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https://mathoverflow.net/questions/294511 | 6 | A topological space $(X,\tau)$ is said to be *homogeneous* if for all $x,y$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$.
Is there an infinite homogeneous Hausdorff space $(X,\tau)$ such that every continous map $f: X\to X$ is either a homeomorphism, or constant?
**EDIT.** I forgot to add "i... | https://mathoverflow.net/users/8628 | Infinite "almost rigid" homogeneous $T_2$-space | Topological groups are homogeneous. In
>
> J. van Mill, "A topological group having no homeomorphisms other than translations," *Transactions of the AMS* **280** (1983), pp. 491-498 ([link](http://www.ams.org/journals/tran/1983-280-02/S0002-9947-1983-0716833-2/S0002-9947-1983-0716833-2.pdf)),
>
>
>
Jan van Mil... | 10 | https://mathoverflow.net/users/70618 | 294529 | 129,525 |
https://mathoverflow.net/questions/294512 | 1 | When thinking up a new TSP heuristic, I encountered the following rational combinatorial optimization problem:
$$\min\_{\alpha \in \lbrace0,1\rbrace^n}\frac{\alpha^T w}{\alpha^T m},\quad w \in \mathbb{R}^n, \ m\in\mathbb{N}^n,\ \|\alpha\|\ne 0 $$
and must admit, that I don't have any good idea of how to go about solvin... | https://mathoverflow.net/users/31310 | Unconstrained Rational Combinatorial Optimization | The 1979 publication [Combinatorial Optimization with Rational Objective Functions](http://theory.stanford.edu/~megiddo/pdf/rational.pdf) of Nimrod Megiddo answers my question sufficiently well.
| 1 | https://mathoverflow.net/users/31310 | 294533 | 129,526 |
https://mathoverflow.net/questions/294532 | 12 | Similar questions have been asked before regarding journals that publish:
* [expository work](https://mathoverflow.net/questions/15366/which-journals-publish-expository-work/269938),
* [recreational mathematics](https://mathoverflow.net/questions/152580/recreational-mathematics-where-to-search),
* [computational resu... | https://mathoverflow.net/users/103722 | Where to publish new mathematical identities? | As Peter Heinig commented, if the mathematics behind the identity is novel and important enough, then you should select a journal like you would select a journal for any other paper—if it's a combinatorial identity, look for a combinatorics journal; if it's a number-theoretic identity, look for a number theory journal,... | 10 | https://mathoverflow.net/users/3106 | 294538 | 129,529 |
https://mathoverflow.net/questions/294542 | -3 | Assuming there are infinitely many twin primes, one can consider a Dirichlet series $ \sum\_{n>0}a\_{n}{n^{-s}} $ and replace the sequence of positive integers with the sequence of twin primes. That way such a "twin prime transform" of the Riemann zeta function would give for $ s=1 $ the so called Brun constant.
Wha... | https://mathoverflow.net/users/13625 | Twin prime based Dirichlet series | It is conjectured that there are $\gg x/\log^2 x$ twin primes up to $x$. If this is the case, then the abscissa of convergence for $\sum\_p p^{-s}$, the sum taken over twin primes, is equal to $1$. If there are much fewer twin primes up to $x$ (but still infinite in total), the abscissa of convergence could be anywhere... | 4 | https://mathoverflow.net/users/11919 | 294544 | 129,530 |
https://mathoverflow.net/questions/294434 | 9 | Fix a field $k$. Given a discrete group $G$, the pro-algebraic completion (or Hochschild-Mostow completion) is the pro-algebraic group $A\_{k}(G)$ with is universal with respect to finite dimensional representations of $G$. We usually omit the field $k$ from the notation when it is clear.
I have encountered pro-algeb... | https://mathoverflow.net/users/121425 | Interesting examples of pro-algebraic completions of groups | You ask specifically about the symmetric groups and the free groups.
This is a bit funny, as a finite group forms its own proalgebraic completion (a slight generalization of this fact is Proposition 1 in Bass-Lubotzky-Magid-Mozes),
while the proalgebraic completion of a non-abelian free group is clearly not a manageabl... | 6 | https://mathoverflow.net/users/89334 | 294548 | 129,531 |
https://mathoverflow.net/questions/294555 | 3 | At page 3 of this article <https://arxiv.org/pdf/1706.03755.pdf> about the new proof of Halasz's Theorem, the authors claim that $$\sum\limits\_{(\log x)^4< p\leq x/2}\frac{\log p}{p\log(x/p)}\ll\log\log x.$$ It should be equivalent to $$\sum\limits\_{(\log x)^4< p\leq x/2}\frac1{p\log(x/p)}\ll\frac{\log\log x}{\log x}... | https://mathoverflow.net/users/110531 | Estimating a sum over prime numbers | Using the fact that
$$\sum\_{y\leq p\leq 2y} \frac{\log p}{p}\ll 1,$$
we get for any $A\geq 2$
$$\sum\_{x/2A\leq p\leq x/A} \frac{\log p}{p\log(x/p)}\ll \frac{1}{\log A}$$
Hence for any positive integer $l$ we have
$$\sum\_{x/2^{l+1}\leq p\leq x/2^l} \frac{\log p}{p\log(x/p)}\ll \frac{1}{l}.$$
Summing this ... | 6 | https://mathoverflow.net/users/101078 | 294562 | 129,535 |
https://mathoverflow.net/questions/294526 | 7 | Let $\mathsf{ZFC}^-$ be the Zermelo-Fraenkel set theory without power set axiom.
For a transitive model $M$ of $\mathsf{ZFC}^-$ and an cardinal $\kappa\in M$ in the sense of $M$, an unary predicate $U$ over $M$ is an $\kappa$-complete ultrafilter if $M$ thinks it is $\kappa$-complete and $U$ is weakly amenable, that is... | https://mathoverflow.net/users/48041 | Consistency strength of the existence of a transitive model of $\mathsf{ZFC}^-$ with a $\kappa$-complete ultrafilter | Let's make some additional assumptions on the ultrafilter $U$. Suppose $M\models{\rm ZFC}^-$ and $\kappa$ is a cardinal in $M$. We say that $U$ is an $M$-*ultrafilter* if $\langle M,\in,U\rangle$ satisfies that $U$ is a $\kappa$-complete normal ultrafilter on $\kappa$. Because $U$ is only $\kappa$-complete for sequence... | 11 | https://mathoverflow.net/users/5984 | 294565 | 129,536 |
https://mathoverflow.net/questions/294167 | 21 | Let $L$ be a first-order language and $M$ be an $L$-structure. Let $D \subseteq M^n$ . Let us say $D$ is definable in $M$ if for some finite set (possibly empty) $A=\{a\_1,...,a\_m\} \subseteq M$ and some formula $\psi[x\_1,...,x\_n,y\_1,...,y\_m]$ , $D=\{(b\_1,...,b\_n)\in M^n : M\vDash \psi[b\_1,...,b\_n,a\_1,...,a\_... | https://mathoverflow.net/users/nan | Are $\mathbb C$ , $\mathbb C[X]$ definable in $\mathbb C[[X]]$? | Emil's idea about quantifier elimination is a good one.
The paper "Model Theory of valued fields" by Raf Cluckers cites the thesis "Quelques propriétés des corps valués" by F. Delon, which I wasn't able to find online, for the claim that the field $\mathbb C((t))$ admits quantifier elimination for the language of Mac... | 11 | https://mathoverflow.net/users/18060 | 294566 | 129,537 |
https://mathoverflow.net/questions/294422 | 9 | MacLane's coherence theorem for a monoidal category states that once the associators for 4-fold products are compatible (i.e., the pentagon axiom holds), it holds for n-fold products, so I can bracket n-fold products in any way I like.
What happens if I forget the unit, i.e. I consider "semigroup categories" as they ... | https://mathoverflow.net/users/121493 | MacLane coherence theorem for "monoidal" category without 1 | Sorry, I noticed this an hour after posting: Yes, it's true, see Theorem 3.1 in MacLane's excellent paper "Natural associativity and commutativity".
It's a mystery to my why this is so scarcely mentioned.
| 8 | https://mathoverflow.net/users/121493 | 294569 | 129,539 |
https://mathoverflow.net/questions/293998 | 2 | Let
$$D\_{tv}(P,Q) = \frac{1}{2}\sum\_{a \in A}|P(a)-Q(a)|$$
and
$$D\_{\infty}(P,Q) = \sup\_{a \in A} \log \max\{\frac{P(a)}{Q(a)}, \frac{Q(a)}{P(a)}\},$$
where $P$ and $Q$ denote probability measures on a finite set $A$.
Can you show that
$$D\_{tv}(P,Q) \leq 1- \exp(-D\_{\infty}(P,Q)) \leq D\_{\infty}(P,Q)?$$
| https://mathoverflow.net/users/83321 | Total variation and relative $\ell_\infty$ metric | The inequality $1- \exp(-D\_{\infty}(P,Q)) \leq D\_{\infty}(P,Q)$ is just an instance of the elementary inequality $e^u\ge1+u$ for all real $u$.
Let us show an improved version of the inequality $D\_{tv}(P,Q) \leq 1- \exp(-D\_{\infty}(P,Q))$, namely,
\begin{equation\*}
D\_{tv}(P,Q) \leq 1- \exp(-D\_{1,\infty}(P,Q))... | 2 | https://mathoverflow.net/users/36721 | 294572 | 129,541 |
https://mathoverflow.net/questions/294541 | 7 | Let $C$ be smooth projective curve defined over a finite field $\mathbb{F}\_q$. Let $$Z(C,u)=\exp(\sum\_{n \ge 1} N\_r(C) u^r/r) \in \mathbb{Z}[[u]]$$ be its zeta function, where $N\_r$ is the number of $\mathbb{F}\_{q^r}$-points on $C$. It is well known (Weil) that $$Z(C,u)=\frac{P\_C(u)}{(1-qu)(1-u)},$$ where $P\_C$ ... | https://mathoverflow.net/users/31469 | Supersingularity and Roots of Unity of Zeta Functions | The names supersingular and superspecial appear in the literature in not a very consistent way.
Passing from $\mathbb{F}\_q$ to $\mathbb{F}\_{q^n}$ changes the roots $\alpha$ to $\alpha^n$. If the roots are roots of unity times $q^{-1/2}$, then passing to a suitable extension, we can assume that they are equal to $q^... | 8 | https://mathoverflow.net/users/2290 | 294577 | 129,542 |
https://mathoverflow.net/questions/294580 | 19 | If A and B are C^\*-algebras that are algebraically isomorphic to each other, does
this imply that they are \*-isomorphic to each other?
| https://mathoverflow.net/users/121439 | Are algebraically isomorphic $C^*$-algebras $*$-isomorphic? | Answering the question in the body of the original post, which seems to be more restricted than the implicit question in the title of the post....
The answer is YES. See
>
> L. Terrell Gardner, *On isomorphisms of $C^\ast$-algebras.*
> Amer. J. Math. **87** (1965) 384–396.
> [MathReview](https://mathscinet.ams.... | 20 | https://mathoverflow.net/users/763 | 294586 | 129,546 |
https://mathoverflow.net/questions/294585 | -1 | I'm studying some papers related to graph partitioning (GP). It is well-known that the GP problem is NP-Complete. Based on my understanding, it means that there is no polynomial time solution to solve this problem, or there is no optimal solution for that.
The following paper mentioned this fact in its introduction:
"... | https://mathoverflow.net/users/106889 | What is the difference between exactness and optimality of an algorithm? | I presume the paper you're referring to is [this one](https://arxiv.org/pdf/0912.1664.pdf) by Hager, Phan and Zhang.
There are no known polynomial-time algorithms for NP-complete problems. This is no exception. I note that at the end of Section 4 it states:
>
> In the worst case, the branch and bound algorithm wi... | 2 | https://mathoverflow.net/users/13650 | 294587 | 129,547 |
https://mathoverflow.net/questions/294581 | 1 | I have some difficulties in understanding the [uni]verse platonic view. How are we to understand the existence of a model of a false theory? what is the relationship of this model to the platonic world of sets?
My personal try is the following, let's assume the existence of a platonic universe $P^{sets}$ of sets, and... | https://mathoverflow.net/users/95347 | Where do models of false theories exist? | I'll answer this from a Platonic viewpoint. Consider a theory $T$ that is false when its primitive concepts are interpreted in the standard way. For example, $T$ could be the theory ZF$\neg$C that you mentioned in this connection. If we interpret the primitive concepts "set" and "element" in the standard way, so that "... | 10 | https://mathoverflow.net/users/6794 | 294589 | 129,549 |
https://mathoverflow.net/questions/294571 | 12 | Where in the literature can I find a naturality statement for Moore-Postnikov towers of maps? Something like the following:
Let $f:X\to A$ and $g:Y\to B$ be maps of connected CW-complexes which both admit a Moore-Postnikov tower of principal fibrations. Then a commuting diagram
$\require{AMScd}$
\begin{CD}
X @>f>> A... | https://mathoverflow.net/users/8103 | Naturality of Moore-Postnikov systems | Working simplicially (in those days called "semi-simplicially") this is surely due to Moore, with details in unpublished 1956 lecture notes and in John C. Moore, Semi-simplicial complexes and Postnikov systems. 1958 Symposium internacional de topología algebraica International symposium on algebraic topology pp. 232–24... | 15 | https://mathoverflow.net/users/14447 | 294592 | 129,550 |
https://mathoverflow.net/questions/294557 | 5 | Given a finite permutation group, i.e. a subgroup of the symmetric group on $n$ symbols in terms of generators, what is the complexity of the word problem? That is, computing if two words in the generators represent the same group element?
| https://mathoverflow.net/users/37580 | Time Complexity of the Word Problem for Finite Permutation Groups | As has been pointed out in comments, you cannot hope to do better in general than $O(n(l\_1+l\_2))$, where $n$ is the degree of the permutation groups and $l\_1$, $l\_2$ are the lengths of the words.
But from a practical point of view it is important to observe that once a base for $G$ has been computed (see my answe... | 10 | https://mathoverflow.net/users/35840 | 294601 | 129,552 |
https://mathoverflow.net/questions/294608 | 8 | What is the maximum total number of zeroes a univariate polynomial $f\in\mathbb{C}[z]$ of degree $d$, together with all of its derivatives, can have at $k$ given points of $\mathbb{C}$?
I am interested in this question for constant $k$, say $k=3$.
One trivial bound is $d \choose 2$, an other is $kd$.
Does there... | https://mathoverflow.net/users/4246 | Number of zeros of the derivatives of a polynomial | The answer is yes. Here is an example for $k=3$. For $m>0$, take $f(z)=z^m(z^2-1)^m$. Then $d=3m$. Of course, $-1$, $0$ and $1$ are roots of $f^{(i)}$ for $0\le i\le m-1$, so we have $3m=d$ roots. However, $f^{(i)}$ is an odd function for $i=m+1,m+3,\ldots, 3m-1$, producing $m$ more roots in $0$. So in this example, th... | 7 | https://mathoverflow.net/users/18739 | 294616 | 129,554 |
https://mathoverflow.net/questions/294619 | 1 | The automorphism group of the Poincare' disk has elements called elliptic, which have a single fixed point in the interior of the disk, and can be represented as a rotation around this fixed point.
It also has parabolic elements, which have a single fixed point in the boundary, and in a sense are a rotation around th... | https://mathoverflow.net/users/48526 | Parabolic elements of the Poincare' disk automorphism group as limit of elliptic ones | Yes. The group of automorphisms of the poincare disc is the Lie group $PSL\_2(\mathbb{R})$, so it has natural topology. To see that the phenomena you consider actually happen, take for example the family of rotations
$\begin{pmatrix}
cos(t^2) & \sin(t^2) \\
-\sin(t^2) & \cos(t^2)
\end{pmatrix}$. This is a family of r... | 5 | https://mathoverflow.net/users/115052 | 294620 | 129,556 |
https://mathoverflow.net/questions/294621 | 0 | Let $H$ be a complex Hilbert space (not necessary separable).
**Spectral Theorem:** Let $A\_1$ and $A\_2$ be two commuting normal operators, then there exists a measure space $(X,\mathcal{E},\mu)$,
two functions $\varphi\_1,\varphi\_2\in L^\infty(\mu)$ and a unitary operator $U:H\longrightarrow L^2(\mu)$, such that e... | https://mathoverflow.net/users/113054 | Semifinite measure and spectral theorem | The answer is yes. Let me explain why.
Let $v \in E$ be any nonzero vector. Then define $E\_0$ to be the closure of the set of vectors of the form $p(A\_1,A\_1^\*, A\_2, A\_2^\*)v$ where $p$ is a complex polynomial. By restricting $p$ to have rational coefficients you can see that $E\_0$ is separable. Now the point i... | 2 | https://mathoverflow.net/users/23141 | 294622 | 129,557 |
https://mathoverflow.net/questions/294300 | 5 | Let $F$ be a compact oriented surface with a foliation $\cal F$ with $k$-prong singularities only (or, if it helps, assume that $\cal F$ admits an invariant measure). Is it true then there exists a pseudo-Anosov $\phi$ such that for every loop or an arc (with endpoints in $\partial F$) $\gamma$, $\phi^n(\gamma)$ is tra... | https://mathoverflow.net/users/23935 | Can every curve be made transversal to a foliation by applying a pseudo-Anosov? | Given a full support measured foliation $\mathcal F$ and given a pseudo-Anosov $\phi$, what you want will work as long as $\mathcal F^u\_\phi$ can be isotoped to be transverse to $\mathcal F$.
The trouble is, this is not always possible. For general choice of $\mathcal F$ and $\phi$, there are a few things one can sa... | 6 | https://mathoverflow.net/users/20787 | 294625 | 129,560 |
https://mathoverflow.net/questions/294446 | 2 | Let $t$ and $x$ be indeterminates, and let $P\in\mathbb Q(t)[x]$ be irreducible. Adjoining a root of $P$ to $\mathbb Q(t)$ we obtain a function field $F/\mathbb Q$. Now suppose that $K$ is an algebraic extension of $\mathbb Q$. Regarding $P$ as an element of $K(t)[x]$, we can adjoin a root of $P$ to $K(t)$ to obtain a ... | https://mathoverflow.net/users/46987 | Genus of a function field after an extension | Let $X$ be the smooth projective curve with function field $F$. Then the base change $X\_K$ is the curve with function field $E$. The genera of $X$ and $X\_K$ are equal because $X\_K$ is regular since $\mathbb Q$ is a perfect field. Hence the genera of $E$ and $F$ are equal.
I would be interested to see a proof rely... | 0 | https://mathoverflow.net/users/46987 | 294631 | 129,561 |
https://mathoverflow.net/questions/294482 | 0 | The sequence [OEIS A056296](https://oeis.org/A056296) can be obtained using
$
a(n)={1\over n}\sum\_{d\backslash n}\varphi(d)\begin{cases}
{n/d+2\brace3}-{n/d+1\brace3}, & \text{$6\backslash d$;} \\
{n/d+2\brace3}-3{n/d+1\brace3}+3{n/d\brace3}, & \text{$3\backslash d$;} \\
2{n/d+1\brace3}-2{n/d\brace3}, & \text{$2... | https://mathoverflow.net/users/14207 | Ordinary Generating Function for OEIS A056296? | It appears that
$ \sum\_{d>0}{\varphi(d)\over6d}\begin{cases}
-6\log(1-3x^d)+6\log(1-2x^d), & \text{$6\backslash d$;} \\
-3\log(1-3x^d)+3\log(1-2x^d)-3\log(1-x^d), & \text{$3\backslash d$;} \\
-4\log(1-3x^d)+6\log(1-2x^d), & \text{$2\backslash d$;} \\
-\log(1-3x^d)+3\log(1-2x^d)-3\log(1-x^d), & \text{else.} \\
\... | 2 | https://mathoverflow.net/users/14207 | 294633 | 129,562 |
https://mathoverflow.net/questions/242674 | 5 | Let $E$ be the total space of a fiber bundle with base $B$ and fiber $F$, where $B$ and $F$ are smooth manifolds.
Under what condition is $E$ unoriented cobordant to $B\times F$?
And what happens if we take $B$, $F$, and $E$ oriented. Is $E$ oriented cobordant to $B\times F$?
Edit : As mentioned in the comment,... | https://mathoverflow.net/users/94123 | Under what condition is a fiber bundle cobordant to the trivial bundle? | Here are three special cases where it is always true:
1. **Sphere bundles of vector bundles.** If $E \to B$ is a vector bundle and $S(E) \to B$ is the associated sphere bundle, then $S(E)$ bounds the disc bundle $D(E)$, so it is nullcobordant. On the other hand, $S^{n-1}\times B$ is nullcobordant as $S^{n-1}$ is, so ... | 4 | https://mathoverflow.net/users/21564 | 294639 | 129,565 |
https://mathoverflow.net/questions/294640 | 2 | Let $f:X\to Y$ be a finite etale Galois morphism of varieties over $\mathbb{C}$. Let $C$ be a smooth quasi-projective connected curve in $X$.
>
>
> >
> > Is $f(C)$ a smooth curve?
> >
> >
> >
>
>
>
| https://mathoverflow.net/users/121594 | Image of curve along a finite etale Galois map | No. Just take two points in $X$ that map to the same point $p$ in $Y$; then a general curve in $X$ containing these two points will have image with a node at $p$.
| 11 | https://mathoverflow.net/users/121595 | 294641 | 129,566 |
https://mathoverflow.net/questions/292960 | 1 | The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW\_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$
We say that a discrete set $\Lambda\subset\mathbb{R}^d$ is sampling for $PW\_\Omega$ if there exists a constant $C>0$ such that for all $f\in PW\_\Omega$ $$\|f\|\_{... | https://mathoverflow.net/users/39180 | Sampling set: relatively dense and uniformly discrete | When asking this question, in principle one should make a distinction between two cases:
1. When $n=1$, the space $PW\_{S}$ is a space of entire functions of exponential type, with all of their unique properties.
2. The case $n\geq2$ is much more involved and as far as I know is not fully understood, some results wer... | 1 | https://mathoverflow.net/users/16040 | 294649 | 129,568 |
https://mathoverflow.net/questions/294659 | 3 | As an exercise for myself I wanted to check GRR in the following situation. Consider $P:X \rightarrow B$ to be an Weierstrass elliptic fibration with a section, and $X\times\_B X$ be the fiber product with $\pi\_{1,2}$ be the corresponding projections to the first and second factor. Then from GRR we have,
$ch(R\pi\_... | https://mathoverflow.net/users/121526 | A question on Grothendieck Riemann Roch | Thanks to Jim Bryan for mentioning my error in writing GRR, and also his hints about the pushforward of cycles, the problem is solved.
$ch(R\pi\_{2\*}\mathcal{O}\_{X\times\_BX})=ch(\pi\_{2\*}\mathcal{O}\_{X\times\_BX})-ch(R^1\pi\_{2\*}\mathcal{O}\_{X\times\_BX})=1-ch(R^1\pi\_{2\*}\mathcal{O}\_{X\times\_BX})$.
Now n... | 8 | https://mathoverflow.net/users/121526 | 294667 | 129,571 |
https://mathoverflow.net/questions/294685 | 6 | In the book Lyndon, Schupp, Combinatorial Group Theory, P.30 in the edition from 2000 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two subgroups are automorphic given their free generators. I searched for papers written by Waldhausen But I didn't find it. Has Wal... | https://mathoverflow.net/users/74457 | An algorithm determining whether two subgroups of a finitely generated free group are automorphic | According to a comment on the math.SE link, the OP wants to understand the question of deciding when two (finitely generated) subgroups of a given free group $F$ are equivalent by an automorphism of $F$.
I believe that this was first solved by Gersten:
>
> S. Gersten, On Whitehead’s algorithm, Bull. Am. Math. Soc... | 7 | https://mathoverflow.net/users/35478 | 294688 | 129,576 |
https://mathoverflow.net/questions/294517 | 3 | Is there a $2$- dimensional foliation of $\mathbb{R}^4\setminus \{0\}$ whose tangent space is contained in $\ker \alpha$ where $\alpha$ is the following non integrable $1$-form?
$$\alpha=(x^2+y^2)dx+(xz+yw)dy+(xz+yw)dz+(z^2+w^2)dw$$
In the matrix language, the $1$-form $\alpha$ can be counted as $\alpha=trace(AA^{t... | https://mathoverflow.net/users/36688 | Is there a $2$ dimensional foliation tangent to this particular $3$ dimensional distribution? | There is no co-dimension 2 foliation of the desired kind in a neighborhood of any point on the line $L\subset\mathbb{R}^4$ defined by $x-w=y=z=0$. Thus, if one wants to find such a foliation, it will be necessary to remove this entire line from the domain.
Here is the reason: Consider the $3$-form $\Upsilon = \alpha ... | 7 | https://mathoverflow.net/users/13972 | 294693 | 129,577 |
https://mathoverflow.net/questions/294700 | 12 | Let $H$ be a Heyting algebra. It is a well-known result that there is a partially ordered set (Kripke frame) X such that there is an embedding of Heyting algebras $f: H \to \mathsf{Up}(X)$, where $\mathsf{Up}(X)$ consists of the up-sets of $X$. Specifically, take $X$ to consist of the prime filters of $H$, ordered by i... | https://mathoverflow.net/users/76274 | Is it possible to completely embed complete Heyting Algebras into upsets of a poset? | No, not in general: for instance, the real interval $([0,1],{\le})$, or any non-atomic complete Boolean algebra, do not have such an embedding. This follows from the following characterization:
>
> **Proposition:** Let $H$ be a complete Heyting algebra. The following are equivalent:
>
>
> 1. $H$ has a complete em... | 14 | https://mathoverflow.net/users/12705 | 294702 | 129,580 |
https://mathoverflow.net/questions/294701 | 1 | I am given the following expression where $f \in L^2(\mathbb{R}^2, \mathbb{R}^{2 \times 2})$
$$\int\_{\mathbb{R}} \int\_{\mathbb{R}} \langle g(x), f(x,y) h(y)\rangle dx dy.$$
The functions $g$ and $h$ are in $L^2(\mathbb{R},\mathbb{R}^2).$
**Question:** Is it then true that
$$\sup\_{\left\lVert g \right\rVert=... | https://mathoverflow.net/users/nan | Recover norm from integral | The inequality $\leq$ is obvious.
The reverse doesn't have to be true.
Let $f(x,y) = \mathrm{Id} \cdot \chi(x,y)$ where $\chi(x,y)$ is the scalar function that equals $1$ on the squares
$$ (x,y) \in [0,1] \times [0,1] \cup [-1,0]\times [-1,0] $$
and zero otherwise.
So
$$ \iint\_{\mathbb{R}\times\mathbb{R}} \la... | 2 | https://mathoverflow.net/users/3948 | 294707 | 129,582 |
https://mathoverflow.net/questions/294645 | 2 | Fix a positive integer $N$ and a real $\epsilon>0$. I'll write $[N]$ for the set $\{1,\dots,N-1,N\}$, $\mathcal{P}X$ to denote the power set of a set $X$, and $\#X$ to denote the number of elements of a finite set $X$.
Suppose $\mathcal{S}\subset \mathcal{P}[N]$ is a set-system which satisfies the following 'sparsene... | https://mathoverflow.net/users/3513 | If we take a small subset of a sparse/uniform set system, how uniform can we arrange for the subset to be? | It does turn out to be possible to take $R$ as roughly $\log N/(-\log k\epsilon)$. More precisely, if you choose $R$ such that $N\epsilon^R {k \choose R}<1$, then you can always find $\mathcal{A}\subset \mathcal{S}$ of size $R$ satisfying the requirements in the question. The conclusion follows quickly from the followi... | 1 | https://mathoverflow.net/users/3513 | 294708 | 129,583 |
https://mathoverflow.net/questions/294651 | 33 | In some of my works I need to prove some results within the internal logic of categories with not much structures (like pretoposes or even just categories with finite limits). The kind of things I want to prove typically involve manipulating inductive type (like a natural number object, a type of binary tree, some W-ty... | https://mathoverflow.net/users/22131 | Proof assistant for working in weaker foundations? | One possibility that's worth thinking about is to use a "meta-proof-assistant" like [Twelf](http://twelf.org/wiki/Main_Page), which implements a meta-theoretic [logical framework](https://ncatlab.org/nlab/show/logical+framework) inside of which you can specify any "object language" you like. It's better-adapted to prov... | 15 | https://mathoverflow.net/users/49 | 294711 | 129,584 |
https://mathoverflow.net/questions/294684 | 7 | Note: this question was updated (2) after GNiklasch's answer was posted, and taking Gro-Tsen's comment into account. The initial question (1) dealt with $\mathbb{Q}\_3$.
---
**Original post (1).** Let's try to solve the equation $x^2 - 2 = 0$ with $x = \frac{a}{b} \in \mathbb{Q}$. We can't have $x^2 \neq 2$, so ... | https://mathoverflow.net/users/1358 | Rational approximations of $\sqrt{2}$ in $\mathbb{R} \times \mathbb{Q}_7$ | Here is a full solution for the modified problem, inspired by Gro-Tsen's valuable comment.
**1.** There are infinitely many rational numbers $a/b\in\mathbb{Q}$ in lowest terms such that
$$ \left|\frac{a^2}{b^2}-2\right|\_\infty\ll\frac{1}{b}\qquad\text{and}\qquad
\left|\frac{a^2}{b^2}-2\right|\_7\ll\frac{1}{b}.$$
To ... | 12 | https://mathoverflow.net/users/11919 | 294719 | 129,588 |
https://mathoverflow.net/questions/294655 | 8 | I am thinking of 3-manifolds as arising from Heegaard splittings which I am thinking about in terms of Heegaard diagrams. I know that 3-manifold groups are rather special in the class of all finitely presentable groups.
I know how to go from a Heegaard diagram ($g$ red curves and $g$ blue curves each forming a cut s... | https://mathoverflow.net/users/99414 | Obstructions to realizing a balanced presentation as a 3-manifold group | $\newcommand{\ZZ}{\mathbb{Z}}$For your first question: the [Baumslag-Solitar](https://en.wikipedia.org/wiki/Baumslag%E2%80%93Solitar_group) group $BS = BS(2,3)$ is not residually finite, so cannot embed in a [three-manifold group](https://mathoverflow.net/questions/61661/residual-finiteness-of-fundamental-group-of-comp... | 3 | https://mathoverflow.net/users/1650 | 294732 | 129,592 |
https://mathoverflow.net/questions/294738 | 7 | Let $R$ be a commutative ring and $M$ be some $R$-module such that $M \otimes -$ is continuous (i.e. preserves all limits). Then one can show that $M$ is f.g. projective.
One way to prove this is to use adjoint functor theorems (anyone you like): We obtain a left adjoint, which has the form $M^\* \otimes -$ by Eilenb... | https://mathoverflow.net/users/98306 | If $M \otimes -$ is continuous, why is $M$ f.g. projective? Alternative proof | I am posting the comment above as *one* answer. Kernels are examples of limits. So if the functor is continuous, then $M$ is flat. Next, consider the natural transformation $M\otimes\_R \text{Hom}\_R(-,R)\Rightarrow \text{Hom}\_R(-,M)$. Both functors convert colimits to limits, and the natural transformation is an isom... | 9 | https://mathoverflow.net/users/13265 | 294740 | 129,593 |
https://mathoverflow.net/questions/294728 | 3 | Given a map from $\big([0,1], \mathcal{B}[0,1], m\big)$ to a Banach space $(X, \|\cdot \|)$. There are **strong measurable** functions (they are the point wise a.e. limit of simple functions) and **weak measurable** functions (for each $u^\* \in X^\*$, we have $t\mapsto \langle u^\*, f(t)\rangle$ is a measurable as a f... | https://mathoverflow.net/users/49551 | $f: [0,1]\rightarrow L^1(\Omega)$ as a (measurable?) function from $[0,1]\times \Omega\rightarrow \mathbb{R}$ | (1) If the Banach space $X$ is separable; and if you use the Lebesgue-measurable sets on $[0,1]$ not the Borel sets; then all three definitions are equivalent.
But of course the main thing of interest is not "measurable function" but "integrable function". When $X$ is not separable, you probably want the Bochner inte... | 4 | https://mathoverflow.net/users/454 | 294741 | 129,594 |
https://mathoverflow.net/questions/294748 | 5 | Given a subset $S$ of the positive integers $\mathbf{N}$, let $\mathrm{d}^\star(S)$ be its upper asymptotic density, that is,
$$
\mathrm{d}^\star(S)=\limsup\_{n\to \infty}\frac{|S \cap [1,n]|}{n}.
$$
Also, for each integer $k \ge 0$, set $S+k:=\{s+k:s \in S\}$.
>
> **Question.** Do there exist $X,Y \subseteq \mathb... | https://mathoverflow.net/users/32898 | Density of intersection with shifted sets | The answer is "yes".
Let $X = \bigcup [4^n, 2 \cdot 4^n)$ and $Y = \bigcup [2 \cdot 4^n, 3 \cdot 4^n)$. Then $d^\star(X), d^\star(Y) \geqslant \tfrac{1}{4}$, but $X \cap (Y + k)$ is a finite set for every $k \geqslant 0$.
| 6 | https://mathoverflow.net/users/108637 | 294751 | 129,596 |
https://mathoverflow.net/questions/294737 | 1 | If $(X,\tau)$ is a topological space, then we say $A\subseteq X$ is a *fiber* if there is $f:X\to X$ continuous and $y\in X$ such that $A = f^{-1}(\{y\})$. For any $T\_1$-space it is clear that fibers are closed.
In $\mathbb{R}$ the converse holds: all closed sets are fibers.
**Question.** Is there a connected $T\... | https://mathoverflow.net/users/8628 | Connected $T_2$-space such that not all closed subsets are fibers | Yes. See [Kannan and Rajagopalan - Constructions and applications of rigid spaces, I](https://www.sciencedirect.com/science/article/pii/0001870878900063) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=501093)), particularly their construction 2.2.4, which gives a strongly rigid connected Hausdorff space $Y$. I... | 6 | https://mathoverflow.net/users/2926 | 294755 | 129,597 |
https://mathoverflow.net/questions/294754 | 4 | Define the Wasserstein-1 metric (or the Earth mover's distance) between two positive measures $\mu\_1$, $\mu\_2$ by
$$
W(\mu\_1, \mu\_2)
=
\inf\_{\gamma \in \Gamma (\mu\_1, \mu\_2)} \int \|x\_1 - x\_2\| \, \mathrm{d} \gamma (x\_1, x\_2)
$$
where $\Gamma(\mu\_1,\mu\_2)$ denotes the collection of all measures on with m... | https://mathoverflow.net/users/112001 | Is the Wasserstein-1 metric translation invariant? | Yes, by Kantorovich--Rubinstein duality $W(\mu\_1,\mu\_2)=\sup\_{f\,\text{is 1-Lip}} \int f d(\mu\_1-\mu\_2)$.
| 7 | https://mathoverflow.net/users/4312 | 294760 | 129,599 |
https://mathoverflow.net/questions/294744 | 5 | If I have two (Serre-)fibrations over the same base, and a weak equivalence of the total spaces that is also a map over the base, could I hope that the induced map on the spaces of sections would also be a weak equivalence?
If I restrict to CW-spaces, this is fairly easy, but I don't really know how to handle the spac... | https://mathoverflow.net/users/10444 | Space of sections of a fibration under weak homotopy equivalence | This is not true in general, unless you assume the base is sufficiently nice (eg a CW-complex). Here is a counter-example.
Let $B = \mathbb{Q}$, the rationals with its topology as a subspace of the reals $\mathbb{R}$. Let $E\_2 = \mathbb{Q}$ as well and let $E\_1 = \mathbb{Q}\_\delta$, the rationals with the discret... | 6 | https://mathoverflow.net/users/184 | 294762 | 129,600 |
https://mathoverflow.net/questions/294778 | 2 | Given measure space $(S, \mathcal{S}, \mu)$, and measurable function $\phi: S \to S$. $\phi$ is measure-preserving if $\forall A \in \mathcal{S}, \mu(A) = \mu(\phi^{-1}(A))$. My confusion is that why we do not define measure-preserving as $\forall A \in \mathcal{S}, \mu(\phi(A)) = \mu(A)$? It seems more natural to me a... | https://mathoverflow.net/users/121667 | Understanding measure-preserving transformation | As pointed out by user LSpice, your definition would be different from the accepted one.
However, it is not a well-constructed definition at all. Indeed, it is possible to have a situation when a function $\phi\colon S \to S$ is $\mathcal{S}$-measurable and $A \in \mathcal{S}$, but $\phi(A)\notin\mathcal{S}$ and hen... | 3 | https://mathoverflow.net/users/36721 | 294782 | 129,607 |
https://mathoverflow.net/questions/294780 | 3 | In Lemma 3.1 and Theorem 3.2 from the article [*Numerical solution of random differential equations: A mean square approach*](https://www.sciencedirect.com/science/article/pii/S0895717706002962), it is stated and proved a Mean Value Theorem for stochastic integrals and derivatives in the mean square sense. See Chapter ... | https://mathoverflow.net/users/121669 | Mean Value Theorem for stochastic processes | Of course, $\xi$ in formula (3.2) must in general depend on $r$ and $t$. Also, there is no reason for the last sentence in the "proof" of Lemma 3.1 in the paper to be true, and I think one can easily construct a relevant counterexample.
It is also easy to construct examples showing that $\xi$ in the equality $\int\_... | 0 | https://mathoverflow.net/users/36721 | 294786 | 129,609 |
https://mathoverflow.net/questions/294787 | 6 | Let $M$ be some large real number and $\delta>0$. I would like to estimate the number of solutions for the inequality
$$|\sqrt{n\_1}+\sqrt{n\_2}-\sqrt{n\_3}-\sqrt{n\_4}|<\delta\sqrt{M},$$
where $n\_i$ are positive integers with $M<n\_i\leq 2M$ for all $1\leq i\leq 4$. Let us denote this quantity by $N(M,\delta)$.
... | https://mathoverflow.net/users/101078 | Number of solutions for the inequality with square roots | For $\delta\geq 1$, the inequality is automatic, hence $N(M,\delta)=M^4$.
For $\delta<1$, the bound by Robert and Sargos is optimal up to the $o(1)$ in the exponent, and actually they mention this right before their Theorem 1. They do not give the proof, but the claim follows easily from their Lemma 1, which is a spe... | 5 | https://mathoverflow.net/users/11919 | 294803 | 129,615 |
https://mathoverflow.net/questions/294791 | 1 | This question is in connection with the question that I've asked at:
[Where do models of false theories exist?](https://mathoverflow.net/questions/294581/where-do-models-of-false-theories-exist)
The answer to that question was that any consistent theory can have its primitives be re-interpreted in such a manner as ... | https://mathoverflow.net/users/95347 | What is against having distinct membership relations on sets in the Platonic realm? | Putting on my Platonist hat for a while (and it's a very comfortable hat), I'd answer the question as follows: There's nothing wrong with having and studying two or more relations, like $\in\_1$ and $\in\_2$ in the question. But I wouldn't want to call both of them "membership"; like many people, I get confused when th... | 5 | https://mathoverflow.net/users/6794 | 294804 | 129,616 |
https://mathoverflow.net/questions/294759 | 5 | Let $F: \mathcal{K} \to \mathcal{C}$ be a functor between $\lambda$-accessible categories, you can assume $\mathcal{C}$ to be Set if needed.
Is it true that $F$ is $\lambda$-accessible if and only if it preserves $\lambda$-directed colimits of $\lambda$-presentable objects?
| https://mathoverflow.net/users/104432 | Can I check the accessibility of a functor on directed colimits of presentable objects? | Yes, and this doesn't require any assumption on $\mathcal C$. This follows from the following basic fact: if $I$ is filtered and $(A\_i)\_{i\in I}$ is a diagram of categories with colimit $A$, then the colimit of any functor $F: A\to\mathcal C$ may be computed as
$$
colim\_{a\in A} F(a) = colim\_{i\in I}(colim\_{a\in A... | 6 | https://mathoverflow.net/users/20233 | 294815 | 129,619 |
https://mathoverflow.net/questions/294816 | 2 | Let $m$ be a maximal ideal of a commutative ring $R$ with $1$. Can we construct a generating set $\{x\_i\}\_{i\in I}$ for the injective envelope $E(R/m) $ of $R/m$ such that $R/m\not\subseteq\langle x\_i\rangle$ for each $i\in I$?
Or is there description for a set of generation set of $E(R/m) $ ?
| https://mathoverflow.net/users/120784 | A generating set for injective envelope | $R/\mathfrak m$ is simple, so $\langle x\_i \rangle \cap R/\mathfrak{m}$ is either $0$, or $R/\mathfrak{m}$. But $R/\mathfrak{m}$ is an essential submodule, so the former cannot happen, assuming that $x\_i \neq 0$. That is, given any set $\{x\_i\}\_i$ of nonzero generators of $E(R/\mathfrak{m})$, we have $R/\mathfrak{m... | 2 | https://mathoverflow.net/users/60903 | 294818 | 129,620 |
https://mathoverflow.net/questions/294276 | 2 | What is an example of a three-dimensional smooth distribution $D$ of $\mathbb{R}^4$ with this property:
>
> Not only $D$ is not integrable but also there is no a two-dimensional foliation $F$ of $\mathbb{R}^4$ such that for every $x\in \mathbb{R}^4$, the tangent space to each leaf at $x$ is contained in $D\_x$.
>
... | https://mathoverflow.net/users/36688 | A strongly non-integrable distribution | Now that [this other question](https://mathoverflow.net/questions/294517/is-there-a-2-dimensional-foliation-tangent-to-this-particular-3-dimensional "this question") as been answered, we can answer this original question: The answer is 'yes' there exists a non-integrable tangent $3$-plane field on $\mathbb{R}^4$ that h... | 4 | https://mathoverflow.net/users/13972 | 294819 | 129,621 |
https://mathoverflow.net/questions/294806 | 8 | What is a good reference for **flat** morphisms of complex-analytic spaces? (The book by Grauert and Remmert doesn't treat them).
Topics I'm interested in: openness of flat maps, descent for coherent analytic sheaves.
| https://mathoverflow.net/users/nan | Reference for flatness in complex-analytic geometry | A reference for the fact that flat maps are open in the complex-analytic category is Theorem 2.12, p. 180 in
C. Bănică, O. Stănăşilă: *[Algebraic methods in the global theory of complex spaces](https://mathscinet.ams.org/mathscinet-getitem?mr=463470)*.
See also my answer to MO question [41158](https://mathoverflow... | 9 | https://mathoverflow.net/users/7460 | 294821 | 129,622 |
https://mathoverflow.net/questions/294811 | 11 | Let $R$ be a ring (say noetherian of finite Krull dimension, possibly with additional hypotheses) henselian along the ideal $(p)$, and let $\hat{R}$ be the $p$-adic completion. Is it true that the étale cohomology of $R[1/p]$ and $\hat{R}[1/p]$ with mod $p$ coefficients coincide? I believe that this should be true (for... | https://mathoverflow.net/users/344 | Etale cohomology of localizations of henselian rings | **TL;DR:** Your expectation is right. In fact, there is a third object to compare with $R[1/p]$ and $\hat R[1/p]$, the affinoid rigid space ${\rm Spf}(\hat R)^{\rm rig}$. The cohomology comparison is given by the Gabber-Fujiwara theorem: see Corollary 6.6.4 in [1].
Let us consider the following general setup: let $(A... | 15 | https://mathoverflow.net/users/3847 | 294834 | 129,626 |
https://mathoverflow.net/questions/294838 | 4 | Let $F\left( n \right) = \sum\limits\_{k = 0}^n {{{\left( {C\_n^k{p^k}{{\left( {1 - p} \right)}^{n - k}}} \right)}^2}} $, prove $F\left( n \right) \ge F\left( {n + 1} \right)$.
UPDATE: More general, denote $F\left( n \right) = \sum\limits\_{k = 0}^n {C\_n^kp\_1^kq\_1^{n - k}C\_n^kp\_2^kq\_2^{n - k}}$, where ${q\_1} =... | https://mathoverflow.net/users/121682 | How to prove the sum of n squared binomial probabilities does not increase as n increases | Denote $q=1-p$, write $[x^a]f(x)$ for a coefficient of $x^a$ in the polynomial (or Laurent polynomial) $f(x)$. We have $$F(n)=[1](px+q)^n(px^{-1}+q)^n=[1]((p^2+q^2)+qp(x+x^{-1}))^n=\\=(2\pi)^{-1}\int\_{0}^{2\pi}((p^2+q^2)+qp(e^{it}+e^{-it}))^ndt.$$
The function $(p^2+q^2)+qp(e^{it}+e^{-it})=p^2+q^2+2pq\cos t$ is non-ne... | 10 | https://mathoverflow.net/users/4312 | 294840 | 129,628 |
https://mathoverflow.net/questions/294835 | 11 | It is well known that, for two functors $F,G : I \to C$ for $I,C$ some $\infty$-categories, the property that a map $\phi: F \to G$ is an equivalence can be checked locally on $I$. Namely, if $\phi(i) : F(i) \to G(i)$ is an equivalence for every $i \in I$, then $\phi$ is an equivalence as a morphism in the
$\infty$-ca... | https://mathoverflow.net/users/115052 | $(\infty,2)$-Categorical Analogue of the Local Nature of Equivalences | **TL DR:** That is not enough. If you let $\psi\_i:G(i)\to F(i)$ be the left adjoint of $\phi\_i$ you also need the condition that for every $f:i\to j$ the canonical morphism
$$\psi\_jG(f)\to F(f)\psi\_i$$
adjoint to $G(f)\to G(f)\phi\_i\psi\_i\cong \phi\_jF(f)\psi\_i$, is an equivalence.
The key word to remember her... | 11 | https://mathoverflow.net/users/43054 | 294841 | 129,629 |
https://mathoverflow.net/questions/294836 | 2 | Let $v(\beta) := \sum\_{n\le X} e(n\beta)$ where $e(\alpha) := e^{2\pi i \alpha}$. It is not hard to show that
$$\log X\ll \int\_0^1 |v(\beta)| d\beta\ll \log X$$
and by considering the underlying Diophantine equation, we have that for $k\ge 0, $
$$\int\_0^1 |v(\beta)|^{2k}d\beta\sim C\_kX^{2k - 1}.$$
For general $p... | https://mathoverflow.net/users/40983 | Moments of certain exponential sum | Yes. Without loss of generality, $X\geq 2$ is an integer. Then we have explicitly
$$ v(\beta)=e\left(\frac{(X+1)\beta}{2}\right)\frac{\sin(\pi X\beta)}{\sin(\pi\beta)},\qquad\beta\not\in\mathbb{Z}.$$
It follows for any $p>0$ that
$$\int\_{|\beta|\leq\frac{1}{4X}}|v(\beta)|^p\,d\beta\asymp\_p\int\_{|\beta|\leq \frac{1}{... | 7 | https://mathoverflow.net/users/11919 | 294843 | 129,631 |
https://mathoverflow.net/questions/294844 | 3 | A graph is [vertex-transitive](https://en.wikipedia.org/wiki/Vertex-transitive_graph) if its automorphism group acts transitively upon its vertices.
A [Hamiltonian path](https://en.wikipedia.org/wiki/Hamiltonian_path) is a path that visits each vertex exactly once.
[Lovász conjecture](https://en.wikipedia.org/wik... | https://mathoverflow.net/users/34538 | Lovász conjecture and 2-connected graphs | The vertex connectivity of a vertex-transitive graph with valency $k$ is at least $2(k+1)/3$ (Mader/Watkins). So if you prove the conjecture for 3-connected graphs, you've done them all.
| 7 | https://mathoverflow.net/users/1266 | 294850 | 129,635 |
https://mathoverflow.net/questions/294847 | 6 | Let $k'/k$ be an extension of algebraically closed fields of characteristic $0$, and $X$ a concentrated (i.e. quasi-compact and quasi-separated) scheme over $k$.
**Question**: is the pullback functor from finite étale covers of $X$ to finite étale covers of $X\_{k'}$ an equivalence?
If $X$ is connected, then by SGA... | https://mathoverflow.net/users/45660 | Finite étale covers of concentrated schemes and extension of base field | Yes, you can reduce to the finite type case by noetherian approximation (Appendix C in Thomason-Trobaugh). Namely, you can write $X=lim\_\alpha X\_\alpha$ where $X\_\alpha$ is of finite type over $k$. Then there is an equivalence of categories $FinEt/X \simeq colim\_\alpha FinEt/X\_\alpha$.
| 5 | https://mathoverflow.net/users/20233 | 294852 | 129,637 |
https://mathoverflow.net/questions/294860 | 0 | Let's assume all axioms of $\text{Z}- \text{Infinity}$.
Now let $F$ be any function that is definable over the whole universe of discourse by a formula in the language of $\text{Z}$.
Now we define $F$ related accessibility $``ACC^F"$ as:
$$X \ ACC^F \alpha \iff \alpha <X \wedge \exists \beta < X \ \big{[}F \bi... | https://mathoverflow.net/users/95347 | What is the consistency strength of F accessibility? | Unless I'm missing something, this scheme is inconsistent.
Take $$F(x)=\bigcup\_{y\in x}(\mathcal{P}(y)).$$ Then for all $X$ with more than one element and **any** $a$ with $a<X$ we have $X$ ACC$^F$ $a$: taking $b=\{\{X\}\}$, we have $\mathcal{P}(X)\subseteq F(\bigcup (a\cup b))$.
Now consider $\varphi(y)$ to be th... | 4 | https://mathoverflow.net/users/8133 | 294862 | 129,640 |
https://mathoverflow.net/questions/294857 | 4 | Suppose I have a (finitely-presented, say) graded module $M$ over $k[x,y]$, and I happen to know the rank $R\_{(a,b),(c,d)}$ of each map $x^{c−a}y^{d−b}:M\_{a,b}→M\_{c,d}$ for each pair of integers with $a\leq c$ and $b \leq d$, as well as the Hilbert function $\dim\_k(M\_{a,b})$.
Recall that the Betti numbers of $M... | https://mathoverflow.net/users/105314 | Combinatorial formula for Betti numbers of a $k[x,y]$-module | It seems unlikely that the Betti numbers can be determined by the data you list (bigraded Hilbert function and ranks of multiplication maps). Consider the following possibility: $\dim M\_{1,0} = \dim M\_{0,1} = 1$, $\dim M\_{1,1} = 2$, and all other $M\_{a,b}$ are zero; and the maps $y : M\_{1,0} \to M\_{1,1}$, $x : M\... | 3 | https://mathoverflow.net/users/88133 | 294866 | 129,643 |
https://mathoverflow.net/questions/294854 | 2 | Let $f:X\rightarrow S$ be a smooth projective morphisms of noetherian schemes and let $\mathcal{O}(1)$ be a relatively ample line bundle. Let $\mathcal{E}$ be a coherent sheaf over $X$ and flat over $S$. Suppose there exists an open subscheme $U\subset X$ such that $\mathcal{E}|\_U$ is locally free and complement of $U... | https://mathoverflow.net/users/nan | determinant of a coherent sheaf, locally free on a big open set | For any sheaf which admits a finite locally free resolution its determinant is defined as the alternating tensor product of the determinants of its terms. Under your assumptions $\mathcal{E}$ admits such a resolution, so this definition applies.
| 4 | https://mathoverflow.net/users/4428 | 294872 | 129,645 |
https://mathoverflow.net/questions/294868 | 5 | Let $(X,||\cdot||)$ be a normed space where $||\cdot||$ is the sup-norm and let $E$ be a convex and compact subset. Let $f:E\to [0,1]$ be continuous and affine, i.e. for all $x,y\in E$ and all $\lambda \in [0,1]$, $f(\lambda x+(1-\lambda)y)=\lambda f(x)+(1-\lambda)f(y)$.
Is $f$ Lipschitz? If not, can you give a simp... | https://mathoverflow.net/users/121674 | affine vs lipschitz | For instance, consider the convex subset $E:=\{x\in\ell\_\infty: 0\le x\_k\le 2^{-k} \text{ for all } k\ge0 \}$ of $\ell\_\infty$. By dominated convergence, $E$ is compact and its relative topology coincides with the topology induced by $\ell\_1$.
In particular, the affine function $f:E\to [0,1]$ defined by $f(x):=\... | 8 | https://mathoverflow.net/users/6101 | 294873 | 129,646 |
https://mathoverflow.net/questions/294660 | 3 | In Appendix A of [this paper](https://arxiv.org/pdf/cond-mat/9904278.pdf), it is claimed that the asymptotic behaviour of
$$\phi\_1(y,\lambda)=\frac{1}{\Gamma(\frac{1-\lambda}{2})}\int\_0^\infty dt~e^{-t}\cos(2y\sqrt{t})t^{-\frac{1+\lambda}{2}},$$
for $\lambda\_2=\text{Im}(\lambda)\to+\infty$ (where $\lambda=\lambd... | https://mathoverflow.net/users/121642 | Asymptotic behaviour of function from integral representation | If we expand $\cos{(2y\sqrt{t})}$ into Taylor series and integrate term by term, we get $$\phi\_1(y,\lambda)=\sum\limits\_{n=0}^\infty\frac{(-1)^n}{(2n)!}\frac{\Gamma\left(n+\frac{1}{2}-\frac{\lambda}{2}\right)}{\Gamma{\left(\frac{1}{2}-\frac{\lambda}{2}\right)}}(2y)^{2n}.$$ Now let us use the following asymptotic expa... | 2 | https://mathoverflow.net/users/32389 | 294874 | 129,647 |
https://mathoverflow.net/questions/294747 | 6 | Unless I'm mistaken the "Free completion under finite limits monad" $C \mapsto C^{lex}$ and the "free co-completion monad" $C \mapsto \widehat{C}$ (the categories of small presheaves) satisfies a distributivity law, and the algebra for the composed monad $C \mapsto \widehat{C^{lex}}$ are the infinitary pretopos.
Has ... | https://mathoverflow.net/users/22131 | References requestion : Pretopos are algebras for a composed monad? | I think Mike is right: in Garner and Lack, [Lex colimits](https://arxiv.org/abs/1107.0778), Proposition 2.3 and the discussion following it show that the free cocompletion 2-monad lifts to the 2-category of finitely complete categories, which is equivalent to the existence of a distributive law between these two 2-mona... | 3 | https://mathoverflow.net/users/2362 | 294878 | 129,649 |
https://mathoverflow.net/questions/294788 | 3 | HR problem deals with a spherical fluid viscous drop falling in a different fluid under influence of gravity. The outer fluid is of uniform speed $U$ in direction of gravity, far away from the drop. The boundary solutions posed are
1. zero velocity of both flows on drop boundary
2. continuity of tangential velocities... | https://mathoverflow.net/users/121671 | Hadamard-Rybczynski problem | Yes, I partially missunderstood Batchelor, leading myself by some other literature. Spherical form of drop is due to conitnuity of normal stresses but in form $-p\_{abs}+2\mu \sigma\_{rr}$ where $p\_{abs}=p+\rho \mathbf{g}\cdot \mathbf{x}$ and this in case when $U$ is the velocity of free falling $U=-\frac{2}{3}\frac{a... | 2 | https://mathoverflow.net/users/121671 | 294880 | 129,650 |
https://mathoverflow.net/questions/294867 | 5 |
>
> **Intro.** *I would be deeply grateful if someone could please clarify the following to me.*
>
>
>
**The question.** (the main point is (4))
Let $X$ be a smooth projective variety over $\mathbf{C}$, and $\mathbf{Z}(n)\_{\mathcal{D}}$ the Deligne complex on $X$.
We have a triangle:
$$\to\Omega^{<n}\_X \to... | https://mathoverflow.net/users/nan | A quite puzzling question on Deligne cohomology sheaves and cycle maps | 1) presumably it's algebraic cycles. The linked announcement works with schemes of finite type over $\operatorname{spec} \mathbb{C}$. For a smooth projective variety, which you're using, Chow's theorem should force analytic cycles to agree with algebraic cycles. Then equivalence relation comes from function fields of s... | 2 | https://mathoverflow.net/users/48362 | 294884 | 129,652 |
https://mathoverflow.net/questions/294885 | 2 | Say $\mathcal{C}$ is a strict monoidal abelian category and $A$ is a *coalgebra* object in $\mathcal{C}$, with left co-modules $M$ and right co-module $N$ (also in $\mathcal{C}$). Then we have a notion of (derived) tensor product $$M\overset{R}{\otimes} {}^{co}\_A N,$$ (note that we are taking a right derived functor b... | https://mathoverflow.net/users/7108 | Variant of co-Tor in a bimodule category | I am not sure that I really understand what you want, but I'd say the relevant structure is that of a module category over a monoidal category. Given a monoidal category $\mathcal E$, one can consider left and right module categories over it. For example, given an associative ring $R$, the category $\mathcal E=R\mathrm... | 2 | https://mathoverflow.net/users/2106 | 294887 | 129,653 |
https://mathoverflow.net/questions/292450 | 6 | I am reading a few papers about counting smooth numbers in the interval $[x, x+\sqrt{x}]$, including the work of [Harman](https://pdfs.semanticscholar.org/04d4/801dc56a086f989df6f9176cefc62b0650de.pdf), and [Matomaki](https://pdfs.semanticscholar.org/229f/f5bc99e60c25365425bf0c6d42ec40ed8eaf.pdf).
Both authors mentio... | https://mathoverflow.net/users/112214 | Counting smooth numbers in short intervals | Suppose that $F(s) = \sum\_{n \sim x} a\_n n^{-s}$ where $a\_n = 1$ if $n$ is $x^{\varepsilon}$ smooth and $a\_n = 0$ otherwise. We want to understand the distribution of the $a\_n$'s in a short interval of length $h$. Then by the usual Perron formula,
$$
\sum\_{x < n < x + h} a\_n = \frac{1}{2\pi i} \int\_{-T}^{T} F(... | 2 | https://mathoverflow.net/users/121697 | 294889 | 129,654 |
https://mathoverflow.net/questions/294807 | 7 | Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure.
I think it should be as follows:
Let $(X,\mathcal{B},\nu)$ be a measure space and let $\mathcal{F}\subset\mathcal{B}$ a sub$-\sigma-$algebra, such that $\nu$ is $\sigma-$finite i... | https://mathoverflow.net/users/50064 | Conditional Expectation for $\sigma$-finite measures | $\newcommand{\N}{\mathbb N}
\newcommand{\R}{\mathbb R}
\newcommand{\B}{\mathcal B}
\newcommand{\F}{\mathcal F}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\renewcommand{\c}{\circ}
\newcommand{\tr}{\operatorname{tr}}$
The definition you quoted is correct.
However, there can be no rea... | 3 | https://mathoverflow.net/users/36721 | 294890 | 129,655 |
https://mathoverflow.net/questions/294886 | 4 | Let $\mathcal{S}$ be a category, fix an object $S \in \mathcal{S}$, and fix two functors $F\_i : \mathcal{C}\_i \to \mathcal{S}\_{/S}$ for $i=1,2$ where $\mathcal{C}\_i$ are arbitrary categories.
Suppose $\mathcal{S}$ admits all colimits indexed by $\mathcal{C}\_i$ and by $\mathcal{C}\_1 \times \mathcal{C}\_2$.
The... | https://mathoverflow.net/users/121694 | Fiber products of colimits | This is just a two-sided way of saying that $\times\_S$ preserves colimits in each variable separately (note that $\mathrm{colim}\_{C\_1\times C\_2} = \mathrm{colim}\_{C\_1} \mathrm{colim}\_{C\_2}$). So in particular it's true whenever $\mathcal{S}\_{/S}$ is cartesian closed, and thus whenever $\mathcal{S}$ is locally ... | 1 | https://mathoverflow.net/users/49 | 294899 | 129,657 |
https://mathoverflow.net/questions/294828 | 28 | It was asked at the [Bulletin of the American Mathematical Society Volume 64, Number 2, 1958](http://www.ams.org/journals/bull/1958-64-02/S0002-9904-1958-10164-0/S0002-9904-1958-10164-0.pdf), as a Research Problem, if a Hurwitz polynomial with real coefficients (i.e. all of its zeros have negative real parts) can be di... | https://mathoverflow.net/users/70464 | All polynomials are the sum of three others, each of which has only real roots | To address the original question about the polynomials with *complex* coefficients.
Given a polynomial $P\in\mathbb C[x]$ of degree $n$, write $P=Q+iR$ with $Q,R\in\mathbb R[x]$, and fix arbitrarily a polynomial $S\in\mathbb R[x]$ of degree $n$ with all roots real and pairwise distinct. As observed in Eremenko's ans... | 23 | https://mathoverflow.net/users/9924 | 294912 | 129,660 |
https://mathoverflow.net/questions/294895 | 7 | One of the answers to [this question](https://math.stackexchange.com/questions/196024/can-someone-explain-the-abc-conjecture-to-me/196123#196123) says:
"In Serge Lang's *Algebra*, he says: "One of the most fruitful analogies in mathematics is that between the integers $\mathbb{Z}$ and the ring of polynomials $F[t]$ ove... | https://mathoverflow.net/users/72288 | An analogy between the ring of polynomials in two variables and another (commutative?) ring | This seems like a strange question. Of course there are many analogies between many different commutative rings. However, if you're looking for a similar analogy, you have to replicate the basic features of Lang's analogy - in particular, the fact that the rings have the same dimension. This immediately kills all your ... | 6 | https://mathoverflow.net/users/18060 | 294914 | 129,661 |
https://mathoverflow.net/questions/294907 | 3 | Let $X,Y$ be infinite-dimensional Banach spaces and $T:X \rightarrow Y$ be a bounded linear operator. Let $M$ be an infinite-dimensional subspace of $X$ ($M$ is not necessarily closed). Let $N$ be an infinite-dimensional closed subspace of $\overline{M}$. My question is the following:
Question. Is there an infinite-d... | https://mathoverflow.net/users/41619 | A question on the norm of the restriction of an operator to subspaces | In general, no. Let $T:X\to Y$ be a bounded linear operator between infinite dimensional Banach spaces, as assumed in the question. Assume further $N:=\ker T$ is a separable subspace of both infinite dimension and infinite co-dimension. Then, there is an infinite dimensional dense subspace $M\subset X$ of $N$ such that... | 1 | https://mathoverflow.net/users/6101 | 294916 | 129,662 |
https://mathoverflow.net/questions/294918 | 13 | If $G$ is a discrete or a Lie Group acting smoothly on a manifold $M$, we can define the algebra of $G$-invariant de Rham classes, $H(M)^G$, and we can also consider the cohomology of the sub-complex of $G$-invariant forms and its cohomology $H(\Omega(M)^G)$ which I note simply $H(M^G)$.
The injection $j:\Omega(M)^G ... | https://mathoverflow.net/users/74372 | Counterexample showing that G-invariant de Rham cohomology different from cohomology of G-invariant sub-complex? | If $G$ is compact, the inclusion $H(M^G) \to H(M)^G$ is an isomorphism. The inverse map is defined as follows: Take a class $\omega$ in $H(M)^G$ and lift it to a closed form $\alpha \in \Omega(M)$. Put $\beta = \int\_{g \in G} g^{\ast} \alpha$, where the integral is with respect to Haar measure normalized to have volum... | 17 | https://mathoverflow.net/users/297 | 294922 | 129,664 |
https://mathoverflow.net/questions/264857 | 14 | For a topological space $X$ and a point $x\in X$, consider the following definitions:
* ([Gerlits and Nagy](http://ac.els-cdn.com/0166864182900657/1-s2.0-0166864182900657-main.pdf?_tid=c2c65c2e-ff4f-11e6-b866-00000aacb35f&acdnat=1488463051_63b7512d2c12e5bb32564f78b774352a)): $X$ is strictly Fréchet at $x\in X$ if for... | https://mathoverflow.net/users/41407 | Strictly Fréchet spaces versus strongly Fréchet spaces | You can find an example of a strongly Fréchet non-strictly Fréchet space in the preprint "[Selective game versions of countable tightness with bounded finite selections](https://arxiv.org/pdf/1603.09715.pdf)" by L. Aurichi, A. Bella and R. Dias (see example 2.10). Basically, it's the one-point compactification of an ap... | 4 | https://mathoverflow.net/users/11647 | 294926 | 129,668 |
https://mathoverflow.net/questions/294906 | 1 | Noting $\mathcal{F}^c$ the cosine transform and $\mathcal{F}^s$ the sine transform defined on real functions by:
$$\mathcal{F}^c [f (x)]=\int\_0^{\infty} f(t) \cos(xt) dt $$
$$\mathcal{F}^s [f (x)]=\int\_0^{\infty} f(t) \sin(xt) dt $$
What can we say about following integral ?
$$\int\_0^{\infty} \mathcal{F}^c [... | https://mathoverflow.net/users/38290 | What gives a "Parseval like" equation mixing cosine and sine Fourier transforms ? | There is no "Parseval-like" equation, in the sense of a single integral over $t$. The best you can do is a double integral over $t,t'$:
$$\int\_0^{\infty} \mathcal{F}^c [f (x)] \mathcal{F}^s [g (x)]\,dx=$$
$$\int\_0^\infty \left(\int\_0^{\infty} f(t) \cos(xt) \,dt\right)\left(\int\_0^{\infty} g(t') \sin(xt')\, dt'\righ... | 2 | https://mathoverflow.net/users/11260 | 294943 | 129,678 |
https://mathoverflow.net/questions/294457 | 21 | Edit: since we seem a bit deadlocked at this point, let me weaken the question. It's fairly easy to see that the set of 8-tuples of reals which can be the eigenvalues of a matrix of the desired form is closed. We know from jjcale and Caleb Eckhardt that its complement is nonempty. Is its complement dense? That is, woul... | https://mathoverflow.net/users/23141 | What are the possible eigenvalues of these matrices? | Let $V$ be the real vector space of the $8\times 8$ matrices of the form given in the question.
Where is an open set in $\mathbb{R}^8$ of possible $8$-tuples of eigenvalues of matrices in $V$.
**Proof :**
Choose $M\_1 \in V$ such that $M\_1$ has no degenerated eigenvalues.
Let $v\_1,...,v\_8$ be an orthonormal ... | 4 | https://mathoverflow.net/users/17261 | 294945 | 129,679 |
https://mathoverflow.net/questions/294920 | 6 | I'm looking for examples of proofs that some Quillen model categories are monoidal, or enriched over an other model category, which are based on explicit computation of the "pushout product" of the generating cofirbations and generating trivial cofibrations, i.e. an explicit combinatorial proofs of a statement like: if... | https://mathoverflow.net/users/22131 | Combinatorial proof that some model categories are monoidal/enriched? | This is far from a comprehensive answer, but since you ask for references, here are two.
The result for simplicial sets is obtained by an explicit computation in Appendix H of Joyal's *The Theory of Quasi-categories and Its Applications*.
For $\Theta\_2$-sets, there is Chapter 3 of Oury's *Duality for Joyal's Categ... | 2 | https://mathoverflow.net/users/12547 | 294959 | 129,686 |
https://mathoverflow.net/questions/294953 | 14 | Is there a simple proof of the fact that:
>
> If $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$
> embedded into $S^3\setminus A$ that such that the circles $A$ and $B$
> are linked with the linking number $1$?
>
>
>
If $A$ is smoothly embedded, than it is easy. The problem is that the g... | https://mathoverflow.net/users/121665 | Linking topological spheres | Let $C = S^3 \setminus A$. Alexander duality says that
$$
H\_1(C) \cong H^1(A) \cong \Bbb Z\, .
$$
Let $\alpha: S^1 \to C$ be any map representing a generator of $H\_1(C)$ (every first homology class is spherical by the Hurewicz theorem). By homotopical approximation we can assume $\alpha$ is a smooth embedding. We ca... | 11 | https://mathoverflow.net/users/8032 | 294960 | 129,687 |
https://mathoverflow.net/questions/294774 | 6 | It is well known that the maximal unramified extension of $\mathbb{Q}\_p$ can be extended by adding the roots of unity of order prime to $p$. Is there any explicit description of the maximal totally ramified extension of $\mathbb{Q}\_p$?
| https://mathoverflow.net/users/121447 | Is there any explicit description of the maximal totally ramified extension of $\mathbb{Q}_p$? | A composite of totally ramified extensions need not be totally ramified:
**Example 1.** (As per LSpice's suggestion) Consider the extensions $\mathbb Q\_p(\sqrt{p})$ and $\mathbb Q\_p(\sqrt{\varepsilon p})$, where $\varepsilon \in \mathbb Z\_p^\times$ is a nonsquare unit (corresponding, for example, to any lift of a ... | 11 | https://mathoverflow.net/users/82179 | 294965 | 129,690 |
https://mathoverflow.net/questions/294975 | 3 | Assume that semi-simplicity of the Galois action on $\ell$-adic cohomology of all smooth projective varieties over finite fields, were known.
Can one deduce that the Galois action on $\ell$-adic cohomology of smooth projective varieties over **number fields** is semi-simple too?
| https://mathoverflow.net/users/nan | Semi-simple Galois actions on étale cohomology | No. For example, it follows from Weil's results in the 1940s that the Galois action on the etale cohomology of abelian varieties over finite fields is semisimple, but it was not until Faltings in 1983 that one knew the similar result over number fields.
| 3 | https://mathoverflow.net/users/121743 | 294976 | 129,694 |
https://mathoverflow.net/questions/294977 | 2 | Let $X$ be an abelian variety over a field $k$.
Let $A^p\_{\rm hom}(X)$ be the $p$-th Chow group of cycles modulo homological equivalence ($\ell$-adic, if $k$ is of char $p$).
Do we have $$A^p\_{\rm hom}(X) = \wedge^p A^1\_{\rm hom}(X)?$$
| https://mathoverflow.net/users/nan | Chow groups modulo homological equivalence for abelian varieties | No. Because Chow groups lie in the even-dimensional homology, their multiplication structure is commutative. So the natural map is from $\operatorname{Sym}^p A^1$ to $A^p$.
Also no for dimensions reasons. For instance $p=2$, $A$ is a product of two isogenous elliptic curves, $A^1$ is three-dimensional (four if they'r... | 5 | https://mathoverflow.net/users/18060 | 294981 | 129,696 |
https://mathoverflow.net/questions/294973 | 1 | Let $G$ be a quasisplit connected reductive group over a $p$-adic field $k$. Identify $G$ with its rational points. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$, both defined over $k$. For $w \in N\_G(T)$, consider the double cosets $BwB$, which are locally closed submanifolds of $G$. Is $B$ the on... | https://mathoverflow.net/users/38145 | Is the Borel subgroup the only closed double coset? | In general, and this holds for algebraically closed field as well as local field of charracteristic 0, at least in the split case, the description of orbit closures is as follows:
Let $\omega \in W\_G$ be a weyl element, and write $\omega = \sigma\_{\alpha\_1} ... \sigma\_{\alpha\_k}$ as a reduced expression in simple... | 5 | https://mathoverflow.net/users/115052 | 294982 | 129,697 |
https://mathoverflow.net/questions/294970 | 5 | Let $X$ be a smooth projective algebraic variety over the complex numbers, and let us name
$$f : X\_{\rm an}\to X\_{\rm Zar}$$
the morphism of sites induced by sending a Zariski open $U\subset X$ to $U^{\rm an}$.
Consider the constant sheaf $\mathbf{Q}(1)$ on $X\_{\rm an}$ whose value group is $2\pi i\cdot\mathbf... | https://mathoverflow.net/users/nan | Spectral sequence in Betti cohomology | No; this spectral sequence usually does not degenerate. It appears that the group in the left hand side are often infinite dimensional. In particular, the $E\_2^{p,p}$-term is the group of codimension $p$ cycles modulo algebraic equivalence (see Corollary 7.4 of the paper Bloch, S., & Ogus, A. (1974). Gersten’s conject... | 2 | https://mathoverflow.net/users/2191 | 294983 | 129,698 |
https://mathoverflow.net/questions/294978 | 3 | Let $W$ be a Coxeter group with associated graph $G$.
Define $$X(G) = \{w \in W : \text{any two simple reflections} \,S\_{\alpha}\, \text{and}\, S\_{\beta} \,\text{appearing in any of the reduced expression of $w$ commute} \} $$.
I have the following questions :
1. Is this set $X(G)$ studied in the literature?
2... | https://mathoverflow.net/users/33047 | Elements of Coxeter group whose simple reflections pairwise commute | If I see it correctly, there is not much going on in the set $X(G)$ from the viewpoint of Coxeter systems: Let $S$ be the simple system and let $G$ be the Coxeter graph of $(W,S)$ with vertex set $S$. Then
$$X(G) = \big\{ s\_1 \cdots s\_k \in W \mid \{s\_1,\ldots,s\_k\} \subseteq S\text{ with } s\_is\_j = s\_js\_i \tex... | 6 | https://mathoverflow.net/users/21291 | 294988 | 129,700 |
https://mathoverflow.net/questions/294989 | 5 | Let $X$ be a Noetherian, integral scheme. Let $\mathcal{F}$ be a torsion free sheaf on $X$ and let $U \subseteq X$ be the open subscheme where $\mathcal{F}$ is locally free.
>
> **Q.** Is it true that $\mathrm{codim}\_X(X\setminus U)\geq 2$?
>
>
>
| https://mathoverflow.net/users/nan | Codimension of the non-locally free locus | The answer is yes, assuming $X$ to be *normal*.
See S. Ishii, *Introduction to singularities* ([Zbl 1308.14001](https://zbmath.org/?q=an:1308.14001)), Proposition 5.1.7 p. 83.
| 6 | https://mathoverflow.net/users/7460 | 294992 | 129,701 |
https://mathoverflow.net/questions/294991 | 3 | Let $p$ be a prime. Let $G=\langle x,y\rangle^{\textrm{pro-}p}$ be the pro-$p$ completion of the free group $\langle x,y\rangle$ generated by symbols $x$ and $y$. Define $G\_{n+1}=[G,G\_n]$ and $G\_1=G$ (more precisely, $G\_{n+1}$ should be the minimal closed subgroup generated by appropriate commutators). Then I natur... | https://mathoverflow.net/users/44005 | Freeness of a quotient group | Let $\Gamma=F/F^4=\langle x,y\rangle$ be the free 3-nilpotent group on 2 generators.
Then $[x,[x,y]]$ and $[y,[x,y]]$ belong to the 3rd term $F^3/F^4$ in the lower central series, and in the quotient, $[x,y]$ is central, so the quotient is 2-nilpotent and hence $[x,[x,y]]$ and $[y,[x,y]]$ generate $F^3/F^4$.
Hence... | 3 | https://mathoverflow.net/users/14094 | 294993 | 129,702 |
https://mathoverflow.net/questions/294985 | 3 | I am studying deformation (as it is introduced in <https://arxiv.org/pdf/math/0611793.pdf> or <http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf>) and rigidity of some infinite dimensional Lie algebras which are defined on field with characteristic zero which its commutation relations are:
$$[J\_m,J\_n]=(m-n)J\_... | https://mathoverflow.net/users/119455 | Lie algebras : Deformations and Rigidity | Let me consider the finite-dimensional case (as YCOR was commenting it is basically impossible to say something in the infinite-dimensional one unless you clarify in a more detailed manner your definitions).
From a geometrical point of view a ``rigid Lie algebra'' is an open subset in the variety of all Lie algebras ... | 4 | https://mathoverflow.net/users/6032 | 294996 | 129,703 |
https://mathoverflow.net/questions/294919 | 221 | Suppose I'm an aspiring mathematician-to-be, who started doing research. Although this is really what I love doing, I found that one disturbing point is that there's always the pressure of discovering something new. When I'm doing mathematical research, there's always the fear in the back of my mind, that maybe, I don'... | https://mathoverflow.net/users/nan | How can a mathematician handle the pressure to discover something new? | This is ancient history, and considering my age, I may have told this story here before. I started at Harvard graduate school in 1957, the same year that Hironaka arrived there to work with Zariski. He was already an accomplished mathematician, even if he didn’t yet have a PhD. Early that year, I must have said to him ... | 173 | https://mathoverflow.net/users/11417 | 295007 | 129,704 |
https://mathoverflow.net/questions/295006 | 2 | Let $X$ be a topological space and $\mathcal{B}$ a base of the topology (i.e. it is closed under intersection and every open set is the union of elements from $\mathcal{B}$). Any functor from $\mathcal{B}$ to, say abelian groups, extends to a sheaf in, at most, one way. I.e., if the functor is not already contradicting... | https://mathoverflow.net/users/59375 | Defining a sheaf from its values on a prebase (plus little more structure) | What people usually call a base of the topology is a family $P$ such that if you have a finite set $U\_i \in P$ then there is a covering of $\cap U\_i$ by elements of $P$.
you do not necessarily need $P$ to be stable under intersection.
This is stronger than the condition you are asking, but this is the correct condi... | 4 | https://mathoverflow.net/users/22131 | 295008 | 129,705 |
https://mathoverflow.net/questions/295012 | 6 | Let $R$ be an integral domain such that every non-constant, irreducible polynomial $f(X) \in R[X]$ has degree $1$.
>
> **Q.** is it true that $R$ is a field?
>
>
>
If $0 \ne a \in R$ , then $X^2-a$ is reducible in $R[X]$. Since this polynomial has content $1$ , we must have a factorization into one degree pol... | https://mathoverflow.net/users/nan | Integral domain over which any non-constant, one variable, irreducible polynomial has degree 1 | The answer is *no*, as shown by the following example.
Take as $R$ the ring $\bar{\mathbb{Z}}$ of [algebraic integers](https://en.wikipedia.org/wiki/Algebraic_integer). Since its fraction field is $\bar{\mathbb{Q}}$, which is algebraically closed, it follows by [Gauss lemma for GCD domains](https://en.wikipedia.org/w... | 13 | https://mathoverflow.net/users/7460 | 295016 | 129,707 |
https://mathoverflow.net/questions/293944 | 7 | I am wondering if there is some general relation between Kazhdan constants of a group and it finite index subgroups?
Let $G$ be a finitely generated group with a generating set $\Sigma$ that satisfies Kazhdan property (T) with constant $\kappa(G,\Sigma)$.
1) if $\Gamma$ is a finite index subgroup of $G$, is there ... | https://mathoverflow.net/users/110128 | Kazhdan constant and finite index subgroups | If $n:=[G:H]$, then $\mathbb C[G] \subset M\_n \mathbb C[H]$, where $g \in G$ maps to a permutation matrix decorated with elements from $H$ and the embedding depends essentially only on a choice of a transveral of the quotient map $G \to G/H$. One can arrange things, so that generators $S \subset G$ map to permutations... | 5 | https://mathoverflow.net/users/8176 | 295017 | 129,708 |
https://mathoverflow.net/questions/295020 | 3 | Let $P\_0$ be a minimal parabolic subgroup of a connected, reductive group $G$ over a $p$-adic field $k$. Let $P$ be a parabolic subgroup containing $P\_0$ with Levi decomposition $P = MN$. Let $N^-$ be a group such that $P^- = MN^-$ is opposite to $P$.
Let $\pi$ be a smooth, irreducible representation of $M$, and c... | https://mathoverflow.net/users/38145 | Functions in the induced space compactly supported in $PN^-$ modulo $P$ | Answer moved from the comments.
1. They exist because $P \times N^- \to G$ is an open embedding (even on the level of varieties). This implies that, for $f \in \mathrm C^\infty\_{\mathrm c}(N^−; \sigma)$, the function $p n^− \mapsto \pi(p)f(n^−)$ has image in the appropriate function space (for $n^− \in N^−$, the ima... | 2 | https://mathoverflow.net/users/2383 | 295028 | 129,712 |
https://mathoverflow.net/questions/295011 | 2 | Let $G$ be a split reductive group over an arbitrary field $k$. By definition, see Jantzen (\*), an ascending chain $$0 = V\_0 \subset V\_1 \subset V\_2 \subset \dots$$ of submodules of a $G$-module $V$ is called a Weyl filtration of $V$ if $V = \bigcup\_{i \geq 0} V\_i$ and if each $V\_i/V\_{i-1}$ is isomorphic to som... | https://mathoverflow.net/users/120232 | Decomposition into Weyl modules | Your conclusion about direct sums is false. It's helpful here to have some examples in mind, such as a typical projective/injective module for the Lie algebra of $G:=\mathrm{SL}\_2$, lifted to $G$. This is *indecomposable* but might have a Weyl filtration with two quotients: for example, the trivial 1-dimensional modul... | 8 | https://mathoverflow.net/users/4231 | 295030 | 129,714 |
https://mathoverflow.net/questions/295019 | 6 | During research involving the Born–Jordan quantization I came across the expression
$$
\frac{d^k}{dx^k}\operatorname{arcsinh}\Big(\frac1x\Big)\Big|\_{x=1}\tag1
$$
for $k\in\mathbb N\_0$. It is not too hard to write this expression as a sum
$$
(1)=\frac{\sqrt{2}}{2^k}\sum\_{j=0}^{k-1} a\_j^k\tag{2a}
$$
for any ... | https://mathoverflow.net/users/116991 | Closed, sum-free form for the $n$-th derivative of $\operatorname{arcsinh}(\frac1x)$ in $x=1$ | Here is a proof that $M\_{nn}=O(n^{-1})$, as the OP conjectured. The function
$$f(z):=\operatorname{arcsinh}(1/z),\qquad |z|<1,$$
is holomorphic in the unit disk, hence by Cauchy's formula
$$ \frac{f^{(k)}(1)}{k!}=\frac{1}{2\pi i}\int\_{|z-1|=r}\frac{f(z)}{(z-1)^{k+1}}\,dz,\qquad 0<r<1.$$
Here and later circles are pos... | 5 | https://mathoverflow.net/users/11919 | 295045 | 129,718 |
https://mathoverflow.net/questions/294896 | 0 | The central limit theorem states that, under certain circumstances, the probability distribution of the scaled mean of a random sample converges to a normal distribution as the sample size increases to infinity. Under stronger assumptions, the Berry–Esseen theorem, or Berry–Esseen inequality, gives a more quantitative ... | https://mathoverflow.net/users/67136 | Berry-Esseen type theorem for Monotonic independence | A Berry Esseen type theorem is indeed plausible. However, the rate of convergence cannot be of order $n^{1/2}$ as in the classical free version of it. In this case one can find an explicit example (centered monotone Poisson) whose rate of convergence is $n^{1/4}$ with respect to the Kolmogorov distance.
Under some m... | 3 | https://mathoverflow.net/users/121781 | 295047 | 129,719 |
https://mathoverflow.net/questions/295044 | 2 | Let $G$ be a profinite group.
Let $K(G,\mathbb Z\_\ell)$ be the Grothendieck group of the derived category of finitely generated $\mathbb Z\_\ell$-modules with continuous $G$-action.
Let $K(G,\mathbb F\_\ell)$ be the Grothendieck group of the derived category of finitely generated $\mathbb F\_\ell$-modules with co... | https://mathoverflow.net/users/45861 | reduction of torsion modules | It is true. Take any finitely generated module $M$ and filter it by powers of $\ell$, namely consider the filtration $M\_i = \ell^iM$. It suffice to prove the claim for the associated graded $\oplus\_i \ell^i M / \ell^{i+1}M$. But all the graded pieces are $\ell$-torsion and for them $M / \ell M \cong M \cong M[\ell]$ ... | 3 | https://mathoverflow.net/users/115052 | 295055 | 129,724 |
https://mathoverflow.net/questions/294846 | 1 | If $R$ is an integral domain satisfying acc on radical ideals (i.e. Noetherian spectrum) and if the fraction field of $R$ is algebraically closed, then is $R$ a field ?
If $R$ is normal (integrally closed in its fraction field) and a factorization domain and the fraction field of $R$ is algebraically closed, then I c... | https://mathoverflow.net/users/nan | Integral domain satisfying a.c.c. on radical ideals and with algebraically closed fraction field | As requested, I am making my comment an answer. For every integer $n\geq 1$, denote by $R\_n$ the power series ring, $$R\_n := \mathbb{C}[[z\_n]],$$ where $z\_n$ is a variable. For every pair of integers $m,n\geq 1$ with $m$ dividing $n$, denote by $f\_{m,n}$ the local $\mathbb{C}$-algebra homomorphism, continuous for ... | 2 | https://mathoverflow.net/users/13265 | 295056 | 129,725 |
https://mathoverflow.net/questions/294949 | 1 | In [this paper](http://www.ams.org/journals/proc/2011-139-04/S0002-9939-2010-10523-3/S0002-9939-2010-10523-3.pdf) E. Formanek says: ``The purpose of this paper is to point out a connection between certain differential equations which have arisen in attempts to establish the two-variable Jacobian Conjecture and the work... | https://mathoverflow.net/users/72288 | Connection between the Jacobian Conjecture and number theory conjectures | An interesting conjecture that has relation with the Jacobian Conjecture is the following:
>
> Let $(\mathcal{O},\mathcal{M},k)$ be a complete discrete valuation ring with k finite.Let $f\_{1},...,f\_{n} \in \mathcal{O}[X\_{1},...,X\_{n}]$
> with the jacobian condition: $\det J\_{f} = 1$. Consider the scheme $$X ... | 6 | https://mathoverflow.net/users/100511 | 295063 | 129,728 |
https://mathoverflow.net/questions/73033 | 14 | Let $p$ be an odd prime and let $F/\mathbb{Q}$ be a Galois extension with Galois group $D\_{2p}$, let $K$ be the intermediate quadratic extension of $\mathbb{Q}$, and $L$ an intermediate degree $p$ extension:
$\;\;\;F$
$\;\;\;\;|\;\;\;\backslash$
$\;\;\;K$ $L$
$\;\;\;\;|{\tiny 2}\;\;/{\tiny p}$
$\;\;\;\m... | https://mathoverflow.net/users/35416 | Class groups in dihedral extensions - some sort of Spiegelungssatz? | I normally don't like to cite my own work on MO, but this time the preprint [arXiv:1803.04064](https://arxiv.org/abs/1803.04064) was written, together with L. Caputo, having the OP's question in mind; and so, first of all, let me thank Alex for having asked it.
The main result is a purely algebro-arithmetic proof tha... | 4 | https://mathoverflow.net/users/18238 | 295074 | 129,732 |
https://mathoverflow.net/questions/295059 | 3 | Let $k$ be an algebraically closed field and $X, Y$ be quasi-projective $k$-scheme.
Let $X\_1, X\_2$ be two irreducible components of $X$ and $f\_i:X\_i \to Y$ be morphims such that $f\_1|\_{X\_1 \cap X\_2}=f\_2|\_{X\_1 \cap X\_2}$. Then, does the morphisms $f\_1$ and $f\_2$ glue to a morphism $f:X \to Y$ such that its... | https://mathoverflow.net/users/43198 | Glue morphism of schemes defined over irreducible components | This is true in a suitable sense: if the closed subscheme $X\_i$ is given by the ideal $\mathcal I\_i$, then $X\_1 \cap X\_2$ is given by $\mathcal I\_1 + \mathcal I\_2$. In this case, it is true that
$$Y = X\_1 \underset{X\_1 \cap X\_2}\amalg X\_2,$$
where $Y \subseteq X$ is the closed subscheme given by $\mathcal I\_... | 6 | https://mathoverflow.net/users/82179 | 295075 | 129,733 |
https://mathoverflow.net/questions/295071 | 3 |
>
> Given $nD$ finite measures $µ\_1, . . ., µ\_{nD}$ in $\mathbb{R}^n$
> does there exists a set of at most D hyperplanes that bisect each of the measures?
>
>
>
How far is this problem explained?
That is closely related to a conjecture in *L. Barba and P. Schnider, Sharing a pizza: bisecting masses with tw... | https://mathoverflow.net/users/121404 | A conjecture: Given $nD$ finite measures that bisects | The [7-day-old paper](https://arxiv.org/abs/1803.02842) by Alfredo Hubard answers this question positively.
*Update on 16 May*: The paper has been revised. It is now coauthored with Roman Karasev and claims to treat only the case when the dimension is of the form $2^t$.
| 6 | https://mathoverflow.net/users/806 | 295078 | 129,735 |
https://mathoverflow.net/questions/295081 | 0 | Let $r\in[0,1]$. We look at the binary represenation of $r$ and say that $r$ is *binarily universal* if every finite binary string appears in at least one place in the binary representation of $r$. Let $U$ be the set of binarily universal members of $[0,1]$.
Is $U$ a Borel set? If yes, what is its Borel measure?
| https://mathoverflow.net/users/8628 | Binarily universal members of $[0,1]$ | $U$ is a $G\_\delta $ set, hence Borel at level 2 of the Borel hierarchy. It has Lebesgue measure 1 and is also comeager, so it is large both in the sense of measure and of Baire category.
| 3 | https://mathoverflow.net/users/4600 | 295082 | 129,736 |
https://mathoverflow.net/questions/295010 | 5 | In the article: <https://arxiv.org/abs/0906.3217> the authors prove in Lemma 1 a formula which helps compute more easily the integral of the Hessian of a function defined on $\Bbb{S}^2$. More precisely, if $h : \Bbb{S}^2 \to \Bbb{R}$ is a $C^2$ function, $Hess(h)(X,Y) = \langle \nabla\_X \nabla h,Y\rangle$ is the Hessi... | https://mathoverflow.net/users/13093 | Simplification of integral on the sphere | There *is* a formula that is roughly of the kind that the OP desires. On a closed, oriented Riemannian surface $(M,g)$ with Gauss curvature $K$, the formula is
$$
\int\_MuH(u)\,\mathrm{d}A = \frac12\int\_M \left(K u |\nabla u|^2
- \nabla^2(u)(J\nabla u,J\nabla u)\right)\,\mathrm{d}A.
$$
Here, $\nabla^2(u)$ is what the... | 7 | https://mathoverflow.net/users/13972 | 295086 | 129,738 |
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