parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/349840 | 5 | In the first section of J. P. May’s *General algebraic approach to Steenrod operations*, May defines for $\pi\subseteq\Sigma\_r$ an integer $q\in\mathbb{Z}$ and a commutative ring $\Lambda$, the $\Lambda\pi$-module $\Lambda(q)=\Lambda$ with sign action $\sigma\lambda = (-1)^{qs(\sigma)}\cdot \lambda$ where $(-1)^{s(\si... | https://mathoverflow.net/users/124042 | Sign in May’s General algebraic approach to Steenrod operations | @FKranhold You mean I got it right? You had me fooled. I should apologize for leaving that detail to the reader, but let me give two excuses. First, one does not actually need that detail to prove Lemma 1.1(iv). It just answers an obvious question the reader might have about the proof. Second, that paper was from the g... | 9 | https://mathoverflow.net/users/14447 | 350739 | 148,382 |
https://mathoverflow.net/questions/350754 | 12 | The power series $\sum\_{n=1}^\infty \ln(n)z^n$ has radius of convergence $1$ and $z=1$ is a singular point. Is $z=1$ an isolated singularity? If yes, what kind of isolated singularity?
I am only able to deduce that $z=1$ cannot be a pole.
Such type of questions appear naturally when one tries to relate the singula... | https://mathoverflow.net/users/151281 | Singularities of power series | Let $f(z)$ be your function. Then $g(z)=f(z)(1-z)$ is equal to
$$
(1-z)\sum\_n \ln(n)z^n=\sum\_{n\geq 2} (\ln(n)-\ln(n-1))z^n
$$
Now, $\ln(n)-\ln(n-1)=\frac{1}{n}+O\left(\frac{1}{n^2}\right)$, which gives us
$$
g(z)=\sum\_{n≥1} \left(\frac{1}{n}+g\_n\right)z^n,
$$
where $g\_n=\ln(n)-\ln(n-1)-\frac{1}{n}=O\left(\fra... | 18 | https://mathoverflow.net/users/101078 | 350759 | 148,389 |
https://mathoverflow.net/questions/350758 | 3 | Let $X\in L^1(\Omega)$ and $\phi\_X$ the corresponding characteristic function.
[We know that](https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)#Properties): $\phi\_X$ is $n$ times differentiable (at $u=0$) iff $\mathbb{E}[X^n]<\infty$. (This depends a bit on if $n$ is even or odd but that's ... | https://mathoverflow.net/users/151291 | Characteristic function and moments | The answers to your questions are no and no.
Question 1: Let $X$ have the standard double exponential distribution, so that the pdf $p\_X$ of $X$ is given by $p\_X(x)=\frac12\,e^{-|x|}$ for real $x$. Then the characteristic function $f\_X$ of $X$ is in $C^\infty$, since $f\_X(t)=\frac1{1+t^2}$ for real $t$. However,... | 4 | https://mathoverflow.net/users/36721 | 350763 | 148,391 |
https://mathoverflow.net/questions/350571 | 8 | It seems that much of the literature in stable homotopy theory seems to study complex orientable cohomology theories. What is the reason of restricting to this class of multiplicative cohomology theories? Is it simply that they are more computable? Is there a good a priori reason that this is an important class of coho... | https://mathoverflow.net/users/136287 | Why do we study complex orientable cohomology theories | There is a sort of *a priori* reason why one would consider the cohomology theory $MU$, without first knowing of its connection to manifold geometry, to formal groups, … .
Since complex-oriented cohomology theories are those cohomology theories with a ring map from $MU$, perhaps having a sufficiently strong interest in... | 24 | https://mathoverflow.net/users/1094 | 350771 | 148,394 |
https://mathoverflow.net/questions/350775 | 2 | This question is mostly about understanding the notation used in the following article:
[Alex Eskin, Andrei Okounkov, *Pillowcases and quasimodular forms*, in: Victor Ginzburg (ed.), *Algebraic Geometry and Number Theory in Honor of Vladimir Drinfeld's 50th Birthday*, Birkhäuser 2006](https://books.google.com.au/books?... | https://mathoverflow.net/users/45170 | 2-quotient of integer partition | I suspect $\alpha\_i$ and $\beta\_i$ refer to heights in the Russian way to describe Young diagrams (rotate the English notation 135 degrees), this makes it into a piecewise linear function, and this interpretation has a few nice applications.
Information regarding quotients, (and its relation to character values in ... | 2 | https://mathoverflow.net/users/1056 | 350782 | 148,398 |
https://mathoverflow.net/questions/350590 | 10 | Let $S\subset \mathbb{C}$ be a finitely generated ring, let $R$ be a finitely generated commutative ring over $S$. Let $G$ be a linear algebraic group over $S$, such that $G\_{\mathbb{C}}$ is reductive. Suppose that Spec$(R)$ is equipped with a $G$-action over $S$.
In this setting, I hope that the following statement... | https://mathoverflow.net/users/151201 | Lifting $G$-invariants from characteristic $p\gg 0$ to characteristic 0 for a reductive algebraic group $G$ | We offer two facts and a Theorem.
Let $S$ be a commutative noetherian ring containing $\mathbb Z$ and let $G=G\_S$ be
reductive over $S$ in the sense of SGA3. That is, $G$ is smooth over $S$
with geometric fibers that are connected reductive.
Let $R$ be a finitely generated commutative $S$-algebra.
Suppose that ... | 5 | https://mathoverflow.net/users/4794 | 350791 | 148,400 |
https://mathoverflow.net/questions/350809 | 3 | Suppose $K=\mathbb{Q}(X\_1,\dots,X\_n)$ is a purely transcendental extension of the rationals on finitely many indeterminates. Can anyone give an example of a rank $1$ valuation on $K$ that fails to be discrete?
If not, is there a theorem that shows that such a rank $1$ valuation must be discrete?
| https://mathoverflow.net/users/1353 | Rank 1 valuations that are not discrete on finite transcendental extensions of the rationals | Example: $val(X\_1^{e\_1}X\_2^{e\_2})=e\_1 + e\_2 \sqrt{2}$.
| 4 | https://mathoverflow.net/users/59248 | 350813 | 148,408 |
https://mathoverflow.net/questions/350801 | 4 | In [this paper](https://www.ams.org/mcom/2008-77-264/S0025-5718-08-02101-7/) on page 8 the author claims that the Taylor expansion for the expression $e^{tD\_V}$ where $D\_V$ is the Lie derivative with respect to a vector field $V$ (defined by $(D\_VG)(x) = \frac{d}{dt}|\_{t=0}V(\phi^t\_V(x))$ and $\phi^t\_V(x)$ is the... | https://mathoverflow.net/users/146998 | Taylor expansion of exponential of a Lie derivative | Maybe I'm missing something, but it seems to me you're overthinking this. Since
$$
\int\_{0}^{1} d\theta (1-\theta ) \theta^{n} = \frac{1}{n+1} -\frac{1}{n+2} = \frac{n!}{(n+2)!}
$$
the third term in your expression is
$$
\sum\_{n=0}^{\infty } \frac{t^{n+2} D\_V^{n+2} }{n!} \frac{n!}{(n+2)!}
= \sum\_{n=2}^{\infty } \fr... | 10 | https://mathoverflow.net/users/134299 | 350815 | 148,409 |
https://mathoverflow.net/questions/350823 | 3 | Consider a proper geodesic $\delta$-hyperbolic space $X$ (in the sense of Gromov). Let ∂ be its Gromov boundary. In the book "Geometric Group Theory" by Cornelia Druţu and Michael Kapovich <https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf>
page 391, for $k>2\delta$ and $a\in X$, they define the shadow topology $\Im \... | https://mathoverflow.net/users/147032 | Topology on the boundary compactification $X^{-}=\partial X\cup X$ of a Gromov-hyperbolic space | To answer your first question, I'm a bit confused because you seem to be okay with the existence of$\rho$ which is a geodesic asymptotic to $\xi$. Actually, $[a,z]$ is nothing else than a geodesic asymptotic to $z$, if $z$ is in the boundary.
Maybe you need a bit of clarification though. I guess it depends on your de... | 3 | https://mathoverflow.net/users/111917 | 350826 | 148,411 |
https://mathoverflow.net/questions/350810 | 5 | Let $\mathbb S\_\kappa$ be the standard forcing for $\square\_\kappa$ by initial segments. This is $(\kappa+1)$-strategically closed.
>
> **Observation:** Let $T \subseteq \kappa^+$ be stationary. If $T$ concentrates on $\mathrm{cof}({<}\kappa)$,then $\mathbb S\_\kappa$ forces $\diamondsuit(T)$. If $T$ concentrates... | https://mathoverflow.net/users/11145 | Forcing square introduces diamond | Regarding Question 2, the assumption of approachability is necessary (or, more precisely, the assumption that $T$ has a stationary subset that is approachable), at least if we have $2^\kappa = \kappa^+$ in the ground model. The reason is that, if $T$ does not have a stationary subset that is approachable, then it becom... | 5 | https://mathoverflow.net/users/26002 | 350827 | 148,412 |
https://mathoverflow.net/questions/350828 | 12 | My question refers to article 301 of section 5 of Gauss's D. A. - there Gauss gives an asymptotic formula for the mean number of classes of forms with positive discriminant ($D>0$):
$$h(D) = \frac{4}{\pi^2}\log (D) + \delta$$
where $\delta$ is the following constant:
$$\delta = \frac{8}{\pi^2}C+\frac{48}{\pi^4}\s... | https://mathoverflow.net/users/118562 | Question about a lesser-known "class number formula" of Gauss | Concerning the historic significance of Gauss's article 301: [Marius Overholt](https://books.google.nl/books?id=kBsHBgAAQBAJ&pg=PA23&lpg=PA23) traces back to this publication on the growth of class numbers the use of local averaging to study the growth of arithmetic functions.
| 9 | https://mathoverflow.net/users/11260 | 350831 | 148,413 |
https://mathoverflow.net/questions/350676 | 3 | Let $K$ be a finite extension of $\mathbb{Q}$. Then there is a surjective homomorphism $\theta:C\_K\to G\_K^{ab}$ from the idele class group to the abelianization of the absolute Galois group of $K$ (considered as a topological group) given by class field theory. The kernel of $\theta$ contains (the cosets of) the sequ... | https://mathoverflow.net/users/nan | The kernel of the global class field theory homomorphism | Well, actually the kernel of $\theta$ is perfectly explicit and it is the connected component of the identity in $C\_K$: see, for instance, Artin-Tate *Class Field Theory*, Chapter IX, §1. Theorem 3 *ibid* says
**Theorem 3** The structure of the connected component of $C\_K$ is that of a direct product of one real li... | 2 | https://mathoverflow.net/users/18238 | 350832 | 148,414 |
https://mathoverflow.net/questions/350173 | 8 | $\DeclareMathOperator\dim{dim}$For a dominant (integral) weight $\lambda$ and any (integral) weight $\mu$ of a simple Lie algebra $\mathfrak{g}$, *Lusztig's $q$-analog of weight multiplicty* $K\_{\lambda,\mu}(q)$ is a $q$-analog of the dimension $\dim V^{\lambda}\_{\mu}$ of the $\mu$-weight space $V^{\lambda}\_{\mu}$ o... | https://mathoverflow.net/users/25028 | Relationship between $q$-Weyl dimension formula and $q$-analog of weight multiplicity? | I think that the best reference for what I mention here is Stembridge's notes ["Kostka-Foulkes Polynomials of General Type"](https://www.aimath.org/WWN/kostka/stembridge.pdf). The picture you describe for type A actually generalizes nicely to all types.
Let's denote by $\Lambda$ the weight lattice, by $\Lambda^{+}$ t... | 5 | https://mathoverflow.net/users/2384 | 350839 | 148,416 |
https://mathoverflow.net/questions/350843 | 9 | It is apparently a result of F. González-Acuña that all closed orientable 3-manifolds contain a fibered knot. (I am not sure exactly where to find a published proof of this result and as an aside I would be interested in hearing about any proofs/references that anyone knows.)
I am wondering to what extent this is tru... | https://mathoverflow.net/users/99414 | Existence of fibered surfaces in arbitrary 4-manifolds? | I suppose that, for a $n$-manifold $M$, containing a "fibered codimension-2 manifold" $N\subset M$ means that $N$ has trivial normal neighbourhood $\nu N = N\times D^2$ and its complement $M \setminus \nu N$ fibers over $S^1$ with a fibration that extends the projection $\partial (\nu N) = N \times S^1 \to S^1$ of its ... | 9 | https://mathoverflow.net/users/6205 | 350844 | 148,418 |
https://mathoverflow.net/questions/350785 | 2 | I have a function $P: \mathbb{R}^n \to \mathbb{R}^n$. This function satisfies:
$$ P(\vec{x}) = J\_P(\vec{x}) \cdot \vec{x}$$
where $\vec{x}\in \mathbb{R}^n$, $J\_P$ is the Jacobian of $P$ and "$\cdot$" is
the matrix-vector product. I would rougly describe it as in the title.
I guess that it is a well known property. ... | https://mathoverflow.net/users/138060 | A function in $\mathbb{R}^n$ is equal to its linearization in each point | As for the symplectic case. Let $J$ be the symplectic matrix $J:=\left[ \matrix{ 0 & I\_n \\ -I\_n & 0 }\right]$. The characteristic lines for the (first order, linear, partial differential, vector) equation
$$P(x)=-J \,{\rm d}P(x)J x, \qquad x\in\mathbb{R}^{2n}\setminus\{0\},$$
are the solutions of the ODE $\dot \xi=... | 3 | https://mathoverflow.net/users/6101 | 350855 | 148,422 |
https://mathoverflow.net/questions/350595 | 5 | Suppose I have $N$ pairs of positive numbers $(a\_1, b\_1), (a\_2, b\_2), \dotsc, (a\_N, b\_N).$ and I want to find a subset of $M$ of them maximizing
$$
\frac{\sum\_{j=1}^M a\_{i\_j}}{\sum\_{j=1}^M b\_{i\_j}}.
$$
Can this be done in polynomial time?
| https://mathoverflow.net/users/11142 | Optimization algorithm sought | The paper ["Dropping Lowest Grades"](http://cseweb.ucsd.edu/~dakane/droplowest.pdf) by Daniel M. Kane and Jonathan M. Kane addresses this question in the context of dropping $r$ quiz grades from a collection of weighted grades.
The solution described there is fundamental the same as that in [David Eppstein's answer](... | 2 | https://mathoverflow.net/users/1079 | 350871 | 148,426 |
https://mathoverflow.net/questions/350284 | 1 | Let $\Sigma$ be a finite alphabet, and consider the free monoid $\Sigma^\*$. Given $w, w' \in \Sigma^\*$ we say that $w$ *overlaps* $w'$ if there exist non-empty words $u, v, u'$ such that $w = uv$ and $w' = vu'$. Given a finite set of words $S$, define the overlap graph $OG(S)$ to be the simple graph with vertex set $... | https://mathoverflow.net/users/145915 | Cliques in overlap graphs for words | At least one of the questions admit counterexamples. Namely, let $\Sigma = \{a, b, c, d \}$, and $S = \{ ab^nc d ab^mc : n > m \}$. Then all words in $S$ are Lyndon, $OG(S)$ is triangle-free and has infinite chromatic number. I can give more details if anyone is interested.
| 1 | https://mathoverflow.net/users/145915 | 350875 | 148,427 |
https://mathoverflow.net/questions/350373 | 2 | Is there tight estimates for the following logarithmic summation ($\gamma,\gamma'\in(0,1)$ and $\mu,\mu'>0$)
$$\log\_2\Bigg(\sum\_{t=\frac{n^{}}2-n^\gamma\sqrt{\mu\ln n}}^{\frac{n^{}}2+n^\gamma\sqrt{\mu\ln n}}\quad\sum\_{\ell=\frac{n^{}}2-n^\gamma\sqrt{\mu\ln n}}^{\frac{n^{}}2+n^\gamma\sqrt{\mu\ln n}}\quad\sum\_{k=\f... | https://mathoverflow.net/users/136553 | Tight sublinear estimates for a triple partial binomial summation | This conjecture does not hold even in the case $\gamma=\gamma'=1/2$.
Indeed, consider the values of $\ell,t,k$ such that
$$|\ell-n/2|\ll\sqrt n,\ |t-n/2|\ll\sqrt n,\ |k-t/2|\ll\sqrt n,$$
where $A\ll B$ or, equivalently, $B\gg A$ means that $|A|\le CB$ for some universal real constant $C>0$; as usual, $A\asymp B$ me... | 1 | https://mathoverflow.net/users/36721 | 350876 | 148,428 |
https://mathoverflow.net/questions/350872 | 13 | Let $R$ be a ring such that all of its elements have a finite number of divisors, ie $\forall r\in R\, |\{x\in R: x|r\}|<\infty$.
Then we can decide whether a polynomial in $R[t]$ is reducible through [Kronecker's method](https://www.encyclopediaofmath.org/index.php/Kronecker_method).
Even in the ring $\mathbb{Q}[x,\... | https://mathoverflow.net/users/146338 | Is there a ring for which the reducibility of a polynomial is undecidable? | Yes, but the answer is a bit unsatisfying. This answer is a summary of the very nice paper [Computable Fields and Galois Theory](https://qcpages.qc.cuny.edu/%7Ermiller/Notices.pdf), Russel Miller, *Notices of the AMS*, 2008.
First of all, if one could not even compute with the elements of the ring $R$ at all, it woul... | 23 | https://mathoverflow.net/users/297 | 350877 | 148,429 |
https://mathoverflow.net/questions/350686 | 4 | Let $A\rightarrow D\leftarrow C$ a diagram of connected pointed toplogical space where $A\rightarrow D$ is a fibration. Denote $P=A\times\_{D}C$. We obtain a homotopy fiber sequence $$ \Omega D\rightarrow P\rightarrow A\times C $$
If we suppose that $D=\Omega X$ for some pointed topological space $X$. Do we obtain a ... | https://mathoverflow.net/users/136909 | pullback and fiber sequence | Yes. Here are some details.
1. The space $P$ sits in homotopy pullback diagram
$\require{AMScd}$
$$
\begin{CD}
P @>>> D \\
@VVV@VVV \\
A\times C @>>> D\times D
\end{CD}
$$
where the the right vertical map is the diagonal. In fact, one can see this by replacing the latter map with the free path fibration $D^I \to D\ti... | 5 | https://mathoverflow.net/users/8032 | 350884 | 148,430 |
https://mathoverflow.net/questions/350890 | 1 | In arithmetic geometry one often encounters continuous representations $\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \operatorname{GL}\_n(\mathbb{Q}\_l)$ for some $n\geq 1$ and some prime number $l$ such that no Tate twist of $\rho$ factors through $\operatorname{GL}\_n(K)$ for any number field $K$. An exampl... | https://mathoverflow.net/users/nan | Galois representations with trivial determinant that do not factor through a number field | Since you mention the term "de Rham" in your question, you are clearly aware of the existence of p-adic Hodge theory; so I am surprised that you do not realise that this theory allows you to write down examples without any effort at all.
The point is that when a representation arises naturally in geometry, then you c... | 5 | https://mathoverflow.net/users/2481 | 350894 | 148,433 |
https://mathoverflow.net/questions/350881 | 14 | It is a well-known result on group theory that if a group has *many pairs of commuting elements* then it is abelian.
This motivated the following pseudo-conjecture.
>
> If a (possibly infinite) set $S$ with a binary operation $\cdot$ is
> such that for *many* triples $a,b,c\in S$ it holds $(a\cdot b)\cdot c
> =... | https://mathoverflow.net/users/147861 | Associativity may fail by little? | The result you quoted appears in this reference: G. Szasz, *Die Unabhängigkeit der Assoziativitätsbedingungen*, Acta. Sci. Math. Szeged 15 (1953), 20-28.
The Szasz theorem requires that the set $S$ have at least four elements, though it is also true for sets of size $3$.
Szasz' proof is constructive and goes as fol... | 15 | https://mathoverflow.net/users/1392 | 350895 | 148,434 |
https://mathoverflow.net/questions/350678 | 1 | The answer of this question might be known but I was not able to find any answer. Let $n\geq 1$ and $S$ be a closed submonoid of $(\mathbb{C}^\*)^n$, that is, a closed and stable by product subset of $(\mathbb{C}^\*)^n$ which also contains the unit $(1,\ldots,1)$. Let also $R\_1,\ldots,R\_d\geq1$, and denote by $A:=\{(... | https://mathoverflow.net/users/142808 | Closed submonoid of $(\mathbb{C}^*)^n$ | My idea in the above comments didn't quite work as stated, but the generality of the result Yemon Choi mentioned bridges the gap.
Define $\log^n |\cdot|: \mathbb{C}^{\*n} \rightarrow \mathbb{R}^n, \log^n |(z\_1, z\_2, \dots, z\_n)| = (\log |z\_1|, \log |z\_2|, \dots, \log |z\_n|)$. This is a map of topological group... | 1 | https://mathoverflow.net/users/44191 | 350905 | 148,438 |
https://mathoverflow.net/questions/350904 | 12 | Let $\mathcal C$ be the free monoidal category generated by an object $X$, and a morphism $X \otimes X \to X$.
This category contains exactly two connected components: that of the monoidal unit $1\in \mathcal C$, and that of $X\in \mathcal C$. (In general, two object $A$ and $B$ of a category are said to be in the sa... | https://mathoverflow.net/users/5690 | Classifying space for Thompson's group F? | What you describe is the so called Squier complex of the semigroup presentation
$\langle x \mid x^2=x\rangle$ (you did not describe the 2-cells, but it is straightforward). The fact that its fundamental group is $F$ was proved by Guba and myself in 1997, "Diagram groups", Memoirs of the AMS,
November, 1997 ([link](http... | 21 | https://mathoverflow.net/users/nan | 350908 | 148,439 |
https://mathoverflow.net/questions/350903 | 1 | Let $T\_1, \ldots, T\_n$ by real symmetric positive definite matrices, with eigenvalues bounded below by $\mu > 0$.
Can I say
$$
\frac{x^T T\_1 T\_2 \ldots T\_n x}{x^T x} \geq \mu^n
$$
If these matrices commute the result is straightforward, but I'm interested in the case where these matrices don't necessarily commute.... | https://mathoverflow.net/users/151366 | Eigenvalues of product of symmetric positive definite matrices | It is true that the product $M=T\_1T\_2$ of two positive definite symmetric matrices has real and positive eigenvalues. And conversely, every matrix $M$ with real positive eigenvalues can be factored $M=T\_1T\_2$ as above. But $x^TMx$ does not need to be positive. Here is an example:
$$M=\begin{pmatrix} 3 & a \\ -a & -... | 3 | https://mathoverflow.net/users/8799 | 350912 | 148,440 |
https://mathoverflow.net/questions/350909 | 1 | Let $X\_1,\ldots, X\_n$ be independent Rademacher random variables (i.e. $\mathbb{P}(X\_i=\pm 1)=1/2$). Consider the random polynomial $$P\_{n}(t)=c+X\_{1}t+X\_2t^2+\cdots+X\_{n}t^n.$$
Is it well known how to get good upper bounds on probabilities of type
$$\mathbb{P}(|\max\_{t\in [0,x]}|P\_{n}(t)|-\mathbb{E}\max\_{... | https://mathoverflow.net/users/24494 | Concentration of maxima of a random polynomial with Rademacher coefficients | Theorem 5.3.2 in [Talagrand's book](https://link.springer.com/book/10.1007/978-3-642-54075-2) states the following (using here somewhat different notations):
>
> Let $X\_1,X\_2,\dots$ be independent Rademacher random variables. Let $U$ be a subset of the closed ball $B(u\_0,s)$ in $\ell^2$ centered at some $u\_0\i... | 2 | https://mathoverflow.net/users/36721 | 350914 | 148,441 |
https://mathoverflow.net/questions/350764 | 2 | I have several questions about geometric morphisms of topoi. It was recommended that I move my question here from Math.Stackexchange, since it would be good to get an expert on topos theory to answer.
1) A geometric morphism $g \dashv f : C \rightarrow D$, with $f : C \rightarrow D$ is called strongly connected if $g... | https://mathoverflow.net/users/30211 | Questions about Geometric Morphisms | 1. I don't know of any significance to this notion. I would explain this by the fact that the notion of topos is not self-dual. Finite limits (including finite products) are an integral part of the notion of topos (and geometric morphism), but finite colimits are not; so we can't expect to always get a meaningful notio... | 3 | https://mathoverflow.net/users/49 | 350916 | 148,443 |
https://mathoverflow.net/questions/350911 | 6 | Fix $\alpha \in \mathbf{R}$. The classical [Dirichlet's approximation theorem](https://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem) states there exist infinitely many rationals $p/q$ such that
$$
\left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2}.
$$
>
> **Question.** Fix $\alpha \in \mathbf{R}$. Is it tru... | https://mathoverflow.net/users/32898 | Sign in Dirichlet's approximation theorem | Yes, this follows from considering the continued fraction of $\alpha$. If $p\_n/q\_n$ is the $n$th convergent to $\alpha$ and $n$ is odd then
$$ 0\leq \alpha - \frac{p\_n}{q\_n} \leq \frac{1}{q\_n^2}.$$
| 10 | https://mathoverflow.net/users/385 | 350918 | 148,444 |
https://mathoverflow.net/questions/350898 | 2 | I am interested in the Allen-Cahn equation in $ R^N$ and one can consider the related energy functional
$$ E(u):= \frac{1}{2}\int\_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int\_{R^N} (u^2-1)^2dx.$$ There has been a lot of work on this equation and in particular on the DeGiorgi conjecture. My question is related to wheth... | https://mathoverflow.net/users/66623 | Finite energy solution for Allen -Cahn equation | Sketch of an argument:
For a fixed $x'$, let $f(x')$ denote the measure of the set $\{ u(x',x\_9) \in (-1/2,1/2) \}$.
Note that for fixed $x'$ you have
$$ \int |\nabla u(x',x\_9)|^2 d x\_9 \geq \int |\partial\_{x\_9} u(x',x\_9)|^2 d x\_9 \geq 1 / f(x') $$
(on the ends of the interval (it is an interval by monoton... | 1 | https://mathoverflow.net/users/3948 | 350920 | 148,445 |
https://mathoverflow.net/questions/350913 | 16 | Let $B \leftarrow A \to C$ be a span of spaces, and consider the homotopy pushout $B \cup\_A C$.
**Question:** When is $B \cup\_A C$ contractible?
This is a pretty open-ended question. I'm interested in necessary conditions or sufficient conditions or interesting examples or special cases, etc.
**Some additional ... | https://mathoverflow.net/users/2362 | When is a homotopy pushout contractible? | Assume all the spaces are connected, $\pi\_1 A \to \pi\_1 B$ is surjective, and $\pi\_1 C$ is abelian.
The pushout, $P = C \sqcup\_A B$, is contractible if and only if $\pi\_1(P)$ and all the $H\_i(P)$ are trivial. However, by Seifert-Van-Kampen, and the hypothesis on $\pi\_1 A \to \pi\_1 B$, we have that $\pi\_1 C$... | 20 | https://mathoverflow.net/users/52918 | 350921 | 148,446 |
https://mathoverflow.net/questions/350752 | 6 | In an answer to this question: [Enriched slice categories](https://mathoverflow.net/questions/259829/enriched-slice-categories), a description of the enrichment of the slice category in an enriched category is given. I'm interested in going a bit further. If we assume that the original enriched category, say $\mathcal{... | https://mathoverflow.net/users/11546 | (Co)tensoring of enriched slice categories | Your guess for the copower (née tensor) is correct. You can check it by giving it the right universal property with respect to the enriched homs as described in the question you linked. Similarly, for the power (née cotensor) you can check that
$g^K$ for $g:B\to X$ can be defined by the pullback of the induced map $B^... | 3 | https://mathoverflow.net/users/49 | 350922 | 148,447 |
https://mathoverflow.net/questions/350924 | 0 | Let $X$ be a Banach space non reflexive and $T$ from $l\_2(X)$ to $l\_2(X)$ a bounded operator defined by:
$$T(x\_n )=\frac{x\_n }{n}.$$
We know that :
$$T^{\*\*-1}(l\_2(X))=\{x\_n^{\*\*} \in l\_2(X^{\*\*}) : \frac{x\_n^{\*\*}}{n} \in l\_2 (X)\}.$$
To prove that $T$ is Tauberian, it suffices to prove that $T^{\*\*−1}(l... | https://mathoverflow.net/users/148688 | Tauberian operators | You can proceed as follows. Let $(x\_n^{\*\*})\in \ell\_2(X^{\*\*})$. Then
$$(x\_n^{\*\*})\in T^{\*\*-1}(\ell\_2(X))\Rightarrow T^{\*\*}(x\_n^{\*\*})=
(\frac{x\_n^{\*\*}}{n})\in \ell\_2(X),$$
hence $(x\_n^{\*\*})\in \ell\_2(X)$.
| 0 | https://mathoverflow.net/users/39421 | 350929 | 148,449 |
https://mathoverflow.net/questions/350900 | 4 | Let $G\_1=\operatorname{GL}\_2(\mathbb C)$ act on $V\_1=\mathbb C^2$ via the standard multiplication. Denote this representation by $\operatorname{St}\_2$. Let $G\_2=\operatorname{SL}\_3(\mathbb C^3)$ act on $V\_2=\mathbb C^6$ via $\operatorname{Sym}^2(\mathbb C^3)$. Then we have the representation of $G\_1\times G\_2$... | https://mathoverflow.net/users/13466 | Orbits of tensor product $\operatorname{St}_2\otimes\operatorname{Sym}^2(\mathbb C ^3)$ | Yes, the number of orbits is finite. Indeed, as Abdelmalek mentioned, this is the question of classification of pencils of conics. The orbits are the following.
First, assume that two conics in the pencil are non-proportional.
1) Assume that at least one of the conics in the pencil is nondegenerate. Then this conic i... | 8 | https://mathoverflow.net/users/4428 | 350933 | 148,450 |
https://mathoverflow.net/questions/350874 | 2 | (Moved from math.stackexchange.)
Is the following proposition true?
>
> Given a multidifferential operator $D$ on $\mathbb{R}^n$ with constant coefficients, i.e. for all functions $f\_1,\dots,f\_k \in C^\infty(\mathbb{R}^n)$ we have
> \begin{align\*}
> D(f\_1,\dots,f\_k)(x) := \sum\_{\alpha\_1,\dots,\alpha\_k} c... | https://mathoverflow.net/users/126256 | Multidifferential operators with vanishing integrals | Denote the space of differential operators $D$ you consider by $\mathcal D^k$, and the subspace where $c\_{\alpha\_1\dots\alpha\_k} = 0$ for $\alpha\_1\neq 0$ by $\mathcal D^k\_{lin}$, i.e. which are $C^\infty(\mathbb R^n)$-linear in $f\_1$. The functional
$$
I:\mathcal D^k\to \operatorname{Hom}(C^\infty\_c(\mathbb R^n... | 1 | https://mathoverflow.net/users/35687 | 350948 | 148,453 |
https://mathoverflow.net/questions/346432 | 2 | I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ${y} \in \mathbb{R}^m$, the image $g(\mathbb{R}^n \times \{y\})$ is an $n$-dimensional $C^1$ manifold, and, similarly, f... | https://mathoverflow.net/users/91442 | A Curved/Warped Version of Fubini's Theorem | I have the answer here: [Fubini's Theorem on Arbitrary Foliations](https://mathoverflow.net/questions/350952/fubinis-theorem-on-arbitrary-foliations)
$$\int\_U f = \int\_{U\_{\eta\_0}} \left(\int\_{U\_\xi} f(\xi,\eta) \frac{|\det DG\_{U\_\xi} (\xi,\eta)| \cdot |\det DG\_{U\_{\eta\_0}} (\xi,\eta\_0)|}{|\det DG(\xi,\et... | 0 | https://mathoverflow.net/users/91442 | 350953 | 148,455 |
https://mathoverflow.net/questions/350321 | 3 | Let $G$ be an undirected graph with no multiple edges or loops. Recall that the independece system $\mathcal{I}(G)$ consists of all those subsets $A$ of the vertex set such that the induced subgraph $G[A]$ is totall disconnected.
The independence system is an abstract simplicial complex and a lot of its topological in... | https://mathoverflow.net/users/7494 | Generalization of independence complex of graphs | Although I am not sure if these concepts been studied much as simplicial complexes, at least the second has been studied quite extensively in graph theory and theoretical computer science (including by me).
The earliest studies I am aware of are
Sampathkumar. Discrete Math., 1993. <https://doi.org/10.1016/0012-365X... | 3 | https://mathoverflow.net/users/9137 | 350959 | 148,458 |
https://mathoverflow.net/questions/350960 | 3 | Which positive integers $n$ can be expressed in the form $n = ac + bd$, where $a,b,c,d$ are positive integers and satisfy $\gcd(a,b) = \gcd(c,d) = 1$? I know that all numbers that are not divisible by 4 or by a prime $p$ of the form $p = 4m + 3$ can be represented this way due to the "primitive" sum of two squares theo... | https://mathoverflow.net/users/151406 | Express an integer as ac + bd | More elementarily, letting a=d=1 turns the problem into representing numbers as b+c with b and c larger than 1, giving all numbers greater than 3 as possibilities.
If only d is allowed to be 1, then let bd be one of 2,4, or 6. For odd n bigger than 9, n-bd is odd and composite,for one of these choices and often not a... | 3 | https://mathoverflow.net/users/3402 | 350965 | 148,461 |
https://mathoverflow.net/questions/350976 | 4 | Let $(M,J,\omega)$ be a symplectic manifold with a compatible almost complex structure, $D$ be the closed unit disk in $\mathbb{C}$, and $u:(D,i)\to (M,J)$ be a $(J,i)$-holomorphic map.
*Question*: Assume $u|\_{\partial D}$ is constant, does this imply $u$ is a constant map?
| https://mathoverflow.net/users/119189 | Pseudo-holomorphic disk which is constant along boundary | Extend $u$ to get a $C^1$ pseudoholpmorphic map defined on $\mathbb{C}$ by setting $u$ constant outside the unit disc. It's $C^1$ because you know the derivative of $u$ along the unit circle vanishes (by assumption), so the Cauchy-Riemann equations satisfied by $u$ on the disc tell you that $du$ vanishes along the unit... | 6 | https://mathoverflow.net/users/10839 | 350978 | 148,465 |
https://mathoverflow.net/questions/350980 | 9 | The following expression is known as Mehta's integral and deeply connected to random matrix theory:
$$\frac{1}{(2\pi)^{n/2}}\int\_{-\infty}^{\infty} \cdots \int\_{-\infty}^{\infty} \prod\_{i=1}^n e^{-t\_i^2/2}
\prod\_{1 \le i < j \le n} |t\_i - t\_j |^{2 \gamma} dt\_1 \cdots dt\_n =\prod\_{j=1}^n\frac{\Gamma(1+j\gam... | https://mathoverflow.net/users/150564 | Scaling in Mehta's integral | Yes, this follows by the [de la Vallée-Poussin necessary and sufficient condition for the uniform integrability](https://www.stat.berkeley.edu/~bartlett/courses/2013spring-stat210b/notes/5notes.pdf). Indeed, suppose that
\begin{equation}
\gamma n^2\to a
\end{equation}
(as $n\to\infty$) for some real $a\ge0$.
Your in... | 10 | https://mathoverflow.net/users/36721 | 350984 | 148,466 |
https://mathoverflow.net/questions/350981 | 6 | Let $X$ be a nice enough topological space, perhaps a complex algebraic variety with its analytic topology. I'm hoping someone could help me prove that the category $\text{Constr}(X)$ of constructible sheaves on $X$ is abelian.
I think one could use that subsheaves and quotients of local systems are local systems. H... | https://mathoverflow.net/users/105661 | Prove category of constructible sheaves is abelian | [Verdier has proved](https://eudml.org/doc/142424) (for multiple classes of spaces, including in particular complex varieties) that for any finite set of analytic subsets, there exists a Whitney stratification for which all these analytic subsets are unions of strata.
In particular, given two constructible sheaves, a... | 12 | https://mathoverflow.net/users/18060 | 350987 | 148,469 |
https://mathoverflow.net/questions/350869 | 3 | In **6.5** of the book by *Kelly*,
>
> **Basic concepts of enriched category theory**. Reprints in Theory and Applications of Categories, No. 10, 2005.
>
>
>
the author claims that the $2$-category $\mathsf{Cat}\_{\mathcal{F}}$ of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors is i... | https://mathoverflow.net/users/104432 | The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors | Let $\mathcal{V}$ be a complete and cocomplete closed symmetric monoidal category. Let $\mathcal{F}$ be a small set of indexing types aka weights (these are just $\mathcal{V}$-functors from a $\mathcal{V}$-category to $\mathcal{V}$) and $\mathsf{cat}\_{\mathcal{F}}$ denote the $2$-category of essentially small $\mathca... | 3 | https://mathoverflow.net/users/2841 | 350997 | 148,470 |
https://mathoverflow.net/questions/351000 | 0 | I am trying to solve the following equation:
$(a\*n + c) \mod (b-n) \equiv 0$
and $n$ must be the lowest value in $[0, b-1]$
for example $a=17$, $c=-59$ and $b=128$, the solution is $n=55$
$n=b-1$ will be always a solution, because $m \mod 1 \equiv 0$
| https://mathoverflow.net/users/148749 | Solve congruence equation where unknown variable is in both sides of congruent operator | Equivalently, you want to solve in integers
$$ y (b-n) = an+c$$
This is equivalent to
$$ (y+a)(b-n) = ab+c $$
You want $b-n$ to be as large as possible subject to $b-n \le b$. Thus you want to
factor $ab+c$ and take its largest divisor $\le b$. In your example, $ab+c = 2117 = 29 \cdot 73$ whose largest divisor $\le 12... | 2 | https://mathoverflow.net/users/13650 | 351013 | 148,476 |
https://mathoverflow.net/questions/351008 | 1 | Let $\mathcal{A}, \mathcal{B}$ be C\*-algebras. A map $\phi \colon \mathcal{A} \rightarrow \mathcal{B}$ is *completely positive* (cp) if it's linear, \* preserving and all of its' coordinatewise extensions to matrices $\phi^{(n)} \colon M\_n(\mathcal{A}) \rightarrow M\_n(\mathcal{B})$ map positive matrices to positive ... | https://mathoverflow.net/users/147609 | When do completely positive maps have a closed image? | This is not a complete answer to the original question, since the original question has a rather open-ended phrasing; but I think it addresses Diego's main points, and shows that even quite well-behaved cp maps won't have the desired properties.
First of all: if we restrict attention to cases where either $A$ or $B$ ... | 5 | https://mathoverflow.net/users/763 | 351015 | 148,477 |
https://mathoverflow.net/questions/351016 | 10 | Let $M$ be a countable model of $ZFC$ and $M[G]$ be a (set) generic extension of $M$. Suppose $N$ is a countable model of $ZF$ with
$$M\subseteq N \subseteq M[G]$$
and that $N=M(x)$ for some $x\in N$; i.e., it is the smallest inner model of $M[G]$ which contains $x$ and $M$.
Is $N$ a symmetric extension of $M$?
... | https://mathoverflow.net/users/9324 | Models of ZF intermediate between a model of ZFC and a generic extension | Yes, if $N=M(x)$ (taking the modern notation over Grigorieff's $M[x]$), then it is a symmetric extension. This is a very recent result of Toshimichi Usuba (see [this](https://arxiv.org/abs/1904.00895) and [that](https://arxiv.org/abs/1912.10246)).
However, by the Bristol model construction, this symmetric extension n... | 10 | https://mathoverflow.net/users/7206 | 351018 | 148,478 |
https://mathoverflow.net/questions/350993 | 6 | Let $X$ be a compact manifold. Denote by $\mathscr{D}^\prime(X \times X)$ the space of tempered distributions on the cartesian product $X \times X$. Given two test functions $\varphi, \psi \in \mathscr{D}(X)$, an element $T \in \mathscr{D}^\prime(X \times X)$ can be evaluated at the function $\varphi \otimes \psi$ on $... | https://mathoverflow.net/users/16702 | Smoothness of family of distributions | Here is a proof using convenient analysis:
By the kernel theorem $\mathscr{D}^\prime(X \times X) = L(\mathscr{D}(X),\mathscr{D}^\prime(X))$ and by
[The Convenient setting of Global Analysis](https://www.mat.univie.ac.at/~michor/apbookh-ams.pdf), 5.18 (which is just the uniform boundedness principle) and 2.14 we have
... | 3 | https://mathoverflow.net/users/26935 | 351023 | 148,480 |
https://mathoverflow.net/questions/351019 | 18 | It is known that adding two numbers and looking at the carrying operation has a link with cocycles in group theory. (<https://www.jstor.org/stable/3072368?origin=crossref>)
When we add two numbers by elementary addition, we choose a basis $b$ for example $b=2$ which corresponds to the cyclic group $C\_2$.
Suppose we ... | https://mathoverflow.net/users/nan | How to add two numbers from a group theoretic perspective? | I think the point is that, forgetting the final carry, the group of $n$-digit binary words is isomorphic to $C\_{2^n}$. In the simplest case, the group of 2-digit binary words is isomorphic to $C\_4$, which is built as a nontrivial extension
$$ 0 \to C\_2 \to C\_4 \to C\_2 \to 0 $$
The 2-cocycle you mention is the one ... | 22 | https://mathoverflow.net/users/5279 | 351025 | 148,482 |
https://mathoverflow.net/questions/351031 | 2 | Consider a 1-Lipschitz function $f: \mathbb R^n \to \mathbb R$ satisfying the inequality
\begin{align\*}
|f(x) - f(y)| \le \|x-y\|\_2, \;\forall x,y \in \mathbb R^n.
\end{align\*}
For $n \ge 2$, can we find a 1-Lipschitz function that saturates the above inequality *on the average*?
To make the notion of "on the aver... | https://mathoverflow.net/users/36687 | Lipschitz functions that saturate the Lipschitz inequality on the average (part 1) | There is no $1$-Lipschitz function $f\colon \mathbb R^n \to \mathbb R$ such that
$$\mathbb E|f(x) - f(y)| \asymp \sqrt{n}.$$
Indeed, for any such function, by the Gaussian concentration for Lipschitz functions (see e.g. [Theorem 2.4, page 31](https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015... | 4 | https://mathoverflow.net/users/36721 | 351032 | 148,486 |
https://mathoverflow.net/questions/350979 | 3 | Given $n$ points on a connected $2$-manifold $M$, I'd like to consider the homotopy classes of paths that "permute" these points.
**Edit** (Clarifying what I mean by this):
Given a set of $n$ distinct points $T=\{x\_{1},\ldots,x\_{n}\}\subset M$, to each point we assign a continuous simple curve $\gamma\_{i}:[0,1]\... | https://mathoverflow.net/users/135817 | Permuting $n$ points in a $2$-manifold | The comments seem to have answered questions 1 and 2i, to show that the group $\operatorname{Mot}\_n(M)$ is indeed the surface braid group $B\_n(M)$.
To answer 2ii, consider a disk $D \subset M$ such that $T \subset D$. Then the inclusion map $D \hookrightarrow M$ induces a homomorphism $B\_n \to B\_n(M)$, so the rel... | 3 | https://mathoverflow.net/users/123931 | 351034 | 148,487 |
https://mathoverflow.net/questions/351029 | 12 | Let $k$ be a field.
A folklore theorem states that dg-categories (over $k$), $A\_{\infty}$-categories (over $k$) and stable ($k$-linear) $(\infty, 1)$-categories are "the same" (see for example
[Stable infinity categories vs dg-categories](https://mathoverflow.net/questions/114251/stable-infinity-categories-vs-dg-ca... | https://mathoverflow.net/users/59235 | "Sameness" of dg and A-infinity categories | By Corollary 9.2.1 in the paper <https://arxiv.org/abs/1410.5675>
the model category of small A\_∞-categories
is Quillen equivalent to the model category of small categories
(with a fixed set of objects for simplicity, but see
also Proposition 9.2.3 for the general case),
where both types of categories are enriched ove... | 8 | https://mathoverflow.net/users/402 | 351044 | 148,490 |
https://mathoverflow.net/questions/351061 | 4 | Let $K$ be a finite extension of $\mathbb{Q}\_p$, and $V$ be a Banach algebra over $K$, then what is the $K$-analytic space corresponding to $V$? What is the definition of $K$-analytic space? This is mentioned in the first paragraph on page 6 in [Laurent Berger's paper](https://arxiv.org/pdf/1405.5430.pdf).
Similarly... | https://mathoverflow.net/users/nan | How does an analytic space correspond to a $p$-adic Banach space | Without reading much beyond the pages adjacent to page 6, but looking at [ST3] (especially seeing the references [BGR] and [NFA]), I believe Berger-Colmez are using $K\_n$-analytic spaces in the sense of Berkovich. Without rehashing the details here, the idea is to extend the ideas introduced by Tate's rigid analytic g... | 6 | https://mathoverflow.net/users/39777 | 351069 | 148,496 |
https://mathoverflow.net/questions/351050 | 9 | **Notation and Setting**: let $\operatorname{Aff}$ denote the category of affine schemes whose objects are covariant representable functors $\operatorname{X}:\operatorname{Ring}\rightarrow\operatorname{Set}$ and $\operatorname{Spec}:\operatorname{Ring^{op}}\rightarrow\operatorname{Func(Ring, Set)} $ be the contravarian... | https://mathoverflow.net/users/142626 | Simple examples of colimits of affine schemes (evaluated in the presheaf category) which are not affine schemes | A basic standard example is the colimit of $\mathbb{A}^0 \to \mathbb{A}^1 \to \mathbb{A}^2 \to \cdots$ with transition maps $x \mapsto (x,0)$. The $R$-valued points are finite sequences in $R$. This functor is not representable.
More generally, let $A$ be a commutative ring with a sequence of ideals $I\_0 \supseteq I... | 10 | https://mathoverflow.net/users/2841 | 351084 | 148,503 |
https://mathoverflow.net/questions/351096 | 6 | Basically, I am trying to compute something with the Adams spectral sequence (as a toy example). The $E^2$ page reduced to computing $Ext^{s,t}\_{\Gamma} (\mathbb{F}\_2, \mathbb{F}\_2)$, where $\Gamma = \mathbb{F}\_2 x/ x^2$ and $deg(x) =1$. Keeping very close track of degree, I found that $Ext^{s,t}\_{\Gamma} (\mathbb... | https://mathoverflow.net/users/41616 | multiplicative structure of Ext | So, maybe I should format this as an answer.
Consider any pair of objects $A, B$ in an abelian category with, say, enough projective objects. Then, the extension group $Ext^i(A,B)$ is defined as $H^i(Hom(A^{\bullet}, B^{\bullet}))$, where $A^{\bullet}, B^{\bullet}$ are resolutions of objects $A, B$ respectively, and ... | 5 | https://mathoverflow.net/users/33286 | 351101 | 148,507 |
https://mathoverflow.net/questions/351089 | 6 | Let $\mathbb{E}=E\_1\times E\_2\times E\_3$ denote the product of three elliptic curves over $\mathbb{Q}$ of prime level $p$ and consider the $p$-adic Galois representation $$V\_p(\mathbb{E})=H^1\_{et}(E\_{1/\bar{\mathbb{Q}}}, \mathbb{Q}\_p)\otimes H^1\_{et}(E\_{2/\bar{\mathbb{Q}}}, \mathbb{Q}\_p)\otimes H^1\_{et}(E\_{... | https://mathoverflow.net/users/151472 | Functional equation of twisted triple product L-function | There is a functional equation for $L(\mathbb{E} \otimes \chi, s)$, but it relates $L(\mathbb{E} \otimes \chi, s)$ to $L(\mathbb{E} \otimes \bar\chi, 4-s)$. If $\chi$ is not trivial or quadratic, then $L(\mathbb{E} \otimes \chi, s)$ and $L(\mathbb{E} \otimes \bar\chi, s)$ are different functions, so you cannot use this... | 5 | https://mathoverflow.net/users/2481 | 351106 | 148,508 |
https://mathoverflow.net/questions/351095 | 14 | Let $k$ be a subfield of $\mathbb{C}$ and let $f\colon \operatorname{SL}\_n(\mathbb{C}) \rightarrow \operatorname{GL}\_m(\mathbb{C})$ be an algebraic homomorphism such that $f(\operatorname{SL}\_n(k)) \subset \operatorname{GL}\_m(k)$. Question: must $f$ be defined over $k$? In other words, are the matrix entries of $f(... | https://mathoverflow.net/users/151477 | Is a representation of $\operatorname{SL}_n$ defined over a field $k$ if its image is contained in $\operatorname{GL}_n(k)$? | Yes, and in fact something much more general is true Let $X$ and $Y$ be affine varieties defined over a field $k$. If the $k$-points of $X$ are Zariski dense, $X$ is reduced, and $f: X\_{\mathbb C} \to Y\_{\mathbb C}$ sends $X(k)$ to $Y(k)$, then $f$ is defined over $k$.
This was inspired by comments of Martin Brande... | 15 | https://mathoverflow.net/users/18060 | 351131 | 148,516 |
https://mathoverflow.net/questions/350700 | 4 | Let $G$ be a Lie group.
In the book Curvature and Characteristic classes, the author (Johan L. Dupont) mentiones in beginning of chapter 5 the following :
>
> The notion of a topological principal $G$-bundle $\pi:E\rightarrow X$ on a topological space $X$ is defined exactly as in Definition $3.1$ (the definition... | https://mathoverflow.net/users/118688 | Chern -Weil map for topological principal G bundles | I think "Chern–Weil theory" here is being used in a slightly nonstandard way. The main result of the chapter, Theorem 5.5, is really just a topological statement that elements of $H^\*(BG,\Lambda)$ correspond exactly to characteristic classes (Definition 5.1), which are defined to be functorial associations of ($\Lambd... | 3 | https://mathoverflow.net/users/4177 | 351144 | 148,522 |
https://mathoverflow.net/questions/351118 | 3 | Consider a finite abelian group $G$ (I'm mostly interested in $\mathbb{Z}\_2^n$).
For two subsets $A$ and $B$ of $G$, one can form a submatrix of the Fourier transform matrix on $G$ by keeping only the rows corresponding to $A$ and the columns corresponding to $B$.
Assuming $A$ and $B$ have some fixed cardinality,... | https://mathoverflow.net/users/112954 | Large Fourier submatrices with small operator norm | Let's normalise the Fourier matrix ${\mathcal F}$ to have entries of magnitude 1. Then the Fourier submatrix ${\mathcal F}\_{A,B}$ corresponding to two subsets $A,B$ has Frobenius norm $(|A| |B|)^{1/2}$ and rank at most $\min(|A|, |B|)$, hence must have operator norm at least $(|A| |B|)^{1/2} / \min(|A|,|B|)^{1/2} = \m... | 6 | https://mathoverflow.net/users/766 | 351145 | 148,523 |
https://mathoverflow.net/questions/351143 | 2 | What is known about a classification of associative (binary) polynomial functions? First of all, it is interesting in two cases: over Integral domain (or even over field) and over ring of integers modulo $n$.
(A polynomial binary function is function $R \times R \to R$ induced by a polynomial in two variables $P$ ove... | https://mathoverflow.net/users/148161 | Classification of associative polynomial functions | Over an infinite integral domain you can classify all polynomials that satisfy the associativity functional equation. A quick degree consideration of both sides tells you that the polynomial is at most degree $1$ in each variable, so the only answers are $x,y, c+x+y$ and $c\_1(x+c\_2)(y+c\_2)-c\_2$. In fact you can cla... | 3 | https://mathoverflow.net/users/2384 | 351149 | 148,524 |
https://mathoverflow.net/questions/351112 | 2 | Consider a chain complex $C$ and a poset $P$ so that there is a filtration by subcomplexes $C^p$ of $C$ where $p\in P$ in such a way that $p<q$ implies $C^p \leqslant C^q$.
As a second option, consider the situation when $C$ is $P$-graded. Then one can set $C^p = \{ x : |x| < p\}$ and $\overline{C}^p = \{ x : |x|\leq... | https://mathoverflow.net/users/21326 | Poset filtrations | There is recent work on the homotopical algebra of the simplicial analoque of your situation. Lurie defined a P-stratified space as being a space over the Alexandroff space corresponding to P. Very recently in his thesis,
<https://arxiv.org/abs/1908.01366>,
Sylvain Douteau has looked at this from the point of view of... | 2 | https://mathoverflow.net/users/3502 | 351156 | 148,528 |
https://mathoverflow.net/questions/350991 | 7 | Let $F:\mathbf{Comp}\to\mathbf{Set}$ be a continuous functor from the category of compact Hausdorff spaces to the category of sets such that $|Fn|\le\mathfrak c$ for any finite ordinal $n$. The continuity of $F$ means that $F$ preserves limits of inverse spectra.
Typical examples of such a functor $F$ are the functor... | https://mathoverflow.net/users/61536 | The diamond principle for functors | The answer to this question is affirmative and follows from a more general
>
> **Theorem.** Let $X\_{\omega\_1}$ be the limit of a continuous well-ordered spectrum $\langle X\_\alpha,p\_\alpha^\beta:\alpha\le \beta<\omega\_1\rangle$ in the category of sets such that each set $X\_\alpha$, $\alpha<\omega\_1$ has car... | 3 | https://mathoverflow.net/users/61536 | 351164 | 148,531 |
https://mathoverflow.net/questions/351163 | 3 | Writing a paper on algebraic surfaces, I was led to consider the finite group $\mathsf{H}(A)$ whose presentation is the following.
I start with an anti-symmetric matrix $A=(a\_{ij})$ of order $2n$ over $\mathbb{Z}\_p$ (the finite group/field with $p$ elements), $p \geq 3$, and I consider the following system of gener... | https://mathoverflow.net/users/7460 | Direct proof (or reference) that a given $p$-group is extra-special | It is clear from standard properties on commutators that $H(A)$ is nilpotent of class at most $2$ with derived group and Frattini subgroup contained in $\langle z \rangle$
(just consider $H(A)/\langle z \rangle$, which is elementary Abelian of order $p^{2n})$.
The general property of commutators that you need is that... | 6 | https://mathoverflow.net/users/14450 | 351168 | 148,533 |
https://mathoverflow.net/questions/351154 | 7 | Does there exist a measurable subset $T$ of $[0, \infty)$ with finite measure and some $\epsilon > 0$ such that for every $r$ with $0 < r < \epsilon$, $nr$ is in $T$ for infinitely many positive integers $n$?
Note: The integers $n$ such that $nr$ lie in $T$ can depend on $r$.
| https://mathoverflow.net/users/132446 | A trapping set with finite measure | No. Denote $T\_k=T\cap [k,k+1)$. Then $\sum |T\_k|<\infty$ (where $|X|$ stands for the measure of $X\subset \mathbb{R}$). Choose a segment $[a,b]\subset (0,\epsilon)$. Note that if $r\in [a,b]$ and $nr\in T\_k$, then $na\leqslant nr< k+1$ and $nb\geqslant nr\geqslant k$, thus $n\in [k/b,(k+1)/a]$. The union of $n^{-1}T... | 10 | https://mathoverflow.net/users/4312 | 351178 | 148,535 |
https://mathoverflow.net/questions/351160 | -1 | $X$ is a separable, completely metrizable topological space equipped with its sigma algebra of Borel sets. $N$ is a countable space.
$X^N$ is the collection of all mappings from $N$ to $X$. It is equipped with product topology.
>
> 1. Is $N^X$ a Polish space?
> 2. Is $X^N$ a Polish space?
> 3. Is $X^X$ a Polish s... | https://mathoverflow.net/users/109527 | $X$ is Polish and $N$ is countable. Is $N^X$ Polish? | If you really mean these spaces $A^B$ to consist of *all* functions $B\to A$, then (1) and (3) are too big to be Polish. They have cardinality $2^{\mathfrak c}$ where $\mathfrak c$ is the cardinal of the continuum. Separable metric spaces have cardinality at most $\mathfrak c$.
On the other hand, (2) is, like any pro... | 7 | https://mathoverflow.net/users/6794 | 351181 | 148,536 |
https://mathoverflow.net/questions/351132 | 5 | For a simple Lie algebra $\frak{g}$, it is usual to scale the inner product so that the shortest simple root has length $2$. With this conventions, where can I find a table (online) of the following values for a general simple $\frak{g}$:
i) $(\alpha\_i,\alpha\_j)$, for all $i,j=1,\dots, r:=\mathrm{rank}(\frak{g})$
... | https://mathoverflow.net/users/125790 | Table of products for Lie algebra inner product of roots and weights | This information is available using the Atlas of Lie Groups and Representations software <http://www.liegroups.org/software>.
Here is an example, comments are in braces.
```
atlas> set G=Sp(4)
Variable G: RootDatum
atlas> set f=G.invariant_form {W-invariant bilinear form}
Added definition [3] of f: (ratvec,ratvec->... | 6 | https://mathoverflow.net/users/6030 | 351197 | 148,541 |
https://mathoverflow.net/questions/351166 | 4 | Let $T$ be a compact oriented torus, let $p\in T$ be a point, and let $T^\*$ be $T - \{p\}$.
In Farb-Margalit's *Primer on mapping class groups*, in the discussion after Proposition 1.5 they say that "the homotopy classes of simple closed curves in $T^\*$ are in bijective correspondence with those in $T$", but they g... | https://mathoverflow.net/users/15242 | Essential simple closed curves on a punctured torus vs those in the torus | There is exactly one simple closed curve in each primitive homology class in $T^\ast$ and that curve is the shortest one in the homology class. Conversely, the simple closed curves on $T$ correspond exactly to the primitive homology classes (primitive = class $(a, b)$ where gcd of $a, b$ (sadly, also denoted by $(a, b)... | 5 | https://mathoverflow.net/users/11142 | 351202 | 148,544 |
https://mathoverflow.net/questions/351088 | 14 | In order to define what it means for a map $f \colon \Omega \subseteq \mathbb R^n \to \mathbb R^n$ to be conformal, it is sufficient to require that $f$ is everywhere differentiable. Does conformality automatically implies that $f$ is $\mathcal C^1$ (hence real-analytic, see below)?
By complex analysis, we know the ... | https://mathoverflow.net/users/25590 | Regularity of conformal maps |
>
> Let $n\geq 3.$ Let $\Omega$ be an open connected subset of $\mathbb R^n,$ and let $f:\Omega\to\mathbb R^n$ be a function having a pointwise derivative $Df(x)$ everywhere satisfying $(Df)^T(Df)=g(x)I$ with $g(x)>0.$ Then $f$ is continuously differentiable.
>
>
>
By the [inverse function theorem](https://terry... | 7 | https://mathoverflow.net/users/112284 | 351206 | 148,546 |
https://mathoverflow.net/questions/351205 | 4 | Theorem 4.19 in Kechris' Classical Descriptive Set Theory says that the space of continuous functions from a compact metric space to a Polish space is Polish. It is therefore obvious that the space of continuous functions from a compact Polish space to a Polish space is Polish.
The space of continuous function is equ... | https://mathoverflow.net/users/109527 | Is the space of continuous functions from Polish space to Polish space Polish? | As already answered here ([$X$ is Polish and $N$ is countable. Is $N^X$ Polish?](https://mathoverflow.net/questions/351160/x-is-polish-and-n-is-countable-is-nx-polish/351182#351182)):
Let us work in a convenient category of topological spaces, ie assume that we have function spaces and sufficiently many nice spaces a... | 4 | https://mathoverflow.net/users/15002 | 351213 | 148,548 |
https://mathoverflow.net/questions/351194 | 6 | Does anyone know examples of a numerically non trivial nef divisor with Iitaka dimension 0? (Unfortunately, this question might be trivial as right now I can't think of any effective divisors with Iitaka dim 0 other than exceptional divisors)
| https://mathoverflow.net/users/150655 | Example of nef divisor with Iitaka dimension 0 | Here's an example that works over an uncountable field.
Let $E$ be an elliptic curve in $\mathbf P^2$ and let $p\_1, \ldots, p\_9$ be 9 points on $E$. Blowing up these points, the proper transform $C$ of $E$ is an irreducible curve with self-intersection 0, hence nef.
Notice that if some multiple $mC$ of $C$ moves... | 6 | https://mathoverflow.net/users/121595 | 351218 | 148,551 |
https://mathoverflow.net/questions/351167 | 3 | Let $X$ be a Banach space and $Y$ be a closed subspace of $X$. For $1<p<\infty$ consider the $p$-th power Bochner Integrable functions which takes values in $X$ and defined on the unit interval $[0,1]$, denoted by $L\_p(I,X)$. It is clear that $L\_p(I,Y)$ is a closed subspace of $L\_p(I,X)$. It is well-known that if $X... | https://mathoverflow.net/users/76412 | $L_p(I,Y)^\perp=L_q(I,Y^\perp)$? | Let's consider the following situation: $E$ and $F$ are Banach spaces, $D\subset E$ is a dense subspace, $Q: E \to F$ is a continuous linear operator, and its restriction $Q\_0$ to $D$ is a quotient map onto its range $R$, i.e., $Q\_0$ takes the open unit ball of $D$ to the open unit ball of $R$. Suppose $R$ is dense i... | 1 | https://mathoverflow.net/users/127871 | 351226 | 148,555 |
https://mathoverflow.net/questions/351233 | 5 | I know that the irreducible modules of $C\_m \wr S\_n$ over $\mathbb{C}$ are parametrised by m-multipartitions. The parts of the multipartition are indexed by the elements of $C\_m$.
My question now is: If $V$ is an irreducible module over $\mathbb{C}$ with multipartition $\underline{\lambda} = (\lambda\_0, \lambda\_... | https://mathoverflow.net/users/151546 | The multipartition of the dual of a $C_m \wr S_n$ module | I assume that by a multipartition you mean here that $\lambda\_i$ is a partition of $k\_i$ and $\sum\_i k\_i = n$. In this case the correspondence you described is indeed the duality correspondence. I believe the easiest way to see this is via Clifford Theory: Consider the short exact sequence $$1\to C\_m^n\to G\to S\_... | 3 | https://mathoverflow.net/users/41644 | 351237 | 148,558 |
https://mathoverflow.net/questions/351188 | 4 | Let $Y\to X$ be a finite surjective morphism of smooth projective geometrically connected varieties over $\mathbb{Q}$. Let $k$ be a number field and consider the induced morphism
$$f:Res\_{k/\mathbb{Q}} Y\_k\to Res\_{k/\mathbb{Q}} X\_k.$$
Let $\Delta\to Res\_{k/\mathbb{Q}}X$ be the diagonal embedding of $X$ into t... | https://mathoverflow.net/users/151501 | The diagonal of the Weil restriction | Welcome to mathoverflow, Pat!
I believe that $f^{-1}(\Delta) \neq Y$. Let $Y$ be a torsor for an elliptic curve $E$ over $\mathbb Q$ and choose $k$ so that $Y\_k \cong E\_k$ or in other words $Y(k) \neq \emptyset$. Choose the torsor so that the diagram $C\_k\cong E\_k\to \mathbb P^1\_k$ commutes. I believe this can b... | 2 | https://mathoverflow.net/users/58001 | 351248 | 148,561 |
https://mathoverflow.net/questions/351246 | 10 | It is well known that there are exactly five 3-dimensional regular convex polyhedra, known as the *Platonic solids*.
In 1852 the Swiss mathematician Ludwig Schlafli found that there are exactly six regular convex 4-polytopes (the generalization of
polyhedra to 4 dimensions) and that, for dimensions 5 and above, there a... | https://mathoverflow.net/users/97532 | What is the convex hull of the quaternionic symmetries of the 3 dimensional cube? | This is the [disphenoidal 288-cell](https://en.wikipedia.org/wiki/Truncated_24-cells#Disphenoidal_288-cell), which is the dual of the [bitruncated 24-cell](https://en.wikipedia.org/wiki/Truncated_24-cells#Bitruncated_24-cell).
This is also mentioned in the "Geometry" section of the Wikipedia article on the 288-cell.
... | 14 | https://mathoverflow.net/users/108884 | 351249 | 148,562 |
https://mathoverflow.net/questions/350188 | 4 | The question is pretty much the title. I'm wondering if anything is known about the smallest size $\kappa$ of a non-measurable subset of the real numbers (regarding the Lebesgue measure). Since we have $\kappa\geq\aleph\_0$ and $\kappa\leq\mathfrak{c}$ with $\kappa=\mathfrak{c}$ at least being consistent (under CH or M... | https://mathoverflow.net/users/138274 | Smallest size of a non-measurable set of reals | I am just going to compile the comments into an answer so i can close this question.
Claim: The smallest size of a non-measurable set is $\text{non}(\mathcal{L})$:
$\geq$: If $A$ is non-measurable, then $A$ is not null.
$\leq$: If $A$ is not null and not of size continuum, then $A$ has to be non-measurable, becau... | 1 | https://mathoverflow.net/users/138274 | 351253 | 148,565 |
https://mathoverflow.net/questions/351241 | 17 | Given a *finite* set of *convex* $d$-dimensional polytopes $\mathcal P$, for some $d\ge 2$.
>
> **Question:** Is it true that there are only *finitely* many different *convex* $(d+1)$-dimensional polytopes whose facets are solely (scaled and rotated versions of) polytopes in $\mathcal P$?
>
>
>
---
**Some ... | https://mathoverflow.net/users/108884 | Can I build infinitely many polytopes from only finitely many prescribed facets? | I think this is indeed true. This should follow from the rigidity of convex spherical polytopes and a few more lemmas. I'll explain how to deal with cases $d\ge 3$.
*Lemma 1, rigidity*. Two convex spherical cobminatorially equivalent polytopes with isomertric faces are isometric.
*Lemma 2, bounded volume.* The boun... | 8 | https://mathoverflow.net/users/943 | 351255 | 148,566 |
https://mathoverflow.net/questions/351238 | 7 |
>
> Suppose $A(p, n)=(a\_{ij}(p))\_{i, j \leq n}$ is an $n\times n$ random matrix over $\mathbb{F\_2}$, with all its entries being i.i.d. and such that $P(a\_{ij}(p) = 1) = p$, where $p$ is some real number from $[0; 1]$. What is the largest possible probability, that $A(p, n)$ is non-singular and with what $p$ is it... | https://mathoverflow.net/users/110691 | What is the largest possible probability that a random matrix over $\mathbb{F}_2$ is non-singular? | For $n\rightarrow\infty$ the probability ${\cal P}\_\infty$ that $A(p,n)$ is nonsingular becomes independent of $p\in(0,1)$, given by
$${\cal P}\_\infty=\prod\_{i=1}^\infty(1-2^{-i})=0.2887880951$$
See theorem 3.2 in [Properties of random matrices and applications](http://www.cs.toronto.edu/~cvs/coding/random_report.p... | 8 | https://mathoverflow.net/users/11260 | 351267 | 148,573 |
https://mathoverflow.net/questions/350977 | 3 | * If $f : X \to Y$ is proper, then specializations lift along $f$, and uniquely.
(This means, if $R$ is a discrete valuation ring with fraction field $K$ and I choose a factorization $\text{Spec}K \to \text{Spec} R \to Y$ through $X$, I get a unique factorization of $\text{Spec} R \to Y$ through $X$ inducing it. )
... | https://mathoverflow.net/users/86614 | When do generizations ("generalizations") lift uniquely? | Your condition is equivalent to stating that, for any map $\operatorname{Spec} R \to X \times\_Y X$, if the special point factors through the diagonal $\Delta: X \to X \times\_Y X$, then the whole map factors through the diagonal.
A sufficient condition is clearly that the diagonal is an open immersion, because if th... | 2 | https://mathoverflow.net/users/18060 | 351274 | 148,574 |
https://mathoverflow.net/questions/351273 | 5 | Let $E$ and $E'$ be elliptic curves over $\mathbb{C}$. I am pretty confident that the only effective divisor $D\subset E\times E'$ with Kodaira dimension zero is the trivial divisor.
How to prove this in a simple manner?
| https://mathoverflow.net/users/151501 | How to properly verify that $E\times E'$ has no non-trivial effective divisors with Kodaira dimension zero | If $D$ has Iitaka dimension zero, then $\dim H^0 ( n E) =1$ for all $n$, because if it were any larger than $\dim H^0(k ne) \geq k+1$.
If we have an automorphism $\sigma$ with $[ n \sigma (D)] = [n D]$ in the divisor class group, we would have two linearly independent sections unless in fact $\sigma(D)= D$.
For $\s... | 7 | https://mathoverflow.net/users/18060 | 351277 | 148,577 |
https://mathoverflow.net/questions/351286 | -1 | In The Consistency of Classical Set Theory Relative to a Set Theory with Intuitionistic Logic in THE JOURNAL OF SYMBOLIC LOGIC Volume 38, Number 2, June 1973 page 316 Harvey Friedman's axiom 8\* $Weak \ Power \ Set$ is:
$(\forall a)(\exists x)(\forall y)(\exists z\in x)(z=y\cap a)$
What do we know about weak power ... | https://mathoverflow.net/users/37385 | Weak power set - what strength may it have? | The asserted set $x$ is just a set that contains all overlap sets between the set $a$ and any set, among its elements. Weak Power as written above is simply:
$(\forall a)(\exists x)(\forall y)(a \cap y\in x) $
Now in classical ZF all subsets of $a$ are overlaps with $a$, i.e. $z \subseteq a \to z \cap a=z$, so all ... | 3 | https://mathoverflow.net/users/95347 | 351292 | 148,579 |
https://mathoverflow.net/questions/351284 | 1 | I am having some difficulty in proving the following inequality:
\begin{equation\*}
\frac{1-e^{-\gamma b}}{b^\eta}-\frac{1-e^{-\gamma s}}{s^\eta}\geq \gamma(1-\eta)\int^b\_sy^{-\eta}e^{-\gamma y}dy
\end{equation\*}
where $\gamma,\ \eta\in(0,1)$, $0\leq s\leq b$ and $b$ satisties:
\begin{equation\*}
\frac{(1-e^{-\gamm... | https://mathoverflow.net/users/151319 | Inequalities involving Gamma function | Let $D(b)$ denote the difference between the left-hand side and the right-hand side of your displayed inequality. Then $D(s)=0$, and the inequality $D'(b)\le0$ can be easily shown to be equivalent to $e^{\gamma b}\ge1+\gamma b$, which latter is true. So, $D(b)\le0$ if $b\ge s\ge0$; that is, the opposite to your propose... | 4 | https://mathoverflow.net/users/36721 | 351294 | 148,580 |
https://mathoverflow.net/questions/350593 | 4 | Consider a conformal homeomorphism $f \colon \Omega \to \Omega'$ between domains $\Omega, \Omega'$ in the plane. Let $E = \mathbb{R}^2 \setminus \Omega$ and $E' = \mathbb{R}^2 \setminus \Omega'$. My understanding is that you can have a situation where $E$ is totally disconnected and yet $E'$ is not. If we think of the ... | https://mathoverflow.net/users/126691 | Conformal mappings for domains whose complement is totally disconnected | The only example I know of a conformal map that "stretches" a point boundary component to a nondegenerate continuum was constructed by Gehring and Martio in *Quasiextremal distance domains and extension of quasiconformal mappings*, J. Analyse Math. 45 (1985), 181–206. See Theorem 4.1.
The authors use earlier results ... | 1 | https://mathoverflow.net/users/1162 | 351300 | 148,581 |
https://mathoverflow.net/questions/351122 | 5 | This question is certainly not research level and in fact quite elementary which is why I asked it on math.stackexchange before: [math.stackexchange](https://math.stackexchange.com/questions/3519251/are-open-mathbbg-m-invariant-subschemes-of-an-affine-scheme-precisely-the-h). However it doesn't seem to get much attenti... | https://mathoverflow.net/users/112369 | Are open $\mathbb{G}_m$-invariant subschemes of an affine scheme precisely the homogeneous radical ideals of its coordinate ring? | Your definition
>
> $\mathbb G\_m \times Y \hookrightarrow \mathbb G\_m \times X \to X$ factors over $Y \hookrightarrow X$
>
>
>
can be simplified a bit. A better one for our purposes is
>
> The two subschemes of $\mathbb G\_m \times X$ defined as the inverse image of $Y$ under the right projection $\mathb... | 2 | https://mathoverflow.net/users/18060 | 351302 | 148,583 |
https://mathoverflow.net/questions/346692 | 4 | I'm reading Bhatt and Scholze's, ["Prisms and prismatic cohomology"](https://arxiv.org/pdf/1905.08229.pdf), and I have a few questions about Lemma 3.1.
1. Why can they choose a finite number of elements generating $A$ such that $IA[1/g\_i]$ is principal? I don't understand how this follows from the fact that $I$ is l... | https://mathoverflow.net/users/69571 | Definition of Zariski localization along a closed subset | Regarding question 2:
My interpretation of the phrase (which fits the proof, i.e. makes the proof of the Lemma work) is the following:
>
> Given a ring $B$ and an ideal $I \subseteq B$, the localization of $B$ along $V(I)$ is $S^{-1}B$ where $S=B \setminus \bigcup V(I)=B \setminus \bigcup\_{\mathfrak{p}\in V(I)} ... | 3 | https://mathoverflow.net/users/60903 | 351306 | 148,584 |
https://mathoverflow.net/questions/351290 | 9 | I'm considering various functors from the category $\text{Vect}$ of real vector spaces to itself, and would like to know that they preserve filtered colimits and possibly even sifted colimits. The functors I'm interested in send $V$ to the power $V^k$, the tensor power $V^{\otimes k}$, the exterior power $\Lambda^k(V)$... | https://mathoverflow.net/users/952 | functors $\text{Vect} \to \text{Vect}$ that preserve filtered and sifted colimits | Maybe the following tools can help.
1. $G: \mathcal{C}\_1 \times \cdots\times \mathcal{C}\_k\to \mathcal{D}$ preserves sifted colimits separately in each variable if and only if it preserves sifted colimits. Indeed, given a diagram $p: K \to \mathcal{C}\_1\times\cdots \times \mathcal{C}\_k$ with $K$ sifted, observe t... | 9 | https://mathoverflow.net/users/6936 | 351307 | 148,585 |
https://mathoverflow.net/questions/351272 | 5 | Let $G$ be a finite group, $S \subset G$ a generating set, $|g|:=|g|\_S=$ word-length with respect to $S$. Let $\phi(g,h)=|g|+|h|-|gh| \ge 0$ be the "defect-function" of $S$. The set $\mathbb{Z}\times G$ builds a group for the following operation:
$$(a,g) \oplus (b,h) = (a+b+\phi(g,h),gh)$$
On $\mathbb{N}\times G$ ... | https://mathoverflow.net/users/nan | A zeta function for the Klein Four group? | Consider a word $w=w\_0w\_1\cdots w\_{n-1}$ with each $w\_i\in\{0,a,b,c\}$.
Following your notation,
$$\lvert w\rvert=\lvert\zeta(w)\rvert=\left\lvert\bigoplus\_{i=0}^\infty m^i\cdot(0,w\_i)\right\rvert=\sum\_{i=0}^{n-1}\lvert 2^i\cdot(0,w\_i)\rvert=\sum\_{i=0}^{n-1} 2^i\lvert w\_i\rvert$$
where $\lvert0\rvert=0$, $\lv... | 2 | https://mathoverflow.net/users/95685 | 351310 | 148,587 |
https://mathoverflow.net/questions/351309 | 11 | The Ramanujan Cos/Cosh Identity, as stated [here](http://mathworld.wolfram.com/RamanujanCosCoshIdentity.html), is
$$\left[1+2\sum\_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+
\left[1+2\sum\_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}=
\frac{2\Gamma^4\left(\frac34\right)}{\pi}.$$
I am l... | https://mathoverflow.net/users/128941 | Reference request: proof of Ramanujan's Cos/Cosh Identity | I expand my comment into an answer.
The key here is the Fourier series for the elliptic function $\operatorname {dn} (u, k) $ given as $$\operatorname {dn} (u, k) =\frac{\pi} {2K}\left(1+4\sum\_{n=1}^{\infty} \frac{q^n} {1+q^{2n}}\cos\left(\frac{n\pi u} {K} \right) \right) $$ where $K$ is the complete elliptic integr... | 25 | https://mathoverflow.net/users/15540 | 351313 | 148,588 |
https://mathoverflow.net/questions/351314 | 1 | I'm interested in bounding the tail probabilities of a quadratic form
$x^t A x$ where $x\in \mathbb{R}^n$ is a sub-Gaussian vector with independent entries. $A\in \mathbb{R}^{n\times n}$ is a matrix. So I'm exactly in the setup of the Hanson-Wright inequality. In fact, I wish I could use it because if it would apply, ... | https://mathoverflow.net/users/151588 | Hanson-Wright inequality with random matrix | Let $x \sim N(0,I\_n)$. For any independent rank-1 projection $A$, conditioned on $A$, we have
$$x^T A x \sim \chi^2\_1.$$ So unconditionally, $x^T A x = O(1)$ with high probability.
Now, let $A = \frac{x x^T}{\|x\|\_2^2}$. Then, $A$ is a rank-1 projection and we have $\mathbb E [A x] = \mathbb E[x] = 0 = \mathbb E[... | 1 | https://mathoverflow.net/users/36687 | 351316 | 148,589 |
https://mathoverflow.net/questions/351326 | 6 | Quantum representations of the mapping class group of a surface are certain representations constructed from the data of a TQFT and described, for example, in and [1](https://webusers.imj-prg.fr/~gregor.masbaum/Masbaum.pdf) and [2](https://arxiv.org/pdf/1812.03888.pdf).
The following sources [3](https://arxiv.org/pdf... | https://mathoverflow.net/users/12395 | Applications of quantum representations of the mapping class group to quantum computers | The context is *topological* quantum computation, where quantum information is stored nonlocally in a physical system, so that it is protected from decoherence by local sources of noise. The nonlocal degree of freedom is a socalled non-Abelian anyon, a particle-like excitation which is described by a (2+1)-dimensional ... | 4 | https://mathoverflow.net/users/11260 | 351331 | 148,593 |
https://mathoverflow.net/questions/351328 | 5 | I posted this question some days ago [at math.stackexchange](https://math.stackexchange.com/questions/3522030/ko-groups-of-mathbbrp-infty-snaiths-theorem-for-ko), but didn't receive an answer.
I have two questions:
* I wonder whether anyone has taken the time to compute $KO\_\*(\mathbb{R}P^\infty)$?
The standard ... | https://mathoverflow.net/users/145064 | $KO_*$ groups of $\mathbb{R}P^\infty$, "Snaiths" theorem for $KO$ | There are many ways to do this. One elementary approach is to use the Adams spectral sequence
$Ext\_A(H^\*ko \wedge RP^\infty, F\_2) \cong Ext\_{A(1)}(H^\*RP^\infty,F\_2) \Rightarrow ko\_\*(RP^\infty) $
and invert the Bott map. An $A(1)$ resolution giving the answer (since $E\_2 = E\_\infty$) can be seen at the b... | 15 | https://mathoverflow.net/users/6872 | 351332 | 148,594 |
https://mathoverflow.net/questions/350682 | 4 | The following series arises in an electrostatics problem for a conducting cylinder:
$$
V=\sum\_{n=1}^\infty\frac{J\_0(k\_n\rho)e^{-k\_nz}}{k\_nJ\_1(k\_n)^2}
$$
where $J\_i$ is the Bessel function of $i^{th}$ order, and $k\_n$ is the location of the $n^{th}$ zero of $J\_0$. $V$ can be proven to converge for $z>0$, and f... | https://mathoverflow.net/users/94200 | Limit for series of Bessel functions evaluated at zeros | Employing the asymptotics of large zeroes of Bessel functions and the large-argument asymptotics of the Bessel functions, it can be shown that the $n$th term of the series behaves like
$$
\frac{1}{\sqrt{2n\rho}}\cos\left(\rho\left(n-\tfrac{1}{4} \right)\pi-\tfrac{\pi}{4}\right)e^{ -\left(n-\tfrac{1}{4}\right)\pi z} + \... | 2 | https://mathoverflow.net/users/35433 | 351337 | 148,595 |
https://mathoverflow.net/questions/351333 | -1 | Suppose I have an arbitrary 256 bit number $m$ another secret number $k$ of the same bit length, and then I multiply them both modulo a 256 bit prime number $p$ to get $c$ as follows:
$$
c = (m\cdot k) \mod p
$$
Is there any way to get $m$ back without knowing $k$?
Is this problem as hard as the discrete log problem?
H... | https://mathoverflow.net/users/147924 | Is there any way to solve this equation without knowing the inverse modulo? | $c$ represents a congruence class, and there are $p$ of them. However both $m$ and $k$ belong to the same [complete residue system](https://en.wikipedia.org/wiki/Modular_multiplicative_inverse), and so for any given $c$ there are $2^{256}$ pairs of $m,k$ that satisfy the equation. So, the answer is no.
| 3 | https://mathoverflow.net/users/70355 | 351340 | 148,596 |
https://mathoverflow.net/questions/351234 | 7 | For any ultrafilter $\mathcal{U}$ on $\omega$ and any finite $k$ we can construct tensor power $\mathcal{U}^{\otimes k}$ which is ultrafilter on $\omega^k$. Does there exist some natural extension of this construction for the ordinal $\omega^\omega$?
**Edit**: my suggestion:
$$
\mathcal{U}^{\otimes\omega}=\{B\subset... | https://mathoverflow.net/users/118366 | Ultrafilter on the ordinal $\omega^\omega$ | The relevant general construction is the *sum* of a family $\{\mathcal V\_i:i\in I\}$ of an indexed family of ultrafilters, with respect to an ultrafilter $\mathcal U$ on the index set $I$. If $\mathcal V\_i$ is an ultrafilter on $X\_i$, then the sum is the ultrafilter $\mathcal W$ on the disjoint union $\bigsqcup\_{i\... | 11 | https://mathoverflow.net/users/6794 | 351344 | 148,597 |
https://mathoverflow.net/questions/351325 | 6 | It's known that order topology is completely normal, so the lexicographic ordering on the unit square is also completely normal. It's also known that the lexicographic ordering on the unit square is not metrizable. I am interested in whether it is perfectly normal. (A space is *perfectly normal* if for any two disjoint... | https://mathoverflow.net/users/119645 | Is the lexicographic ordering on the unit square perfectly normal? | As Nate Eldredge points out in the comments, the book *Counterexamples in Topology* states that this space is not perfectly normal, but does not provide a proof. Here is the idea for a proof.
A *perfectly normal* space is a normal space in which every closed set is a $G\_\delta$. So to prove this space is not perfect... | 5 | https://mathoverflow.net/users/70618 | 351350 | 148,601 |
https://mathoverflow.net/questions/351280 | 15 | Fix a complex projective scheme $X$ and a closed point $x\in X$.
Let $d\_x$ denote the dimension of the Zariski tangent space at $x$.
This is the local embedding dimension of $X$ at $x$ -- the minimal dimension of a smooth scheme containing an open neighbourhood of $x$.
In a paper I blithely asserted that $d(X)... | https://mathoverflow.net/users/7653 | Local versus global embedding dimension | It seems there is a counterexample. This is based on Jason Starr's suggestion in the comments.
If we have a surface $S$ with two smooth disjoint curves $C\_1$ and $C\_2$, which are isomorphic, and let $X$ be obtained by gluing $C\_1$ and $C\_2$ along that isomorphism $i: C\_1\to C\_2$, then $X$ is projective if there... | 8 | https://mathoverflow.net/users/18060 | 351363 | 148,608 |
https://mathoverflow.net/questions/351224 | 3 | Problem description
-------------------
Suppose we have a finite alphabet $\mathcal{X}$, where each letter $X \in \mathcal{X}$ indexes into some fixed set of distributions, $\{P\_{1},\ldots,P\_{|\mathcal{X}|}\}$. For example, you might like to think of each $P\_{X}$ as Gaussian with different means.
A letter $X \in... | https://mathoverflow.net/users/137476 | Lower bounding decoding error in a noisy adversarial channel | Consider the special case where $|\mathcal X| = 2$ and $k=1$, that is, if you pick $P\_1$ the adversary picks $P\_2$ and generates a sample $Y^n = (Y\_1,\dots,Y\_n)$ drawn i.i.d. from $P\_2$, and vice versa. Then your question turns into the minimal error in a (Bayesian) binary hypothesis test. By Le Cam's lemma:
$$
\i... | 2 | https://mathoverflow.net/users/36687 | 351364 | 148,609 |
https://mathoverflow.net/questions/351367 | -2 | Is the following provable in MK?
$\not \exists S: \\ \text{ } \\1. \ \ \forall s \in S \exists a,b (s=\langle a,b \rangle) \\ \text{ } \\2. \ \ \forall x (set(x) \to \exists! X (\neg set(X) \land \{x\} \times X \subset S \land \forall k(\langle x,k \rangle \in S \rightarrow k \in X))) \\ \text{ } \\3. \ \ \forall X (... | https://mathoverflow.net/users/95347 | Does MK prove internally that there are more proper classes than sets? | Maybe I'm missing something, but isn't this just the diagonal argument?
---
Suppose $S$ were as in your question. Let $E=\{\langle x,y\rangle: set(x)\wedge y\in x\}$, and consider the set $$Z=\{\langle \langle x,0\rangle, y\rangle: \langle x,y\rangle\in S\}\cup\{\langle\langle x,1\rangle, y\rangle: \langle x,y\ra... | 2 | https://mathoverflow.net/users/8133 | 351370 | 148,610 |
https://mathoverflow.net/questions/351369 | 9 | In [this recent question](https://mathoverflow.net/q/351280/82179) (which now has an answer), Richard Thomas asked whether any projective $k$-scheme $X$ of (local) embedding dimension $d(X)$ can be embedded in a smooth $k$-scheme of dimension $d(X)$. If $i \colon X \hookrightarrow Y$ is such an embedding, then in parti... | https://mathoverflow.net/users/82179 | Global obstructions for being a quotient of a rank $d$ vector bundle | Let me explain a simple example.
Let $C \subset \mathbb{P}^3$ be a twisted cubic curve. It is a locally complete intersection of codimension 2, hence its ideal $I\_C$ is locally generated by two sections. Let me show that there are no surjections $E \twoheadrightarrow I\_C$ from a locally free sheaf $E$ of rank 2.
... | 12 | https://mathoverflow.net/users/4428 | 351371 | 148,611 |
https://mathoverflow.net/questions/351368 | 16 | Motivation:
-----------
Most dimensionality reduction algorithms assume that the input data are sampled from a manifold $\mathcal{M}$ whose intrinsic dimension $d$ is much smaller than the ambient dimension $D$. In machine learning and applied mathematics circles this is typically known as the *manifold hypothesis*.
... | https://mathoverflow.net/users/56328 | Physical interpretation of the Manifold Hypothesis | Q: Might there be a reason why stable physical processes would tend to have low-dimensional phase spaces.
Yes. One reason is physical processes have dissipation. E.g., turbulence is "known" to be chaotic dynamics on a low dimensional manifold (i.e., strange attractor) in the infinite dimensional phase space (of $L^2$... | 6 | https://mathoverflow.net/users/30684 | 351381 | 148,616 |
https://mathoverflow.net/questions/351378 | 2 | Let $G$ be a finite $p$-group and $(\mathbb{F},\mathcal{O},\mathbb{K})$ a sufficiently large $p$-modular system. Furthermore let $\mu\_{p^\infty}\leq \mathcal{O}^\times$ the group of roots of unity of $p$-power order.
If $\alpha$ is a 2-cocyle with values in $\mu\_{p^\infty}$ does there always exist a one-dimensional... | https://mathoverflow.net/users/3041 | Are there always 1-dimensional projective representations | Ah, dammit. The answer is no and the first group I would have looked at next is the counterexample... the extra special group $G=\langle x,y,z \mid x^p=y^p=[x,y]=z, z^p=1\rangle$ fits into an non-trivial extension $1\to C\_p \to G \to C\_p\times C\_p\to 1$ and the corresponding 2-cocycle $\alpha$ gives us the counterex... | 4 | https://mathoverflow.net/users/3041 | 351387 | 148,618 |
https://mathoverflow.net/questions/351376 | 16 | A *plane partition* of $n$ is an table of integers $A=(a\_{ij})$ which add up to $n$ and non-increase in rows and columns. For example,
$$A= \begin{matrix} 331 \\
32 \ \ \\
11 \ \
\end{matrix}
$$
is a plane partition of $(3+3+1)+(3+2)+(1+1)=14$.
One can view plane partitions as an arrangement of cubes stacked in th... | https://mathoverflow.net/users/4040 | Plane partitions with equal margins | Hopefully I understood the definitions and computed correctly. How about $n = 13$ with
$$A = \begin{matrix} 3 3 1 \\ 2 1 1 \\ 2 \ \ \end{matrix}$$
and
$$B = \begin{matrix} 3 2 2 \\ 3 1 \ \\ 1 1 \ \end{matrix}$$
where all margins are $(7,4,2)$.
| 7 | https://mathoverflow.net/users/51668 | 351388 | 148,619 |
https://mathoverflow.net/questions/351091 | 44 | I am trying to use a modification of group cohomology to **prove the existence and uniqueness** of Haar measure on a compact Hausdorff group.
---
I think the best way of introducing the idea I am pursuing is via analogy.
Let $G$ be a finite group and let $A = \mathbb{C}[G]$ be the group algebra. For each $G$-set ... | https://mathoverflow.net/users/30211 | Existence and uniqueness of Haar measure on compacta; a cohomological approach | Fix a compact group $G$ and consider its category of Banach representations:
the objects are (complex) Banach spaces $X$ endowed with a $G$-action by automorphims (not necessarily isometries) such that the action maps $G\times X\to X$ are jointly continuous and the morphisms are bounded maps $X\to Y$ commuting with the... | 26 | https://mathoverflow.net/users/89334 | 351405 | 148,625 |
https://mathoverflow.net/questions/351402 | 9 | I am stuck at the following :
Let $G$ be a graph and $A$ is its adjacency matrix.
Let the eigenvalues of $A$ be $\lambda\_1\le \lambda\_2\leq \cdots \leq \lambda\_n$.
If we remove some edges from the graph $G$ and form the graph $H$ keeping the number of vertices same, is there any result how the smallest eigenvalu... | https://mathoverflow.net/users/141429 | What happens to eigenvalues when edges are removed? | The smallest eigenvalue can go up or down when an edge is removed.
For "down": $G=K\_n$ for $n\ge 3$.
For "up": Take $K\_n$ for $n\ge 1$ and append a new vertex attached to a single vertex of the original $n$ vertices. Now removing the new edge makes the smallest eigenvalue go up.
Both of these follow from the fa... | 16 | https://mathoverflow.net/users/9025 | 351411 | 148,628 |
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