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https://mathoverflow.net/questions/252478 | 5 | I wonder whether it is true that *the composition of two [GIT-quotients](https://en.wikipedia.org/wiki/GIT_quotient) is another GIT-quotient*. It should be an analogue of a set-theoretic formula $X/(G \times H)\simeq (X/G)/H$ but with GIT-quotients instead.
---
**Background. Mumford's theorem (GIT, thm. 1.10).** ... | https://mathoverflow.net/users/43639 | Supposed generalization of $X/(G \times H)\simeq (X/G)/H$ for GIT-quotients | *My question should have consisted of two.* I will formulated both using linearization via line bundles, not embeddings into projective space, as it makes them more clear.
**Question 1.** Suppose that $\pi: X \to Y$ is a quotient by $G$, that $H$ acts on $X$ commuting with $G$ (so that $H$ also acts on $Y$) and that ... | 0 | https://mathoverflow.net/users/43639 | 264935 | 119,058 |
https://mathoverflow.net/questions/264925 | 2 | Given the linear Diophantine equation:
$ a\_1 x\_1 + a\_2 x\_2 + ... + a\_n x\_n = b$
and its particular solution $(x\_1^\*, x\_2^\*, ..., x\_n^\*)$.
How to write down all the solutions of this equation?
| https://mathoverflow.net/users/106256 | All the solutions of linear Diophantine equation | First, complete the vector $(a\_1, \dotsc, a\_n)$ to a matrix $A$ with determinant $1.$ Then the solutions (of the homogeneous system) are all integer linear combinations of the second through $n$ *columns* of $A^{-1}.$ That you can complete the vector with gcd 1 to a unimodular matrix is basic to geometry of numbers, ... | 2 | https://mathoverflow.net/users/11142 | 264936 | 119,059 |
https://mathoverflow.net/questions/264946 | 2 | Let $\mathcal{A}$ be an Abelian category, $X$ be a complex, $F$ be a contravariant exact functor. I am wondering whether F preserves the homology of X, that means whether $H^{i}(FX)=F(H^{-i}(X)),\ \forall i$? (Obviously, this is true for covariant functors)
| https://mathoverflow.net/users/106253 | homology under exact functors | Yes. There are two ways to describe $H^iX$: it is the cokernel of $X^{i-1}\to ker(X^i\to X^{i+1})$, and it is also the kernel of $coker(X^{i-1}\to X^i)\to X^{i+1}$. From the first of these it is clear that $F(H^iX)$ is the kernel of $coker(FX^{i+1}\to FX^i)\to FX^{i-1}$. (And from the second it is clear that $F(H^iX)$ ... | 4 | https://mathoverflow.net/users/6666 | 264947 | 119,064 |
https://mathoverflow.net/questions/264942 | 1 | I encounter the following problem of the type of degree theory in Hilbert spaces.
Consider the Hilbert space $L^2(\mathbb T)$ with the natural inner product.
Taking Fourier expansion with bases $e^{i2\pi k x}$, we have a direct sum decomposition into $E^-\oplus E^0\oplus E^+$, where $E^-,E^0,E^+$ correspond to su... | https://mathoverflow.net/users/15214 | Topological degree in Hilbert spaces | I am not sure that I understand you correctly, but it seems that your question is answered by the following result (you can find it as Corollary 3.5 in Y.Benyamini, J.Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1. American Mathematical Society Colloquium Publications, 48. American Mathematical Society,... | 3 | https://mathoverflow.net/users/37822 | 264951 | 119,065 |
https://mathoverflow.net/questions/264854 | 1 | Let $A$ be an algebra over a field k. $D$ is the standard duality functor. A module $\_AM$ is called a generator if $add(A) \subseteq add(M)$, a cogenerator if $add(D(A)) \subseteq add(M)$. $M$ is n-rigid if $Ext\_A^i(M,M)=0$ for $1 \leq i \leq n$. Now suppose $\_AM$ is a generator-cogenerator which is n-rigid and neit... | https://mathoverflow.net/users/83554 | How to get that one module is tilting iff the other one is? | From the comments, it seems that you know how to prove equivalence for the first two conditions for being a tilting module.
For the last condition you can use $\mathfrak{C}(D(\Lambda))= \mathfrak{C}(\operatorname{Hom}\_A(M^-,M))$, since the final condition for a $\Lambda^{\mathrm{op}}$-module $T$ to be tilting is tha... | 2 | https://mathoverflow.net/users/21483 | 264960 | 119,069 |
https://mathoverflow.net/questions/264950 | 4 | Conside the One dimentional Schrodinger Operator
$$
-\frac{d^2}{dx^2} + ( V(x) + E )
$$
Where the *Potential Function* $V$ is of the form $V(x) = ax^2 + b^2x^4$ , $a,b \in \mathbb{R} $.
What is known about the Spectral growth of the above Operator for large enough value of $E$,$|a|$ and $b$.Can we describe the sin... | https://mathoverflow.net/users/83956 | Spectral growth of One dimensional Schrodinger Operator | First of all, it is sufficient to consider only one parameter: making a change of the independent variable $z\mapsto kz$, with appropriate $k$ one can eliminate either $a$ or $b$. Let us eliminate $b$ and consider
$$-y''+(x^4+ax^2)y=\lambda y.$$
Then eigenvalues (in $L^2(R)$) become functions of $a$, and the asymptotic... | 4 | https://mathoverflow.net/users/25510 | 264965 | 119,070 |
https://mathoverflow.net/questions/264948 | 1 | Is there an explicit example of a function in the (topological) support of the law of Brownian motion (with respect to the topology of uniform convergence of continuous functions)?
(You can take "explicit" to mean "doesn't invoke the axiom of choice".)
| https://mathoverflow.net/users/29961 | Deterministic function in the support of Brownian motion | As clarified, you're asking about the law of Brownian motion on a bounded interval $[0,T]$, as a probability measure $\mu$ on $C([0,T])$ (Wiener measure).
The zero function is in the topological support. This amounts to showing that $P(\sup\_{t \in [0,T]} |B\_t| < \epsilon) > 0$ for all $\epsilon$ and you can find se... | 3 | https://mathoverflow.net/users/4832 | 264966 | 119,071 |
https://mathoverflow.net/questions/264962 | 9 | Let $f(x,y)$ be a binary quadratic form with co-prime integer coefficients. We say that $f$ is a *proper subform* of $g(x,y)$ if there exists an integer matrix $A = \left(\begin{smallmatrix} a\_1 & a\_2 \\ a\_3 & a\_4 \end{smallmatrix}\right)$ with $|\det A| > 1$ such that
$$\displaystyle f(x,y) = g(a\_1 x + a\_2 y,... | https://mathoverflow.net/users/10898 | On binary quadratic forms which are not proper subforms of another binary quadratic form | Yes. It is about equivalence over $SL\_2 \mathbb Z.$ Your form $f$ (primitively) represents some value not divisible by the fixed prime $p.$ Indeed, from original coefficients $\langle a,b,c \rangle$ we know that at least one of $a,c, a+b+c$ is not divisible by $p.$ We may therefore demand $a \neq 0 \pmod p$ in $\langl... | 8 | https://mathoverflow.net/users/3324 | 264970 | 119,073 |
https://mathoverflow.net/questions/264564 | 8 | Convincing numerical evidence prompts me to ask:
>
> **Question.** Is $\sum\_{k=0}^n\sum\_{j=0}^k\binom{k}j^2\binom{2j}j(2j+1)^2$ divisible by $(n+1)^2$?
>
>
>
| https://mathoverflow.net/users/66131 | Divisibility of a binomial sequence | The answer is yes, and the proof can be found in V. J. W. Guo, J.-C. Liu, Proof of some conjectures of Z.-W. Sun on the divisibility of certain double sums, Int. J. Number Theory 12 (2016), 615-623. An arXiv version is also [available](https://arxiv.org/abs/1412.5415).
| 3 | https://mathoverflow.net/users/11919 | 264980 | 119,076 |
https://mathoverflow.net/questions/49002 | 9 | I'm currently at a Differential Geometry meeting and there is a mini-course on positively curved Riemannian manifolds. There, we were told that a technique to construct such manifolds is a Cheeger deformation, which (if I understood correctly) is a generalization of a one-parameter family of surfaces of revolution give... | https://mathoverflow.net/users/10328 | What is a Cheeger deformation? | Cheeger deformations were first introduced in Jeff Cheeger's 1973 paper [Some examples of manifolds of nonnegative curvature](http://projecteuclid.org/euclid.jdg/1214431964), and were inspired by the Berger sphere construction. The basic idea is to consider a Lie group acting by isometries on a Riemannian manifold, and... | 15 | https://mathoverflow.net/users/106283 | 264981 | 119,077 |
https://mathoverflow.net/questions/264983 | 4 | My question is expressed by means of Quine's definition of rational numbers in Set Theory and its Logic, chapters 17 and 18.
Let pairing of natural numbers be represented as by his definition 17.1 $x;y=\_{def}x+(x+y)^2$, let as by 18.1 $x/y=\_{def} \{z;w|z,w\in\mathbf{N}\wedge x\cdot w > y\cdot z \}$ and as in 18.10 l... | https://mathoverflow.net/users/37385 | Is the set of rational numbers recursive? | With this definition, each rational number is a set of natural numbers, so you're asking about the status of a **set of sets** of natural numbers. Every $\Sigma\_1^0$ subset of $\mathcal P(\mathbb N)$ is open in the usual topology (the product topology on $2^{\mathcal N}$), but this version of $\mathbb Q$ is not open. ... | 5 | https://mathoverflow.net/users/6794 | 264984 | 119,078 |
https://mathoverflow.net/questions/264985 | 8 | Let $H$ be a separable Hilbert space with a fixed orthonormal basis $\{e\_n\}\_n$. For a bounded operator $T$ on $H$, the *diagonal* of $T$ is the unique operator $D\_T$ on $H$ which is diagonal with respect to the above basis, and whose diagonal entries are given by $d\_n=\langle T(e\_n),e\_n\rangle$. It is well know ... | https://mathoverflow.net/users/97532 | If the diagonal of a positive operator is compact, is the operator itself compact? | Nope. For each $n$ let $T\_n$ be the $n\times n$ matrix all of whose entries are $\frac{1}{n}$. This is a rank $1$ projection. So $T = \bigoplus T\_n$ is a projection with infinite dimensional range, and hence is not compact. But its diagonal entries go to zero as $n \to \infty$, which means that $D\_T$ is compact.
| 14 | https://mathoverflow.net/users/23141 | 264994 | 119,082 |
https://mathoverflow.net/questions/264979 | 9 | Let $\Delta\_+$ denote the category of finite ordinal numbers with monotonic maps (the subscript indicates that $0$ is included, so this is the [augmented simplex category](https://ncatlab.org/nlab/show/simplex+category)). This has a monoidal structure (given by the sum), which is not symmetric. But we can make it symm... | https://mathoverflow.net/users/98306 | What is this symmetric simplex category, concretely? | $\Delta\_+$ is the monoidal category generated from the associative operad, considered as a non-symmetric operad. Similarly, $(\Delta\_+)\_{{\rm sym}}$ is the symmetric monoidal category generated from the associative operad, this time considered as a symmetric operad. This category can then be explicitly described as ... | 11 | https://mathoverflow.net/users/51164 | 264998 | 119,085 |
https://mathoverflow.net/questions/265029 | 1 | There is a lot of work done on low rank updates to inverses or SVDs (or similar decompositions), but I am wondering what can be said about the inverse or SVD
$(A + c \times B)$
in terms of $(A + d \times B)$ for some other $d$. That is, in general full-rank updates, but along very particular dimensions.
$c$ and ... | https://mathoverflow.net/users/39430 | update SVD of linear combination after changing scalar weight | I don't think there is any computational advantage in general. For any pair of symmetric $M$, $N$ and given $c\neq d$, one can find symmetric $A, B$ such that $M=A+cB$, $N=A+dB$, so essentially you are asking "if I already know the eigendecomposition of a symmetric matrix $M$, can I get the one of any other completely ... | 1 | https://mathoverflow.net/users/1898 | 265030 | 119,096 |
https://mathoverflow.net/questions/265032 | 0 | In the following article
<https://www.math.cornell.edu/~hatcher/Papers/3Msurvey.pdf>
Hatcher mentioned that there is only one prime
closed 3-manifold with infinite cyclic fundamental group, which is
$S^1\times S^2$.
This implies that the 3-torus $T^3$ can be decomposed as a direct sums of three $S^1\times S^2$. Is it... | https://mathoverflow.net/users/17787 | Decomposition of 3-dimensional mapping torus into connected sum | The first part of this answer is merely an elaboration on Marc Kegel and T. Amdeberhan's comments, and addresses the question before the edit. As they say, $T^3$ is prime by virtue of being irreducible (a stronger condition) -- since $\pi\_2(T^3)$ is trivial, every embedded 2-sphere is null-homotopic in $T^3$ and it fo... | 7 | https://mathoverflow.net/users/353 | 265038 | 119,098 |
https://mathoverflow.net/questions/265035 | 3 | A [torus bundle](https://en.wikipedia.org/wiki/Torus_bundle) is labeled by an element $M$ of $SL(2,\mathbb{Z})$ -- the mapping class group of a torus. How to compute the fundamental group of a torus bundle from the 2-by-2 matrix $M \in SL(2,\mathbb{Z})$? Any references?
| https://mathoverflow.net/users/17787 | The fundamental group of torus bundle | A quick google search on "fundamental group of a mapping torus" gives many hits - eg:
<https://math.stackexchange.com/questions/39589/fundamental-group-of-mapping-torus?rq=1>
| 3 | https://mathoverflow.net/users/1650 | 265040 | 119,100 |
https://mathoverflow.net/questions/264765 | 7 | Let $TOP$ be the stable homeomorphism space, with $TOP(n) = Homeomorphisms(\mathbb{R}^n)$. What is known about its $\mathbb{Z}\_2$-homology $H\_\*(TOP, \mathbb{Z}\_2)$? In particular I am interested in the map $H\_\*(TOP, \mathbb{Z}\_2) \rightarrow H\_\*(G, \mathbb{Z}\_2)$, where $G$ is the stable space of homotopy aut... | https://mathoverflow.net/users/106191 | Homology $H_*(TOP, \mathbb{Z}_2)$ of the stable homeomorphism space | I first comment on the specific case on the kernel of $H\_{14}TOP\to H\_{14}G$ when restricted to the image of Hurewicz homomorphism $\pi\_{14}TOP\to H\_{14}TOP$ (all homology groups are $\mathbb{Z}/2$-homology). I think it is a consequence of Sullivan's decomposition that at the prime $2$ the space $G/TOP$ decomposes ... | 5 | https://mathoverflow.net/users/51223 | 265051 | 119,104 |
https://mathoverflow.net/questions/265015 | 6 | Here length means 1-Hausdorff measure. This seems to be known, what is the reference? Or very short proof?
| https://mathoverflow.net/users/4312 | Connected planar compact set with finite length is path connected | See Exercise 3.5 in "The geometry of fractal sets" by K. J. Falconer.
It says that any such set is an image of rectifiable curve.
For a short proof, check "Rectifiable curve" in [my collection](https://arxiv.org/abs/0906.0290).
P.S. The earliest reference I found: Theorem 2 in *Continua of finite linear measure.... | 7 | https://mathoverflow.net/users/1441 | 265053 | 119,105 |
https://mathoverflow.net/questions/265048 | 1 | Let $F\_1(\mathbf{x}, \mathbf{y}), \ldots, F\_r(\mathbf{x}, \mathbf{y})$ be bihomogeneous polynomials with rational coefficients with bidegree $(d\_1, d\_2)$, which means
$$
F\_i( s x\_1, \ldots, s x\_{n\_1} ; t y\_1, \ldots, t y\_{n\_2} ) = s^{d\_1} t^{d\_2} F\_i(\mathbf{x} ; \mathbf{y} ).
$$
Let $V$ be the algebraic... | https://mathoverflow.net/users/84272 | Question about properties of affine varieties defined by bihomogeneous polynomials | 1. Since each $F\_i(u\_1,\dotsc,u\_{n\_1},v\_1,\dotsc,v\_{n\_2})=0$, then $F\_i(su\_1,\dotsc,su\_{n\_1},tv\_1,\dotsc,tv\_{n\_2}) = s^{d\_1}t^{d\_2} F\_i(u\_1,\dotsc,u\_{n\_1},v\_1,\dotsc,v\_{n\_2})=s^{d\_1}t^{d\_2}0 = 0$. So certainly the point $(su\_1,\dotsc,su\_{n\_1},tv\_1,\dotsc,tv\_{n\_2}) \in V$. The map $\mathbb... | 4 | https://mathoverflow.net/users/88133 | 265054 | 119,106 |
https://mathoverflow.net/questions/265042 | 4 | Consider the group $G:=\prod\_p C\_p$ where the product is taken over all primes, endowed with the product topology.
I'm trying to classify the compact subgroups of $G$. Is there any subgroups of $G$ besides direct products of finite groups?
| https://mathoverflow.net/users/106317 | subgroups of $\prod_p C_p$ | Let $G$ be your group and $C$ a compact subgroup. Both $G$ and $C$ are profinite groups. The Sylow subgroups of $G$ are $C\_p$ and any Sylow subgroup of $C$ is contained in a Sylow subgroup of $G$. Therefore, it is either trivial or $C\_p$. Since $G$ is abelian so is $C$ and thus, $C$ is a direct product of its Sylow s... | 6 | https://mathoverflow.net/users/5034 | 265061 | 119,110 |
https://mathoverflow.net/questions/265063 | 1 | Let $T$ be a bounded operator. Then, the operators $\left\lvert T \right\rvert:=\sqrt{T^\*T}$ and $\left\lvert T^\* \right\rvert:=\sqrt{TT^\*}$ are well-defined.
Is there a way to write
$$(\left\lvert T \right\rvert -i)^{-1} -(\left\lvert T^\* \right\rvert -i)^{-1}$$
in terms of operators that are more accessible f... | https://mathoverflow.net/users/105722 | Resolvent difference of absolute values! | I think the answer is "no". Consider the operator $\bar{T} = \left[\matrix{0&0\cr T&0}\right]$ acting on $H \oplus H$. Then $|\bar{T}| = \left[\matrix{|T|&0\cr 0&0}\right]$ and $|\bar{T}^\*| = \left[\matrix{0&0\cr 0&|T^\*|}\right]$. So $$(|\bar{T}| - i)^{-1} - (|\bar{T}^\*| - i)^{-1} = \left[\matrix{(|T| - i)^{-1} - i&... | 2 | https://mathoverflow.net/users/23141 | 265064 | 119,112 |
https://mathoverflow.net/questions/122997 | 10 | This is a reformulation of this [MO question](https://mathoverflow.net/questions/122857) which recieved little or no attention due to the fact that the OP gave no motivation whatsoever. I found the question quite interesting and decided to give it another try (if this is not OK please let me know and I´ll delete this p... | https://mathoverflow.net/users/17836 | Is there a compact space with no countably generated dense subspace? | An $\eta\_1$-set is a linearly ordered set $(Q, \leq)$ with the property that if A and B are countable subsets of $Q$ satisfying $a < b$ for all $a \in A, b \in B$, then there is an $x \in Q$ satisfying $a < x < b$ for all $a \in A, b \in B$. (This definition and the notation are from Gillman and Jerison.) Let $(X, \le... | 3 | https://mathoverflow.net/users/89233 | 265067 | 119,115 |
https://mathoverflow.net/questions/227720 | 13 | Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as geodesics in a very large dimensional manifold.
My question is, is there any research studying this idea?
Moreover are there any papers linking differential geometry to stochastic Calculus?
Any links would be greatly apprec... | https://mathoverflow.net/users/36886 | Geometric characterization of martingales | I do not think Hsu's book is a good place to start with, although it has some strong holds-like details in calculation, neither its depth nor its clarity is comparable to
>
> Stroock, Daniel W. An introduction to the analysis of paths on a
> Riemannian manifold. No. 74. American Mathematical Soc., 2005.
>
>
>
... | 7 | https://mathoverflow.net/users/25437 | 265068 | 119,116 |
https://mathoverflow.net/questions/263880 | 3 |
>
> Let $C^n=\{0,1\}^n$ be a metric space (Hamming Cube). The distance on $C^n$ is defined by
> $$
> d(\varepsilon,\varepsilon'):=|\{j:\varepsilon\_j\ne\varepsilon'\_j\}|,
> $$
> $\varepsilon=(\varepsilon\_1,\dots,\varepsilon\_n)$.
>
>
>
Let $s,k$ be integers such that $sk=n$. We divide each $\varepsilon$ into... | https://mathoverflow.net/users/80191 | Intuition for inequality involving permutation and Hamming Cube | If we took $\phi\_i(\varepsilon, (1))$, where $(1)$ is the identity permutation, we'd have $d(f(\varepsilon), f(\varepsilon\_{I\_i}))$: the distance we go when we change the $i$-th block of $\varepsilon$.
But there's nothing special about the order of the coordinates of $\varepsilon$, nor about the way we partition ... | 2 | https://mathoverflow.net/users/106323 | 265076 | 119,119 |
https://mathoverflow.net/questions/265102 | 1 | Let us take the 1D case: for an n-step random walk on a line confined between two boundaries at positions t and s, can we determine the average time (number of steps) the walker spends off the boundaries given the distance between the boundaries is d? That is, the time spent on all sites but t+1 (neighbouring site of t... | https://mathoverflow.net/users/nan | Simple finite random walks with reflective boundaries | For a random walk on $\bf Z$ where you can jump only to your two closest neighbors, this can be computed explicitely using martingale, Markov chains or renewal theory. For more complex random walks, there are algorithms but the formulas become more complicated.
So for example, for a random walk with absorbing barrie... | 1 | https://mathoverflow.net/users/6129 | 265103 | 119,129 |
https://mathoverflow.net/questions/265073 | 4 | I encountered the following identity in a paper on number theory,
$$\int\_{-\infty}^{\infty}\frac{dW}{(W+i)^{\frac{3}{2}}(W^2+1)^s}=\frac{e^\frac{-3\pi i}{4}\sqrt{2}\pi \Gamma(2s+\frac{1}{2})}{2^{2s}\Gamma(s+\frac{3}{2})\Gamma(s)},$$
with $Re(s) > 0$ and $i=\sqrt{-1}$.
Since the author did not give the proof fo... | https://mathoverflow.net/users/100719 | An integral identity relate to the Gamma function or the Beta function | To evaluate $\int\_{-\infty}^\infty {dx\over (1+ix)^a\,(1-ix)^b}$, view this as $\int\_{-\infty}^\infty \widehat{f\_a}(x)\,\overline{\widehat{f\_{\overline{b}}}(x)}\,dx$ where $f\_c$ are functions whose Fourier transforms are $(1+ix)^{-c}$. Use the Gamma-function identity
$$
\int\_0^\infty e^{-ty}\,t^c\;{dt\over t}\;=\... | 5 | https://mathoverflow.net/users/15629 | 265106 | 119,131 |
https://mathoverflow.net/questions/265100 | 2 | Let $A$ be an algebra over a field k. A module $\_AM$ is called a generator if $\textrm{add}(A) \subseteq \textrm{add}(M)$, a cogenerator if $\textrm{add}\big(D(A)\big) \subseteq \textrm{add}(M)$. $M$ is $n$-rigid if $\textrm{Ext}\_A^i(M,M)=0$ for $1 \leq i \leq n$.
Now suppose $\_AM$ is a generator-cogenerator whic... | https://mathoverflow.net/users/83554 | How to get that $\Omega^2_{\Lambda}(N) \cong \textrm{Hom}_A(M,Y)$? | This is an application of the Yoneda lemma. There is an exact sequence
$$0\to\Omega^2\_\Lambda(N)\to P\_1\to P\_0\to N\to0$$
with $P\_i$ projective. By definition of $\Lambda$, it follows that $P\_i=\operatorname{Hom}\_A(M,M\_i)$ for some $M\_i\in\operatorname{add}{M}$. (Here I am making the additional assumption t... | 3 | https://mathoverflow.net/users/21483 | 265107 | 119,132 |
https://mathoverflow.net/questions/265077 | 1 | **Background**
>
> This question follows up on the previous question I had asked [here](https://mathoverflow.net/questions/222793/how-large-can-a-subset-of-1-ldots-n-be-if-all-pairwise-lcms-of-its-elemen) That question had a nice answer, for the specific parameter choice $M=N+1,$ now I'm interested in $M=N\log N.$
... | https://mathoverflow.net/users/17773 | Maximum possible size of subset of $\{1,\ldots,N\}$ with LCM of members bounded below | You cannot do much better then in the construction of Woett. To see this note that the size of $A$ is bounded above by the maximal size of a set $A$, such that $gcd(a, a')<\frac{N}{\log N}$ for all $a\neq a'$ in $A$. Suppose that $A$ is a set with the latter property. For each $a\in A$, let $d(a)$ be the largest diviso... | 1 | https://mathoverflow.net/users/37555 | 265114 | 119,135 |
https://mathoverflow.net/questions/265037 | 8 | Let $\mathbb{F}\_p$ be a finite field, $A=\{a\_1,\dots,a\_k\}\subset\mathbb{F}\_p^\*$ a $k$-element set, for $k<p$. $\mathfrak{S}\_k=$permutation gp.
>
> **Question.** Is it true there is always a $\pi\in\mathfrak{S}\_k$ such that the following are pair-wise distinct in $\mathbb{F}\_p$?
> $$a\_{\pi(1)}, \,a\_{\pi(... | https://mathoverflow.net/users/66131 | sum-sets in a finite field | It took me some effort to find the references you were requesting in your comment, but here they are eventually:
* [Ordering subsets of the cyclic group to give distinct partial sums](https://mathoverflow.net/questions/164300/ordering-subsets-of-the-cyclic-group-to-give-distinct-partial-sums/203111#comment503832_203... | 10 | https://mathoverflow.net/users/9924 | 265120 | 119,138 |
https://mathoverflow.net/questions/264387 | 2 | ***21/03/2017:*** *I have decided to accept Denis Serre's answer, even though it does not exactly answer my question, however I like its simplicity and I'd say it is close enough to the desired claim. Of course, I would still love to see the answer to my original question.*
We are given two subspaces $M$ and $N$ of $... | https://mathoverflow.net/users/15129 | Simultaneous extensions of strictly convex functions | Here is an elementary proof that the answer is positive, at least if you relax the condition that the extension be $C^2$.
>
> Lemma. Given $x\_0\in C\_M$, there exists a linear form $\lambda$ such that $f(x\_0)+\lambda(x-x\_0)\ge0$ over the graphs of both $f$ and $g$.
>
>
>
To see this, consider the tangent sp... | 2 | https://mathoverflow.net/users/8799 | 265124 | 119,140 |
https://mathoverflow.net/questions/265117 | 3 | I'm trying to understand a certain function on a specific adic space I'm stuck on something silly and is probably due to my lack of understanding of the points in this case. This is Proposition 3.3.5 of Scholze's MSRI lecture notes.
Let $A=\mathbb{Z}\_p [[T]]$, and let $Y=Spa(A,A)^{an}$ . Then one can define a unique... | https://mathoverflow.net/users/106351 | Relative position of elements in adic space | Presumably you are taking the $(p,T)$-adic topology on $A$. In this topology the element $T$ is topologically nilpotent, so for any $x \in \mathrm{Spa}(A,A)$ you have $|T(x)|<1$.
| 3 | https://mathoverflow.net/users/84144 | 265128 | 119,142 |
https://mathoverflow.net/questions/265126 | 8 | There is a theorem due to Sato which says that any vector bundle over $\mathbb CP^{\infty}$ decomposes as a direct sum of line bundles (which is a generalization of Grothendieck's result over $\mathbb CP^1$). I am wondering if there is a similar theorem for vector bundles over $\mathbb RP^{\infty}$ or a criterion for s... | https://mathoverflow.net/users/100955 | Vector bundles over $RP^{\infty}$ | You are essentially asking about the set $[B\mathbb{Z}/2,BSO(2n)]$. This bijects with the set of conjugacy classes of homomorphisms from $\mathbb{Z}/2$ to $SO(2n)$, or in other words real, oriented representations of $\mathbb{Z}/2$. Essentially the same thing works for $[BP,BG]$ whenever $P$ is a finite $p$-group and $... | 12 | https://mathoverflow.net/users/10366 | 265129 | 119,143 |
https://mathoverflow.net/questions/264910 | 10 | I have frequently seen results like:
There are 4 isomorphism types of finite subgroups of $SL\_2(\mathbb{Z})$, namely $\mathbb{Z}\_2,\mathbb{Z}\_3,\mathbb{Z}\_4,\mathbb{Z}\_6$.
I wonder what is known of we replace $\mathbb{Z}$ by the ring of integers $\mathcal{O}\_K$ of an algebraic number field $K$. I've found the... | https://mathoverflow.net/users/22709 | Finite subgroups of Lie group over algebraic ring of integers | I'll comment on the last question, with the group $G = {\rm SL}(2,5).$ All $2$-dimensional irreducible complex representations of $G$ are real valued, so all have Schur index dividing $2$ by a standard theorem ( Speiser?). Since $G$ has a quaternion subgroup of order $8$, the Schur index of such a representation is nec... | 5 | https://mathoverflow.net/users/14450 | 265134 | 119,145 |
https://mathoverflow.net/questions/265135 | 6 | Let $k$ be a commutative ring. Let $Ch(k)$ denote the monoidal category of chain complexes. I need a reference, including a proof, for the following "folklore fact" (see for example [here](https://arxiv.org/abs/0807.1471v1) after Definition 1.2.1):
>
> A chain complex $A \in Ch(k)$ is dualizable if and only if it i... | https://mathoverflow.net/users/98306 | Reference for dualizable chain complexes | This is Proposition 1.6 of 'Duality, Trace and Transfer' by Dold and Puppe, <http://www.maths.ed.ac.uk/~aar/papers/doldpup2.pdf>.
| 10 | https://mathoverflow.net/users/16785 | 265136 | 119,146 |
https://mathoverflow.net/questions/265098 | 20 | I asked [the same question](https://math.stackexchange.com/q/2181688/660) a week ago on Mathematics Stackexchange but got no answer.
What would be a simple example of an additive functor $F:\mathcal C\to\mathcal C'$ of abelian categories such that the right derived functor
$$
RF:\text D(\mathcal C)\to\text D(\mathcal... | https://mathoverflow.net/users/461 | Example of an additive functor admitting no right derived functor | Let ${\cal C}$ be the category of finite dimensional ${\bf Z}/2$-vector spaces equipped with a ${\bf Z}/2$ action, let ${\cal C'}$ be the category of finite dimensional ${\bf Z}/2$-vector spaces and let $F: {\cal C} \to {\cal C'}$ be the fixed subspace functor $V \mapsto V^{{\bf Z}/2}$. Consider the chain complex $X$ s... | 15 | https://mathoverflow.net/users/51164 | 265142 | 119,148 |
https://mathoverflow.net/questions/261459 | 3 | Put $P=[0,1]$. Is there a compact subset $L$ of the hyper space of $P$ such that the pair $(P, L)$ satisfies the following axioms of projective geometry. Furthermore the obvious maps from the configuration space of $P$ to $L$ and configuration space of $L$ to $P$ would be continuous?
[Non-isomorphic projective planes... | https://mathoverflow.net/users/36688 | Continuous projective geometry on the interval | If $(P,L)$ is an abstract projective plane, then for any point $p\in P$ and any line $\ell\in L$ not incident to $p$ there is a bijection between the set of points incident to $\ell$ and the set of lines incident to $p$. Under a reasonable definition of "topological projective plane" this bijection should be a homeomor... | 4 | https://mathoverflow.net/users/6666 | 265145 | 119,149 |
https://mathoverflow.net/questions/265132 | 4 | A standard formulation of the one-dimensional variational problem is to find necessary and sufficient conditions for the functional $x:\mathbb R\rightarrow \mathbb R$ that minimizes
$
\int\_0^1 L[t, x\_t, \dot x\_t] dt
$
For a given $L: \mathbb R^3\rightarrow \mathbb R$ that is sufficiently well-behaved.
I am en... | https://mathoverflow.net/users/9118 | Double calculus of variations | Let
$$S (x) := \iint\_{[0,1]^2} \mathcal{L} (t\_1, t\_2, x (t\_1), x (t\_2), \dot x (t\_1), \dot x (t\_2)) \, \mathrm d t\_1 \mathrm d t\_2$$
Hence,
$$\begin{array}{rl} \delta S := S (x + \delta x) - S (x) &= \displaystyle\iint\_{[0,1]^2} \partial\_3\mathcal{L} (t\_1, t\_2, x (t\_1), x (t\_2), \dot x (t\_1), \dot... | 1 | https://mathoverflow.net/users/91764 | 265147 | 119,151 |
https://mathoverflow.net/questions/264856 | 8 | Minkowski's Linear Forms Theorem is often stated about linear forms with real coefficients. However, in Narkiewicz's *Elementary and Analytic Theory of Algebraic Numbers*, the following generalization of Minkowski's Linear Forms Theorem is stated (the theorem is actually stated for a general lattice in real space, but ... | https://mathoverflow.net/users/106225 | Minkowski's Linear Forms Theorem With Complex Coefficients | Credit where it's due to @so-calledfriendDon who found a source before I figured this out. However, I figure I'll post the proof because Google Books can be a finicky thing.
First, note that we can assume that all $\epsilon\_j = 1$. If not, replace $L\_j$ by $\frac{L\_j}{\epsilon\_j}$ and note that all of the hypothe... | 2 | https://mathoverflow.net/users/106225 | 265149 | 119,152 |
https://mathoverflow.net/questions/265085 | 6 | This question is related to the [question](https://mathoverflow.net/questions/262063/large-gaps-in-singer-planar-difference-sets) I asked earlier.
For a natural number $n$, a set $D$ of integer numbers is called a *$n$-cyclic difference set* if each integer number $x\notin n\mathbb Z$ can be uniquely represented as t... | https://mathoverflow.net/users/61536 | Large gaps in Singer's difference sets | Firstly, in my previous answer I showed that the gaps are bounded by $O(n^{\frac 34} \log n)$ (and one can remove the $\log$ with a little more care). I expect that this is the best known bound on the gap.
Problem 2 is definitely an open problem. Suppose one can show that there is a gap of size $C\sqrt{n}$. By trans... | 3 | https://mathoverflow.net/users/38624 | 265152 | 119,153 |
https://mathoverflow.net/questions/265153 | 3 | Let $A$ be an algebra. We know that $Soc M =\oplus \_A Soc M\_{\alpha}$ and $Rad M =\oplus \_A Rad M\_{\alpha}$ if $M= \oplus \_A M\_{\alpha}$ as $A$-modules.
Now let $M,N$ be $A$-modules, $\Lambda:=End\_A(N)$, whether $Soc\_{\Lambda}(Hom\_A(N,M)) =Hom\_A(N,Soc\_A(M))$ and $Rad\_{\Lambda}(Hom\_A(N,M)) =Hom\_A(N, Rad\... | https://mathoverflow.net/users/83554 | Whether hom functor and socle (radical) are commutative? | Take a symmetric Nakayama algebra with Kupisch series [3,3]. (see for example the book by assem simson skowronski in the chapter about nakayama algebras for the meaning) Then $Hom(e\_0 A, soc(e\_1A))=Hom(e\_0 A, S\_1)=0$, but $Hom(e\_0A , e\_1A)$ is nonzero and thus has nonzero socle.
Next try: Let $A=K[x]/(x^2)$ wi... | 7 | https://mathoverflow.net/users/61949 | 265154 | 119,154 |
https://mathoverflow.net/questions/265160 | 7 | By a theorem of Landau, the number of integers $n\leq x$ whose prime divisors belong to only arithmetic progressions $a\_1,\dots,a\_r$ mod $q$, with $r\leq\varphi(q)$ and $a\_i$ coprime to $q$ for each $i$, is
$$Cx (\log x)^{r/\varphi(q)-1} + O(x (\log x)^{r/\varphi(q)-2}),$$
for some constant $C$ depending on $r$ ... | https://mathoverflow.net/users/48554 | Density of numbers whose prime factors belong to given arithmetic progressions | This result has been generalised a fair bit. The main generalisation is that you can replace congruence conditions by so-called "Frobenian conditions", namely conditions of the type which arise in the Chebotarev density theorem.
There have also been some improvements in the error term, but substantial improvements are ... | 5 | https://mathoverflow.net/users/5101 | 265164 | 119,157 |
https://mathoverflow.net/questions/265176 | 3 | Dihedral groups are Coxeter groups of type $I\_m$, $m \geq 3$. The Coxeter matrix of $I\_m$ is
\begin{align}
\left( \begin{matrix} 1 & m \\ m & 1 \end{matrix} \right).
\end{align}
When $m=3,4,6$, $I\_m$ are Coxeter groups of types $A\_2,B\_2,G\_2$ respectively. They have the Cartan matrices
\begin{align}
\left( \beg... | https://mathoverflow.net/users/11877 | What is the Cartan matrix for a dihedral group? | This is a little awkward to answer, because $I\_m$ isn't crystallographic for $m \not \in \{2,3,4,6 \}$, which makes it unclear how to normalize the lengths of the roots. Let $\alpha\_1$ and $\alpha\_2$ be the simple roots and $\alpha\_1^{\vee}$ and $\alpha\_2^{\vee}$ be the corresponding co-roots. Let $d\_i$ be the po... | 3 | https://mathoverflow.net/users/297 | 265179 | 119,161 |
https://mathoverflow.net/questions/265177 | 7 | Is there a compact $n$-dimensional manifold $M$ or, more generaly, a compact $n$-dimensional topological space $M$ with the following property?
"For every continuous map $f:M \to \mathbb{R}^{n}$ there are points $a,b,c \in M $ with $f(a)=f(b)=f(c)$."
This is motivated by the following obvious consequence of the Bor... | https://mathoverflow.net/users/36688 | A generalization of the Borsuk Ulam theorem | If M is allowed to be a simplicial complex, five $n$-simplices with a common $(n-1)$-dimensional face should do the job. If $M$ should be a manifold, take any closed non-orientable one.
The multiplicity of maps between manifolds can be studied with the help of characteristic classes, see [this preprint of Roman Karas... | 16 | https://mathoverflow.net/users/98590 | 265180 | 119,162 |
https://mathoverflow.net/questions/265047 | 7 | I'm interested in the topological properties of certain real algebraic curves in high-dimensional spaces. I want to visualize these curves (say, [like this](https://mathematica.stackexchange.com/questions/5968/plotting-implicitly-defined-space-curves)), and so I'm pursuing dimensionality reduction into $\mathbb{R}^3$. ... | https://mathoverflow.net/users/29873 | Does generic projection into $\mathbb{R}^3$ preserve real-algebraic-curve-ness? | The following is pretty much the standard argument due, I think, to Whitney (from the proof of his embedding theorem in the "stable range"). Suppose that $V\subset {\mathbb C}^N$ is an affine complex-algebraic subset defined over the real numbers (I.e. by polynomials with real coefficients), where $dim(V)=m$ and $2m+1<... | 5 | https://mathoverflow.net/users/21684 | 265183 | 119,163 |
https://mathoverflow.net/questions/263941 | 6 | Assume $\kappa > \aleph\_1$ is regular and let $P=(P\_\alpha, \dot{Q}\_\beta: \alpha \leq \kappa^+, \beta< \kappa^+)$ be an iteration such that:
1) For even $\beta, \dot{Q}\_\beta$ is forced to be $\aleph\_1$-closed, $\kappa$-c.c. and a subset of $V\_\kappa$.
2) For odd $\beta, \dot{Q}\_\beta$ is forced to be $\kap... | https://mathoverflow.net/users/11115 | Iterated forcing and projections | You can drop the hypothesis of $P^E$ being $\kappa$-c.c.
So take $(p\_1,p\_2)\in P^E\times P^O$, and set $p=\pi(p\_1,p\_2)$. Let $q\in P$ be such that $q\leq p$.
We shall build $q\_1\in P^E$ and $q\_2\in P^O$ such that $(q\_1,q\_2)\leq(p\_1,p\_2)$ and $\pi(q\_1,q\_2)\leq q$. We define $q\_1\upharpoonright\beta$ and... | 1 | https://mathoverflow.net/users/29916 | 265187 | 119,164 |
https://mathoverflow.net/questions/265192 | 4 | It is widely known that on a finite-dimensional vector space over a complete field, every norm is equivalent. However, I'm looking for a counterexample over a field which is not complete.
I have found no such counterexample neither by myself nor in the internet, but it is frequently stated that such result does not h... | https://mathoverflow.net/users/106387 | Non-equivalent norms on finite dimensional vector spaces over a non-complete field | Field $\mathbb Q$ with the usual absolute value $|\cdot|$ from the real numbers.
Two norms on $\mathbb Q^2$ ...
$$
\|(x,y)\|\_1 = |x|+|y|
$$
and
$$
\|(x,y)\|\_2 = \left|\,x+\sqrt{2}\;y\,\right|
$$
In both of these, $|\cdot|$ is still the usual absolute value for the real numbers.
| 8 | https://mathoverflow.net/users/454 | 265200 | 119,168 |
https://mathoverflow.net/questions/264941 | 12 | It is known that almost every pair of elements in a connected compact Lie group (topologically) generates the group.
Obviously this isn't true for non-connected groups but
>
> Given a compact Lie group $G$, is it true that almost every pair of elements of $G$ generates a subgroup containing the connected componen... | https://mathoverflow.net/users/97477 | Does almost every pair of elements in a compact Lie group generates the connected component? | The closed subgroups of $G$ not containing the identity component lie in countably many conjugacy classes of subgroups. So it is sufficient to show that for each closed subgroup $H$ not containing the identity component, the probability that Haar-random $g\_1$ and $g\_2$ both lie in some conjugate of $H$ vanishes.
Su... | 15 | https://mathoverflow.net/users/18060 | 265209 | 119,172 |
https://mathoverflow.net/questions/265208 | 4 | Suppose $R$ is a commutative ring spectrum, and let $f,g: X \to BGL\_1(R)$ be $E\_1$-maps. Then their Thom spectra are $E\_1$ ring spectra. If $f$ and $g$ are homotopic via $E\_1$-maps, does it follow that their Thom spectra are equivalent as $E\_1$ ring spectra? (And not, for example, just as ring spectra up to homoto... | https://mathoverflow.net/users/106395 | Invariance of Thom spectra | Yes this is true. This follows implicitly from the fact that the $E\_1$-structure on $Mf$ can be identified with the canonical $E\_1$-structure on $\mathrm{colim}\_Xf$ (see for example [here](https://arxiv.org/abs/1411.7988)) and that is invariant under equivalences of $E\_1$-map.
| 5 | https://mathoverflow.net/users/43054 | 265211 | 119,174 |
https://mathoverflow.net/questions/265216 | 0 | We work on a Polish group $G$ and consider finite signed measures. Let $\theta\_x\mu(B) := \mu(x^{-1}B)$. Fix a $\mu$. It is clear that $x \mapsto \theta\_x\mu$ is continuous when the codomain is given the weak\* topology when identified with a subset of $C\_b(G)^\*$. (This would be called convergence in distribution i... | https://mathoverflow.net/users/24840 | Is shifting $x \mapsto \theta_x\mu$ Borel measurable wrt total variation topology? | It isn't. Let $\mu =\delta\_e$ be Dirac mass at the identity, and set $F(x) = \theta\_x \delta\_e = \delta\_x$. Let $A$ be any non-Borel subset of $G$ and let $B = \{\delta\_x : x \in A\}$. Then $B$ is closed in the total variation topology (for any $x \ne y$ we have $\|\delta\_x - \delta\_y\| = 2$). But $F^{-1}(B) = A... | 2 | https://mathoverflow.net/users/4832 | 265219 | 119,177 |
https://mathoverflow.net/questions/265218 | 5 | In Theorem 2 of the paper "A polynomial divisor problem" by Friedlander and Iwaniec, Theorem 2 states that $$\sum\_{a^6 + b^2\le x} \Lambda(a^6 + b^2)\sim cx^{2/3}$$ for some constant $c > 0$ (in the paper itself, they give a precise error term and make $c$ explicit).
They then say that "The proof of Theorem 2 will b... | https://mathoverflow.net/users/40983 | Locating a certain result on primes represented by a certain polynomial | They prove it in section 14 ("An Application") of their paper
* John Friedlander & Henryk Iwaniec, "The Illusory Sieve" (2005)
As pointed out by Lucia in the comments, the result is conditional on the existence of exceptional characters, an hypothesis which is generally not expected to hold.
| 7 | https://mathoverflow.net/users/43108 | 265220 | 119,178 |
https://mathoverflow.net/questions/264928 | 1 | Let $M$ be a smooth manifold, $U \subset M$ an open set, $f : U \to M$ a $C^1$ diffeomorphism onto its image and $\Lambda \in U$ a hyperbolic set for $f$.
Fix a sufficiently small $\gamma > 0$ and consider a $(\lambda, \mu)$-splitting and the family of horizontal cones
$$H\_x^\gamma = \{ u + v : E^u\_x, v\in E^s\_x,... | https://mathoverflow.net/users/70760 | Showing that $Df_x H_x^\gamma \subset H_{f(x)}^{\lambda \mu^{-1} \gamma}$, where $H_x^\gamma$ is a family of horizontal cones | Since $E^u, E^s$ are equivariant, the projections of $df\_x (u + v)$ according to $E^u\_{f x}, E^s\_{fx}$ are exactly $df\_x u$ and $df\_x v$, respectively. So,
$$
\frac{\| df\_x v \|}{ \|df\_x u\|} \leq \frac{\| df\_x|\_{E^s\_x}\|}{m( df\_x|\_{E^u\_x})} \frac{\|v\|}{\|u\|} \, ,
$$
where $m(A) = \min \{ \| A v \| : \| ... | 0 | https://mathoverflow.net/users/40264 | 265232 | 119,181 |
https://mathoverflow.net/questions/265225 | 5 | A finite transitive permutation group $G$ can always be ``decomposed'' into primitive permutation groups, called its primitive components, although the decomposition is not unique. See Chapter 1 of *Finite Permutation Groups* by Wielandt.
Suppose $H$ is a primitive component of $G$, and $T$ is a composition factor o... | https://mathoverflow.net/users/4162 | composition factors of primitive components | Here is a proof of the claim I made in my comment: if $m(G)$ denotes the minimal degree of a faithful permutation representation of a finite group $G$ and $S$ is a composition factor of $G$, then $m(S) \le m(G)$. I think that answers your questions. (Note that, since $m(H) \le m(G)$ for any $H \le G$, this implies $d(S... | 3 | https://mathoverflow.net/users/35840 | 265254 | 119,188 |
https://mathoverflow.net/questions/265255 | 9 | Let $\kappa$ be a measurable cardinal (or other large cardinal) and let $j:V\longrightarrow M$ be a witness. We know that $j(\kappa)$ has large cardinal properties in $M$, but what about $j(\kappa)$ in $V$?
Let me give a couple tentative nonstandard definitions:
* Call a measurable cardinal $\kappa$ *weakly compact... | https://mathoverflow.net/users/38866 | Large cardinal properties of $j(\kappa)$ | If $\kappa$ is measurable and there is a weakly compact cardinal $\lambda$ above, then there is an elementary embedding $j:V\to M$ with critical point $\kappa$ and $j(\kappa)=\lambda$. The reason is that one may simply iterate a normal measure, which pushes $j(\kappa)$ higher, until it hits that $\lambda$. The same arg... | 8 | https://mathoverflow.net/users/1946 | 265258 | 119,189 |
https://mathoverflow.net/questions/265256 | 18 | One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:
$$\label{p}\tag{P}
f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,b).
$$
We know that a convex function $g:(a,b)\to \mathbb{R}$ is locally Lipschitz and its second derivative $g''$ e... | https://mathoverflow.net/users/53175 | How bad can the second derivative of a convex function be? | The second derivative of a convex function, in the distributional sense, is a non-negative bounded measure. And conversely. If this measure $\mu$ contains a sum $\sum\_na\_n\delta\_{x=x\_n}$, where $(x\_n)\_n$ is dense and $a\_n>0$, $\sum\_na\_n<\infty$, then $\mu|\_I\not\in L^1(I)$ for every non-void open interval $I$... | 16 | https://mathoverflow.net/users/8799 | 265261 | 119,191 |
https://mathoverflow.net/questions/265214 | 11 | I'm wondering if the notion of an orientable/non-orientable manifold has any reasonable extension that allows for a similar classification of finite geometries.
For example, the real projective plane is non-orientable. Does this somehow mean that a finite projective plane is "non-orientable" too?
I would also be in... | https://mathoverflow.net/users/25121 | Is there a well-known notion of orientability for finite geometries? | To do that you would need a notion of a non-orientable and an orientable linear transformation, i.e., essentially a notion of a "positive" and "negative" determinant, where "positive" determinants would form a subgroup *not* containing the element $-1$. This works for the field $\mathbb{Z}/p\mathbb{Z}$ if and only if t... | 8 | https://mathoverflow.net/users/28128 | 265263 | 119,192 |
https://mathoverflow.net/questions/256258 | 3 | The finite-dimensional representations over $\mathbb C(q)$ of $U\_q(\mathfrak{sl}(2))$ are all highest weight. There are two irreducible modules of each dimension. In one, the highest weight vector $v$ transforms under the Cartan element $K$ as $K v = + q^{\dim V - 1}v$. In the other, $Kv = -q^{\dim V - 1}v$. In partic... | https://mathoverflow.net/users/78 | What is the name of the Hopf algebra whose comodules are the "positive" highest weight modules of $U_{q}(sl(2))$? | It usually goes by "quantized function algebra" of $SL\_2$, or "regular function algebra" of $SL\_q(2)$, of course with various minor variations. In Kassel's book it's denoted by $SL\_q(2)$ (see Section VII.5). In Chari-Pressley book it's $\mathcal{F}\_q(SL\_2(\mathbb{C}))$ (Chapter 13; though they restrict to positive... | 3 | https://mathoverflow.net/users/9942 | 265266 | 119,193 |
https://mathoverflow.net/questions/265246 | 8 | $
\newcommand{\Span}{\mathbf{Span}}
\newcommand{\cE}{\mathcal E}
$Given a category $\cE$ with plenty of limits, let $\Span(\cE)$ denote the bicategory of spans in $\cE$. It is known that monads in $\Span(\cE)$ are the same thing as internal categories in $\cE$ (this is not quite true at the level of natural transformat... | https://mathoverflow.net/users/18702 | Algebras in a bicategory of spans | In general, it doesn't exist. That is, $\mathbf{Span}(\mathcal{E})$ doesn't have all EM-objects. If we embed $\mathbf{Span}(\mathcal{E})$ into the bicategory $\mathbf{Prof}(\mathcal{E})$ of internal categories and profunctors, then the EM-object of an internal category $C$ regarded as a monad in $\mathbf{Prof}(\mathcal... | 7 | https://mathoverflow.net/users/49 | 265268 | 119,194 |
https://mathoverflow.net/questions/265253 | 3 | An urn is filled with n black and n white balls. Do the following Markov process: 1. Draw ball, memorize color, throw it back. 2. Draw another ball (might be the same!), color it with memorized color, throw it back. 3. Rinse and repeat until only one color remains, when the process stops. 4. Count number of steps. What... | https://mathoverflow.net/users/11504 | Drunkards Uphill Walk | Let $f(i)$ be the expected stopping time with $i$ white balls and $n-i$
black balls. Then $f$ satisfies the difference equation
$$f(i-1)-2f(i)+f(i+1)=-\frac{n^2}{i(n-i)}$$
The general solution to this equation is
$$\begin{split} f(i) & = \sum\_{j=1}^{i-1} -(i-j) \frac{n^2}{j(n-j)} + Ai+B \\
& = \sum\_{j=1}^{i-1} n \lef... | 4 | https://mathoverflow.net/users/93798 | 265276 | 119,196 |
https://mathoverflow.net/questions/265022 | 3 | I'm looking for resources giving the PDFs for the $\ell^2$-norm of various spherically symmetric, continuous multivariate distributions.
For instance, the PDF for the $\ell^2$-norm of a multivariate standard normal distribution can be shown to be the chi distribution. I'm looking for similar results regarding any or... | https://mathoverflow.net/users/106305 | What is known about the PDFs for the $\ell^2$-norm of these multivariate distributions? | Given a spherically symmetric distribution $P(\vec{x})=f(y)$ of an $n$-dimensional vector $\vec{x}$, of length $y=|\vec{x}|$, then the distribution $P(u)$ of the $\ell^2$-norm $u=|\vec{x}|^2=\sum\_{i=1}^n x\_i^2$ is just
$$P(u)\propto \frac{1}{u} u^{n/2}\,f(\sqrt{u}),$$
omitting a normalization constant. So for the... | 2 | https://mathoverflow.net/users/11260 | 265285 | 119,201 |
https://mathoverflow.net/questions/265281 | 10 | In the paper ["The Volume Conjecture and Topological Strings"](https://arxiv.org/abs/0903.2084) it is said that the mirror Calabi-Yau threefold is given by
$X := \{ (x,y,u,v) \in \mathbb{C^\* \times\mathbb{C^\*} \times \mathbb{C} \times \mathbb{C}} $ : uv = A(x,y) },
where A(x,y) is the A-polynomial for a given knot... | https://mathoverflow.net/users/101705 | Calabi-Yau manifolds and knot theory | $X$ is certainly Calabi-Yau in the algebraic sense: the canonical line bundle is trivialized by the holomorphic volume form
$\Omega=\frac{dx}{x} \wedge \frac{dy}{y} \wedge \frac{du}{u}$
(this comes from the fact that $X$ is a conic bundle over $(\mathbb{C}^{\*})^2$, degenerate over the curve $A=0$. The form $\Omeg... | 11 | https://mathoverflow.net/users/25309 | 265298 | 119,205 |
https://mathoverflow.net/questions/265260 | 12 | P. Hall gave a formula for the number of generators of $G^n$ for any finite simple group $G$. One famous example is the fact that $A\_5^{19}$ is 2-generated, but $A\_5^{20}$ is not. The question of computing $d(G^n)$ has extensively been studied, starting with the work of Wiegold. However, what we need is a bound for t... | https://mathoverflow.net/users/37555 | How many generators does a direct product of alternating groups need? | To summarize what was indicated in the comments:
* as @DerekHolt points out, $d(A\_{k\_1}^{e\_1}\times\dots\times A\_{k\_r}^{e\_r})=\max\_i d(A\_{k\_i}^{e\_i})$. This is a simple
observation: if one has $d$ generators for $A\_{k\_i}^{e\_i}$
for all $i$, this is a surjective homomorphism $\phi\_i:F\_d\to A\_{k\_i}^{e\... | 14 | https://mathoverflow.net/users/1345 | 265300 | 119,206 |
https://mathoverflow.net/questions/265302 | 0 |
>
> **Question.** Let $a, b, c\geq0$ be integers. Does this inequality hold?
> $$\binom{(a+b+2)(a+c+3)+1}{c+3}\geq\binom{a+c+3}{c+3}(a+b+3)^{c+3}.$$
>
>
>
This inequality happens to appear in some intermediate step involved in [this work](https://www.math.temple.edu/~tewodros/Catalan_Convexity.pdf). Thank you f... | https://mathoverflow.net/users/66131 | inequality with binomials by breaking ups | I think we can do this combinatorially, but it's messier than I thought originally and with my current crude version I can't treat the full range. After relabeling, the inequality becomes
$$
{mn+1 \choose k} \ge {n\choose k} (m+1)^k ,
$$
for $n\ge k\ge 3$, $m\ge n-k+2$. So $m\ge 2$, and I can handle $m\ge 4$, which I w... | 3 | https://mathoverflow.net/users/48839 | 265309 | 119,209 |
https://mathoverflow.net/questions/265310 | 88 | If I swap the digits of $\pi$ and $e$ in infinitely many places, I get two new numbers. Are these two numbers transcendental?
| https://mathoverflow.net/users/nan | If I exchange infinitely many digits of $\pi$ and $e$, are the two resulting numbers transcendental? | Nice question, Erin. Here is one quick easy thing to say.
If $\pi$ and $e$ disagree in infinitely many digits, then there are continuum many choices of the particular subset of those digits to swap, and so we get continuum many different numbers this way. Since there are only countably many algebraic numbers, it woul... | 60 | https://mathoverflow.net/users/1946 | 265311 | 119,210 |
https://mathoverflow.net/questions/265312 | 2 | It's known that the class of well-founded linear orders is not axiomatizable in FO logic. But I can't understand the relation between the aforesaid argument and the compactness theorem.
Let $\mathcal{A}$ be a well-founded linear order (which means it contains no infinite descending chain for $a\_1, a\_2, ...$ such th... | https://mathoverflow.net/users/106458 | an application of compactness theorem | You just add to $\Sigma$ the assertions $c\_{n+1}<c\_n$, where these are new constant symbols. Any finite collection of the resulting theory is consistent, since there are well-orders with a finite descending sequence of any particular finite length, but there can be no well-order realizing all of these statements. So ... | 4 | https://mathoverflow.net/users/1946 | 265313 | 119,211 |
https://mathoverflow.net/questions/265316 | 7 | Let $k\geq 1$ and let $f=\sum\_{n\geq 1}a(n)q^{n}$, $a(n)\in\mathbb{R}$, be a normalised cuspidal Hecke eigenform of weight $2k$ for $\Gamma\_{0}(N)$ without complex multiplication. So the result of Barnet-Lamb et al. tells us that the numbers $\frac{a(p)}{2p^{k-1/2}}$ are $\mu$-equidistributed in $[-1,1]$, where $\mu$... | https://mathoverflow.net/users/30104 | Sato-Tate conjecture when Fourier coefficients are complex numbers | First, note that if $f = \sum\_{n \geq 1} a(n) q^{n}$ is a cuspidal Hecke eigenform lying in the new subspace of $S\_{k}(\Gamma\_{0}(N), \chi)$, where $\chi$ is a Dirichlet character modulo $n$, then one can determine the argument of $a(n)$ in terms of the values of $\chi$. In particular, for $\gcd(n,N) = 1$, then
$$ \... | 6 | https://mathoverflow.net/users/48142 | 265317 | 119,213 |
https://mathoverflow.net/questions/265324 | 2 | The following has strong experimental evidence.
>
> **Question.** For $n\geq3k$, is this identity true? Proof?
> $$\sum\_{j=0}^{\lfloor\frac{k}2\rfloor}\binom{n-2k+j}{j,k-2j,n-3k+2j}=\sum\_{j=0}^{\lfloor\frac{k}2\rfloor}\binom{n-k-2j-1}{n-2k-1,k-2j}.$$
>
>
>
| https://mathoverflow.net/users/66131 | In search of a binomial identity proof | I use a common notation $[x^ay^b]f(x,y)$ for a coefficient of $x^ay^b$ in the series $f(x,y)$.
LHS equals $[t^kz^{n-2k}]\sum\_{j=0}^{k/2} (t^2+tz+z)^{n-2k+j}$, and we may extend the summation for $j$ from $2k-n$ to $\infty$ as our monomial $t^kz^{n-2k}$ does not appear in other summands. That is,
$$LHS=[t^kz^{n-2k}]... | 3 | https://mathoverflow.net/users/4312 | 265333 | 119,217 |
https://mathoverflow.net/questions/265299 | 26 | There are several conventions for the definition of the Fourier transform on the real line.
1 . No $2\pi$. Fourier (with cosine/sine), Hörmander, Katznelson, Folland.
$ \int\_{\bf R} f(x) e^{-ix\xi} \, dx$
2 . $2\pi$ in the exponent. L. Schwartz, Trèves
$\int\_{\bf R} f(x) e^{-2i\pi x\xi} \, dx$
3 . $2\pi$ sq... | https://mathoverflow.net/users/6129 | The $2\pi$ in the definition of the Fourier transform | The version number 2 is the only one that makes the Fourier transform both a unitary operator on $L^2$ and an algebra homomorphism from the convolution algebra in $L^1$ to the product algebra in $L^\infty $.
It is not, however, of widespread use in analysis as far as I know. From the point of view of semiclassical a... | 24 | https://mathoverflow.net/users/50356 | 265337 | 119,220 |
https://mathoverflow.net/questions/265165 | 1 | I am reading some of Colmez papers about $(\varphi,\Gamma)$-modules over the Robba ring $\mathcal{R}\_L$, where $L$ is a finite extension of $\mathbb{Q}\_p$. Now I would like to understand this ring better.
In some proofs he uses the logarithm of an element of $\mathcal{R}\_L$. I could not yet find a source where a ... | https://mathoverflow.net/users/104544 | Logarithm function on the Robba ring $\mathcal{R}_L$ and some properties of $\mathcal{R}_L$ | I don't think that Colmez uses the log of a general element of $\mathcal{R}\_L$. The ring $\mathcal{R}\_L$ is some localization/completion of the ring of coordinates on a Lubin-Tate formal group, and the log that appears in Colmez' papers is the formal group log, ie $\log\_{LT}(X)$ if you choose a coordinate $X$ on you... | 2 | https://mathoverflow.net/users/5743 | 265338 | 119,221 |
https://mathoverflow.net/questions/265332 | 2 | Suppose that ${\cal C}$ is a set of subsets of $\{1,\ldots,n\}$ with the following properties:
1. $\{1,\ldots,n\}\notin {\cal C}$,
2. for all $x,y\in \{1,\ldots, n\}$ there is $A\in {\cal C}$ such that $\{x,y\}\subseteq A$, and
3. $|A\cap B| \leq 1$ for all $A\neq B\in{\cal C}$.
For $j\in \{1,\ldots,m\}$ we set the... | https://mathoverflow.net/users/8628 | Minimizing the maximum degree of a set of sets | Consider the 9-point affine plane $\ F\_3\times F\_3,\ $ over the 3-element Galois field $\ F\_3.\ $ Then $\ n=9,\ $ while the respective $\ m(C)=4 < n-1$.
Here $\ C\ $ is the set of affine lines, i.e. sets described by the linear equations (homogeneous and non-homogenous, just like in an elementary school).
The ex... | 4 | https://mathoverflow.net/users/8385 | 265348 | 119,224 |
https://mathoverflow.net/questions/265351 | 0 | Let $G=(V,E)$ be a finite simple graph such that between any two vertices there is a path of length at most $2$, and suppose that every vertex has at least $2$ neighbors.
Does $G$ have a Hamiltonian path (by which I mean a path visiting all vertices exactly once)?
| https://mathoverflow.net/users/8628 | Hamiltonian path in graphs of diameter and minimal degree $2$ | **No.** Glue a bunch of triangles together at the same vertex. For a $2$-connected example, take the previous example and add a universal vertex.
For an example that is $n$-connected take $K\_{n, n+2}$.
| 3 | https://mathoverflow.net/users/2233 | 265357 | 119,228 |
https://mathoverflow.net/questions/265371 | 3 | Let $K$ a totaly real number field, $\mathcal{O}\_K$ its ring of integre and $h$ the narrow class number of $K$. Let $\mathbf{f}$ a collection $(f\_1, ..., f\_h)$ of Hilbert cusp forms $f\_\lambda$
$(\lambda =1, ..., h)$ of weight $k=(k\_1,\dots,k\_n)$ with respect to $\Gamma\_{\lambda}(\mathcal{N}),$ $$R\_{\mathbf{f}}... | https://mathoverflow.net/users/44319 | Poles of the Rankin-Selberg zeta function associated to Hilbert cusp forms | (Thanks again to GH-from-MO for explaining the convention in the relevant paper.) I think this is easier to understand from a (genuinely) adelic viewpoint, since that makes the proof(s) be the same as for the classic Rankin-Selberg story from 1939, in the same way that Iwasawa-Tate's viewpoint on Hecke L-functions make... | 6 | https://mathoverflow.net/users/15629 | 265378 | 119,236 |
https://mathoverflow.net/questions/265381 | 1 | For start, is $\int \limits\_{2}^{x} \frac{e^{-0.3\sqrt{\ln(t)}}}{\ln^2(t)} dt \leq x\ln(\ln(x)) \frac{e^{-0.3\sqrt{\ln(x)}}}{\ln^2(x)}$ ?
If the above is true, what is a better bound for the integral ?
| https://mathoverflow.net/users/106239 | What is the upper bound for $\int \limits_{2}^{x} \frac{e^{-0.3\sqrt{\ln(t)}}}{\ln^2(t)} dt$? | Write $f(x)=\frac{e^{-0.3\sqrt{\log t}}}{\log^2 t}$. On the interval $[2, xf(x)]$ bound the integral trvially by $xf(x)$. On $[xf(x), x]$ the integrand is close to constant. More precisely, we have
$$
\frac{f(xf(x))}{f(x)} \leq e^{-0.3\left(\sqrt{\log xe^{-0.4\sqrt{\log x}})}-\sqrt{\log x)}\right)}\frac{\log^2 x}{\log^... | 4 | https://mathoverflow.net/users/37555 | 265383 | 119,238 |
https://mathoverflow.net/questions/265386 | 0 |
>
> Let $A \in \mathbb{F}^{m \times n},$ $m<n,$ and let $b \in \mathbb{F}^{m \times n}$ be such that the system $Ax=b$ is consistent. Does it follow that the set $X$ of minimal support solutions of this system $$ X:= \arg \min\limits\_{x:Ax=b, x \in \mathbb{F}^n}\{\|x\|\_0\}$$ is such that $|X| < \infty?$
>
>
>
... | https://mathoverflow.net/users/85776 | Can there be underdetermined linear systems whose set of minimal support solutions is infinite? | If the minimal $\|x\|\_0$ for a solution is $k$, any solution with $\|x\|\_0 = k$ has support one of the ${n \choose k}$ subsets of $n$ with cardinality $k$.
The $m \times k$ submatrix $A\_S$ of $A$ consisting of these columns has the property that $A\_S x = b$ has a solution with all entries nonzero, but no
solution ... | 2 | https://mathoverflow.net/users/13650 | 265392 | 119,240 |
https://mathoverflow.net/questions/195592 | 12 | I would like to find a pedagogical reference where the classification, up to isomorphism, of principal $SU(2)$ bundles over a four-dimensional compact, oriented manifold is explained. In particular I am interested in the torus $T^4$ case. Is there a similar classification when the base manifold is non-compact, in parti... | https://mathoverflow.net/users/66688 | Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds | Let $X$ be an $(n-1)$-connected CW complex with $\pi\_n(X) = G$. By attaching cells of dimensions at least $n+2$, we obtain a CW complex $Y$ the same $(n+1)$-skeleton as $X$, but with $\pi\_i(Y) = 0$ for $i > n$. For $i \leq n$, we have, by cellular approximation,
$$\pi\_i(Y) = \pi\_i(Y^{(n+1)}) = \pi\_i(X^{(n+1)}) ... | 8 | https://mathoverflow.net/users/21564 | 265399 | 119,243 |
https://mathoverflow.net/questions/265389 | 15 | Suppose $(O, G, \alpha)$ is a triple where $O$ is some mathematical object, $G$ is a group and $\alpha : G \rightarrow Aut(O)$. Many different areas of mathematics study such triples. However, I only know of a couple of examples of a new object that encodes this system.
1) If $G$ is a group, $N$ a normal subgroup of ... | https://mathoverflow.net/users/76593 | Are there other semidirect product/crossed products in other areas | Such triples, involving two objects -or even categories- and some kind of an "action" of one of them on the other (respecting some or all of its structure maps), are quite general and are met -as has already been indicated in the other answers and the comments as well- in various different branches of mathematics and m... | 15 | https://mathoverflow.net/users/85967 | 265402 | 119,244 |
https://mathoverflow.net/questions/265401 | 4 | I vaguely remember that I once attended a seminar or conference talk in which it was mentioned that the following question is open.
*Is there a (smooth) surface bundle over a surface $\Sigma\_h \to E \to \Sigma\_g$ that does not admit a flat structure? (Equivalently, one can ask whether there is a homomorphism $\pi\_... | https://mathoverflow.net/users/14233 | Surface bundles over surfaces with(out) flat structure | It's still open for all values of $g$ and $h$. One reference for it is
M. Bestvina, T. Church, and J. Souto,
Some groups of mapping classes not realized by diffeomorphisms,
Comment. Math. Helv. 88 (2013), no. 1, 205–220.
It is stated as an open problem at the end of the introduction.
One interesting positive res... | 8 | https://mathoverflow.net/users/317 | 265405 | 119,247 |
https://mathoverflow.net/questions/265377 | 4 | Let $k$ be a finite field, and let $G$ be the absolute Galois group of $k$, which is isomorphic to $\widehat{\mathbb{Z}}$. Let $\mathcal{C}$ be the category of $G$-modules. Then, we have the following:
For a finite $G$-module $N$, we have
$$
Ext^r\_{\mathcal{C}}(N, \mathbb{Z}) \simeq H^{r-1}(G,~N^D),
$$
where $N^D:=... | https://mathoverflow.net/users/46108 | Galois cohomology of finite fields | As noted in R. van Dobben de Bruyn's answer, since $N$ is killed by some positive integer we can reformulate the problem as that of constructing natural isomorphisms ${\rm{Ext}}^i\_G(N, \mathbf{Q}/\mathbf{Z}) \simeq {\rm{H}}^i(G, N^D)$ for any $i \ge 0$ and any finite discrete $G$-module $N$ (with $G := {\rm{Gal}}(k\_s... | 6 | https://mathoverflow.net/users/81332 | 265418 | 119,251 |
https://mathoverflow.net/questions/265422 | 0 | Let $X\neq \emptyset $ be a finite set and suppose that ${\cal C}$ is a set of subsets of $X$ with the following properties:
1. $X\notin {\cal C}$, and
2. for all $x,y\in X$ there is $A\in {\cal C}$ such that $\{x,y\}\subseteq A$.
Let $m=|{\cal C}|$. Is there a bijection $f: \{1,\ldots, m\}\to {\cal C}$ such that f... | https://mathoverflow.net/users/8628 | Walking "withouth gaps" through a set of sets | Still no. You may choose three subsets which cover all pairs of elements and add many mutually disjoint subsets to this collection.
| 3 | https://mathoverflow.net/users/4312 | 265432 | 119,256 |
https://mathoverflow.net/questions/265430 | 2 | Let $e\_n(x\_1,x\_2,x\_3,\dots)$ denote the $n$-th [elementary symmetric function](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial) in the infinite variables $x\_1,x\_2,x\_3,\dots$.
Let $u$ and $v$ be the roots of $z^2-6z+1=0$.
>
> **Question.** Let $x\_j=\frac1{j^8}$. The following seems to be true... | https://mathoverflow.net/users/66131 | special values of symmetric functions at powers of $\frac1j$ | As Gro-Tsen suggests in the comments, we have to expand the infinite product $$f(t)=\prod\_j \left(1-\frac{t^8}{j^8}\right)=\prod\_{j;\,w^4=1} \left(1-\frac{\omega t^2}{j^2}\right)=\prod\_{w^4=1}\frac{\sin\pi\sqrt{w}t}{\pi\sqrt{w}t},$$
we expand product of four sines as an alternating sum of cosines $$\sin a\sin b\sin ... | 11 | https://mathoverflow.net/users/4312 | 265436 | 119,257 |
https://mathoverflow.net/questions/265438 | 14 |
>
> **Question.** The following is always an integer. Is it not?
> $$\frac{(2^n-1)(2^n-2)(2^n-4)(2^n-8)\cdots(2^n-2^{n-1})}{n!}.$$
>
>
>
John Shareshian has supplied a cute proof. I'm encouraged to ask:
>
> **Question.** Can you give alternative proofs, even if they are not particularly as short?
>
>
>
... | https://mathoverflow.net/users/66131 | $n!$ divides a product: Part I | It is, because $S\_n$ embeds in $GL\_n({\mathbb F}\_2)$.
| 47 | https://mathoverflow.net/users/36466 | 265440 | 119,258 |
https://mathoverflow.net/questions/265411 | 0 | Let me define each half space i as:
$${H\_i}:{c\_i}{\bf{x}} \le {b\_i}$$
The intersection of all such ${H\_i}$ gives a polyhedron (bounded or not). Suppose I am interested in if ${H\_i}$ is active (corresponding to a facet) in such polyhedron and if so, what is all of its vertices and rays, since we know that such ea... | https://mathoverflow.net/users/40780 | algorithms and tools available for a particular polytope computation | I've had luck with [polymake](https://polymake.org) and I think its basic functions can easily do all you ask for in your question. I think [this tutorial page](https://polymake.org/doku.php/tutorial/apps_polytope) covers all you want (except getting the rays -- for that [this page](https://polymake.org/doku.php/tutori... | 1 | https://mathoverflow.net/users/353 | 265461 | 119,266 |
https://mathoverflow.net/questions/265443 | 8 | Every separable Banach space is a linear quotient of $\ell\_1$, however not every separable Banach algebra is a Banach-algebra quotient of $\ell\_1(G)$ for some group $G$ (these are the so called unitary Banach algebras).
Is every separable Banach algebra a quotient of $\ell\_1(S)$ for some countable semigroup?
I'... | https://mathoverflow.net/users/106520 | What algebras are quotients of $\ell_1(\mathbf{N})$? | (What follows is largely the result of digging around online, based on knowing a few more magic words than the OP.)
**Answer to the first question (I think).**
Let $V$ be a separable Banach space. The standard proof that $V$ is isometrically isomorphic to a linear quotient of $\ell\_1$ works by choosing a countable... | 8 | https://mathoverflow.net/users/763 | 265463 | 119,267 |
https://mathoverflow.net/questions/265466 | 3 | Let $E>1$ and consider an annulus in $\mathbb{R}^2$ with outer radius $R=\sqrt{E}$ and inner radius $R=\sqrt{E-1}$.
How many unit cubes do I need to cover the annulus?
The area of a $2$-dimensional annulus does not depend on the outer and inner radius, so one could think that the number of needed cubes depends on... | https://mathoverflow.net/users/106525 | How many unit cubes to cover an annulus? | Gerhard already mentioned what I'm going to say in a comment, but let me make it explicit anyway: To cover a line of slope $0\le m\le 1$ and length $L\gg 1$, you need about $L/\sqrt{1+m^2}$ unit squares with sides parallel to the axes.
So the part of the circle of radius $R$ with angle between $\alpha$ and $\alpha+d\... | 1 | https://mathoverflow.net/users/48839 | 265475 | 119,272 |
https://mathoverflow.net/questions/265480 | 9 | This is a follow up on [another MO question](https://mathoverflow.net/questions/265438/n-divides-a-product).
>
> **Question.** For $n\geq2$, the following is always an integer. Is it not?
> $$\frac{(2^n-2)(2^{n-1}-2)\cdots(2^3-2)(2^2-2)}{n!}.$$
>
>
>
| https://mathoverflow.net/users/66131 | n! divides a product: Part II | As $S\_n$ also embeds in $GL\_{n-1}({\mathbb F}\_2)$, you only need to check that $n!$ is not divisible by $2^n$.
| 17 | https://mathoverflow.net/users/36466 | 265482 | 119,274 |
https://mathoverflow.net/questions/265453 | 2 | Suppose I have a rational projective variety $X$ and a quadric bundle $Q \to X$ such that the total space of $Q$ is rational. Assume now that I operate on $X$ with a finite group $G$ and that the quotient $X/G$ is still a rational projective variety. Assume that the quadric bundle structure is invariant w.r.t. the $G$-... | https://mathoverflow.net/users/4096 | Rational quadric bundles and group quotients | What precisely do you mean by "quadric bundle"? If you mean a "family of quadric hypersurfaces", then that fails already for the plane conic bundle $Q/G$ over the projective plane $X/G$ associated to a pair of a smooth cubic threefold $Y\subset \mathbb{P}^4$ and a line $L$ in $Y$.
Let $X\to L$ be the projectivized n... | 3 | https://mathoverflow.net/users/13265 | 265489 | 119,276 |
https://mathoverflow.net/questions/265474 | 4 | **Problem.** Is every finite Abelian $p$-group $G$ isomorphic to the additive group of a local commutative ring $R$ whose residue field $R/{\mathbf m}$ has rank, equal to the rank of the group $G$?
Here $\mathbf m$ stands for the unique maximal ideal of the ring $R$.
The *rank* of a finite Abelian $p$-group $G$ is... | https://mathoverflow.net/users/61536 | Does every finite abelian $p$-group $G$ admit a local ring structure with residue field of the same rank as $G$? | This won't be true if $G=\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.
If $G$ has a local ring structure with maximal ideal $\mathfrak{m}$, and quotient field $G/\mathfrak{m}$ isomorphic to $\mathbb{F}\_4$, then $\mathfrak{m}/\mathfrak{m}^2\cong\mathbb{Z}/2\mathbb{Z}$ as an abelian group.
But this is impossi... | 6 | https://mathoverflow.net/users/22989 | 265514 | 119,286 |
https://mathoverflow.net/questions/265486 | 0 | I want to know how to compute this function:
$f : \mathbb{Z}\_m \rightarrow \mathbb{N}$
$f(z) = |\{ (x, y) \in \mathbb{Z}\_m^2 \mid xy \equiv z \}|$
| https://mathoverflow.net/users/105528 | How many pairs of numbers between 0 and n-1 are equal to z mod n? | Although this is by far not research level, I cannot keep myself from posting a very simple solution which somehow has not been mentioned in the comments above.
I use the basic fact that the congruence $ax\equiv b\pmod m$ has exactly $(a,m)$ solutions if $(a,m)\mid(b,m)$, and does not have any solutions otherwise. As... | 4 | https://mathoverflow.net/users/9924 | 265524 | 119,289 |
https://mathoverflow.net/questions/265525 | 2 | Given any bijection $\varphi$ between the irrationals and $\omega^\omega$, and a subset $A \subseteq \mathbb{R} \smallsetminus \mathbb{Q}$ of size $\mathfrak{d}$ , under which properties $\varphi(A)$ is dominating?
What about the question in the title?
I guess we should assume $\mathfrak{d} < \mathfrak{c}$.
| https://mathoverflow.net/users/70149 | Is there a subset of irrationals of size $\mathfrak{d}$ whose image, under any bijection to the Baire space, remains dominating? | For the title question, the answer is no, when d is less than c, because you could map your set into the 2-valued functions and map the complement to the rest. So the image would not be dominating.
| 3 | https://mathoverflow.net/users/1946 | 265526 | 119,290 |
https://mathoverflow.net/questions/265507 | 11 | A non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can be constructed by taking the fixed points of a non-trivial diagram automorphism (outer automorphism). Then how are their Grassmannians rel... | https://mathoverflow.net/users/100955 | Diagram folding of simple Lie algebras | Most maximal parabolics of $SL\_{2n}$ are not $\sigma$-invariant, not even up to conjugation. So $(G/P)^\sigma$ does not make sense. The correct statement is: Let $I\subseteq\{1,\ldots,2n-1\}$ be a symmetric subset, i.e., with $i\in I\Leftrightarrow 2n-i\in I$. Let $P\_I\subseteq SL\_{2n}$ be the corresponding paraboli... | 11 | https://mathoverflow.net/users/89948 | 265535 | 119,293 |
https://mathoverflow.net/questions/265530 | 1 | The title speaks of itself. How far is an arbitrary finite diffuse measure space from being **almost** isomorphic to a product of $[0;1]$ with another diffuse measure space? What would be reasonable sufficient conditions? I need only almost isomorphism because I am interested in analysis of $L^p$-functions. Thank you.
... | https://mathoverflow.net/users/89313 | Diffuse measure space as a product of $[0;1]$ and another diffuse measure space | Here is a result that covers "most" finite measure spaces encountered in applications. Let $(\Omega,\Sigma)$ be a *standard Borel space*, that is $\Sigma$ is the Borel $\sigma$-algebra for some separable completely metrizable topology on $\Omega$. Let $\mu$ be any finite diffuse measure on $(\Omega,\Sigma)$, we can tak... | 2 | https://mathoverflow.net/users/35357 | 265539 | 119,295 |
https://mathoverflow.net/questions/265533 | 5 | A key lemma to Kirszbraun's theorem for $\mathbb{R}^2$ states the following:
Given any two finite collections of points $x\_1,\dots,x\_n$ and $x\_1',\dots,x\_n'$ in $\mathbb{R}^2$ such that $|x\_i'x\_j'|\le |x\_ix\_j|$ for all $i,j=1,\dots,n$ and any $x\in \mathbb{R}^2$, it's always possible to find $x'$ in the conve... | https://mathoverflow.net/users/95073 | Possible generalization to Kirszbraun's theorem for $\mathbb{R}^2$ | Finally I understood your question; here is a counterexample:
Consider a rhombus $x\_1x\_3x\_2x\_4$ and let $x$ be the midpoint of the diagonal. Let $x\_1'x\_3'x\_2'x\_4'$ be a rhombus with the same side lengths such that $|x\_1'-x\_2'|<|x\_1-x\_2|$.
| 3 | https://mathoverflow.net/users/1441 | 265547 | 119,300 |
https://mathoverflow.net/questions/265505 | 2 | Let $S$ be an integral scheme with function field $K = K(S)$. Let $\mathscr{A}, \mathscr{B}$ be Abelian schemes over $S$. Let $L/K$ be a separable field extension. Given $f\_L \in \mathrm{Hom}(\mathscr{A}\_L,\mathscr{B}\_L)$, why does there exist an étale cover $T \to S$ with function field $L'$, $L/L'/K$ and an extens... | https://mathoverflow.net/users/nan | extending homomorphisms of Abelian schemes | Since the OP suggested it, I am posting my comments as an answer. By the construction of Hilbert and Quot schemes, there is a relative Hom scheme, $\text{Hom}\_S(\mathcal{A},\mathcal{B})$ over $S$ whose connected components are quasi-projective over $S$. The claim is that these components are proper over $S$. By the va... | 5 | https://mathoverflow.net/users/13265 | 265553 | 119,302 |
https://mathoverflow.net/questions/265400 | 5 | Let $A \subseteq B$ be two (associative with $1$) $k$-algebras, where $k$ is a field of characteristic zero, and let $f$ be a $k$-automorphism of $A$.
I am interested to know 'when' one can extend $f$ to a $k$-automorphism of $B$.
Three nice answers:
(1) [This question](https://mathoverflow.net/questions/192281/inn... | https://mathoverflow.net/users/72288 | Extending an automorphism from a sub-algebra to the algebra | The short answer to your question is: almost never. Many counterexamples have already been constructed in the comments. Let me make another easy counterexample:
**Example.** Let $k$ be algebraically closed, and let $A = k[t]$, $B = k[\sqrt{t}] = k[x]$. Consider the automorphism $t \mapsto t+1$ of $A$. It cannot be ex... | 4 | https://mathoverflow.net/users/82179 | 265559 | 119,304 |
https://mathoverflow.net/questions/265560 | 4 | Let $X \to Y$ be a cyclic etale cover of smooth projective geometrically connected curves over some field $k$. Then the map is classified by an element of the cohomology group $H^1\_{et}(Y\_{k\_s}, \mu\_n)$; in other words the data of the covering is equivalent to giving a line bundle on $Y$ together with a trivializat... | https://mathoverflow.net/users/106554 | Line bundles and Cyclic Covers of Curves |
>
>
> >
> > **Theorem.** Let $X \to Y$ be an étale Galois cover with group $G$ of proper geometrically integral schemes over any field $k$. Then we have an exact sequence
> > $$0 \to \operatorname{Hom}(G,k^\times) \to \operatorname{Pic}(Y) \to \operatorname{Pic}(X)^G.$$
> > In particular, the kernel has size at m... | 7 | https://mathoverflow.net/users/82179 | 265563 | 119,306 |
https://mathoverflow.net/questions/265528 | 14 | Given Ramanujan's famous $\frac1{\pi}$ formula $$\frac 1\pi=\frac {2\sqrt2}{99^2}\sum\_{k=0}^\infty\frac {(4k)!}{k!^4}\frac {26390k+1103}{396^{4k}}$$
which is a level 2 Ramanujan-Sato series. It can also be expressed as
$$\frac{1}{\pi} =\frac{192 \sqrt 2}{(396^2)^{3/2}} \sum\_{k=0}^\infty \tbinom{2k}{k}\tbinom{2k}{k... | https://mathoverflow.net/users/12905 | Numerology with Ramanujan's pi formula | (*Too long for a comment*.) After staring hard at my question and recalling an old [MSE post](https://math.stackexchange.com/questions/1952204/the-chudnovsky-pi-formula-1-pi-revisited) of mine, I made an inspired guess and found,
>
> Level 8
>
>
>
$$\frac{1}{\pi}=\frac{192\sqrt{2}}{(396^2+4\color{blue}\alpha)^... | 7 | https://mathoverflow.net/users/12905 | 265573 | 119,311 |
https://mathoverflow.net/questions/265587 | 1 | Let $C(D) \cap H(\bar D)$ denote the inner product space of functions these are analytic in unit disk $D$ and continuous in $\bar D$, equipped with the inner product $(f,g)= \frac{1}{2 \pi} \int\_{0}^{2 \pi} f(e^{i \theta}) \overline {g(e^{i \theta})} d \theta$, is it a Hilbert space?
| https://mathoverflow.net/users/106574 | A problem on completeness of a specific space of complex functions | No, it is not complete. Completion of this space is called $H^2(D)$.
It consists of all analytic functions for which
$$\| f\|^2:=\sup\_r\frac{1}{2\pi}\int\_0^{2\pi}|f(re^{i\theta})|^2d\theta$$
is finite, or alternatively
$$f(z)=\sum\_0^\infty a\_nz^n,\quad \sum\_0^\infty|a\_n|^2<\infty,$$
or alternatively, all those fu... | 1 | https://mathoverflow.net/users/25510 | 265594 | 119,314 |
https://mathoverflow.net/questions/265595 | 13 | Let $L$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$. The Möbius function $\mu$ for $L$ is defined recursively by: for $\forall a,b \in L$ with $a<b$, $\mu(b,b) = 1$ and $\mu(a,b) = -\sum\_{a<c\le b}\mu(c,b)$.
The Möbius number of $L$ is defined by $\mu(\hat{0},\hat{1})$.
Define the Möbius num... | https://mathoverflow.net/users/34538 | The Möbius number of the nonabelian finite simple groups | The answer to your question is ``yes". Something more general was proved by Hawkes, Isaacs and \"Ozaydin in a 1989 paper in the Rocky Mountain Journal of Mathematics.
CORRECTION: As Sebastien Palcoux notes below, this result is due to Kratzer and Th\'evenaz
More precise results for the groups $PSL\_2(p)$ were known... | 17 | https://mathoverflow.net/users/36466 | 265597 | 119,315 |
https://mathoverflow.net/questions/265584 | 27 | The Jacobi theta function $\theta(z) = 1 + 2 \sum\_{n = 1}^\infty q^{n^2}$, with $q = e^{\pi i z}$ is a (twisted) modular form with weight $1/2$. It has an associated $L$-function $L(\theta, s) = \sum\_{m =1}^\infty \frac{1}{(m^2)^s} = \sum\_{m=1}^\infty \frac{1}{m^{2s}} = \zeta(2s)$, which stands roughly in the same r... | https://mathoverflow.net/users/56878 | Is there a "Langlands philosophy" reason for the fact that the L-function of the Jacobi theta function is (almost) the Riemann zeta function? | Good question. I don't understand fully what's happening, but here is an idea.
Let $f=\sum a\_n q^n$ be a modular form of weight $k+1/2$, nebentypus $\chi$. Assuming $k \geq 1$, the Shimura correspondence attaches to $f$ a
modular form of integral weight $2k$, $g = \sum b\_n q^n$ such that
$$L(g,s) = L(\chi',s-k+1) \... | 15 | https://mathoverflow.net/users/9317 | 265611 | 119,320 |
https://mathoverflow.net/questions/265610 | 4 | A Hausdorff topological space $X$ is called strongly
zero-dimensional whenever for every closed subset $A$ of $X$ and
every open subset $U$ of $X$ such that $A \subseteq U$, there
exists a **clopen** subset $V$ of $X$ such that $A \subseteq V
\subseteq U$.
Now, let $X'$ be a Hausdorff topological space such that for
... | https://mathoverflow.net/users/106584 | Strongly zero-dimensional topological spaces and a simillar condition | (Having posted this, I saw that all of it is in the comment by Gro-Tsen)
Taking, in the definition of \*-space, $C=\text{closure of $O$}$ shows that closure of an open set must be clopen. This is clearly also sufficient. So \*-spaces are exactly extremally disconnected Hausdorff spaces - those with closures of open s... | 1 | https://mathoverflow.net/users/41291 | 265612 | 119,321 |
https://mathoverflow.net/questions/265577 | 3 | Fix some countable language $\Sigma$, and some reasonable way of interpreting reals as $\Sigma$-structures with domain $\omega$. Let $T$ be a complete $\Sigma$-theory with continuum-many isomorphism types of countable models (**EDIT**) which is not a counterexample to Vaught's conjecture. Then of course there is a perf... | https://mathoverflow.net/users/8133 | Perfectly transversable theories | The property you are asking for is a very strong condition on $T$. Let met try to rephrase the question more carefully:
The set of countable $\Sigma$-structures with universe $\omega$ is naturally a Polish space $\mbox{Mod}(\Sigma)$ which has no isolated points (if it has isolated points, then necessarily $\Sigma$ wo... | 4 | https://mathoverflow.net/users/26705 | 265615 | 119,324 |
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