolean algebra and the notion of an independent family of partitions.
Suppose that $B$ is a Boolean algebra. Then a subset $R\subseteq B$ is said to be indep... | 5 | https://mathoverflow.net/users/22277 | 233624 | 108,390 |
https://mathoverflow.net/questions/233633 | 16 | In a recent paper a quite unexpected result about a new pattern in prime numbers emerged:
[Unexpected biases in the distribution of consecutive primes](http://arxiv.org/abs/1603.03720)
by Oliver, R. J. L.; Soundararajan, K. (Submitted on 11 Mar 2016)
>
> While the sequence of primes is very well distributed in ... | https://mathoverflow.net/users/1047 | Could this unexpected bias in the distribution of consecutive primes have any impact on the security of encryption algorithms? | This discovery is beautiful, but it's unlikely to have any impact on cryptography. Of course this lack of relevance is a matter of speculation/opinion, not proven mathematical fact, but these sorts of correlations between consecutive primes just haven't come up in cryptography to my knowledge, because there seems to be... | 20 | https://mathoverflow.net/users/4720 | 233636 | 108,391 |
https://mathoverflow.net/questions/233644 | 8 | Let $u:[0,1]\to\mathbb{R}^n$ be a bounded Borel function.
It is well-known that if, for any compact interval $I\subseteq [0,1]$,
$$ \int\_I|u-u\_I|^2\le C|I|^{1+\alpha} $$
for some $C,\alpha>0$ (here $u\_I:=\frac{1}{|I|}\int\_I u$), then $u$ is in fact $\frac{\alpha}{2}$-Holder continuous: this was first proved by Camp... | https://mathoverflow.net/users/36952 | Does infinitesimal variance imply continuity? | Since $u$ is assumed bounded, your condition is equivalent to
$$
\frac{1}{|I|} \int\_I |u-u\_I|\, dx =o(1)
$$
as $|I|\to 0$, uniformly in $I$, and this is the condition that defines VMO.
So you are asking if functions in $VMO\cap L^{\infty}$ are continuous, and this is known to be false. [This classical paper by Sara... | 8 | https://mathoverflow.net/users/48839 | 233657 | 108,398 |
https://mathoverflow.net/questions/233643 | 6 | Numbers $x\_1,x\_2,\ldots,x\_n$ are drawn independently and uniformly from the interval $[0,1]$. Order them as $y\_1\ge y\_2\ge\dots\ge y\_n$, and let $S$ be their sum. Let $k$ be the smallest index such that $y\_1+\dots+y\_k\geq S/2$.
What is $E[k]$, and what are other known properties about $k$ (e.g., variance)? I... | https://mathoverflow.net/users/88952 | Lowest index giving half of the sum | Here's an approximate answer: let $U\_1,U\_2,\ldots,U\_n$ be the i.i.d. uniform random variables. I will consider them ordered from smallest to largest because it makes the arithmetic cleaner. We're then asking adding from smallest to largest, how many do we need to sum in order that $U\_{i\_1}+\ldots +U\_{i\_k}>\frac ... | 1 | https://mathoverflow.net/users/11054 | 233662 | 108,400 |
https://mathoverflow.net/questions/233614 | 3 | I have the problem for computing the j-derivative of a logarithm, with $j\gg1$
\begin{equation}
c\_j=\left.\frac{\partial^j}{\partial s^j}\log\left(1+Ae^s+Be^{2s}\right)\right|\_{s=0},
\end{equation}
being A and B real numbers ($0<A,B<1$). What I have done is to perform a Taylor expansion of the logarithm, finding
\beg... | https://mathoverflow.net/users/88906 | aproximate sum involving binomial coefficients | Just a rough idea. Let $\alpha, \beta$ be the zeros of $1+Ax+Bx^2$, then for $j\geq 1$
$$c\_j = \left.\left(\frac{\partial}{\partial s}\right)^j \log( B(\alpha - e^s)(\beta-e^s) )\right|\_{s=0} = \left.\left(\frac{\partial}{\partial s}\right)^j \log(\alpha-e^s) + \left(\frac{\partial}{\partial s}\right)^j \log(\beta-e^... | 4 | https://mathoverflow.net/users/7076 | 233665 | 108,401 |
https://mathoverflow.net/questions/233675 | -1 | Do you know some reference about the notion of parallel transport for simplicial manifolds?
| https://mathoverflow.net/users/41970 | Parallel transport on simplicial manifold? | Defining parallel transport is equivalent to defining a connection. Connections on simplicial manifolds are discussed in chapter 6 of Dupont's book "Curvature and characteristic classes": <http://link.springer.com/book/10.1007%2FBFb0065364>
| 3 | https://mathoverflow.net/users/39082 | 233676 | 108,402 |
https://mathoverflow.net/questions/233683 | 2 | For $t\in \mathbb{Z}\times\mathbb{Z}$ and $A\subseteq\mathbb{Z}\times\mathbb{Z}$ we set $t+A :=\{t+a: a\in A\}$.
Call $A\subseteq\mathbb{Z}\times\mathbb{Z}$ *tileable* if there is $T\subseteq\mathbb{Z}\times\mathbb{Z}$ such that
1. $t\_1\neq t\_2\in T$ implies $(t\_1+A)\cap (t\_2+A) =\emptyset$;
2. $\bigcup\{t+A: ... | https://mathoverflow.net/users/8628 | Tileable subsets of $\mathbb{Z}\times\mathbb{Z}$ | Yes. Uncountable. For each set $P\subset \mathbb Z$ take the tile $A = (\{0\}\times \mathbb Z) \cup (\{1\} \times P) \cup (\{-1\} \times (\mathbb Z \setminus P))$.
| 5 | https://mathoverflow.net/users/27742 | 233687 | 108,405 |
https://mathoverflow.net/questions/233635 | 3 | Assume $K$ is an imaginary quadratic extension of $\mathbb{Q}$, and $E$ an elliptic curve defined over $\mathbb{Q}$.
Let $p\neq l$ be primes in $\mathbb{Q}$ where $E$ has good reduction. Assume $p$ splits in $K/\mathbb{Q}$ and $l$ stays inert.
Denote by $K\_{l}$ the localization of $K$ at the place $l$.
Is there always... | https://mathoverflow.net/users/70751 | Cohomology of elliptic curves | More generally let k be a local field of residual characteristic $\ell$ and $E/k$ an elliptic curve with good reduction. Then for any prime $p\neq\ell$, there is a perfect duality between $E(k)/p^n$ and $H^1(k,E)[p^n]$ discovered by Tate. As both are finite abelian groups, they are indeed isomorphic. Knowing the struct... | 8 | https://mathoverflow.net/users/5015 | 233688 | 108,406 |
https://mathoverflow.net/questions/233658 | 7 | It is known that AC implies LEM constructively, and also that AC implies the ultrafilter principle. Is there a similar relationship between the ultrafilter principle and classical logic? In other words, are there any inference rules of propositional logic which are classically-but-not-constructively admissible, but are... | https://mathoverflow.net/users/62519 | Which Heyting algbras arise out of some elementary topos which satisfies the ultrafilter principle? | I think your formulation of the ultrafilter principle implies the de Morgan law.
Let $U$ be any proposition, consider the boolean algebra $A$ freely generated by an element $v$.
so $A = \{ 0,1,v,\neg v \}$.
Consider the following equivalence relation on $A$:
$$ \{(0,0),(1,1),(v,v),(\neg v, \neg v) \} \cup U \ti... | 5 | https://mathoverflow.net/users/22131 | 233689 | 108,407 |
https://mathoverflow.net/questions/233400 | 7 | Let $H \subset G$ be a closed subgroup of a lie group and $G/H$ the homogeneous coset space. There's an exact sequence of adjoint representations of $H$:
$$0 \to \mathfrak{h} \to \mathfrak{g} \to \mathfrak{g/h}\to 0 $$
The canonical principal $H$-bundle $G \to G/H$ gives an exact functor from representations of $H$... | https://mathoverflow.net/users/22810 | Tangent bundle of a homogeneous space and the euler exact sequence | The isomorphism $G\times\_H(\mathfrak g/\mathfrak h)\to T(G/H)$ is induced by the map $G\times (\mathfrak g/\mathfrak h)$ mapping $(g,X+\mathfrak h)$ to $T\_gp\cdot L\_X(g)\in T\_{gH}(G/H)$.
The Euler sequence corresponds to an exact sequence for the restriction of the standard representation of $G$ to $H$. This is ... | 10 | https://mathoverflow.net/users/64141 | 233692 | 108,410 |
https://mathoverflow.net/questions/233616 | 6 | I have a technical question about unbounded chain complexes. I couldn't think of a descriptive title for it.
Let $P$ be a chain complex of contravariant functors on $\mathbf{R}$ (the real numbers regarded as an ordered set), valued in the category of abelian groups.
Suppose that for every real number $s$, both of t... | https://mathoverflow.net/users/1048 | On the ordered set of real numbers, does sheaf+cosheaf imply constant? | Here is one possible way to proceed.
(Caveat lector: I haven't checked all the details carefully.)
Denote by S the sphere spectrum and by N the (contractible) spectrum that implements a nullhomotopy for S.
A map A→B of spectra is a weak equivalence if and only if
for any map Σ^k(S)→A with a nullhomotopy Σ^k(N)→B for ... | 2 | https://mathoverflow.net/users/402 | 233700 | 108,414 |
https://mathoverflow.net/questions/233699 | 0 | For any set $X$ we define $[X]^2 =\big\{\{a,b\}: a, b \in X\text{ and }a\neq b\big\}$.
Let $$E = \big\{\{(a\_1, a\_2), (b\_1, b\_2)\}\in[\omega\times\omega]^2: |a\_i-b\_i| = 1\text{ for some } i\in\{1,2\}\big\}.$$
Is there a topology $\tau$ on $\omega\times\omega$ such that $A\subseteq (\omega\times\omega)$ is conn... | https://mathoverflow.net/users/8628 | Topology on $\omega\times\omega$ such that topologically connected equals graph-connected | No. There is no topology such that sets with even just two elements are connected if and only if there is an edge between the two vertices for your graph.
To see this consider that a set of two elements $\{ a,b \}$ is topologically connected if and only if all open sets containing $a$ also contain $b$ or all open set... | 4 | https://mathoverflow.net/users/89005 | 233731 | 108,427 |
https://mathoverflow.net/questions/233726 | 1 | Let $\delta, \epsilon>0$ and $f: \mathbb{T} \rightarrow \mathbb{C}$ such that (i) $f(0)=1$ and (ii) $|f(x)| \le \epsilon$ for all $|x| > \delta$. What is the smallest $L = L(\delta, \epsilon)$ (asymptotically) such that there exists such a $f$ with $\widehat{f}(m) =0$ for all $m \not \in [-L,L]$.
| https://mathoverflow.net/users/10858 | extremal kernels for functions on the torus | It suffices to approximate uniformly a function $g$ with $g(0)=1$, $g(x)=0$ for $|x|>\delta$ by a trigonometric polynomial $p\_L$ of degree $\le L$, with error $\le \epsilon/2$. By [Jackson's Theorem,](https://en.wikipedia.org/wiki/Jackson%27s_inequality) the error can be made
$$
\|g-p\_L\|\_{\infty} \le \frac{C\_n}{L^... | 2 | https://mathoverflow.net/users/48839 | 233752 | 108,435 |
https://mathoverflow.net/questions/233758 | 1 | I just encountered the following statement: *any subsymmetric basic sequence is either weakly null or equivalent to the unit vector basis of $\ell\_1$.*
It's the first time I realize that. I do see the case in which it is equivalent to the unit vector basis of $\ell\_1$. However, I cannot prove that if that's not the... | https://mathoverflow.net/users/36832 | Classification of subsymmetric basic sequences | In the theorem you quote, part of the definition of subsymmetric includes the hypothesis that the sequence is unconditionally basic. If you do not include this in the definition of subsymmetric, then the summing basis for $c\_0$ is an example of a weakly Cauchy subsymmetric basis that is not equivalent to the unit vect... | 2 | https://mathoverflow.net/users/2554 | 233761 | 108,437 |
https://mathoverflow.net/questions/233743 | 5 | I am trying to understand how the [tensor product of presentable categories](https://ncatlab.org/nlab/show/tensor+product+of+presentable+%28infinity%2C1%29-categories) works: let $\otimes\colon {\cal A}\times {\cal B}\to {\cal A}\otimes{\cal B}$ the universal bilinear functor corresponding to $\text{id}\_{{\cal A}\otim... | https://mathoverflow.net/users/7952 | On the tensor product of presentable categories | 1. Yes. You can present $\mathcal A \otimes \mathcal B$ as a certain localization of the free cocompletion of $\mathcal A \times \mathcal B$. By "generate under colimits" I of course mean that you are allowed to take transfinitely-iterated colimits: you can take a colimit whose entries are the values of colimits whose ... | 3 | https://mathoverflow.net/users/78 | 233762 | 108,438 |
https://mathoverflow.net/questions/233719 | 5 | How can I prove the following:
$d^{ij}(m,k) > d^{ji}(m,k)$ for all $k < \frac{1}{2}\binom{m}{2},$
where $d^{ij}(m,k)$ denotes the number of permutations of $(1,\ldots,i,\ldots,j,\ldots,m)$ with $k$ inversions and where $i$ is on the left of $j.$ Similarly $d^{ji}(m,k)$ denotes the number of permutations where $i$ i... | https://mathoverflow.net/users/89007 | Counting the number of permutations of $(1,\ldots,i,\ldots,j,\ldots,m)$, where $i < j$ and number of inversions is $k$ | Denote $\Delta^{ij}(m,k)=d^{ij}(m,k)-d^{ji}(m,k)$. By reverting the permutation we observe that $d^{ij}(m,k)=d^{ji}(m,{m\choose 2}-k)$, so
$$
\Delta^{ij}(m,k)=-\Delta^{ij}(m,{m\choose 2}-k).
\qquad (\*)
$$
Now we prove the statement by the induction on $m$. Firstly, we present the step, and then we establish the b... | 4 | https://mathoverflow.net/users/17581 | 233789 | 108,445 |
https://mathoverflow.net/questions/233765 | 0 | Let $S$ be the unit circle and for any $x,y \in S$ let $d(x,y)$ be the lenght of the smallest arc between $x$ and $y$. A **bijective** map $\phi : S\longrightarrow S$ is called 1-iso if the following holds:
$$ \forall x,y \in S \;\; d(x,y)=1 \longleftrightarrow d(\phi (x), \phi (y) ) =1$$
Now the question: Is there a ... | https://mathoverflow.net/users/85969 | On 1-iso maps and subsets of the unit circle | For any point $a\in S$, there is an infinite (in both directions) chain $\dots,a\_{-1},a\_0,a\_1,a\_2,\dots$ of points such that $d(a\_i,a\_{i+1})=1$, and no other points are at distance $1$ from the points of this chain. The whole circle is partitioned into (continuum of) such chains; for every chain we choose its rep... | 2 | https://mathoverflow.net/users/17581 | 233794 | 108,448 |
https://mathoverflow.net/questions/233219 | 15 | Let $f: \mathcal{E} \rightarrow \mathcal{T}$ be a geometric morphism between two (Grothendieck) toposes (or maybe more generally a bounded geometric morphism between elementary toposes).
It is well known that if $f$ is an open or proper surjection then objects (and even locales) descend along $f$.
What I want to kn... | https://mathoverflow.net/users/22131 | Descent of Higher categorical structures along geometric morphisms | I have been able to gather all the elements for the case of $2$-categorical descent along open surjections, so I will write it as an answer.
It is worth noting that the proof below follows the exact same path as the proof of descent for locales and objects along open surjections given in the Elephant.
So what follow ... | 5 | https://mathoverflow.net/users/22131 | 233796 | 108,450 |
https://mathoverflow.net/questions/232584 | 4 | Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, and the symbols $+$ and $\cdot$ as standing for ordinary addition and multiplication.'' This comment is found in section ... | https://mathoverflow.net/users/28128 | Looking for a source for Intended Interpretation | Here are quotes from three well-known sources.
Shoenfield, *Mathematical Logic* (1967), page 23:
>
> We construct a model of $N$ by taking the universe to be the set of natural numbers and assigning the obvious individuals, functions, and predicates to the nonlogical symbols of $N$. This model is called the *sta... | 11 | https://mathoverflow.net/users/5442 | 233809 | 108,454 |
https://mathoverflow.net/questions/233778 | 6 | The commutativity degree $d(G)$ of a finite group $G$ is defined as the ratio
$$\frac{|\{(x,y)\in G^2 | xy=yx\}|}{|G|^2}.$$It is well known that $d(G)\leq5/8$ for any finite non-abelian group $G$. If $P(G)$ is the monoid of subsets of $G$ with respect to the usual product of group subsets, then the commutativity degree... | https://mathoverflow.net/users/17565 | A question on the commutativity degree of the monoid of subsets of a finite group | @Derek Holt is completely right: as $|G|\to \infty$, the fraction of subsets $(A,B)$ with $AB=G$ tends to $1$, so there is no such $c$.
Indeed, assume that $|G|=n$. Let us choose the subsets $A$ and $B$ uniformly and independently. Fix any $g\in G$; there are $n$ pairs $(a,b)$ with $ab=g$, each pair belongs to $A\tim... | 5 | https://mathoverflow.net/users/17581 | 233810 | 108,455 |
https://mathoverflow.net/questions/233803 | 10 | Let $U$ be a representation of $S\_m$ and $V$ a representation of $S\_n$. Then the representation $\operatorname{Ind}\_{S\_m\wr S\_n}^{S\_{mn}}(U^{\otimes{n}}\otimes V)$ has a nice interpretation in terms of symmetric functions. If $\operatorname{ch}(U)$ and $\operatorname{ch}(V)$ are the Frobenius characteristics of $... | https://mathoverflow.net/users/10273 | Super-plethysm? | This amounts to study composition of "linear species" in the category of complexes.
The correct way to handle these computations using plethysm is
to introduce an auxiliary variable $t$ and to weight the cohomology $H^i$ with the weight $(-t)^i$. The plethysm must act on $t$ by $p\_n(t)=t^n$.
I do not know a writte... | 6 | https://mathoverflow.net/users/10881 | 233819 | 108,460 |
https://mathoverflow.net/questions/233014 | 9 | I came up with the following coloring concept when studying neural networks (which are often modelled using directed graphs). No idea whether there is already an established name for it.
If $X$ is a non-empty set, we say that $M\subseteq X$ is a *majority* if $|M| > |X\setminus M|$.
Let $G=(V,E)$ be a finite direct... | https://mathoverflow.net/users/8628 | Majority coloring for directed graphs | Let me answer Question 2. The answer is in negative: there exists an upper bound for majority coloring numbers of all tournaments. I will not care about the sharpness of the bound.
Let $G$ be a tournament on $n$ vertices, and let $d\_1\leq d\_2\leq \dots\leq d\_n$ be the in-degrees of its vertices (denote the vertice... | 7 | https://mathoverflow.net/users/17581 | 233821 | 108,461 |
https://mathoverflow.net/questions/233826 | 5 | I've been looking for references/answers to this problem for several days and I couldn't find anything.
If we consider the closed unit ball $B$ in $\mathbb C^2$ then for any point $(z\_1,z\_2)\notin B$ we can find a polynomial $P$ of two variables with $$|P(z\_1,z\_2)|>\sup\_{(w\_1,w\_2)\in B}|P(w\_1,w\_2)|.$$
In ... | https://mathoverflow.net/users/89073 | Symmetric polynomial separating points | I think combining the two tricks in your post answers the question. Let $(x\_1, x\_2)$ be a point not in $B$. Set
$$\ell(z\_1, z\_2) = \frac{\overline{x\_1}}{\sqrt{|x\_1|^2+|x\_2|^2}} z\_1 + \frac{\overline{x\_2}}{\sqrt{|x\_1|^2+|x\_2|^2}} z\_2.$$
Then $\ell$ is a linear polynomial with $\ell(x\_1,x\_2)>1$ but $|\ell(y... | 6 | https://mathoverflow.net/users/297 | 233832 | 108,469 |
https://mathoverflow.net/questions/233834 | 3 | Given any Borel measure $\mu$ on $\mathbb{R}$, define a map that sends any $f\in C\_c(\mathbb{R})$ to $$T\_\mu(f)(y)=\int \langle\exp(-i x \lambda),f(x)\rangle\exp(iy\lambda)d\mu(\lambda).$$
Here $\langle\cdot,\cdot\rangle$ denotes the inner product of functions with respect to the Lebesgue measure.
Is (a multiple of... | https://mathoverflow.net/users/18261 | Restrictions on spectral measure | No. This becomes clear if you use more suggestive notation: you are sending $f$ to the (inverse) Fourier transform of $\widehat{f}\mu$. Exponential decay can be obtained from (a variant of) the Paley-Wiener Theorem: if $\widehat{g}$ is holomorphic on a strip $|\textrm{Im}\, z|<c$ and
$$
\sup\_{|y|<c}\int\_{-\infty}^{\i... | 2 | https://mathoverflow.net/users/48839 | 233841 | 108,472 |
https://mathoverflow.net/questions/233840 | 10 | Let $C$ be a category. The *groupoid completion* of $C$ is the free groupoid on $C$, i.e. the category $C[C^{-1}]$ obtained by localizing at everything. Recall that the classifying space $\mathbf{B}C$ of $C$ is the geometric realization of the nerve of $C$. It's well-known that there is an equivalence of groupoids $C[C... | https://mathoverflow.net/users/2362 | Can the groupoid completion of a topological category be recovered from its classifying space? | I'm not sure how well-posed this question is, what exactly is the topological structure of $\mathcal{C}[\mathcal{C}^{-1}]$?
However, the question certainly makes good sense for simplicial categories and the answer is affirmative. Dwyer and Kan proved that if $\mathcal{C}$ is a cofibrant simplicial category, then $\ma... | 5 | https://mathoverflow.net/users/12547 | 233843 | 108,473 |
https://mathoverflow.net/questions/233849 | 1 | In functional analysis, there is a concept of a self dual space (Hilbert) and a self double dual space (reflexive). I am curious as to whether a generalization exists or not (and if it exists, whether or not it is actually useful). Say a space is not reflexive, but if we keep on taking the dual many times, say $X$, $X^... | https://mathoverflow.net/users/89083 | Structure of chain of duals in functional analysis | No. The sequence $c\_0$, $c\_0^\* \cong l^1$, $c\_0^{\*\*} \cong l^\infty$, $\ldots$, doesn't stabilize.
Let $E$ and $F$ be Banach spaces. The following two lemmas are standard, and good exercises.
Lemma 1. If $E$ embeds isometrically in $F$, then $E^\*$ is a quotient of $F^\*$.
Lemma 2. If $E$ is a quotient of $... | 4 | https://mathoverflow.net/users/23141 | 233850 | 108,475 |
https://mathoverflow.net/questions/233839 | 2 | The input of my problem is a set of positive values $a=\{a\_1,...,a\_n\}$ where $n\geq 3$.
I want to construct an $n$-gon where the lengths of the $n$ facets are the values $a\_i$ for $i=1,...,n$.
My questions are:
* I suppose this problem has been studied before. Is it possible to have some references?
* From a ... | https://mathoverflow.net/users/85242 | Constructing a polygon of $n$ facets from a set of positive values representing the length of the facets | No, it is not always possible, because it is not always possible to partition the (cyclic) sequence of edge-length into 3 portions of adjacent edges, for whose sums the triangle inequality holds; e.g. 1,1,3 doesn't allow the construction of a polygon.
Your condition on existence suffices, as has been already demonst... | 1 | https://mathoverflow.net/users/31310 | 233854 | 108,476 |
https://mathoverflow.net/questions/233833 | 25 | For $X$ a topological space, from the short exact sequence
$$ 0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0 $$
we get a Bockstein homomorphism
$$H^i(X, \mathbb{Z}/2) \rightarrow H^{i+1}(X, \mathbb{Z}/2)$$
This is also known as the Steenrod square $Sq^1$.
Now sup... | https://mathoverflow.net/users/84144 | Steenrod operations in etale cohomology? | You maybe want to have a look at
* P. Brosnan and R. Joshua. Comparison of motivic and simplicial operations in mod-$\ell$ motivic and étale cohomology. In: Feynman amplitudes, periods and motives, Contemporary Math. 648, 2015, 29-55.
There are two sequences of cohomology operations in motivic and étale cohomology... | 26 | https://mathoverflow.net/users/50846 | 233862 | 108,477 |
https://mathoverflow.net/questions/233867 | 3 | (1) If $N^k$ is a submanifold in a compact Riemannian manifold $M^{k+m},\ m\geq 1$ s.t. each $p\in N$ has the following property : There exists independent set $\{ X\_i\}\_{i=1}^k$ tangent to $T\_pN$ s.t. $$\sum\_{i=1}^k( \nabla\_{X\_i}n,{X\_i})=0$$ for any $n$ which is any unit normal to $N$, then $N$ is not totally g... | https://mathoverflow.net/users/36572 | Some manifold which is not totally geodesic in a compact manifold | Let us consider the case of hypersufaces for the sake of simplicity. The higher codimension case follows by taking products with $(S^1)^k$.
>
> *Observation :* Minimal submanifolds satisfy both of your conditions. In fact in this case the vectors $X\_i$ can be taken orthonormal. In fact if you ask for $X\_i$ to be... | 3 | https://mathoverflow.net/users/8887 | 233873 | 108,481 |
https://mathoverflow.net/questions/233659 | 1 | I am not familiar with Robinson's construction as I do not have access to his text or to precise accounts of this, but I have come to understand that the proof predicate of Robinson arithmetic is non-standard e.g. in that it does not fulfill all of the Hilbert-Bernays-Löb derivability conditions. Can I rest assured tha... | https://mathoverflow.net/users/37385 | A question on the provability predicate of Q | Robinson arithmetic is $\Sigma^0\_1$ complete - every true $\Sigma^0\_1$ sentence is provable in Q. If you are asking whether $Q \vdash A$ implies $Q \vdash \text{Pr}(A)$, that completeness results says that the answer is yes, because $\text{Pr}(A)$ is a true $\Sigma^0\_1$ sentence.
If I remember correctly, Robinson... | 3 | https://mathoverflow.net/users/5442 | 233881 | 108,485 |
https://mathoverflow.net/questions/233894 | 1 | Consider a finite measure space $(X,\Sigma,\mu)$.
Consider the function $d:\Sigma \times \Sigma \to [0,1]$ given by
$$d(\sigma\_1,\sigma\_2) = \mu \left\{ (\sigma\_1^c \cap \sigma\_2) \cup (\sigma\_1 \cap \sigma^c\_2) \right\}.$$ One can verify that $d$ is a pseudometric (where $d(\sigma\_1,\sigma\_2) = 0$ means that $... | https://mathoverflow.net/users/16852 | Is there a name for this metric on a Borel sets | The book [Dictionary of distances](https://books.google.com/books?id=I-PQH8gcOjUC&lpg=PA25&ots=5bY662ttFI&dq=counting%20symmetric%20difference%20metric&pg=PA25#v=onepage&q=counting%20symmetric%20difference%20metric&f=false) by Deza and Deza lists several names for this object (and its induce metric on the quotient when... | 5 | https://mathoverflow.net/users/3948 | 233896 | 108,488 |
https://mathoverflow.net/questions/233899 | 2 | I am currently looking at *stratified Mukai flop*. Roughly speaking, this is a construction that, starting with a grassmannian $G$ inside a hyperkahler $X$ produces a birational manifold $X^\*$ (with a dual grassmannian $G^\*$ inside). The easiest example is the standard Mukai flop, namely the case in which the grassma... | https://mathoverflow.net/users/54269 | Grassmannian inside a hyperkahler manifold | This is just the general case of the example from my comment above. Let $S\subset \mathbb{P}^g\_{\mathbb{C}}$ be a K3 surface of degree $2g-2$ that contains no curve that spans a $\mathbb{P}^r$ with $r<g-1$. Any such curve would be contained in (many) hyperplane sections of $S$, hence would have degree $< 2g-2$. So if ... | 3 | https://mathoverflow.net/users/13265 | 233911 | 108,491 |
https://mathoverflow.net/questions/233909 | 7 | In noncommutative algebraic geometry a commonly studied family of objects are quantum projective spaces. Theses are certain deformations of the homogeneous coordinate ring of $\mathbb{CP}^n$. For example, see this [mathoverflow post](https://mathoverflow.net/questions/109347/point-modules-of-quantum-projective-space-ma... | https://mathoverflow.net/users/89074 | Quantum Grassmannians? | apparently, [quantum grassmannians](https://www.encyclopediaofmath.org/index.php/Quantum_Grassmannian) come in many variations --- this may be what you are looking for:
[Graded quantum cluster algebras and an application to quantum Grassmannians](http://arxiv.org/abs/1301.2133), Grabowksi & Launois, 2010.
>
> Amo... | 2 | https://mathoverflow.net/users/11260 | 233915 | 108,494 |
https://mathoverflow.net/questions/233889 | 0 | **Background** : If a compact Riemannian manifold $M$ with a no curvature condition has disjoint two submanifolds $N\_i$, then the distance between them is attained by some minimizing geodesic $c$.
If $c'(0)$ is orthogonal to $T\_{c(0)} N\_1$, then $\exp\_{c(0)}\ tv$ with $|v|=1,\ v\perp T\_{c(0)}N\_1$ goes to where ... | https://mathoverflow.net/users/36572 | Projection of geodesic is geodesic | Counterexample: Let $M$ be the round sphere $S^2$, let $N$ be the equator. Let $c$ start from the equator, going nearly north, miss the north pole and come back again. The normal projection $\alpha(t)$ to the equator is along longitudinal lines. The projection moves first very slow along the equator. If $c(t)$ is near ... | 3 | https://mathoverflow.net/users/26935 | 233918 | 108,496 |
https://mathoverflow.net/questions/233917 | 4 | Let $E$ be an elliptic curve defined over a number field $F$ and $F\_\infty$ be the cyclotomic $\mathbb{Z}\_p$-extension of $F$. Is it true that the $p$-primary subgroup of $E$ over $F\_\infty$ i.e. $E[p^\infty](F\_\infty)$ is finite ?
Proofs or references are welcome.
| https://mathoverflow.net/users/44637 | Finiteness of the $p$-primary subgroup of an elliptic curve over the cyclotomic $\mathbb{Z}_p$-extension | Yes. If $E$ has potentially good reduction, this is due to H. Imai, Proc Japan Adac Math Sci 51 (1975). A non-standard proof is Theorem A.2.8 in Coates-Sujatha's "Galois cohomology of elliptic curves"
| 4 | https://mathoverflow.net/users/5015 | 233919 | 108,497 |
https://mathoverflow.net/questions/233807 | 21 | The complete level modular curve $X(p)$ does not have potentially good reduction at $p$ for any $p \neq 2,3,5,7,13$ because then there are cusp forms on $X\_0(p)$ showing up in the cohomology of $X(p)$, whose associated Galois representations have unipotent local monodromy at $p$, which contradicts potentially good red... | https://mathoverflow.net/users/18060 | Does X(13) have potentially good reduction at 13? | Regarding Will's question on $X(13)$, it follows from Michael Stoll's answer and the following lemma that $X(13)$ does not have potentially good reduction.
Lemma. *Let $X\to Y$ be a finite morphism of smooth projective geometrically connected curves over a number field $K$. Suppose that $Y$ has non-zero genus. If $X$... | 12 | https://mathoverflow.net/users/4333 | 233920 | 108,498 |
https://mathoverflow.net/questions/233921 | 47 | Apologies in advance if this question is too elementary for MO. I didn't find an explanation of these ideas in any algebraic geometry books (I don't know French).
The following is an excerpt from [this archive](http://www.mta.ca/~cat-dist/archive/2003/03-3):
>
> Thierry Coquand recently asked me
>
>
> "In your ... | https://mathoverflow.net/users/69037 | Grothendieck says: points are not mere points, but carry Galois group actions | Suppose $k$ is a field, not necessarily algebraically closed. $\text{Spec } k$ fails to behave like a point in many respects. Most basically, its "finite covers" (Specs of finite etale $k$-algebras) can be interesting, and are controlled by its absolute Galois group / etale fundamental group. For example, $\text{Spec }... | 45 | https://mathoverflow.net/users/290 | 233922 | 108,499 |
https://mathoverflow.net/questions/232563 | 4 | Suppose we have an exterior algebra over $\mathbb{F\_2}$, say $R = \Lambda\_{\mathbb{F\_2}}V$, where $V$ is an $n$-dimensional $\mathbb{F}\_2$ vectorspace. Let $x\_1,\ldots,x\_n$ be a basis of that vectorspace.
This ring is a local ring whose maximal ideal $I$ is generated by $x\_1,\ldots,x\_n$.
The square of any ... | https://mathoverflow.net/users/3969 | Ideals in exterior algebras over the field with two elements | The following work is joint work with Omar Antolin-Camarena.
We want to prove that if $r\in I^2$, then $\dim(r)<2^{n-1}$ (Dimension as an $\mathbb{F}\_2$-vectorspace).
We will prove this by induction on the number of variables $n$ and the length of $r$ (number of monomials that are summed up). Rename the variables su... | 2 | https://mathoverflow.net/users/3969 | 233923 | 108,500 |
https://mathoverflow.net/questions/233924 | 6 | *I posted this on math.stackexchange to no avail, so I hope it's appropriate to post here despite that it might not be research-level. I expect the answer to this is well-known to people studying non-associative algebras, but I cannot find it in my references and the more thorough literature on the topic is expensive! ... | https://mathoverflow.net/users/14835 | Does the Cayley-Dickson construction preserve isomorphism of quaternion algebras? | Suppose $K$ is a field, and $B$ your quaternion algebra $K$. The octonion algebra $C$ made from $B$ using the Cayley-Dickson construction depends on an auxiliary choice of an element $c \in K^{\times}$. Namely, $C$ is the set of pairs $(u,v)$ with $u,v \in B$ with addition
$(u\_1,v\_1) + (u\_2, v\_2) = (u\_1 + u\_2,... | 6 | https://mathoverflow.net/users/25514 | 233929 | 108,504 |
https://mathoverflow.net/questions/233491 | 6 | *Edit: In my original post I failed to require the group to be a manifold group. The answer below from @BenLinowitz works in that case. I am really interested though in when the group is torsion-free, so have edited this to reflect that.*
For a non-elementary finite-covolume Kleinian group $\Gamma$,
let $\mathbb{Q}(\... | https://mathoverflow.net/users/14835 | For an arithmetic hyperbolic 3-manifold group, when is its trace field not its invariant trace field? | For cusped 3-manifolds that are link complements in $\mathbb{Z}/2\mathbb{Z}$ homology spheres, the trace field and invariant trace field are equal (see Neumann and Reid [Arithmetic of Hyperbolic Manifolds](http://www.math.columbia.edu/~neumann/preprints/nrarith.pdf) Corollary 2.3), so it makes sense to look at manifold... | 3 | https://mathoverflow.net/users/27453 | 233938 | 108,506 |
https://mathoverflow.net/questions/231278 | 1 | I have a Laplace tranform in the form given below
$\mathcal{L}\_I(s)=\text{exp}(-\pi\lambda \Gamma(1+\frac{2}{\alpha})\Gamma(1-\frac{2}{\alpha})P^{2/\alpha}s^{2/\alpha})$
Can some one help me to find the inverse Laplace transform of it?
Here, $\alpha$ can take values like 1,2,3,4,5...
$P$ and $\lambda$ are cons... | https://mathoverflow.net/users/61400 | How to find the Inverse Laplace Transform of the following? | In addition to Carlo Beenakker's answer. <http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/924/735> (The Kohlrausch function: properties and applications, by R.S. Anderssen, S.A. Husain and R.J. Loy),
besides the Doetsch's result for $g\_{1/2}(t)$ cited by Carlo, cites the following results ($s\_0=1$ is ... | 0 | https://mathoverflow.net/users/32389 | 233947 | 108,511 |
https://mathoverflow.net/questions/233945 | 3 | Let $G$ be a Cayley graph, and $H$ a graph cospectral with $G$. Must $H$ be a Cayley graph? Does a counterexample exist? If $G$ is a circulant graph, does a counterexample exist?
| https://mathoverflow.net/users/75264 | Graphs cospectral with Cayley graphs | To rectify joro's answer; he is certainly correct about the general direction, although it is not clear from his answer whether any of these 35-vertex SRGs is Cayley.
However, on 25 vertices there are a number of SRGs of degree 12, one of them is a certainly a Cayley graph (the Paley graph), and several SRGs with the... | 4 | https://mathoverflow.net/users/11100 | 233953 | 108,514 |
https://mathoverflow.net/questions/233828 | 4 | The Borel--Bott--Weil Theorem is usually stated for the complete flag manifold of $SU(N)$. Does an analogue hold for the other flags, for example the Grassmannians?
More precisely, suppose $G(\mathbf C)$ is a complex reductive group, and $P(\mathbf C)$ is a parabolic subgroup. Characters $\lambda$ of $P(\mathbf C)$ g... | https://mathoverflow.net/users/89074 | Borel--Bott--Weil for the Grassmannians | Kostant, *Lie algebra cohomology and the generalized Borel-Weil theorem*, Ann. Math. 74 (1961), 329-387.
W. Schmid, *Homogeneous complex manifolds and representations of semisimple Lie groups*, **Proceedings of the International Congress of Mtahematicians: Helsinki 1978** (ed. O. Lehto) 195-208.
You could also look... | 3 | https://mathoverflow.net/users/13268 | 233964 | 108,518 |
https://mathoverflow.net/questions/233965 | 0 | Can anyone please tell me The relationship between $p$-solvable Group
and solvable group.and find an example of a $p$-solvable group that is not solvable group or vice-versa.
| https://mathoverflow.net/users/89140 | The relationship between $p$-solvable Group and solvable group | A finite group is solvable if an only if it is $p$-solvable for every prime $p$. An example of a $p$-solvable group which is not solvable is the semidirect product $G = VSL(2,5)$, where $V$ is an elementary Abelian group of order $121$, and $SL(2,5)$ acts faithfully and irreducibly as a group of linear transformations ... | 4 | https://mathoverflow.net/users/14450 | 233967 | 108,519 |
https://mathoverflow.net/questions/233975 | 1 | Let G be a compact group, suppose $G=AB$ where $A$,$B$ are profinite subgroups of $G$. Is it true that G is profinite group?
| https://mathoverflow.net/users/89147 | Set product of profinite subgroups of a compact group is profinite | Yes. Let $f$ be a finite-dimensional continuous unitary representation of $G$. Then $f(G)=f(A)f(B)$. Since $A$, $B$ are profinite and $f(G)$ is a compact Lie group, $f(A)$ and $f(B)$ are both finite. Hence $f(G)$ is finite; in particular the unit component $G^0$ is contained in the kernel of $f$. Since this holds for e... | 3 | https://mathoverflow.net/users/14094 | 233981 | 108,523 |
https://mathoverflow.net/questions/233984 | 7 | How many pure math papers are published a year? I vaguely remember seeing a figure of 10,000 but that might be old, and I may be wrong.
| https://mathoverflow.net/users/39865 | How many papers are posted a year? | At [SCImago](http://www.scimagojr.com/index.php) you can find pretty much the entire [statistics](http://www.scimagojr.com/countrysearch.php?area=2600&country=&w=world):
The first graph gives the total number of math papers per year.
The second graph breaks it down per subject area (so you can distinguish "pure" from... | 24 | https://mathoverflow.net/users/11260 | 233994 | 108,526 |
https://mathoverflow.net/questions/233985 | 4 | I have the following problem: let $G$ be a finite directed graph with $V$ vertices $v\_i$ and $E$ (directed) edges $e\_j$. I know that if an edge $e\_k = (v\_i, v\_j)$ is in the graph, then the opposite edge $-e\_k = (v\_j, v\_i)$ is not in the graph. I also know that the graph contains at least one cycle. My goal is t... | https://mathoverflow.net/users/89151 | Removing cycles in a directed graph by swapping edges orientation | This is closely related to a standard (NP-complete) optimization problem, the minimum feedback edge set. Normally, this problem is defined as asking for the minimum set of edges $F$ to remove from a digraph $G$ to make $G-F$ acyclic. However, reversing the same set of edges $(G-F+F^R)$ also gives a DAG.
To see this, ... | 7 | https://mathoverflow.net/users/440 | 234023 | 108,536 |
https://mathoverflow.net/questions/233983 | 1 | I'm considering the diffraction problem described in section 3.16 of "Linear and quasilinear elliptic equations" of Ladyzhenskaya and Uraltseva (1968). Let $\Omega$ be an open bounded subset in $\mathbb{R}^n$, and suppose the surface $\Gamma$ partition $\Omega$ into sub-regions $\Omega\_1$ and $\Omega\_2$ where $\Omega... | https://mathoverflow.net/users/70847 | Why are the tangential derivatives in this diffraction problem zero? | You can exchange the partial derivatives on $\nu = \partial\_{x\_s} \eta$. Taking one partial of $u$ is allowed in the whole $\Omega$ since $u \in H^1\_0(\Omega) \subset W^1\_0(\Omega)$. So $\partial\_{x\_s} u$ makes sense there. Mixed second partials of $u$ are justified by equation 16.10, which you didn't mention; it... | 2 | https://mathoverflow.net/users/4923 | 234027 | 108,538 |
https://mathoverflow.net/questions/234009 | 1 | Let $E→M$ be a plane bundle endowed with an almost complex structure $J.$
$J$ induces a natural positive definite inner product in the associated bundle $End(E)→M$,denoted by $<,>$. More precisely if $A,B \in End(E\_p)$
then chose a frame $e\_1,e\_2$ in $E\_p$ such that
$$ J(e\_1)=−e\_2, J(e\_2)=e\_1.$$
This can alwa... | https://mathoverflow.net/users/70498 | Triviality of a circle fibration induced by an almost complex structure | There is a decomposition $\mathrm{End}(E)\cong P\oplus\underline{\mathbb R^2}$, where the second summand is trivial and spanned by $\mathrm{id}$ and $J$. The bundle $P$ carries a natural orientation: if $0\ne A\in P\_p$, then $J\circ A\in P\_p$, let $A$, $J\circ A$ form an oriented frame. An oriented real vector bundle... | 4 | https://mathoverflow.net/users/70808 | 234035 | 108,540 |
https://mathoverflow.net/questions/234045 | 2 | Let $X$ be a normal, projective variety and $U$ be the regular locus of $X$. Let $\mathcal{F},\mathcal{G}$ be reflexive sheaves on $X$ and $f:\mathcal{F} \to \mathcal{G}$ be a morphism. Suppose that the restriction of $f$ to $U$ is surjective i.e., $f|\_U:\mathcal{F}|\_U \to \mathcal{G}|\_U$ is surjective.
Is it tru... | https://mathoverflow.net/users/58203 | On a morphism between reflexive sheaves | No, that is not true. Let $X$ be a normal, quasi-projective variety over a field $k$. Let $U\subset X$ be the regular locus. Denote by $Z\subset X$ the closed complement of $U$ with its reduced, induced structure. Assume that $Z$ is not empty. Denote by $\mathcal{I}\subset \mathcal{O}\_X$ the ideal sheaf of $Z$. Since ... | 4 | https://mathoverflow.net/users/13265 | 234046 | 108,544 |
https://mathoverflow.net/questions/234044 | 2 | I am looking for a solution for a conjecture as follows.
>
> *In Cartesian plane, no exist an equilateral triangle such that three vertices are integer numbers.*
>
>
>
I hope that you like the question and let me a answer.
| https://mathoverflow.net/users/76698 | Conjectute: no exist an equilateral triangle such that all vertices are integer numbers | Let A,B,C be the vertices.Use the fact that the area is $E=\frac{1}{2}\cdot |det(\vec{AB},\vec{AC})|$
Since $det(\vec{AB},\vec{AC})$ is an integer and the area must be of the form
$AB^2\cdot \frac{\sqrt3}{4}$ which is not an integer you can see the contradiction
| 3 | https://mathoverflow.net/users/38851 | 234050 | 108,547 |
https://mathoverflow.net/questions/233804 | 4 | Whenever a certain type of (Schauder) basis is defined, it is natural to ask where that type lies in the scheme of other types of bases. This involves finding counter-examples of one type of basis which is not another. For example, given that
\begin{equation}\text{symmetric basis }\Rightarrow\text{ subsymmetric basis }... | https://mathoverflow.net/users/73784 | Non-equivalence of admitting different types of bases in Banach spaces | For Q2, consider the original Tsirelson space. It has an unconditional, then quasi greedy, basis. However, it does not contain a democratic basis (see Remark 5.8 in DKK2003).
| 2 | https://mathoverflow.net/users/89195 | 234057 | 108,550 |
https://mathoverflow.net/questions/233158 | 1 | Let $X\_k$ be a symmetric (discrete time) random walk on $\mathbb{Z}$ and let $m,n\in\mathbb{N}$. I want to chose uniformly from the paths of $X\_k$, which
1. start at $0$
2. stay in $[0,n]\cap\mathbb{Z}$ for precisely $m$ steps.
3. (optional: chosing only among those that leave to the left or chosing only among thos... | https://mathoverflow.net/users/88695 | Choose uniformly from fixed-length paths in $[0,n]\cap\mathbb{Z}$ with fixed start and end | You can use a "dynamic programming" solution. For anyone unfamiliar with this terms, the basic idea is that there are an exponential in $n,m$ number of possible paths, so it takes too long to enumerate them all. But we can express everything we need to know about the problem using a small "state" consisting of where th... | 1 | https://mathoverflow.net/users/29697 | 234063 | 108,552 |
https://mathoverflow.net/questions/233477 | 3 | I am interested in realizing commensurability classes of hyperbolic $3$-manifolds whose quaternion algebra (note: not invariant quaternion algebra) is isomorphic to one of the form $\Big(\frac{a,b}{F(\sqrt{-d})}\Big)$,
where $F\subset\mathbb{R}$
and $a,b,d\in F^+$.
(The reason why is rather lengthy but if you really wa... | https://mathoverflow.net/users/14835 | How many non-commensurable non-arithmetic manifolds have a quaternion algebra like this? | The comments from @IanAgol above lead to an affirmative answer in the compact case. This paper gives an affirmative answer in the non-compact case: <http://www.math.umt.edu/chesebro/AIMCLC.pdf>
| 1 | https://mathoverflow.net/users/14835 | 234076 | 108,556 |
https://mathoverflow.net/questions/234066 | 17 | Recall that an finite-dimensional algebra $A$ over a field $k$ is central simple iff there is an iso
$A \otimes\_k A^{op} \cong M\_n(k)$
where $A^{op}$ is the opposite ring and $M\_n(k)$ is the matrix ring.
On the other hand, a finite field extension $K / k$ is Galois iff there is an iso
$K \otimes\_k K \cong K... | https://mathoverflow.net/users/18116 | Non-commutative Galois theory | Let $k$ be a field. Say that a $k$-algebra $A$ is **separable** if any of the following equivalent conditions holds (it is not obvious that they are equivalent):
1. $A$ is projective as an $(A, A)$-bimodule.
2. $A$ is geometrically semisimple in the sense that $A \otimes\_k L$ is semisimple for any field extension $L... | 16 | https://mathoverflow.net/users/290 | 234090 | 108,559 |
https://mathoverflow.net/questions/234029 | 5 | For $n\ge3$ let $a(n)$ be the minimum number of [Hamiltonian paths](https://en.wikipedia.org/wiki/Hamiltonian_path) in a strong (i.e., [strongly connected](https://en.wikipedia.org/wiki/Strongly_connected_component)) [tournament](https://en.wikipedia.org/wiki/Tournament_(graph_theory)) of order $n.$
>
> Where is $a... | https://mathoverflow.net/users/43266 | The minimum number of Hamiltonian paths in a strongly connected tournament of order $n$ | [Arthur H. Busch](http://www.combinatorics.org/ojs/index.php/eljc/article/view/v13i1n3) proved that the answer is about $5^{n/3}$.
| 4 | https://mathoverflow.net/users/4312 | 234093 | 108,560 |
https://mathoverflow.net/questions/234072 | 1 |
>
> **I. Elliptic curves**
>
>
>
Given integers $a,b,m\_k$. Let,
$$x^2+a = m\_1u\_1^2\\x^2+b = m\_1u\_2^2\tag1$$
If there is a rational point $x\_i$, then the pair (after a transformation) is birationally equivalent to an *elliptic curve*, call it $E\_1$, and frequently has infinitely many rational points. A... | https://mathoverflow.net/users/12905 | On elliptic curves, $\sqrt{x^2-101y^2} ,\sqrt{x^2+101y^2}$, and their ilk | The possible $m\_k$, when assumed to be squarefree, must be divisors of the resultant of $x^2 + a$ and $x^2 + b$, which is $(a-b)^2$; so $m\_k \mid a-b$. (Note that bot factors must be in the same square class; if a prime $p$ divides the squarefree representative of this class, then the binary forms $x^2 + a z^2$ and $... | 4 | https://mathoverflow.net/users/21146 | 234107 | 108,563 |
https://mathoverflow.net/questions/233812 | 5 | Has anyone studied a version of [Day convolution](https://ncatlab.org/nlab/show/Day+convolution) for an enriched presheaf category $V^{A^{\mathrm{op}}}$ where the monoidal structure of $V$ is "twisted" on one side by an action of $A$? I'm thinking of modifying the standard formula
$$(F \otimes G)(a) = \int^{b,c} F(b)... | https://mathoverflow.net/users/49 | Twisted Day convolution | The following is a proof that the twisted monoidal structure $\otimes^\rho$ is a convolution on $[A, V]$, where $A$ is a monoidal category, $V$ the cosmos on which $A$ is enriched, and $\rho\colon A\times V\to V$ a *monoidal action* (in the sense that $(v^a)^b \cong v^{a\cdot b}$, $v^i \cong v$ where $i$ is the monoida... | 3 | https://mathoverflow.net/users/7952 | 234109 | 108,564 |
https://mathoverflow.net/questions/234078 | 6 | Theorem 3 of the [nLab article "Full field algebra"](https://ncatlab.org/nlab/show/full+field+algebra) states that
>
> **Theorem 3.** Two vertex operator algebras $V$ may appear as the left and right chiral halfs of a full conformal field theory precisely if their modular tensor categories of representations have t... | https://mathoverflow.net/users/69505 | When two vertex (operator) algebras can be patched-up to a full CFT on a genus 0 surface? | If your VOA $V$ is not rational, then it is quite unlikely that its category of representations is a modular tensor category. That is, you can safely conclude that Theorem 3 contains an unstated assumption that $V$ is rational. At this point, we only know the modular tensor property when $V$ is rational, $C\_2$-cofinit... | 5 | https://mathoverflow.net/users/121 | 234110 | 108,565 |
https://mathoverflow.net/questions/234042 | 3 | I was reading the paper "Planar separators" by Alon, Seymour and Thomas (available on the first author's webpage). They consider a planar triangulation, that is, a maximally planar graph $G$ drawn in the plane so that the boundary of every face is a triangle. They construct a cycle $C$ on vertices $v\_0, \ldots , v\_{2... | https://mathoverflow.net/users/37432 | Menger's Theorem for planar triangulations | The standard version of [Menger's theorem](https://en.wikipedia.org/wiki/Menger's_theorem) says that in any finite graph, the maximum number of vertex-independent paths between two vertices $a$ and $b$ is the same as the size of the smallest vertex cut separating $a$ and $b$. (Two paths are *vertex-independent* if they... | 2 | https://mathoverflow.net/users/23297 | 234112 | 108,566 |
https://mathoverflow.net/questions/234051 | 44 | I am interested in applications of algebraic geometry to machine learning. I have found some papers and books, mainly by Bernd Sturmfels on algebraic statistics and machine learning. However, all this seems to be only applicable to rather low dimensional toy problems. Is this impression correct? Is there something like... | https://mathoverflow.net/users/89192 | Applications of algebraic geometry to machine learning | One useful remark is that dimension reduction is a critical problem in data science for which there are a variety of useful approaches. It is important because a great many good machine learning algorithms have complexity which depends on the number of parameters used to describe the data (sometimes exponentially!), so... | 26 | https://mathoverflow.net/users/4362 | 234114 | 108,567 |
https://mathoverflow.net/questions/234108 | 45 | Prime gaps studies seems to be one of the most fertile topics in analytic number theory, for long and in lots of directions :
* lower bounds (recent works by Maynard, Tao et al. [1])
* upper bounds (recent works by Zhang and the whole Polymath 8 project [2])
* statistics on most frequent gaps ("jumping champions" [3... | https://mathoverflow.net/users/43737 | Why such an interest in studying prime gaps? | Since you ask about zeta zeros, [Riemann hypothesis implies](https://en.wikipedia.org/wiki/Prime_gap) the gap is
$O(\sqrt{p\_n} \log p\_n)$.
Larger gap will give you nontrivial zero off the critical line,
disproving RH.
On the other hand, bounding the gap by $O(polylog(p\_n))$
will solve the open problem for [dete... | 31 | https://mathoverflow.net/users/12481 | 234119 | 108,569 |
https://mathoverflow.net/questions/234121 | 0 | Let $C$ be a curve in a projective homogeneous variety $X$.
Fixed a general point $x$ in $X$, does there exist a curve $V$ in $X$ passing
through $x$ and such that $C$ and $V$ have the same homology class?
For instance when $X$ is a Grassmannian this is true.
Thanks.
| https://mathoverflow.net/users/nan | Curves in homogeneous varieties | I think that this problem might not be appropriate for MathOverflow, since this follows immediately from homogeneity. On the other hand, perhaps the OP is indirectly asking why every projective homogeneous variety is homogeneous under the action of a **connected** group. This is a standard fact in the theory of algebra... | 6 | https://mathoverflow.net/users/13265 | 234124 | 108,571 |
https://mathoverflow.net/questions/232905 | 3 | From Harvey Friedman's [manuscript](https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/) on "Order Invariant Relations and Incompleteness":
>
> DEFINITION 4.4. A $\Pi\_1^0$ sentence is a sentence asserting that some given Turing machine never halts at the empty input tape. A $\Pi\_2^0$ se... | https://mathoverflow.net/users/20781 | Can Turing machines clarify mathematical, philosophical, and physical existence? |
>
> The question whether some postulated structure (like ZFC) has mathematical existence is equivalent to a $\Pi\_1^0$ sentence. The link between consistency and existence is provided by the model existence theorem of first order logic (i.e. the completeness theorem). ... No, the model existence theorem is not equiva... | 3 | https://mathoverflow.net/users/20781 | 234133 | 108,575 |
https://mathoverflow.net/questions/201041 | 4 | I want to prove the following: Let $R$ be a perfectoid ring and $\varpi$ a pseudo uniformizer in $R$ which admits all $p$-th power roots, then a module over $R^\circ$ is flat if and only if it has no $\varpi$ nontrivial torsion.
(I know this is true when $R$ is a perfectoid field.)
| https://mathoverflow.net/users/68892 | Flatness over a perfectoid ring | Even in the case $R$ and $S$ are perfectoid algebras over a perfectoid field, it's not the case that $S$ is $R$-flat in many situations, eg. perf $R$, take a higher rank point in $\text{Spa}(R,R^0)$, look at the completed res field $\kappa$, at the $\kappa$-normalization $\kappa^+$ of $R^0$, and finally at the map $R^0... | 1 | https://mathoverflow.net/users/nan | 234136 | 108,577 |
https://mathoverflow.net/questions/234130 | 6 | Let $V$ be a finite dimensional vector space over a field $K$ of characteristic zero. Assume that we are given a set of (not necessarily homogeneous) elements $f\_1,\ldots f\_n$ in the tensor algebra $T(V)$. How can we decide if the ideal $I$ generated by the elements $f\_i$ is the entire ring $T(V)$ or not? Is there a... | https://mathoverflow.net/users/41644 | Trivial algebras given by generators and relations | It is well known that the analogous question is undecideable for Groups. (By Encoding a Turing machine into the Generators and relations and using undecideablity theorems like the undecideabilty of the halting Problem there.)
You can expect that the question for algebras is undecideable as well, as you can for examp... | 5 | https://mathoverflow.net/users/88855 | 234137 | 108,578 |
https://mathoverflow.net/questions/234104 | 7 | Let $(X,\tau), (Y,\sigma)$ be topological spaces with $|X|$ infinite and suppose $\varphi:X\to Y$ is a bijection such that for all $x\in X$ we have that $(X\setminus\{x\}) \cong (Y\setminus\{\varphi(x)\})$.
Does this imply that $(X,\tau) \cong (Y,\sigma)$?
| https://mathoverflow.net/users/8628 | Reconstructibility of topological spaces | The answer is ``No''. Let $X$ be the Cantor set and let $Y = X \setminus \{p\}$, where $p$ is any element of $X$. Let $\varphi \colon X \to Y$ be any bijection. Since the Cantor set with one point removed is homeomorphic to the Cantor set with two points removed, the given condition is satisfied. However, $X$ is compac... | 11 | https://mathoverflow.net/users/89233 | 234138 | 108,579 |
https://mathoverflow.net/questions/234148 | 10 | Let $M$ be a real symmetric integer valued positive definite matrix with $\det(M) \geq 1$. I would like write code to compute
$$S\_M= \sum\_{x \in \mathbb{Z}^n} e^{-x^TMx}.$$
One option is to simply iterate over the vectors $x$ starting with ones with small coefficients and stop if things seem to be converging. Apa... | https://mathoverflow.net/users/45564 | How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently | You are trying to compute a multi-dimensional theta function, and this question is studied in depth in [this 2003 Math. Comp. article by Deconinck, Heil, Bobenko, van Hoeij,and Schmies](http://www.ams.org/journals/mcom/2004-73-247/S0025-5718-03-01609-0/S0025-5718-03-01609-0.pdf).
| 14 | https://mathoverflow.net/users/11142 | 234150 | 108,582 |
https://mathoverflow.net/questions/234153 | 1 | Let $X$ be a projective non-normal scheme, say over the complex numbers, endowed with an ample line bundle $L$. Let $\nu \colon \hat{X} \to X$ be its normalization. Is it possible that $(X,L)$ and $(\hat{X},\nu^\*L)$ have the same Hilbert polynomial?
(In the case of curves the answer is negative.)
| https://mathoverflow.net/users/48866 | Hilbert polynomial of the normalization | No. One has an exact sequence $0\to\mathcal{O}\_X\to \nu\_\*\mathcal{O}\_{\widehat{X}}\to F\to 0$ and $F\neq 0$ if $X\neq \widehat{X}$. If the two Hilbert polynomials are equal, it follows that $\chi(F\otimes L^n)=0$ for all $n$ and thus for large $n$, we must have $h^0(F\otimes L^n)=0$, which is impossible since $F\ne... | 11 | https://mathoverflow.net/users/9502 | 234154 | 108,584 |
https://mathoverflow.net/questions/234139 | 2 | In proving the the existence of the Reedy model structure on the category of simplicial objects in a model category $\mathcal{C}$, Goerss-Jardine prove there is a pullback square induced by a map of simplicial objects $f:X\to Y$, namely
$$\begin{array}{ccc}
Y\_n\times\_{M\_{n,k+1}}M\_{n,k+1}X& \to & X\_{n-1}\\
\downar... | https://mathoverflow.net/users/67149 | pullback square in Goerss-Jardine | I would understand this proof as describing the limit of a "deleted 3-cube" in two different ways.
We have a square involving maps $X\_n\to X\_{n-1}$, $X\_n\to M\_{n,k}X$, $X\_{n-1}\to M\_{n-1,k}X$, and $M\_{n,k}X\to M\_{n-1,k}X$; this square maps to a similar square with $X$ replaced with $Y$. Altogether it is a cu... | 4 | https://mathoverflow.net/users/437 | 234170 | 108,587 |
https://mathoverflow.net/questions/234161 | 3 | I am looking for a second order proof that ordinals are well-ordered up to $\epsilon\_0$.
So, starting with encoding those ordinals in numbers, defining the < relation on those encoded ordinals and then prove the well-ordering and as consequence, that transfinite induction is allowed.
Preferably the proof is such ... | https://mathoverflow.net/users/5917 | Reference request, proof about well-ordering of ordinals | The usual proof that the ordinal *notations* for ordinals below $\varepsilon\_0$ is well-ordered takes place in set theory and uses the fact that actual ordinals (i.e., not notations) are well-ordered.
The proof relies on Cantor Normal Form: that every ordinal $\alpha$ can be written in a unique way in the form $$\al... | 10 | https://mathoverflow.net/users/2000 | 234173 | 108,589 |
https://mathoverflow.net/questions/234158 | 4 | What I want to ask is about the structure of the Goldbach function that defined by
$$ R(x)=\#\{ p \mid x-p \in \mathbb{P} , \ p\leq x/2\}$$
for $x\in 2\mathbb{N}$, where $\mathbb{P}$ is the set of prime numbers. As I understand, the methods which are developed to prove the Goldbach type problems have difficulties in th... | https://mathoverflow.net/users/49625 | Relation between the binary Goldbach problem and binary version of Mobius sum | Many analytic number theory questions about the primes are really questions about estimating sums involving the [von Mangoldt function](https://en.wikipedia.org/wiki/Von_Mangoldt_function) $\Lambda$. For instance, the Goldbach conjecture (when viewed from the perspective of analytic number theory) is basically asking f... | 7 | https://mathoverflow.net/users/766 | 234177 | 108,591 |
https://mathoverflow.net/questions/234167 | 0 | Let $X$ be a Banach space with basis $(e\_n)\_{n=1}^\infty$, and suppose that $(x\_i)\_{i=1}^\infty$ is a normalized block basic sequence of $(e\_n)\_{n=1}^\infty$. In addition assume that $(x\_i)\_{i=1}^\infty$ is subsymmetric, weakly null, and such that $\sup\_i \left\vert \text{supp}(x\_i) \right\vert = \infty$.
M... | https://mathoverflow.net/users/36832 | Extracting a subsequence for which $\sup_j \left\vert \text{supp}(x_{n_j}) \right\vert < \infty$ | No, take, for instance, a reflexive Orlicz sequence space $\ell\_M$ not isomorphic to $\ell\_p$ and a block sequence $(x\_i)$ equivalent to $\ell\_p$ basis that space contains. Every block sequence in the space with uniform bound on the support will be equivalent to the unit vector basis of $\ell\_M$.
| 3 | https://mathoverflow.net/users/3675 | 234181 | 108,592 |
https://mathoverflow.net/questions/234183 | 13 | **In short:** What can we say about the collection of all solutions of an ODE when we don't have uniqueness?
When we teach a first course in ODE's, we look at the equation
$f:D\to \mathbb{R}, \quad D\subseteq \mathbb{R}^2,$
$y'(x) = f(x,y),\quad y(x\_0 ) = y\_0, \quad (x\_0,y\_0 )\in D $
and prove two theorems
... | https://mathoverflow.net/users/42864 | Solutions-set first order ODE's without uniqueness | 1. In most common applications, uniqueness and the Lipschitz condition are violated only at certain points which form a curve which is a "singular solution". The keyword is [Clairault Equation](https://en.wikipedia.org/wiki/Clairaut%27s_equation) The simplest example is $y'=y^{1/3}$, the singular solution
is $y=0$. Suc... | 29 | https://mathoverflow.net/users/25510 | 234198 | 108,595 |
https://mathoverflow.net/questions/234128 | 17 | When building $kU/2$ via its Postnikov tower, there are some interesting Massey products that show up in the Steenrod algebra, and I'd like to understand them. I bet these appear somewhere in the litterature, but I have not been able to find a reference. I will call $X = kU/2$, $H = H\mathbb{F}\_2$ and I'll use $X\_{\l... | https://mathoverflow.net/users/7183 | Massey products in the Steenrod algebra | As near as I've been able the find, the primary reference for a proof is probably Kristensen and Madsen's "[On the structure of the operation algebra for certain cohomology theories](http://www.maths.ed.ac.uk/~aar/surgery/uicc/krismads.pdf)." This result (in fact, its generalization to all the Milnor primitives) occurs... | 8 | https://mathoverflow.net/users/360 | 234208 | 108,599 |
https://mathoverflow.net/questions/234199 | 2 | Let $k$ be a field of characteristic $\neq 2$ and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. Let $X$ be the set of all places of $F$. Let $S = \{\infty\} \subset X$ where $\infty$ is the unique place of $F$ such that $\text{ord}\_\infty(T) < 0$. Note that $O\_S = k[T, \sqrt{T^3 + 1}]$... | https://mathoverflow.net/users/nan | What is the cokernel of $O_S \to F_\infty/O_\infty$? | I realize now that this is the type of question that should be solved by the OP. However, since my guess above is incorrect (which I should have realized before I clicked submit), I feel obliged to submit a correct answer.
The cokernel is a one-dimensional $k$-vector space, generated by the class of $u^{-1}$. By Hens... | 2 | https://mathoverflow.net/users/13265 | 234210 | 108,600 |
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