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/223864 | 7 | **Motivation**: Let $\ell$ be an odd prime. There is a conductor-preserving correspondence between primitive Dirichlet characters of order $\ell$
and cyclic, degree $\ell$ number fields $K/\mathbb{Q}$.
The proof of this correspondence can be found in Chapter 3 of Washington's ``Introduction to Cyclotomic Fields".
*... | https://mathoverflow.net/users/56667 | Hecke characters and Conductors | Let $k$ be a number field. By class field theory, any Hecke character $\chi$ of $k$ of order $\ell$ determines a cyclic extension $k'/k$ of degree $\ell$. Moreover, the set of Hecke characters determining this cyclic extension $k'/k$ equals $\{\chi,\chi^2,\dots,\chi^{\ell-1}\}$. These $\ell-1$ Hecke characters have the... | 6 | https://mathoverflow.net/users/11919 | 223873 | 104,784 |
https://mathoverflow.net/questions/160878 | 0 | I am currently researching discrete geometry and I am in need of an upper bound on a generalized kissing number in 3-dimensions dependent upon a parameter $\eta$ which is the radii of spheres touching a central unit ball.
That is, center the unit ball $B \subset \mathbb{R}^3$ at the origin and define $k\_{s}(\eta)$ t... | https://mathoverflow.net/users/20343 | Generalized Sphere Kissing Problem | I happened to write a paper which eventually answered my problem, although it happens to be hid inside the $A^{\text{LP}}(3,\theta)$ term in the bound of Theorem 1 of my paper which is under review at the Journal of Geometry:
"Upper Bounds for Non-Congruent Sphere Packings" ([arXiv link](http://arxiv.org/abs/1510.004... | 0 | https://mathoverflow.net/users/20343 | 223874 | 104,785 |
https://mathoverflow.net/questions/223877 | 18 | The moduli space of graphs $MG\_n$ is the quotient of Culler-Vogtmann's outer space $X\_n$ by the action of $\mathrm{Out}(F\_n)$. It can be thought of as the space of metric graphs homotopy equivalent to a wedge of $n$ circles with all vertices of valence $\geq 3$. A metric graph can be thought of as simply a graph wit... | https://mathoverflow.net/users/9417 | Is the moduli space of graphs simply connected? | Yes, it is. If $G$ is a discrete group acting on a simply-connected simplicial complex $X$, then a theorem of M. A. Armstrong says that there is a short exact sequence
$$1 \longrightarrow H \longrightarrow G \longrightarrow \pi\_1(X/G) \longrightarrow 1,$$
where $H$ is the subgroup of $G$ generated by elements that... | 19 | https://mathoverflow.net/users/317 | 223879 | 104,786 |
https://mathoverflow.net/questions/223848 | 2 | Let $K$ be a field of characteristic zero complete with respect to a non-Archimedean absolute value. Suppose that $A$ and $B$ are two affinoid $K$-algebras. I'd like a reference that will answer the following question:
>
> If $A$ and $B$ are both reduced then is the completed tensor product $A\widehat{\otimes}\_K B... | https://mathoverflow.net/users/35330 | Product of reduced affinoid spaces over a field is reduced (reference request) | For all those kind of properties, I recommend looking at Ducros's paper "Les espaces de Berkovich sont excellents" (Ann. Inst. Fourier 59 (2009), no. 4, 1407-1516, <http://aif.cedram.org/item?id=AIF_2009__59_4_1443_0>). It is written in the language of Berkovich spaces, so it is even slightly more general than what you... | 4 | https://mathoverflow.net/users/4069 | 223884 | 104,790 |
https://mathoverflow.net/questions/223888 | 6 | Let $O(n)$ be the $n$-th orthogonal group and $O$ be the direct limit of $O(n)$ with respect to $n$. Let $BO(n)$ and $BO$ be the classifying spaces.
**Question:**
Why $BO$ is an $H$-space? My supervisor said "$BO=\Omega^\infty\mathbb{E}$ where $\mathbb{E}$ is a spectrum." What does this mean?
| https://mathoverflow.net/users/41075 | classifying space of orthogonal groups | $BO$ is the connected component of the zeroth space of a spectrum called the [real K-theory spectrum](http://ncatlab.org/nlab/show/K-theory+spectrum). This spectrum represents a cohomology theory, namely real K-theory, and this means that $BO$ has much more structure than an H-space: it is in fact an [infinite loop spa... | 12 | https://mathoverflow.net/users/290 | 223889 | 104,791 |
https://mathoverflow.net/questions/223885 | 9 | Let $f:X \to Y$ be a morphism of schemes. I am interested in sufficient conditions on $f$ which would ensure that the induced map $\pi\_1^{et}(X) \to \pi\_1^{et}(Y)$ of etale fundamental groups is surjective. According to this question
<https://math.stackexchange.com/questions/491305/etale-fundamental-group>
SGA 1 X ... | https://mathoverflow.net/users/12395 | Under what conditions is the induced map of etale fundamental groups surjective? | There is a very general criterion for a map on $\pi\_1$ to be surjective. Recall that for $X$ connected, the category of finite étale covers of $X$ is equivalent to the category $\pi\_1(X)\text{ -}\operatorname{Set}\_f$ of finite sets with a continuous $\pi\_1(X)$-action. Under this correspondence, the $Y \to X$ finite... | 11 | https://mathoverflow.net/users/82179 | 223892 | 104,793 |
https://mathoverflow.net/questions/223584 | 49 | I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be *reductive*.
Hence my question : why is *reductive* interesting?
Of course I have already seen some formal definitions, l... | https://mathoverflow.net/users/43737 | Which philosophy for reductive groups? | Harish-Chandra told the following (paraphrased) little story that he heard from Chevalley: "When God and the Devil were creating the universe, God gave the Devil a free hand in building things but told him to keep off certain objects to which He would attend. Semisimple groups were among those special items."
From th... | 56 | https://mathoverflow.net/users/81332 | 223895 | 104,794 |
https://mathoverflow.net/questions/223883 | 6 | Suppose that $G$ and $H$ are groups (not isomorphic) and $G\ast H$ the free product. Let $Aut(G)$, $Aut(H)$ be the automorphism groups of $G$ and $H$. What is $Aut(G\ast H)$ ?
| https://mathoverflow.net/users/82229 | Automorphism group of a free product | This is not an answer, but it's too long for a comment and makes an important point that I hope is useful.
It's not enough to think of your group just as some free product. To understand its automorphism group you need to think of it as a free product in a *canonical* way, and for this you need the Grushko decomposit... | 10 | https://mathoverflow.net/users/1463 | 223906 | 104,800 |
https://mathoverflow.net/questions/223913 | 7 | According to several sources, it is conjectured (or at least believed)
that the rational points of curves over the rationals of genus $g > 1$
are uniformly bounded by $g$. E.g. [here p. 1](http://www.ma.utexas.edu/users/voloch/Preprints/unbounded.pdf).
Assuming the curve is irreducible, the singular points (which are... | https://mathoverflow.net/users/12481 | What is the exact statement about uniform boundedness of rational points on curves of genus greater than one? Singular points can be unbounded | The precise statement of the conjecture is:
>
> **Uniformity Conjecture.** Let $K$ be a number field and $g\geq2$ an integer. There exists a number $B(K,g)$ such that for any smooth curve
> $X$ of genus $g$ defined over $K$
>
>
> $$|X(K)|\leq B(K,g)$$
>
>
>
and the original reference for it is:
* Lucia Ca... | 7 | https://mathoverflow.net/users/43108 | 223915 | 104,803 |
https://mathoverflow.net/questions/223887 | 4 | Let $\Sigma\_k$ be the $k$-th symmetric group and $B\Sigma\_k$ be its classifying space. How to prove:
for any $n\geq 1$ and the $n$-skeleton $sk\_n (B\Sigma\_k)$, there exists a finite dimensional $CW$-complex $K$ such that
(i). $sk\_n(B\Sigma\_k)\subseteq K\subseteq B\Sigma\_k$;
(ii). $H^\*(K;\mathbb{Q})$ is tr... | https://mathoverflow.net/users/76736 | rational cohomology of symmetric groups | Take the $n$-skeleton. It has trivial rational homology except possibly in degree $n$. Now add enough $n+1$-cells from the $n+1$-skeleton to kill this top homology. You won't have created any $n+1$-dimensional homology since the boundary operator $\partial\_{n+1}\colon C\_{n+1}\to C\_n$ is a rational isomorphism onto $... | 7 | https://mathoverflow.net/users/9417 | 223922 | 104,806 |
https://mathoverflow.net/questions/214950 | 0 | Let $V(x\_1,..,x\_n)$ be the Vandermonde matrix induced by $x\_1,..,x\_n$, and
let $\tilde{V} := V(\frac{x\_1}{h},...,\frac{x\_n}{h})$.
My intuition says that the condition number should be invariant under such scaling of the nodes at least for some special cases of node configurations. My question is then:
1. Are ... | https://mathoverflow.net/users/41258 | The condition number of a scaled Vandermonde matrix | For any configuration of nodes, as $h\to \infty$, $\tilde{V}$ tends to a singular matrix while remaining bounded in norm, so $\mathcal K(\tilde{V})\to \infty$.
| 1 | https://mathoverflow.net/users/13360 | 223923 | 104,807 |
https://mathoverflow.net/questions/223925 | 1 | In my question [Existence or otherwise of a set of "sufficiently intricate" open sets](https://mathoverflow.net/questions/223847/existence-or-otherwise-of-a-set-of-sufficiently-intricate-open-sets/223850?noredirect=1#comment552389_223850), I asked about whether it is possible to partition Lebesgue-almost all of $\mathb... | https://mathoverflow.net/users/15570 | Existence or otherwise of a set of "sufficiently intricate" open cells | Take the three-dimensional example, which comes from a grid. Find a maximal tree inside the grids corresponding to $V\_1$ and $V\_2$, so pick a subset of the set of edges that connects each node to every other one, but creates no loops. Now, "cut" the cubes corresponding to the remaining edges in $V\_1$, $V\_2$ by taki... | 1 | https://mathoverflow.net/users/70808 | 223941 | 104,812 |
https://mathoverflow.net/questions/223911 | 5 | Let $P$ be the convex hull of a finite set of points in $\mathbb Z^d$, and $p(n) = \#\{nP \cap \mathbb Z^d\}$ be its **Ehrhart polynomial,** which is also the Hilbert polynomial of the corresponding projective toric variety.
If we write it as $\sum\_{k=0}^d c\_k {n \choose k}$, then since $p$ is $\mathbb Z$-valued, t... | https://mathoverflow.net/users/391 | Coefficients of Ehrhart polynomials, in the binomial-coefficient basis | One possible answer: if you write the Ehrhart polynomial as $\sum\_{k=0}^{d} c\_k\binom{n-1}{k}$, then the coefficients $c\_k$ are called the $f^{\ast}$-vector by Felix Breuer in the paper [Ehrhart f\*-coefficients of polytopal complexes are non-negative integers](http://arxiv.org/abs/1202.2652). He gives a combinatori... | 8 | https://mathoverflow.net/users/82938 | 223943 | 104,813 |
https://mathoverflow.net/questions/223938 | 16 | The question is pretty much in the title: What is the maximal subgroup of $S\_d$ of maximal index (so minimal size)? A slight variant (I am not sure if it leads to a different answer) is: what if we restrict to *transitive* subgroups? (Of course, in both cases, there might be massive ties, but at least the index is wel... | https://mathoverflow.net/users/11142 | Minimal maximal subgroup of the symmetric group | Maximal intransitive groups will be $S\_a\times S\_b$ and have comparatively small index. I think once $d>6$ (otherwise there is a small number effect) the maximal subgroup of maximal index will always be transitive: If $d$ is prime, $AGL(1,d)$ is transitive, maximal, of larger index; if $d=a\cdot b$, then $S\_a\wr S\_... | 15 | https://mathoverflow.net/users/59303 | 223947 | 104,814 |
https://mathoverflow.net/questions/223940 | 1 | Let given torsion free abelian group $A$ of finite rank. Let for prime number $p$, given that $\cap\_i p^iA =\{0\}$. Is it true that for any $p$- torsion abelian group $B$, $\text{Hom}\_{\mathbb{Z}}(A, B)$ is torsion $\mathbb{Z}$ module.
| https://mathoverflow.net/users/82143 | Abelian group of finite rank | Let $p$ be any prime. There exists a subgroup $A$ of $\mathbf{Z}[1/p]^2$ containing $\mathbf{Z}^2$ such that $\bigcap\_n p^nA=\{0\}$ and $A/\mathbf{Z}^2$ is infinite (isomorphic to the quasi-cyclic group $P\_p=\mathbf{Z}[1/p]/\mathbf{Z}$). Then $\mathrm{Hom}(A,P\_p)$ contains $\mathrm{Hom}(P\_p,P\_p)\simeq\mathbf{Z}\_p... | 3 | https://mathoverflow.net/users/14094 | 223952 | 104,817 |
https://mathoverflow.net/questions/223954 | 16 | A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the *additive energy* of $A$ is defined to be the quantity $E(A):=|\{(a,b,c,d) \in A^4 : a+b=c+d \}|$.
>
>
> >
> > Where in the literature did the term "additive energy... | https://mathoverflow.net/users/630 | Where did the term "additive energy" originate? | Van Vu and I coined the term in our book because there did not seem to be a widely adopted name for it previously. (Gowers, for instance, refers to "number of additive quadruples" rather than "additive energy", but this seemed to be too unwieldy to use for our purposes.) I think we settled on "energy" due to the vaguel... | 34 | https://mathoverflow.net/users/766 | 223962 | 104,822 |
https://mathoverflow.net/questions/223946 | 0 | Is there a simple and **explicit** continuous function $f\colon[0,\infty)^2\to\mathbb C$ such that $f$ is analytic on $(0,\infty)^2$ and $|f(x+iy)|/(x+y)\to1$ as $x+y\to\infty$, where $(x,y)\in[0,\infty)^2$?
It seems not too hard to construct such a function $f$ as the sum of a series of polynomials, as is done in t... | https://mathoverflow.net/users/36721 | Explicit analytic function with modulus asymptotic to $\Re z+\Im z$ | Such function does not exist, "explicit" or not.
Consider $u(z)=\log|f(z)/z|$. Your condition implies that $u$ is harmonic and
bounded when $|z|>r$ for some $r>0$ and $z$ is in the first quadrant. Moreover it tends to $0$
on positive real and imaginary axes. But on the line $x=y$ it tends to $\log\sqrt{2}>0$. This cont... | 6 | https://mathoverflow.net/users/25510 | 223963 | 104,823 |
https://mathoverflow.net/questions/223934 | 8 | Let $M$ be a smooth Riemannian manifold (without boundary). Let $X\subset M$ be a smooth compact submanifold with boundary, $\dim X=\dim M$.
**Under what conditions $X$, equipped with the induced intrinsic metric, is an Alexandrov space with curvature bounded below?**
A simple sufficient condition is some convexit... | https://mathoverflow.net/users/16183 | When a Riemannian manifold with boundary is an Alexandrov space? | Yes, this condition also necessary.
Assume $X$ is an Alexandrov space.
At any point of $\partial\_MX$ (the relative boundary of $X$ in $M$)
the space of directions is a half-sphere;
therefore $\partial\_MX$ is also boundary of $X$ as it is defined for Alexandrov spaces.
If curvature $\ge 0$ then the distance functi... | 5 | https://mathoverflow.net/users/1441 | 223968 | 104,827 |
https://mathoverflow.net/questions/223939 | 4 | Every positive integer can be written as the sum of 4 squares $n = a\_1^2 + a\_2^2 + a\_3^2 + a\_4^2$ however, if we only allow [sum of 3 squares](http://oeis.org/A000408) some numbers have to be left out:
>
> $n = a^2 + b^2 + c^2$ $\longleftrightarrow$ $n \equiv 4^a (8k+7)$
>
>
>
Excuse me for using the same ... | https://mathoverflow.net/users/1358 | Proving Legendre's Sum of 3 Squares Theorem via Geometry of Numbers | A proof of the three squares theorem by the geometry of numbers was given by Ankeny in 1957. His paper is available [here](http://www.ams.org/journals/proc/1957-008-02/S0002-9939-1957-0085275-8/S0002-9939-1957-0085275-8.pdf).
**P.S.** Also, I think Legendre's proof was incomplete: he assumed Dirichlet's theorem about... | 12 | https://mathoverflow.net/users/11919 | 223969 | 104,828 |
https://mathoverflow.net/questions/223703 | 5 | I am looking for a reference for the following. Say we have a $G$-space $X$ whose homology groups (in field coefficients $k$) are non-zero only in dimension zero and for a fixed $n>0$. Let $M$ denote the top homology group of $X$. Then in the cohomology spectral sequence for the fibration $X\rightarrow (EG\times X)/G\r... | https://mathoverflow.net/users/82685 | Transgression in terms of k-invariant for chain complexes | I think the difficulty is that you are assuming that $X$ only has homology in two degrees, but are then looking at the cohomology spectral sequence. (To get sensible answers I seem to have to take cohomology to be negatively graded.)
Suppose instead that $H^0(X;k)=k$ and $H^{-n}(X;k)=N$ are the only two non-trivial c... | 1 | https://mathoverflow.net/users/318 | 223979 | 104,832 |
https://mathoverflow.net/questions/223977 | 4 | Let $X$ be a nice space, maybe a manifold, and let $Y$ be a based space.
*What sort of conditions must we impose on $Y$ (and $X$ if need be) to get a homotopy equivalence*
$$
\mathcal C\_c(X,Y) \simeq \mathcal C\_0(X,Y)$$
*between the space of continuous maps with compact support and that of maps which vanish (in the... | https://mathoverflow.net/users/39713 | Homotopy equivalence of maps with compact support and maps which vanish at infinity | Let's assume $X$ is locally compact Hausdorff, so that $\hat X$ is compact Hausdorff and you can indeed use $C(\hat X,Y)$.
If the inclusion $i:C\_c(X,Y)\to C(\hat X, Y)$ is a homotopy equivalence for all $Y$, in particular for $\hat X$, then the identity map $\hat X\to \hat X$ is based homotopic to a map that takes ... | 3 | https://mathoverflow.net/users/6666 | 223981 | 104,833 |
https://mathoverflow.net/questions/223975 | 4 | Consider the class map $$cl:CH^i(X)\to H^{2i}\_{cont}(X,\mathbb{Z}\_l(i))$$ where the RHS is the continuous etale cohomology(defined by Jannsen in his paper "Continuous etale cohomology"). In this paper he mentiones that the kernel $CH^i(X)\_l^0$ might depend on $l$.
What is known about this issue? Is there an exampl... | https://mathoverflow.net/users/39304 | $l$-dependence of the group of homologically zero cycles | the way it's stated, with integral cohomology, it certainly can depend on $\ell$. This is because of the occurrence of torsion classes Clearly a $p$-torsion class is sent to $0$ by this map for $\ell\neq p$, so it is sufficient to find an $\ell$-torsion class that is not sent to zero under this map.
But the Kummer ex... | 5 | https://mathoverflow.net/users/18060 | 223982 | 104,834 |
https://mathoverflow.net/questions/223990 | -3 | In Infinite dimensional Lie algebras book by Victor G Kac, In prop.3.6 He proves that, any integrable $g(A)$ - module $V$ is direct sum of finite dimensional, irreducible, $h$ - invariant $g\_{(i)}$ modules. He has proved only that $V$ is sum of such irreducibles, but he hasnt given the proof for the sum is direct . Ho... | https://mathoverflow.net/users/33047 | If a g-module is sum of irreps then is direct sum of irreps | Yes, this is a standard fact. Zorn's Lemma implies there is a submodule $W$ of $V$, such that $W$ is a direct sum of irreducibles, but no larger submodule of $V$ is a direct sum. The maximality implies that $W = V$. Otherwise, there is some irreducible $L$ that is not contained in $W$, so $L + W$ contradicts the maxima... | 1 | https://mathoverflow.net/users/68305 | 223993 | 104,837 |
https://mathoverflow.net/questions/223919 | 3 |
>
> Let $G$ be a finitely generated Fuchsian group.
>
>
>
(i.e. a discrete subgroup of $\mathrm{PSL}\_2(\mathbb{R})$).
>
> Is it true that $d(G) < 2\beta\_{2}^1(G) + 1$ ?
>
>
>
Here, $\beta\_{2}^1(G)$ stands for the first $L^2$-Betti number of $G$, and $d(G)$ is the smallest cardinality of a generating ... | https://mathoverflow.net/users/38889 | An inequality for Fuchsian groups? | A priori the inequality seems unlikely because for a group containing an index k surface subgroup the L^2-Betti numbers are those of the surface group divided by k, so the right hand side gets approximately divided by k, but one wouldn't expect the same for the left hand side.
To get an explicit counterexample look a... | 5 | https://mathoverflow.net/users/39082 | 223995 | 104,838 |
https://mathoverflow.net/questions/223992 | 5 | Let $S^n$ be the $n$-sphere and consider a $2$-sheeted covering
$$
S^n\longrightarrow\mathbb{R}P^n.
$$
We have an associated vector bundle
$$
\xi: \mathbb{R}^2\longrightarrow S^n\times\_{\mathbb{Z}/2}\mathbb{R}^2\longrightarrow \mathbb{R}P^n
$$
where the nontrivial element of $\mathbb{Z}/2$ acts on $\mathbb{R}^2$ by re... | https://mathoverflow.net/users/76736 | covering map from spheres to projective spaces and the associated vector bundle | Theorem 7.4 of J. F. Adams, Vector fields on spheres, Ann. of Math. 75 (1962), 603–632
says that $$\tilde{KO}({\mathbb R} P^n)=\mathbb Z\,/\,2^{\phi(n)},$$ generated by $\xi-1$, where $\phi(n)$ is the number of integers $s$ such that $0 < s\le n$ and $s$ is congruent to 0,1, 2 or 4 modulo 8, and $\xi$ is the canonical ... | 7 | https://mathoverflow.net/users/78588 | 223998 | 104,840 |
https://mathoverflow.net/questions/223996 | 3 | If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of torsion free coherent sheaves on a surface. Here $M$ is a line bundle, $E$ a vector bundle of rank 2 and $N$ the quotient. The exact sequence corresponds to the Harder Narasimhan filtration of $E$ with respect to an ... | https://mathoverflow.net/users/70211 | How do we get the quotient $Ext^1(N,M)/Hom(N,M)$? | The action is trivial, as you wrote. Generally if $G/S$ acts trivially on $X/S$ and $G$ is abelian then any object in the quotient stack $[X/G]=B\_X G$ has an automorphism group canonically isomorphic to $G$ (for $G$ non abelian, it could be a conjucagy form of $G$). Here it is easy to check that if
$$0\to M\xrightarro... | 3 | https://mathoverflow.net/users/11682 | 223999 | 104,841 |
https://mathoverflow.net/questions/224003 | 0 | I know how to estimate the integral\* (see the update)
\begin{gather}
\int f(Ub)d\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [2]
\end{gather}
where $f:S^{n-1}(\mathbb{R})\to \mathbb{R}$ and $f(x)=\sum\limits\_{i=1}^{n}\frac{1}{x\_i^2}$ and $d\mu$ is the Haar measure over orthogonal g... | https://mathoverflow.net/users/54526 | Change of variable for integration with respect to Haar measure | the two integrals are not the same, here is a simple example for $n=2$:
$$U=\begin{pmatrix}
\cos\alpha & -\sin\alpha\\
\sin\alpha & \cos\alpha
\end{pmatrix},\;\;d\mu(U)=d\alpha$$
$$M=\frac{1}{2}\begin{pmatrix}
i+\sqrt 2 & 1\\
-1 & -i+\sqrt 2
\end{pmatrix},\;\;b={1\choose 2}$$
$$\int g(UMb)d\mu(U)=5.80664$$
$$\i... | 1 | https://mathoverflow.net/users/11260 | 224011 | 104,844 |
https://mathoverflow.net/questions/224018 | 2 | I notice that $T^2=S^1\times S^1$ can be embedded in $\mathbb{R}^3$ as a hypersurface (submnaifolds of codimension 1).
In general,
(1). could the product of spheres $S^{m\_1}\times\cdots\times S^{m\_n}$ be embedded in Euclidean space as a hypersurface?
(2). could $T^n=\prod\_n S^{1}$ be embedded in Euclidean sp... | https://mathoverflow.net/users/65800 | embeddings of product of spheres in Euclidean spaces | This is [asked on MSE](https://math.stackexchange.com/questions/161293/product-of-spheres-embeds-in-euclidean-space-of-1-dimension-higher), and answered (see [Jim Belk's answer](https://math.stackexchange.com/a/161313), which is **NOT** the accepted answer).
| 9 | https://mathoverflow.net/users/11142 | 224022 | 104,847 |
https://mathoverflow.net/questions/224029 | 7 | Let me state a standard result first. Let a $A\subset \mathbb{R}^d$ be a set of fixed volume. Define $A\_t$ to be the set of all points at distance at most $t$ from $A$. Then the volume of $A\_t$ is minimal if $A$ is a ball of the prescribed volume.
Another way to define $A\_t$ is by $A\_t=A+B(0,t)$, where $B(0,t)$ i... | https://mathoverflow.net/users/24494 | Unusual isoperimetry and maximizing the measure of unions of translates of a set | Fix large $N$.
Take $A$ to be union of $\varepsilon$-balls which centers have integer coordinates between $\pm N$.
(You have to ajust $\varepsilon$ to get the needed volume.)
In this case $U(A)$ is a union $\varepsilon$-balls which centers have integer coordinates between $\pm (N+1)$.
Therefore $\mathrm{vol}[U(A)]$ ... | 8 | https://mathoverflow.net/users/1441 | 224032 | 104,851 |
https://mathoverflow.net/questions/223994 | 6 | I am looking for some references on modular representation theory of the orthogonal groups $O\_{2n+1}(2)$, $O\_{2n}^{+}(2)$, or $O\_{2n}^{-}(2)$ over $\mathbb{F}\_2$.
| https://mathoverflow.net/users/82685 | Representations of orthogonal groups over the field of two elements | The question is brief, but there are a great many unknown features of the modular representation theory of these families of groups (for the prime $p=2$). The approach via algebraic groups pioneered by Steinberg is conceptually attractive but far from able to deal effectively with small primes at this point. There migh... | 6 | https://mathoverflow.net/users/4231 | 224038 | 104,854 |
https://mathoverflow.net/questions/223754 | 7 | I'm reading bits and pieces of Kolar, Michor, & Slovak's *Natural Operations in differential Geometry*, and I'm having "doubt" about some of the definitions. All I'm trying to do is sheafify some of the concepts and see how natural they seem to me. I don't know any differential geometry so please don't kill me.
Let $... | https://mathoverflow.net/users/69037 | Natural operators in differential geometry - why are they natural? | This is a long comment on some of the questions from the second batch. Before I start, here is a typical example of a natural operator. Take $F=\Lambda^kT^\*$, $G=\Lambda^{k+1}T^\*$ to be the natural bundles of differential forms of degrees $k$ and $k+1$. They are clearly local, and they behave naturally under pullback... | 3 | https://mathoverflow.net/users/70808 | 224039 | 104,855 |
https://mathoverflow.net/questions/224037 | 7 | Let $A\_1,A\_2,\ldots, A\_n$ be distinct in the plane. For every $1\le i \le n$, let
$S\_i=\sum\limits\_{j=1}^n d(A\_i,A\_j)$ be the sum of distances from $A\_i$ to all the other points.
Assume that $S\_i=S\_j$ for all $1\le i<j\le n$. Is it true that the points $A\_1,A\_2,\ldots, A\_n$ must be the vertices of a conv... | https://mathoverflow.net/users/36060 | metric condition forcing convex position | Suppose that the points are not in convex position, then there is a point (WLOG let it be $A\_n$) that is a convex combination of other points:
$$A\_n = \sum\_{i=1}^{n-1} \lambda\_i A\_i\text,$$
where $\lambda\_i\ge0$ and $\sum\_i\lambda\_i=1$. Now consider the function $f\_i(X) = d(A\_i,X)$. This function is conve... | 7 | https://mathoverflow.net/users/20186 | 224040 | 104,856 |
https://mathoverflow.net/questions/224061 | 1 | Let ***A*** be non singular matrix of order **N** and inverse of ***A*** is known. Is it possible to ***find/approximate*** inverse of ***A*** if only one element of ***A***; **a(i,j)**; is replaced by number approaching infinity(***M*** = big number like 99999999) with less computations.
In this case, **1** row and ... | https://mathoverflow.net/users/83008 | Inverse of Matrix with one element approches infinity | The inverse of the sum of an invertible matrix and a matrix of rank $1$ can be computed using the [Sherman-Morrison formula](https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula).
| 2 | https://mathoverflow.net/users/13650 | 224074 | 104,863 |
https://mathoverflow.net/questions/224076 | 1 | Is it possible to construct a stochastic process $X\_t$ where the limit
$\lim\_{\Delta \rightarrow 0} \rm{Var}\left(\frac{X\_{t\_0+\Delta}-X\_{t\_0}}{\Delta}\right)$
does not exist but the sample paths are still differentiable?
| https://mathoverflow.net/users/83017 | Differentiability of stochastic process | consider $\sum X\_n e^{int}$ where the $X\_n$ have the property that $X\_n$ are independent and eventually 0. Then every sample path is a trig polynomial and infinitely differentiable, however by arranging that $\sum var(X\_n) < \infty, \sum var(nX\_n) = \infty $ you'll probably get what you want.
| 1 | https://mathoverflow.net/users/74268 | 224087 | 104,869 |
https://mathoverflow.net/questions/224102 | 2 | I was reading [this paper](http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2425490) when I came across something called the edge-perspective degree distribution in a network. Consider a graph $G$, the degree distribution of whose nodes is $f(d)$. They say the edge-perspective degree distribution is $\frac{d f(d)}{\s... | https://mathoverflow.net/users/42371 | Edge-perspective degree distribution | Its meaning should be something like "degree distribution seen from a random edge". Indeed, let us first think of an unoriented edge as of two oriented (in opposite directions) ones. Assume that there are $N$ nodes, where $N$ is large. Then, the total number of (oriented) edges should be roughly $N\sum\_d d f(d)$. So, ... | 3 | https://mathoverflow.net/users/81488 | 224110 | 104,878 |
https://mathoverflow.net/questions/224109 | -1 | On the set $[n]:= \{1,\ldots,n\}$ we consider the set $${\cal P}\_2([n]) = \big\{\{a,b\}: a,b \in [n], a\neq b\big\}.$$
Since $$|{\cal P}\_2([n])| =2^{n \choose 2}$$ there are exactly $2^{n\choose 2}$ graphs on $n$ points.
We set the clique number $\omega(G)$ to be the largest $n$ such that the complete graph $K\_n... | https://mathoverflow.net/users/8628 | Do graphs with $\omega(G) = \chi(G)$ grow "common" as $|V|$ grows large? | I would say that this limit is zero. Most of the graphs on $n$ vertices have $\sim n^2/4$ edges. For such graph, the expected number of complete subgraphs of size $k\sim2\log\_2n$ is
$$
{n\choose k}\cdot \frac1{2^{k\choose 2}}\approx C\left(\frac{ne}{k2^{(k-1)/2}}\right)^k\to 0.
$$
So most of our graphs have $\omega(G... | 6 | https://mathoverflow.net/users/17581 | 224118 | 104,883 |
https://mathoverflow.net/questions/224113 | 8 | Consider a sequence $V\_N$ of subspaces of $\ell^N\_1$ so that $\dim V\_N = N- n$ and $n$ is $\mathsf{o}(N)$. Is it true that these spaces are "thick" (unofficial terminology), i.e. are there constants $K\_1,K\_2>0$ so that for $N$ large enough,
$$
\sup\_{x \in V, \|x\|\_1 \leq K\_1} \|x\|\_\infty \geq K\_2
$$
where $\... | https://mathoverflow.net/users/18974 | Thin large subspaces of $\ell^N_1$ | Improved version of my answer. The following version of Kashin's
(1977) result is needed here: For any $\alpha\in(0,1)$ there
exists $C=C(\alpha)$ such that for any $N$ there is an
$\lceil\alpha N\rceil$-dimensional subspace $L$ of $\ell^N\_1$
satisfying
$$\forall x\in L\quad
\frac1{\sqrt{N}}||x||\_1\le||x||\_2\le\frac... | 6 | https://mathoverflow.net/users/37822 | 224122 | 104,886 |
https://mathoverflow.net/questions/224125 | 2 | I was given an open problem as a birthday present recently. While I can probably handle spoilers at this point, what I really want are literature and other references. Also acceptable would be suggestions for approaches. Ideally, search terms for the web are most welcome. (It is my lack of imagination for search terms ... | https://mathoverflow.net/users/3402 | Looking for (information about) long diamonds | Here is a simple short proof of one of the first main questions.
>
> **Claim.** $C\_p < 2$.
>
>
>
*Proof.* Let $C\_p$ be above. We look at two cases: (i) $1 < p < 2$, and (ii) $p > 2$.
Case (i): From [this short note](https://web.math.princeton.edu/~naor/homepage%20files/inequality.pdf) we know that
\begin{e... | 2 | https://mathoverflow.net/users/8430 | 224131 | 104,888 |
https://mathoverflow.net/questions/224137 | 7 | Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite undirected graph. More strongly, for any finite group $G$ there exist infinitely many non-isomorphic simple connected g... | https://mathoverflow.net/users/19885 | Frucht's type theorem for Riemann surface | You seem to be asking about the group of isometries, not the fundamental group. If so, for every $n$ and every finite group $G$ there is a compact hyperbolic manifold of dimension $n$ whose isometry group is $G.$ See [Belolipetsky and Lubotzky.](http://arxiv.org/abs/math/0406607) They actually do $n\geq 4,$ but the pap... | 10 | https://mathoverflow.net/users/11142 | 224139 | 104,891 |
https://mathoverflow.net/questions/224143 | 16 | Suppose that we interpret the output tape of a Turing machine as an assignment of true or false to all sentences of PA, taking the $n$th output bit as the truth value of the sentence with Goedel number $n$. (We can ignore bits when no sentence has number $n$, or we can choose an encoding which is an onto function; it d... | https://mathoverflow.net/users/83038 | Can a stochastic Turing machine output a consistent extension of PA with positive probability? | The answer is no, but it's almost yes. A stochastic Turing machine can find a diagonally non-computable ($\textsf{DNC}$) function ($f$ with $f(x)\ne\varphi\_x(x)$ for all $x$) and finding a complete extension of PA is equivalent to finding a $\textsf{DNC}$ function with $f(x)\in\{0,1\}$ for all $x$.
Antonin Kucera, i... | 11 | https://mathoverflow.net/users/4600 | 224145 | 104,894 |
https://mathoverflow.net/questions/224138 | 9 | Let $k$ be a field. I would like a reference for realization functors from Morel-Voevodksy's stable category $SH(k)$ to the derived categories of $Gal(\bar{k}/k)$-modules. Has something like this been written down?
Thanks!
| https://mathoverflow.net/users/24706 | Realization Functor From $SH$ to Derived Category of $Gal$-Modules | The most general functor of this form was constructed by Ayoub in *[La réalisation étale et les opérations de Grothendieck](http://user.math.uzh.ch/ayoub/PDF-Files/Realisation-Etale.pdf)*.
Ayoub considers the ∞-category $DA^{et}(S,\Lambda)$ which is defined exactly as $SH(S)$ except that (1) spectra are replaced by c... | 13 | https://mathoverflow.net/users/20233 | 224148 | 104,895 |
https://mathoverflow.net/questions/224154 | 8 | While studying some seemingly unrelated topological questions, I have experimentally discovered what appears (to me) to be a remarkable sum over partitions. I was wondering if anyone knows how to prove it.
Fixing $n \geq 1$, it can be stated as follows:
$$1=\sum\_{(a\_1^{k\_1},\ldots,a\_p^{k\_p}) \vdash n} \left(\f... | https://mathoverflow.net/users/83045 | A remarkable sum over partitions | Here is a more informative version of this identity. Let $Z\_n$ denote the *cycle index polynomial* of the symmetric group $S\_n$, namely
$$Z\_n = \frac{1}{n!} \sum\_{\sigma \in S\_n} z\_1^{c\_1(\sigma)} z\_2^{c\_2(\sigma)} \dots $$
where $c\_i(\sigma)$ denotes the number of $i$-cycles of $\sigma$. The observations... | 15 | https://mathoverflow.net/users/290 | 224159 | 104,898 |
https://mathoverflow.net/questions/224163 | 0 | This was first posted to SE, but now I think its better to be posted here.
For what positive real numbers $\alpha$, the sequence $a\_n = \frac{\lfloor n\alpha\rfloor}n $ is (not necessary strictly) increasing for sufficiently large indexes ? ($\lfloor x\rfloor$ is the integer part of $x$).
| https://mathoverflow.net/users/72273 | An increasing sequence of real numbers | Only if $\alpha$ is an integer (in which case the sequence is constant). Suppose $\alpha$ is not an integer. By subtracting $\lfloor \alpha \rfloor$, we may assume $0 < \alpha < 1$. Then there are arbitrarily large $n$, such that $0 < \lfloor n\alpha \rfloor = \lfloor (n+1)\alpha \rfloor$, so $$ \frac{\lfloor n\alpha \... | 4 | https://mathoverflow.net/users/68305 | 224164 | 104,900 |
https://mathoverflow.net/questions/224147 | 3 | Let $\widehat{F\_2}$ be the free profinite group of rank 2. Is its Frattini subgroup $\Phi(\widehat{F\_2})$ trivial?
I know that the Frattini subgroup of a pro-$p$ group is open. On the other hand, the Frattini subgroup of the abstract free group $F\_2$ is trivial.
The general profinite case $\widehat{F\_2}$ howeve... | https://mathoverflow.net/users/15242 | Is the Frattini subgroup of a free profinite group trivial? | It is trivial. This is a special case of Corollary 8.7.5 of the book of Ribes and Zalesskii. It uses that every proper open normal subgroup of a closed normal subgroup of a free profinite group is free profinite and that the Frattini subgroup is pronilpotent.
| 4 | https://mathoverflow.net/users/15934 | 224174 | 104,903 |
https://mathoverflow.net/questions/224046 | 6 | To show that the positive existential theory of $\mathbb{C}[t, e^{\lambda t} \mid \lambda \in \mathbb{C}]$ in the language $\{+, \cdot , ' , 0 , 1, t\}$ is undecidable we have to prove the following: $$n \in \mathbb{N} \leftrightarrow \exists x \left (n \in \mathbb{C} \land tx'=nx \land x(1)=1\right )$$
So do we hav... | https://mathoverflow.net/users/52805 | Show that the positive existential theory is undecidable | I like this question very much. Before answering, let me try to
explain the question in my words.
You are considering the structure $\mathbb{C}[t,e^{\lambda t}]\_{\lambda\in\mathbb{C}}$,
which is the ring of all polynomial expressions over $\mathbb{C}$ in the
indeterminate variable $t$ and $e^{\lambda t}$, for any $\... | 11 | https://mathoverflow.net/users/1946 | 224178 | 104,904 |
https://mathoverflow.net/questions/224181 | 7 | See David Speyer's answer [here](https://mathoverflow.net/a/20115/80013).
>
> I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)$ and $c(t) \in \mathbb{C}[t]$ such that
> $$a(t)^3 + b(t)^3 = c(t)^3.$$... | https://mathoverflow.net/users/nan | No nonconstant coprime polynomials $a(t)$, $b(t)$, $c(t) \in \mathbb{C}[t]$ where $a(t)^3 + b(t)^3 = c(t)^3$ | (With corrections noted by @GH from MO): The map $t\to (a(t),b(t),c(t))$, if at least one of the ratios $a(t)/c(t)$ or $b(t)/c(t)$ is non-constant, would extend to a non-constant map from a lower-genus (compact connected) curve ($\mathbb P^1$) to a higher-genus such, the elliptic curve defined by $a^3+b^3=c^3$. (Maybe ... | 12 | https://mathoverflow.net/users/15629 | 224183 | 104,905 |
https://mathoverflow.net/questions/224170 | 2 | Motivated by the [following RG question](https://www.researchgate.net/post/Does_anyone_know_that_it_is_possible_to_add_a_row_to_set_of_convex_and_compact_full_rank_matrices_such_that_rank_of_any_matrix_increases) we ask a related question as follows:
We identify $\mathbb{R}^{n} \otimes \mathbb{R}^{m}$ with $\mathbb{R... | https://mathoverflow.net/users/36688 | The action of $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ on $\mathbb{R}P^{(mn-1)}$ | By the evident isomorphism of $\mathbb R^m\otimes \mathbb R^n$ with $M\_{m\times n}(\mathbb R)$, the orbits of the action of $Gl(\mathbb R^m)\otimes Gl(\mathbb R^n)$ corresponds to orbits of the action of $Gl(\mathbb R^m)\times Gl(\mathbb R^n)$ on $M\_{m\times n}(\mathbb R)$ by
left and right multiplication:
$$(A,B)\c... | 5 | https://mathoverflow.net/users/51663 | 224184 | 104,906 |
https://mathoverflow.net/questions/224088 | 11 | Does the quaternionic Hopf fibration possibly represent a *non-torsion* element in the $G$-equivariant stable homotopy groups of spheres, for $G$ a finite subgroup of $SO(3)$ and in RO(G)-degree being its canonical 3d representation?
For the complex Hopf fibration the analog is true, as far as I see: In
* Shôrô Ara... | https://mathoverflow.net/users/381 | equivariant stable class of quaternionic Hopf fibration in RO(G)-degrees of ADE-type | I think the Hopf construction gives a non-torsion class when G is dihedral or exceptional, but probably not when G is cyclic.
Non equivalently, we can perform Hopf constructions on the 0, 1, 3, and 7 spheres. Only the 0 sphere gives a non torsion class in stable homotopy.
Suppose $G$ acts on our sphere, preserving ... | 6 | https://mathoverflow.net/users/437 | 224185 | 104,907 |
https://mathoverflow.net/questions/224171 | 6 | Let $\ f\_n \ $ be a sequence of Morse functions on $\mathbb{R}^d$, adequately converging (in the $C^2$-topology, say) to a limit Morse function $\ f$:
$$ f\_n \to f \ .$$
At any critical point $\ p\ $ of the limit function $\ f$, it can be proved that there exists an open neighborhood $\ U\ $ such that, for $n \geq ... | https://mathoverflow.net/users/7519 | Stable manifolds of a sequence of Morse functions | The answer is no. Of course, in a neighbourhood of a critical point,
one has convergence.
Assume that the stable manifolds $W^s(p\_n)$ converge to some set $W$ in the Hausdorff distance.
It can happen that there is a sequence $q\_n\in W^s(p\_n)$ that converges
to a critical point $q\in W$ that lies in the interior of... | 5 | https://mathoverflow.net/users/70808 | 224187 | 104,909 |
https://mathoverflow.net/questions/224173 | 3 | Are there examples of smooth closed manifolds (not necessarily oriented) that admit an open book decomposition but that are not the boundary of any compact smooth manifold?
| https://mathoverflow.net/users/67031 | open book decompositions and being a boundary | In dimensions > 6 a simply connected manifold is an open book if and only if its signature is zero. (winkelnkemper '73)
On the other hand, for example the 8-dimensional bordism group is $Z\oplus Z$ (generated by the simply connected manifolds $CP^4$ and $CP^2\times CP^2$), so it necessarily has some nontrivial (and s... | 6 | https://mathoverflow.net/users/39082 | 224189 | 104,910 |
https://mathoverflow.net/questions/198352 | 3 | Is there an online repository for zeros of the prime zeta function? I looked at the Yahoo group
[Prime numbers and primality testing](http://groups.yahoo.com/group/primenumbers/files/Pari-GP%20code/pzeros.txt) listed on the MathWorld notebook for the prime zeta function, but the group is restricted. Is there any open ... | https://mathoverflow.net/users/45057 | Prime zeta zeros - reference | It seems they were hidden deep in the notebook download from the Mathworld page all along! [*Here is a link to them*](https://github.com/martinq321/Prime_Zeta_Zeros/blob/master/zero_values) for ease of reference.
| 0 | https://mathoverflow.net/users/45057 | 224198 | 104,914 |
https://mathoverflow.net/questions/224132 | 4 | I'm new in the field, so I'm sorry in advance if my question is too naive.
Let's consider $S$ a surface of genus $g\ge 2$ with an hyperbolic metric $g$. Let's call $\mathcal{S}(S)$ the set of closed geodesics on $S$ with respect to $g$.
Usually on $\mathcal{S}(S)$ there is the topology induced by the hausdorff metr... | https://mathoverflow.net/users/83036 | Compact open topology on the space of geodesics | Consider the set $\tilde{\mathcal G}(S)$ of all unit speed parametrised geodesics, closed or not.
Then regard the space of closed parametrised geodesics $\tilde{\mathcal S}(S)$ as a subspace with the respective subspace topology. Denote the quotients by $\mathbb R$ as $\bar{\mathcal G}(S)$ and $\bar{\mathcal S}(S)$, so... | 1 | https://mathoverflow.net/users/70808 | 224202 | 104,916 |
https://mathoverflow.net/questions/224056 | 5 | Let $F$ be the free *profinite* group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL\_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also happy to consider $\text{Out}(F)$ rather than $\text{Aut}(F)$, though there shouldn't be much difference.)
Here, the i... | https://mathoverflow.net/users/15242 | abelian and nonabelian parts of Aut($\widehat{F_2}$) | $\newcommand{\Zhat}{\widehat{\mathbb{Z}}}$
Here are some partial answers, though I would welcome any additional input.
There is a natural map
$$p : F\rightarrow F^\Delta\times\Zhat^2$$
The kernel of $F\rightarrow\Zhat^2$ is just $[F,F]$, and since all finite quotients of $F^\Delta$ are perfect, $[F,F]$ surjects ont... | 2 | https://mathoverflow.net/users/15242 | 224206 | 104,919 |
https://mathoverflow.net/questions/224210 | 8 | Let $f\colon X\to Y$ be a surjective continuous map between two topological spaces such that $X,Y$ are path-connected and such that every fibre $f^{-1}(y)$ is connected, for each $y\in Y$. Is there always a continuous section $s\colon Y\to X$ (i.e. a continuous map $s$ such that $f\circ s$ is the identity on $X$)?
If n... | https://mathoverflow.net/users/23758 | Existence of a continuous section | No, even if $Y=[0,1]$. The piecewise linear continuous nondecreasing surjection $f:[0,1] \rightarrow [0,1]$ which maps $[1/3,2/3]$ to $1/2$ and is otherwise 1-1 and linear has no continuous section.
| 12 | https://mathoverflow.net/users/17029 | 224219 | 104,921 |
https://mathoverflow.net/questions/224231 | 12 | Recently I saw an interesting lemma:
For any $s>0$, the closed unit ball in $H^s$ is also closed in the $L^2$ norm. That is, suppose $u\_j\in H^s$ and $\|u\_j\|\_{H^s}\le 1$. Suppose $u\_j\to u$ in $L^2$. Then $u\in H^s$ and $\|u\|\_{H^s}\le 1$.
A possible proof of the above lemma is to take the Fourier transform a... | https://mathoverflow.net/users/37103 | When is the closed unit ball in a smaller Banach space closed in a larger Banach space? | Suppose that you have two Banach spaces $X$ and $Y$, and a (bounded) operator $TX:\to Y$. The operator $T$ is called *semi-embedding* if $T$ is injective and $T(B\_X)$ is closed in $Y$. So your are asking when natural maps between classical Banach spaces are semi-embeddings.
I do not know characterizations of semiem... | 12 | https://mathoverflow.net/users/39421 | 224234 | 104,926 |
https://mathoverflow.net/questions/224222 | 5 | An object $G$ of a category $\mathcal{C}$ is a *dense generator* if every object $X$ is the colimit of the canonical diagram of copies of $G$ mapping to $X$. (This canonical diagram is indexed by the full subcategory of the slice $\mathcal{C}\_{/X}$ on the objects of the form $G \to X$.)
An object $G$ is called a *co... | https://mathoverflow.net/users/644 | What is an example of a colimit-dense generator which is not dense? | $\Bbb R$ is colimit-dense in the category of real vector spaces but not dense
(see 6.F, 6.34 in my book with J. Adámek ["Locally presentable and accessible categories"](http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511600579)).
| 9 | https://mathoverflow.net/users/73388 | 224240 | 104,927 |
https://mathoverflow.net/questions/224232 | 20 | I suspect that the curve $x^5 + y^5=7$ has no $\mathbb Q$ points, and a brief computer search verifies this hypothesis for denominators up to $10^4$. What techniques can be used to show that there are no solutions?
| https://mathoverflow.net/users/29961 | Rational points on the "quintic circle" $x^5 + y^5 = 7$ | There is an action of $\mu\_5$, the group of fifth roots of unity, on your curve,
given by $\zeta \cdot (x,y) = (\zeta x, \zeta^{-1} y)$. The quotient by this
group action is the hyperelliptic curve
$$C \colon Y^2 = X^5 + \frac{49}{4},$$
the map being given by $(X, Y) = (-xy, x^5 - \frac{7}{2})$. So it is enough
to fin... | 41 | https://mathoverflow.net/users/21146 | 224242 | 104,929 |
https://mathoverflow.net/questions/223211 | 1 | Let $C$ be a site and $F$ an abelian presheaf on $C$. Suppose that for each object $U$ in $C$ there is a covering $\{ U\_i\to U \}$ such that $F(U\_i)=0$. Is it true that $F^{sh}=0$?
This should be true, and is trivially if the presheaf is separated, but I don't see how to prove it in general.
edit: slight clarifica... | https://mathoverflow.net/users/82627 | If presheaf is zero on a covering is the sheaf zero? | Indeed in the situation above
$$F^\nmid=\varinjlim\_{\lbrace U\_i\to U\rbrace} \ker \left(\prod\_i F(U\_i)\rightrightarrows \prod\_{i,i'} F(U\_i\times\_U U\_{i'})\right)=:H(\lbrace U\_i\to U\rbrace, F)$$
is already zero. Fix a covering $\lbrace U\_i\to U \rbrace$. By assumption we find $\lbrace V\_{ij}\to U\_i\rbr... | 2 | https://mathoverflow.net/users/82627 | 224244 | 104,930 |
https://mathoverflow.net/questions/224225 | 5 | Let $\Gamma = PSL(2,\mathbb{Z}) = \langle S,T \ | \ S^2=(ST)^3=1 \rangle$. Let $G$ be some mystery normal subgroup of $\Gamma$ that we happen to think may be congruence. Recall that a subgroup of $\Gamma$ is a congruence subgroup of level $N$ if it contains the principal congruence subgroup of level $N$.
Are there me... | https://mathoverflow.net/users/83081 | Upper bound on level of a congruence subgroup of the modular group | If you only look at upper bounds, you can use the result by Lubotzky: For every algebraic group G, there is some $c$, such that the level of a congruence subgroup $U$ is bounded by the $c(G:U)$.
If you want to determine the level algorithmically, you can look at the image of the subgroup in $PSL(\mathbb{Z}/p^n\mathbb... | 4 | https://mathoverflow.net/users/37555 | 224245 | 104,931 |
https://mathoverflow.net/questions/223926 | 6 | I'm trying to understand the ergodic theory approach to statistical mechanics, namely how ergodic measure preserving dynamics lead to the Gibbs measure.
I have a compact space $X$, a probability measure $\mu$ on $X$ and a one parameter family of transformations $T\_t:X\rightarrow X$ ergodic on $\mu$. As I understand,... | https://mathoverflow.net/users/82928 | Ergodic theory: from Dynamics to Gibbs measure | That's an excellent but highly unresolved question. The problem is that the *physicists* tend to be not so interested in mathematical foundations once a theory (statistical mechanics in this case) is successful and there is some plausible heuristic justification for it, whereas *ergodic theorists* quite often have litt... | 6 | https://mathoverflow.net/users/23297 | 224246 | 104,932 |
https://mathoverflow.net/questions/224247 | 6 | It is known that every Mahlo cardinal $\kappa$ is hyper $\kappa$-inaccessible. It the converse true, namely: every cadinal $\kappa$ which is hyper $\kappa$-inaccessible is a Mahlo cardinal ?
| https://mathoverflow.net/users/69236 | Mahlo cardinal and hyper k-inaccessible cardinal | The answer is no. Mahloness is much stronger than this.
Every Mahlo cardinal $\kappa$ is a limit of such cardinals. One can see this, because there is a club of $\gamma<\kappa$ with $V\_\gamma\prec V\_\kappa$, and by Mahloness, we can find such a $\gamma$ that is inaccessible. Since the degrees of hyper-inaccessibil... | 8 | https://mathoverflow.net/users/1946 | 224248 | 104,933 |
https://mathoverflow.net/questions/224259 | 6 | Let $X$ be a $\sigma$-compact topological space and $(Y,d)$ be a metric space.
Let $\{K\_n\}$ be a sequence of compact subsets of $X$ whose union is $X$.
Define $\rho\_n(f,g):=\sup \{d(f(z),g(z)): z\in K\_n\}$ and $\rho(f,g)=\sum\_{n=0}^\infty (\frac{1}{2})^n \frac{\rho\_n(f,g)}{1+\rho\_n(f,g)}$ for all $f,g\in C(X... | https://mathoverflow.net/users/83098 | How do I prove that compact-open topology is metrizable? | I don't think this does induce the compact-open topology as stated.
Let $X = \{1,1/2, 1/3, \dots, 0\}$ with its usual Euclidean metric (so $X$ is a compact metric space). Let $K\_0 = \{0\}$ and $K\_n = \{1/n\}$, so that $K\_n$ is compact and $X = \bigcup\_n K\_n$. Let $Y = \{0,1\}$ with the obvious metric. Now the co... | 5 | https://mathoverflow.net/users/4832 | 224264 | 104,937 |
https://mathoverflow.net/questions/224243 | 23 | I have read in many places that the noetherian hypothesis is often overkill - both in commutative algebra and in ($\overset?=$) algebraic geometry. In particular, I've read that coherence and finite presentation are the *really* important properties. This is nicely stated in the foreword to Quitté and Lombardi's [Commu... | https://mathoverflow.net/users/69037 | Examples of Noetherian overkill | 1) It is sometimes stated that a finitely generated module $M$ over a Noetherian commutative ring $R$ is projective if for all maximal ideals $\mathfrak m\subset R$ the localized module $M\_\mathfrak m$ is free over $R\_\mathfrak m$.
However the noetherianity of $R$ is unnecessary if you add the hypothesis that $M$ ... | 13 | https://mathoverflow.net/users/450 | 224277 | 104,942 |
https://mathoverflow.net/questions/224283 | 7 | Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $A$ be an abelian variety over $k$ such that $A[p](k) = 0$, i.e., such that $A$ has $p$-rank $0$. If I am not mistaken, this implies that $\mathrm{Ker}(F) \subset A[p]$, where $\mathrm{Ker}(F)$ is the Frobenius kernel of $A$. Is it true that $A[... | https://mathoverflow.net/users/63877 | $p$-torsion of an abelian variety of $p$-rank $0$ | No, this is another entry in the list of ways in which elliptic curves can be a poor guide to the higher-dimensional case. The kernel of $F\_{A/k}:A \rightarrow A^{(p)}$ is *always* contained in $A[p]$ since $\ker F\_{A/k}$ is an infinitesimal commutative group scheme whose own Frobenius morphism vanishes (and all such... | 9 | https://mathoverflow.net/users/81332 | 224290 | 104,948 |
https://mathoverflow.net/questions/224296 | 7 | I need to count the number of perfect matchings of a certain family of graphs. This family of graph is non planar and a type of snark. For the initial cases, it seems that this number is growing exponentially. My request is different from the one [here](https://en.wikipedia.org/wiki/FKT_algorithm) because right now, I ... | https://mathoverflow.net/users/83122 | Algorithm to count the number of perfect matchings in non planar graph | Counting the number of perfect matchings in arbitrary graphs (i.e. non planar, non bipartite...) seems to be quite more difficult than the restricted cases that FKT-type algorithms can handle.
In particular, Valiant proved that the problem is in $\mathrm{\text{#}P}$.
With that in mind, you can find information on a... | 8 | https://mathoverflow.net/users/43108 | 224298 | 104,950 |
https://mathoverflow.net/questions/224316 | 3 | We know that, if we have a surface $z=f(x,y)$ with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form):
$$g=\begin{bmatrix}
1+\left ( \frac{\partial f(x,y)}{\partial x} \right )^2 & \frac{\partial f(x,y)}{\partial x}\frac{\partial f(x,y)}{\partial y} & \\\
\frac{\partial f(x,y)}{\... | https://mathoverflow.net/users/21753 | Prescribing an induced metric | Assume $b\ne0$.
Then you know $\alpha=\tfrac{\partial f}{\partial x}$ and $\beta=\tfrac{\partial f}{\partial y}$ up to sign.
You should check $$\frac{\partial \alpha}{\partial y}=\frac{\partial \beta}{\partial x}.$$
If this is true you can restore $f$ by integrating.
| 2 | https://mathoverflow.net/users/1441 | 224320 | 104,953 |
https://mathoverflow.net/questions/224300 | 2 | Let $a < b$ be two natural numbers. I will use these as an example:
\begin{align\*}
a & = 2^5 \cdot 3^2 \cdot 5^2 = 7200\\\
b & = 2^3 \cdot 3^5 \cdot 7^1 = 13608
\end{align\*}
I seek to "morph" $a$ to $b$ via $a{=}n\_0,n\_1,n\_2,\ldots,n\_k{=}b$
such that
* Each step is *upward*: $n\_{i-1} < n\_i < n\_{i+1}$ (monoton... | https://mathoverflow.net/users/6094 | Gradual monotonic morphing between two natural numbers | If I understand the question right, then $k$ is simply the number of divisors $d$ of $\text{lcm}(a,b)$ such that $a\le d\le b$ and $d\mid\gcd(a,b)$. (So in finding a longest chain, we may assume that $a$ and $b$ are relatively prime.)
In your example, a longest chain would be
\begin{equation}
7200\to 7560\to 7776\to ... | 2 | https://mathoverflow.net/users/18739 | 224322 | 104,954 |
https://mathoverflow.net/questions/224346 | 2 | This question is in reference to the following Mathoverflow [question](https://mathoverflow.net/questions/218481/ck-one-parameter-family-of-metrics) and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g\_t$ perturbs real analytically in time, so does the Laplacian $\Delta\_t$ ... | https://mathoverflow.net/users/81039 | A clarification regarding analytic perturbation of metrics and Laplacian | This seems to be answered by Rafe Mazzeo in this question: [Analytic dependence on the metric](https://mathoverflow.net/questions/131744/analytic-dependence-on-the-metric) (the accepted answer elaborates on Rafe's).
| 2 | https://mathoverflow.net/users/11142 | 224353 | 104,963 |
https://mathoverflow.net/questions/224188 | 8 | I'm familiar with sympy, the matlab symbolic package, reduce, and have tried out a few other computer algebra systems. However, as far as I can tell, none of them seem to be able to do algebra on variable sized matrices - they can only work with fixed sized matrices.
Are there any that can do algebra for variable siz... | https://mathoverflow.net/users/49223 | Computer Algebra Systems that support variable sized matrices | SymPy has a [matrix expressions](http://docs.sympy.org/latest/modules/matrices/expressions.html) module that does this. Example:
```
>>> from sympy import MatrixSymbol, Matrix, symbols
>>> n, m = symbols('n m', integer=True)
>>> X = MatrixSymbol('X', n, m)
>>> Y = MatrixSymbol('Y', m, n)
>>> (X*Y).T
Y'*X'
```
Matr... | 9 | https://mathoverflow.net/users/11781 | 224354 | 104,964 |
https://mathoverflow.net/questions/224352 | 4 | Let $\mathcal{A}$ be an uncountable almost disjoint family (not necessarily maximal) of infinite subsets of $\mathbb{N}$. Denote by $\mathcal{A}\_{\subseteq}=\{ B\subseteq\mathbb{N}:|B|=\omega \wedge \exists A\in\mathcal{A}(B\subseteq A) \}$.
Question 1: Must there exist a $B\in\mathcal{A}\_{\subseteq}$ such that for... | https://mathoverflow.net/users/16107 | A property of uncountable almost disjoint families | Yes to both: Just take a condensation point $A$ of $\mathcal{A}$. This means every clopen neighborhood of $A$ contains uncountably many members of $\mathcal{A}$.
| 3 | https://mathoverflow.net/users/83154 | 224355 | 104,965 |
https://mathoverflow.net/questions/222021 | 7 | Let $p:X\rightarrow Y$ be a double cover of curves, denote by $$SU\_n:=(p\_\*SL\_n(\mathcal O\_X))^{\tilde{\sigma}}$$
i.e. the $\tilde{\sigma}-$invariant part, the action of $\tilde{\sigma}$ is given by $$\tilde{\sigma}(g)=\,^t(g\circ\sigma)^{-1}$$ where $\sigma$ is the involution induced by the double cover. $SU\_n$ i... | https://mathoverflow.net/users/66528 | Etale fundamental of a parahoric group scheme | There is an exact sequence
$$ \pi\_1 ((SU\_n)\_{\overline{\eta}}) \to \pi\_1((SU\_n)\_{\eta}) \to \operatorname{Gal}(\overline{\eta}|\eta)$$
Over $\overline{\eta}$, $p\_\* SL\_n$ is just $SL\_n \times SL\_n$. The involution $\tilde{\sigma}$ acts by switching the two $SL\_n$s and then doing an performing some automo... | 2 | https://mathoverflow.net/users/18060 | 224356 | 104,966 |
https://mathoverflow.net/questions/224334 | 1 | I have a question regarding the sums $\sum\_{i=1}^{n}v\_{j}\left(i\right)$ where $v\_j$ are eigenvectors of adjacency matrix $A$ which have been normalized to unit length.
Ordering the eigenvectors by their eigenvalues and computing these sums on both random and real-world graphs I noticed that $\sum\_{i=1}^{n}v\_{1... | https://mathoverflow.net/users/83141 | Sum of Eigenvectors Entries of an Adjacency Matrix | This is not true in general. Consider first a graph with two components, one a complete graph on $4$ vertices and the other a cycle on $n-4$ vertices.
Then $v\_1$ is the eigenvector for the largest eigenvalue of the complete graph, and sums to $2$, while $v\_2$ is the eigenvector for the largest eigenvalue of the cy... | 3 | https://mathoverflow.net/users/405 | 224375 | 104,972 |
https://mathoverflow.net/questions/224361 | 4 | **Question:** Let $f(x) \in x\mathbb{C}[[x]]$. What is the (asymptotically) fastest algorithm for calculating the coefficient of $x^n$ in $e^{f(x)}$?
**Naive Solution 1:** Using fast polynomial multiplication (I assume it takes $O(n \log n)$ time to multiply two polynomials of degree $n$), one can solve this problem ... | https://mathoverflow.net/users/31469 | Extraction of Coefficients in the Exponential Function of a Series | It can be done with $O(1)$ polynomial multiplications, i.e. $O(n \log n)$ with your assumptions. One uses Newton's method to invert the logarithm of a power series. The logarithm is computed using $O(1)$ polynomial multiplications as
$$\log(f(x)) = \int \frac{f'(x)}{f(x)}.$$
More generally, one can use similar tech... | 4 | https://mathoverflow.net/users/4854 | 224379 | 104,973 |
https://mathoverflow.net/questions/224395 | 1 | Let $C\_1,C\_2,\ldots,C\_n$ be the conjugacy classes of a finite group $G$. It is possible that there are two non-conjugate subgroups $K \leq G$ and $H \leq G$ such that $|H \cap C\_i|=|K \cap C\_i|$ for all $i = 1,\ldots,n$?
| https://mathoverflow.net/users/23661 | Determining conjugacy class of a subgroup from the sizes of its intersections with the conjugacy classes | Yes, this is precisely the definition of a *Gassmann triple*. Your condition is equivalent to the permutation representations $\mathbb{C}[G/K]$ and $\mathbb{C}[G/H]$ being isomorphic. I believe that the smallest example (in terms of $[G:K]$) is $G=SL\_3(\mathbb{F}\_2)$, $K$ the stabilizer of a line in $\mathbb{F}\_2^3$... | 7 | https://mathoverflow.net/users/40821 | 224396 | 104,978 |
https://mathoverflow.net/questions/224226 | 2 | Let $X=Jac(C)$ be an abelian surface over $\mathbb{C}$, the Jacobian of a genus 2 curve. Let $L$ be a symmetric line bundle. Let $Y$ be the Kummer surface, quotient of $X$ by the action of involution. Then $L^2$ is totally symmetric, hence there is a line bundle $L'$ on $Y$ which pulls back to $L^2$. Further since $L^2... | https://mathoverflow.net/users/70211 | Is there a unique line bundle in the Kummer surface which pulls back to a totally symmetric line bundle? | 1) Yes. If there is another one, it differs from $L'$ by a line bundle $M$ with $M^{2}\cong \mathcal{O}\_Y$. Consider the resolution $\pi :\hat{Y}\rightarrow Y$ obtained by blowing up the double points $p\_1,\ldots ,p\_{16}$. Since $\hat{Y}$ is simply connected, we have $\pi^\* M\cong \mathcal{O}\_{\hat{Y}}\ $.
Thus... | 1 | https://mathoverflow.net/users/40297 | 224400 | 104,979 |
https://mathoverflow.net/questions/224414 | 21 | I am trying to understand bits and pieces of Lawvere's article [*Continuously Variable Sets; Algebraic Geometry = Geometric Logic*](https://www.dropbox.com/s/p8739l5bk01dpdc/Lawvere%20F.W%20-%20Continuously%20Variable%20Sets%3B%20Algebraic%20Geometry%20%3D%20Geometric%20Logic.pdf?dl=0). I'm not doing very well.
I kno... | https://mathoverflow.net/users/69037 | Joyal's construction of the spectrum of a commutative ring | Since I don't know precisely which parts of Lawvere's article you have difficulties with, this answer is a bit a long and tries to give a bit of context. If you want me to be more specific at some point, just say so.
Classically, the spectrum of a ring $A$ can be defined as the set of its prime ideals equipped with t... | 40 | https://mathoverflow.net/users/31233 | 224415 | 104,982 |
https://mathoverflow.net/questions/224413 | 7 | What is the asymptotics of the number of the maximal subgroups of $S\_n$ (as a function of $n$)? This must be written down somewhere...
**EDIT** I am actually more interested in the number of *conjugacy classes* of maximal subgroups (the difference is graphically illustrated by Derek's and Gerry's comments.)
| https://mathoverflow.net/users/11142 | number of maximal subgroups of the symmetric group | With regard to the conjugacy class question, you should refer to this:
>
> Liebeck, Martin W.; Shalev, Aner *Maximal subgroups of symmetric groups*.
> J. Comb. Theory, Ser. A 75, No.2, 341-352 (1996).
>
>
>
The following is a quote from the ZBMath review by W. Knapp:
>
> The purpose of this paper is to gi... | 13 | https://mathoverflow.net/users/801 | 224429 | 104,987 |
https://mathoverflow.net/questions/224370 | 4 | Let $K$ be a number field and consider the maximal abelian extension $K^{ab}$ of $K.$ For a finite prime $p,$ letting $K\_p$ be the completion of $K$ at $p,$ we have an extension $K\_p \subset K\_p K^{ab}$ where the latter denotes the compositum of the fields.
Is $K\_p K^{ab} = K\_p^{ab}?$ That is, can I get the max... | https://mathoverflow.net/users/nan | Is the localization of the maximal abelian extension still a maximal abelian extension? | Your notations $N(C\_K^{\rm{ab}})$ and $N(C\_{K^{\rm{ab}}})$ are meaningless as written (and the first was probably a typo, meant to be the latter), and your comments about norms from $(K\_p)^{\rm{ab}}$ are also meaningless as written, since there is no sense of "norm" from an infinite-degree extension or useful notion... | 3 | https://mathoverflow.net/users/81332 | 224430 | 104,988 |
https://mathoverflow.net/questions/223933 | 5 | Suppose $K\subset \mathbb{R} ^n$ is a closed convex set whose interior contains the origin. We can assign a gauge function to $K$ as $g\_{K}(x):=\inf\{\lambda>0 \mid x\in\lambda K\}$. $g\_K$ has all the properties of a norm on $\mathbb{R}^n$ except for $g\_K(-x)=g\_K(x)$, and it is a norm when $K=-K$. We can think of $... | https://mathoverflow.net/users/35800 | Questions about the regularity of the "norm" associated to a convex set | Although the answers to these regularity questions are not addressed explicitly in the paper cited in the comments to the question, they all follow from the formulas derived in that paper.
1) The answer is yes. The boundary $\partial K$ is the radial graph of its gauge function over the unit sphere. A hypersurface is... | 3 | https://mathoverflow.net/users/613 | 224432 | 104,989 |
https://mathoverflow.net/questions/224425 | 6 | I have a question about the topological space underlying a Banach space.
A topological space $X$ is realcompact iff it is homeomorphic to a closed subset of an infinite product of the form $\mathbb R^\kappa$. Closed subsets of realcompact spaces are realcompact.
A classical result in infinite topology states that e... | https://mathoverflow.net/users/58628 | Which Banach spaces are realcompact? | Every metric space of nonmeasurable cardinality is realcompact. [1] 15.24. Thus, if there are no measurable cardinals, then every metric space is realcompact
As you noted...To get a Banach space that is not realcompact: Let $X$ be a set with measurable cardinal, and then the discrete topology on it is not realcompac... | 8 | https://mathoverflow.net/users/454 | 224433 | 104,990 |
https://mathoverflow.net/questions/224405 | 8 | The [partial μ-recursive functions](https://mathoverflow.net/questions/164694/are-all-functions-in-bishops-constructive-mathematics-continuous) which may or may not be provably total seem to have some direct relation to the initial motivations for intuitionistic mathematics. (Following Kronecker, one motivation might h... | https://mathoverflow.net/users/20781 | Did Bishop, Heyting or Brouwer take partial functions seriously? | I don't believe that Bishop explicitly assumed all functions are continuous. I think that "no discontinuous function can be proved to be total in Bishop's constructive mathematics" is actually a very good way of summarizing the situation.
For example, consider how we would state "All functions from $\mathbb{R}$ to $... | 10 | https://mathoverflow.net/users/5442 | 224438 | 104,992 |
https://mathoverflow.net/questions/223891 | 4 | Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with minimal distance (NP-hard in some formulation, right?)
I wonder what is known in case $p\in\Lambda^\*$ is in the dual ... | https://mathoverflow.net/users/22709 | Closest point to a dual lattice point (in particular for root lattices!) | For a general $\Lambda$ the assumption $p \in \Lambda^\*$
doesn't help, because $\Lambda$ can be rescaled to make any $p$
arbitrarily close to a dual lattice point.
The special case of a scaled root lattice is easy, because you can just
subtract roots from a coset representative until it can be made no smaller.
But a... | 2 | https://mathoverflow.net/users/14830 | 224451 | 104,997 |
https://mathoverflow.net/questions/224410 | 2 | Observe $\mathbb{Z}\_q^n = \mathbb{Z}\_q \times \cdots \times\mathbb{Z}\_q$ as a module over $\mathbb{Z}\_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$.
I am interested in the following questions:
How many submodules of size $q^k$, $k\leq n$, does it have? How many of them are free? Can something be said about the ... | https://mathoverflow.net/users/83189 | The number of submodules of $\mathbb{Z}_q^n$ | Aren't you just asking for the number of subgroups of $\mathbb{Z}\_q^n$
of size $q^k$?
There isn't going to be a simple answer. The classification and
enumeration of subgroups of finite abelian groups goes back to Garrett
Birkhoff. See
<http://plms.oxfordjournals.org/content/s2-38/1/385.citation>. Some more
recent work... | 4 | https://mathoverflow.net/users/2807 | 224463 | 105,001 |
https://mathoverflow.net/questions/224408 | 1 | Given $\alpha\in\Bbb N$, can there be more than $(\log N)^4$ divisors (composites allowed) of $N$ in $\big[\frac\alpha2,\alpha\big]$ when $\sqrt N\in\big[\frac\alpha2,\alpha\big]$?
What is the maximum number of divisors (composites allowed) asymptotically?
| https://mathoverflow.net/users/nan | On the number of divisors in a given range | This is certainly possible. I will construct an example with $\alpha=\sqrt{N}$. That is, I will exhibit a square number $N$ with more than $(\log N)^4$ divisors lying in $[\tfrac{1}{2}\sqrt{N},\sqrt{N}]$.
Let $x>2$ be a large parameter. Consider
$$ M:=\prod\_{p\leq x}p \qquad\text{and}\qquad N:=M^2\lfloor\tfrac{2}{3... | 8 | https://mathoverflow.net/users/11919 | 224465 | 105,002 |
https://mathoverflow.net/questions/224411 | 4 | *Asked on MSE without response [here](https://math.stackexchange.com/questions/1534467/goldbach-for-certain-classes-of-n).*
$\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$.
The [Wiki article on the Goldbach conjecture](https://en.wikipedia.org/wiki/Goldbach's_conjecture) sta... | https://mathoverflow.net/users/45057 | Goldbach for certain classes of $n$ | Well, "specific" is a vague term. For example, any number of the form $p+q$ (with $p$ and $q$ primes), such as $p+3$, is trivially a sum of two primes. But the answer to your (vague) question is certainly no, at least I don't know any reasonable characterization (independent of the primes) that would yield a representa... | 2 | https://mathoverflow.net/users/11919 | 224469 | 105,005 |
https://mathoverflow.net/questions/224446 | 3 | For one-dimensional random walk, it is well-known that if the walk goes for $n$ steps, with constant probability it ends within $\pm\sqrt{n}$.
What is the bound, in terms of $n$, such that if the walk goes for $n$ steps, with constant probability it *always stays* within that bound? It is probably no longer on the or... | https://mathoverflow.net/users/83212 | Bound that random walk stays within with constant probability? | My interpretation of the problem is that you want a function $f(n)$ so that a walk of $n$ steps stays within $f(n)$ of the origin with probability $c+o(1)$ for some $0\lt c \lt 1$. If so, the right form of $f$is still $c'\sqrt{n}$ since the rescaled limit as $n\to \infty$ is Brownian motion, and for any $a$, Brownian m... | 4 | https://mathoverflow.net/users/2954 | 224472 | 105,006 |
https://mathoverflow.net/questions/224494 | 4 | The above Diophantine equation is unknown to have any further integer solutions other than $(x, y, z) = (1, 1, 1)$ and $(4, 4, -5)$.
I am a prospective undergraduate mathematics student in Zimbabwe and I recently obtained a new result that the above equation indeed has no further integer solutions such that $ \frac{1... | https://mathoverflow.net/users/nan | A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$ | My opinion is:
**Step 1**.
Repeat the arguments you use in your proofs again and again.
If you are still sure that you have some new (valid) things to say about the equation then go to
**Step 2**.
Learn how to use [LaTeX](http://arxiv.org/) ,write your paper and submit it to the [arXiv](http://arxiv.org/... | 7 | https://mathoverflow.net/users/38851 | 224503 | 105,017 |
https://mathoverflow.net/questions/224493 | 4 | I would be interested to know the answer to the above question for the constructible bounded derived category on complex analytic or complex algebraic manifolds (or some other context). A reference would be helpful.
| https://mathoverflow.net/users/16183 | Does the nearby cycle functor commute with the Verdier duality? | According to Dimca's "Sheaves in topology" Proposition 4.2.10, the (perverse) nearby and vanishing cycle functors commute non-canonically with the Verdier duality functor if you have field coefficients. For a proof he refers to Brylinski's "Transformations canoniques, dualité projective, théorie de Lefschetz, transform... | 4 | https://mathoverflow.net/users/1310 | 224505 | 105,018 |
https://mathoverflow.net/questions/224501 | 12 | I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them.
**First definition**: Let $\mathbf{C}$ be a simplicial model category and $\mathrm{A}$ be a set of morphism. A map $f\colon V\to W$ is an $A$-local eq... | https://mathoverflow.net/users/12204 | Simple question: different definitions of Bousfield localization | Yes, they are the same. In order to prove it, we should show that they have the same new weak equivalences (this is enough, because both have the same cofibrations). Have a look at Barwick's paper *On Left and Right Model Categories and Left and Right Bousfield Localization.* In 4.45 he defines what "Enriched left Bous... | 4 | https://mathoverflow.net/users/11540 | 224506 | 105,019 |
https://mathoverflow.net/questions/224491 | 0 | It is well known that Schubert polynomials form a basis for the polynomial ring $\mathbb{Z}[x\_1,x\_2,x\_3,...]$.
I am interested in knowing how to express a particular polynomial into sum of Schubert polynomial.
Does anyone can give me reference on program( say Macaulay2) expressing a particular polynomial in ter... | https://mathoverflow.net/users/41979 | expressing in terms of sum of (double) schubert polynomial | First break $p$ into its homogeneous components, since Schubert polynomials are homogeneous.
Now find the lex-last term $c \prod\_i x\_i^{L\_i}$ of $p$ -- look for the largest $n$ such that $x\_n$ occurs in a term, then take the terms with the highest power of that $x\_n$, then of those take the terms with the highes... | 3 | https://mathoverflow.net/users/391 | 224508 | 105,021 |
https://mathoverflow.net/questions/224252 | 5 | Let $G$ be a finitely generated hyperbolic group, and let $H \leq G$ be a subgroup whose profinite completion is finitely generated. Must $H$ be finitely generated?
>
> In view of Ian Agol's answer, I am ready to assume that $G$ is
> residually finite.
>
>
>
| https://mathoverflow.net/users/38889 | Subgroups of hyperbolic groups | Let $A$ be a finitely generated group, and let $\beta \colon A \to A$ be an injective homomorphism which is not surjective. Freely construct a group $G$ generated by $A$ and some formal element $t$ such that the equality $tat^{-1} = \beta(a)$ holds in $G$ for each $a \in A$. $G$ is called the strict ascending HNN exten... | 5 | https://mathoverflow.net/users/38889 | 224511 | 105,022 |
https://mathoverflow.net/questions/224496 | 21 | This is a rather soft question. I was wondering if someone could explain on a fundamental and intuitive level, what the Okounkov-Vershik approach to representation theory of $S\_n$ is all about. It's one thing to read a few chapters of a book or a paper, but it's another to try and understand the wider picture of the t... | https://mathoverflow.net/users/82976 | Okounkov-Vershik approach to representation theory of $S_n$ | First, I'll note that they provide a pretty clear explanation of their motivation on page two of that paper. So it would strengthen your question if you indicated you had read that, and what about it you found unsatisfying.
My own feeling is that the key observation of their approach is that the space of elements of... | 15 | https://mathoverflow.net/users/66 | 224517 | 105,024 |
https://mathoverflow.net/questions/224512 | 8 | I know that such questions may be better suited for math.stackexchange, but I believe that that the topic of simplicial homotopy theory is advanced enough for mathoverflow.
Besides, I know that there are a lot of people working in homotopy theory who have probably at least used the book as a reference.
I know that, obv... | https://mathoverflow.net/users/82193 | P.G.Goerss, J.F.Jardine, "Simplicial Homotopy Theory" prerequisites | As the commenters already argued, I would not regard this book as a self-contained introduction. For instance, from a brief browse through the introductory chapters:
* The reader is assumed to be familiar with CW-complexes and several of the major theorems about them already which will be generalized (e.g. the Whiteh... | 18 | https://mathoverflow.net/users/360 | 224532 | 105,031 |
https://mathoverflow.net/questions/224484 | 6 | (1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset X$ is trivial. Then is it true that there is an exact triangle of complexes $K^\* (Z)\to K^\* (X)\to K^\* (U) \to$? Here ... | https://mathoverflow.net/users/7108 | K theory long exact sequence | Regarding the first question:
If $X$ is quasi-compact quasi-separated and $U \to X$ is a quasi-compact open immersion, then Thomason-Trobaugh showed that there is a "proto-localization sequence", i.e. a fibre sequence of spaces
$$ K\_{Z}(X) \to K(X) \to K(U) $$
where $K\_{Z}(X)$ is the K-theory of perfect complexes ... | 9 | https://mathoverflow.net/users/2503 | 224550 | 105,036 |
https://mathoverflow.net/questions/224175 | 1 | Suppose $\rho:G \_{\mathbb{Q}} \rightarrow GL\_n(\mathbb{Q}\_p)$ is a Galois rep. It has a uniquely defined (up to semisimplification residual rep $\bar{\rho}$. $\bar{\rho}$ is unramified where $\rho$ is. I have examples where it becomes good at a prime where $\rho$ is bad. Can it become more ramified then $\rho$? What... | https://mathoverflow.net/users/6084 | How much extra ramification in a residual representation | All this is completely worked out in an old paper of Carayol whose name escapes me -- just a few pages, probably mid to late 80s or early 90s. In short: assuming you're not talking about ramification at $p$ then the Swan conductors of $\rho$ and its reduction coincide, so the only change in conductor can be "tame". No,... | 0 | https://mathoverflow.net/users/43076 | 224556 | 105,039 |
https://mathoverflow.net/questions/224586 | 0 | Given a graph $G$ with $m$ edges, what is the maximum chromatic number $\chi(G)$ that the graph can have?
My guess is that $\chi(G) \leq r(m)$ where $r(m) := \max\{k\in \mathbb{N}:
\frac{k(k-1)}{2} \leq m\}$, but I can't prove this.
(This is motivated by the fact that the largest complete graph you can form with ... | https://mathoverflow.net/users/8628 | Maximal chromatic number with a fixed number of edges | Your guess is correct. Denote $\chi(G)=k$. Then $G$ contains a subgraph with all degrees at least $k$ (proof: if degree of vertex $v$ is less than $k$, then $\chi(G\setminus v)=k$. Indeed, if $\chi(G\setminus v)<k$, color $G\setminus v$ with $k-1$ colors and then extend this coloring to $v$. So, minimal subgraph of $G$... | 3 | https://mathoverflow.net/users/4312 | 224587 | 105,047 |
https://mathoverflow.net/questions/224572 | 2 | Let $X \sim \text{Bin}(n,p)$. Wikipedia claims
$$\mathbf P[X \leq (p-\epsilon)n ] \leq e^{ - 2 \epsilon^2 n}.$$
This follows from Hoeffding's inequality (<https://en.wikipedia.org/wiki/Hoeffding%27s_inequality#General_case>).
Is this tight? Could a more clever application of Hoeffding, or perhaps Chernoff bounds yie... | https://mathoverflow.net/users/52896 | Tight binomial left tail bound | This is tight, at least when $p=\frac12$. You simply need to approximate $\log\big({n \choose \frac{n}{2}-\varepsilon n} \frac1{2^{n}}\big)$ using Stirling's formula and you'll see that the leading coefficient is indeed $-2\varepsilon^2 n$.
| 3 | https://mathoverflow.net/users/1061 | 224589 | 105,048 |
https://mathoverflow.net/questions/218513 | 1 | Is there an efficient algorithm to create a [transitive reduction](https://en.wikipedia.org/wiki/Transitive_reduction) from a single [linear extension](https://en.wikipedia.org/wiki/Linear_extension) of a given partial order?
**Update:** I'm aware of the [time complexity of computing a transitive reduction](https://e... | https://mathoverflow.net/users/80376 | Transitive reduction from a linear extension of a partial order | The time can be reduced by at least half. A partial order $P$ with a given linear extension allows certain optimizations for transitive reduction that are not available without the extension. These optimizations proceed from the fact that we can immediately produce an upper triangular matrix for the incidence relation ... | 1 | https://mathoverflow.net/users/81605 | 224594 | 105,049 |
https://mathoverflow.net/questions/224569 | 24 | The [nlab](http://ncatlab.org/nlab/show/smooth+manifold#patching_as_idempotent_splitting) has a particularly interesting thing to say about the category of smooth manifolds: *it is the idempotent-splitting completion of the category of open sets of Euclidean spaces and smooth maps*.
After proving this, the following ... | https://mathoverflow.net/users/69037 | Smooth manifolds as idempotent splitting completion | The theorem is the following: (from 1.15 of [here](http://www.mat.univie.ac.at/~michor/dgbook.pdf))
* Theorem: Let $M$ be a connected manifold and
suppose that $f:M\to M$ is smooth with $f\circ f= f$. Then the
image $f(M)$ of $f$ is a submanifold of $M$.
Proof: We claim that there is an open neighborhood $U$ of
$f(... | 12 | https://mathoverflow.net/users/26935 | 224595 | 105,050 |
https://mathoverflow.net/questions/224560 | 2 | Let $F$ be a finite extension of $\mathbb Q\_p$, where p is an odd prime. Let $G$ be a connected reductive group defined over $F$. Let $M, H$ be closed $F$-subgroups of $G$ (in particular, I'm interested in the case when $M$ is a Levi subgroup of $G$, and $H$ is the group of fixed points of an $F$-involution of $G$.)
... | https://mathoverflow.net/users/74112 | F-points of product of closed subgroups vs. product of F-points, F a local field, reference? | I am just posting my comment above as an answer. Let $K/F$ be a finite Galois extension, and let $m \in M(K)$ and $h\in H(K)$ be elements such that the element $g=m\cdot h^{-1}$ is Galois invariant, i.e., $g$ is an element of $G(F)$. Consider the non-Abelian $1$-cocycles in $G(K)$ for $\Gamma = \text{Aut}(K/F)$, $( m^{... | 2 | https://mathoverflow.net/users/13265 | 224600 | 105,053 |
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