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/263612 | 2 | The dual to the category of commutative unital C\*-algebras is equivalent to the category of compact Hausdorff spaces, a concrete category. Can the dual to the category of unital C\*-algebras also be concretized, yielding a concrete notion of noncommutative topological space? If so, is there a known description of this... | https://mathoverflow.net/users/83073 | Is the dual of the category of unital C*-algebras concretizable? | The opposite of any concretizable category is concretizable: if $U : C \to \text{Set}$ is a faithful functor, then so is the composite
$$C^{op} \xrightarrow{U^{op}} \text{Set}^{op} \xrightarrow{\text{Hom}(-, 2)} \text{Set}.$$
Really we can pick any concretization of $\text{Set}^{op}$ we want; the one above correspo... | 5 | https://mathoverflow.net/users/290 | 263613 | 118,572 |
https://mathoverflow.net/questions/263588 | 2 | This is a follow up on [the MO question here](https://mathoverflow.net/questions/263482/sinc-ing-integral). I kept being fascinated and bemused by these functions.
Denote $\text{sinc}(x)=\frac{\sin x}x$. Experiments suggest that
$$\sum\_{n=1}^{\infty}\text{sinc}^j\left(\frac{n}{\pi}\int\_{\mathbb{R}}\text{sinc}^j(x)\... | https://mathoverflow.net/users/66131 | more on "sinc-ing" integrals and sums | Results like this hold for small values of $j$, but are eventually false. Put $f(x)$ to be the indicator function of the interval $[-1/2,1/2]$, and put $f\_j(x)$ to be the convolution of $f$ with itself $j$ times. Then the Fourier transform of $f\_j$ is
$$
{\hat f\_j}(\xi) = \int\_{-\infty}^{\infty} f\_j(x) e^{-2\pi ... | 10 | https://mathoverflow.net/users/38624 | 263615 | 118,573 |
https://mathoverflow.net/questions/263516 | 2 | Let $S$ be a set, or equivalently the topological space with discrete topology and $2$ two point set with discrete tiopology. $βS$ be the Stone–Čech compactification of $S$. By Tychonoff theorem the topology on $2^S$ is compact with respect to the product topology.
Is the compact-open topology on $2^{\beta S}$ the p... | https://mathoverflow.net/users/73577 | Compact open topology on $2^{\beta S}$ with $S$ a set | Let me try to put some things in order here.
For a topological space $X$ we denote here by $2^X$ the set of all continuous functions $X\to \{0,1\}$.
Fixing a set $S$ we consider the set $2^{\beta S}$ where $\beta S$ denotes the Stone–Čech compactification of $S$.
For $A,B\subset \beta S$ let me write
$$U(A,B)=\{f : ... | 5 | https://mathoverflow.net/users/89334 | 263626 | 118,577 |
https://mathoverflow.net/questions/263624 | 1 | I was wondering whether the following interpolation between $L^1$ and $L^2$ spaces is true:
Let $f \in \mathbb{R}^n$ be such that
$$ \alpha\_1:= \int\_{\mathbb{R}} \left\lVert f(x\_1,\cdot,....\cdot) \right\rVert\_{L^2(\mathbb{R}^{n-1})} dx\_1$$
up to
$$ \alpha\_n:= \int\_{\mathbb{R}} \left\lVert f(\cdot,\cdo... | https://mathoverflow.net/users/105584 | Interpolation between $L^1$ and $L^2$ spaces | Yes for $n = 1$, and no for $n \geq 2$. Indeed, if we had an inequality $||f||\_1 \leq C(\alpha\_1 + \dots + \alpha\_n)$ then we could set $f = \mathbb{1}\_{[0,K]^n}$ and get
$$
K^n \leq C n K^{\frac{n+1}{2}},
$$
and we get $n \leq 1$ by letting $K$ tend to infinity.
| 1 | https://mathoverflow.net/users/21724 | 263627 | 118,578 |
https://mathoverflow.net/questions/244351 | 9 | I would like to see an example of a spectral triple $(A,H,D)$ such that the underlying algebra $A$ is commutative but this spectral triple is not $\theta$-summable in the sense that $e^{-tD^2}$ is never traceclass operator.
EDIT: Spectral triple is defined as a unital $\*$-algebra $A$ represented faithfully on some ... | https://mathoverflow.net/users/24078 | Spectral triples which are not $\theta$-summable | So, under this generality, it seems to me that we can choose $A$ in a clever way: if we make $A$ smaller, then it gets "easier" to satisfy the condition. So why not take $A$ to be the scalar multiples of the identity-- then $D$ commutes with all of $A$ on the nose.
Now let $H=\ell^2$ and let $D$ be multiplication by ... | 5 | https://mathoverflow.net/users/406 | 263629 | 118,579 |
https://mathoverflow.net/questions/263590 | 6 | Does anyone know of any papers which give results on the existence of Minkowski units in totally real cyclic fields of prime degree? F. Marko has some results for composite degree in his paper "On the existence of Minkowski units in totally real cyclic fields" but I'm more interested in the prime degree case. See <http... | https://mathoverflow.net/users/56362 | Existence of Minkowski units in totally real cyclic fields of prime degree | See this [short note](https://projecteuclid.org/download/pdf_1/euclid.jmsj/1259943179) by Brumer and the book [Elementary and Analytic Theory of Algebraic Numbers](https://archive.org/details/ElementaryAndAnalyticTheoryOfAlgebraicNumbers3rdEdition) by Narkiewicz, in particular Section 3.3.
| 3 | https://mathoverflow.net/users/3503 | 263632 | 118,580 |
https://mathoverflow.net/questions/263451 | 8 | Given the group $\mathrm{SL}(2,\mathbb{Q}\_p)$ one can describe precisely the number of vertices in the Bruhat-Tits building of distance $k$ from a fixed vertex $v$, where the distance between two vertices $v$ and $v'$ is the minimal number of edges in a path connecting the two. Denoting this quantity by $N\_k$, we see... | https://mathoverflow.net/users/105554 | Closed formula for vertices distance k in building of SLn | We consider the Bruhat-Tits building of $\text{SL}\_n(\mathbb{Q}\_p)$.
As usual, we identify the vertices with *lattices*, that is subgroups of the of $\mathbb{Q}^n$ commensurated with the *standard lattice* $\Lambda\_0=\mathbb{Z}\_p^n$, defined up to homothety, namely multiplication by $p^k$ for some $k\in \mathbb{Z}$... | 1 | https://mathoverflow.net/users/89334 | 263633 | 118,581 |
https://mathoverflow.net/questions/263639 | 8 | I searched on the internet, but I could not find anything useful about applications of forcing in constructive set theories.
Are there any developments of forcing in CZF or IZF?
Thanks in advance.
| https://mathoverflow.net/users/83598 | Forcing in Constructive Set Theories | See
[Forcing for IZF in sheaf toposes](https://www.researchgate.net/publication/228380851_Forcing_for_IZF_in_sheaf_toposes)
Also
[Toposes from Forcing for Intuitionistic ZF with Atoms.](https://arxiv.org/abs/1702.03399)
---
Edit:
Maybe more references:
[Heyting-valued models for intuitionistic set the... | 8 | https://mathoverflow.net/users/11115 | 263649 | 118,585 |
https://mathoverflow.net/questions/263647 | 0 | Suppose I have a function $Q(z)$ of a complex variable $z\in\mathbb P^1$, possessing square root type branch points at the positions $\left\{z\_i\right\}\_{i=1}^{2M}$. I know that the Riemann surface $\mathcal M$ on which $Q$ lives has genus $0$.
I have the following questions:
* Aside from the knowledge of the poi... | https://mathoverflow.net/users/18961 | Uniformizing variable for branched covering of the Riemann sphere | 1. Besides the position of ramification points $z\_j$ you need monodromy of $Q$ to determine $M$. Once $M$ is defined, you need a normalization of your uniformizing function: it is defined up to a conformal automorphism of the sphere $L^1$.
2. No general formula exists. To find the uniformization in the case of genus $... | 0 | https://mathoverflow.net/users/25510 | 263653 | 118,587 |
https://mathoverflow.net/questions/263643 | 6 | For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h\_{\square}$ and $c\_{\square}$ denote the [hook length and content](http://www-math.mit.edu/~rstan/transparencies/hooks.pdf) of $\square$, respectively.
R Stanley [proved the following](http://www-math.mit.edu/~rstan/papers/hooks.pdf) in Theor... | https://mathoverflow.net/users/66131 | hooks and contents: Part I | Specialize the Cycle Index Formula to get
$$\prod\_{n=1}^\infty \exp \bigl( \frac{a\_i}{i}z^i \bigr) = \sum\_{n=0}^\infty \frac{z^n}{n!} \sum\_{\pi \in \mathfrak{S}\_n} a\_1^{\mathrm{cyc}\_1(\pi)} a\_2^{\mathrm{cyc}\_2(\pi)} \ldots, $$
by setting $a\_i = t^2$ if $i$ is even and $a\_i = t$ if $i$ is odd. We get
$$... | 13 | https://mathoverflow.net/users/7709 | 263658 | 118,589 |
https://mathoverflow.net/questions/261935 | 10 | Marvin J. Greenberg provided an elementary proof of the Kronecker–Weber theorem [here](https://www.jstor.org/stable/2319208) (Amer. Math. Monthly, 81 (1974), no. 6, 601-607).
An argument in the lemma 4 was found to be wrong as noticed in *[Correction to "An Elementary Proof of the Kronecker-Weber Theorem"](http://www.j... | https://mathoverflow.net/users/84923 | Where exactly is the flaw in this proof of the Kronecker–Weber theorem? | As promised, here are my calculations of the higher ramification groups in certain cyclotomic extensions.
Let $G$ denote the Galois group of the field $L$ of $p^n$-th roots of unity, and consider the Hilbert subgroups for the completely ramified prime $p$. The decomposition group and the inertia both are equal to $G$... | 6 | https://mathoverflow.net/users/3503 | 263662 | 118,591 |
https://mathoverflow.net/questions/263589 | 5 | My claim is as follows:
>
> Let $\Gamma$ be a discrete subgroup of $\operatorname{Isom}(\Bbb{H}^{n})$, the isometries of hyperbolic $n$-space. If $\Gamma$ is a lattice in $\operatorname{Isom}(\Bbb{H}^n)$ then the limit set of $\Gamma$ is $\partial \Bbb{H}^{n}$.
>
>
>
I'm inclined to believe that the statement ... | https://mathoverflow.net/users/100971 | What is the limit set of a hyperbolic lattice? | This question is probably not really research level given that the answer exists in many textbooks. Nevertheless, it may be useful to have different viewpoints.
An elementary geometric argument follows from Chapter 8 of [Thurston's notes](https://dl.dropboxusercontent.com/u/8592391/ThurstonNotes.pdf). In fact, the ar... | 4 | https://mathoverflow.net/users/1345 | 263668 | 118,594 |
https://mathoverflow.net/questions/263659 | 8 | I need a reference for the following assertion (if it's true). Let $X$ be a minimal elliptic surface over the field of complex numbers. Assume that its Kodaira dimension $\kappa(X)=1$. Then $X$ does not contain rational curves. Thanks!
| https://mathoverflow.net/users/9658 | Rational curves on elliptic surfaces | I don't think this is true.
Take a general pencil of cubics in $\mathbb{P}^2$, and blow-up the 9 fixed points to get an elliptic fibration $f:S\rightarrow \mathbb{P}^1$ which admits a section (at least 9 in fact). Pull back by a degree $n\geq 3$ covering $\mathbb{P}^1\rightarrow \mathbb{P}^1$ branched along two points... | 7 | https://mathoverflow.net/users/40297 | 263669 | 118,595 |
https://mathoverflow.net/questions/263607 | 11 | Let $\Sigma$ be a closed genus $g$ surface. Assume that $\mathcal{C}$ is a smooth and proper dg- (or $A\_\infty$-) category which admits a faithful action
$$ MCG(\Sigma) \to Auteq(\mathcal{C})$$
by the mapping class group by auto-equivalences.
1. What good properties about $\mathcal{C}$ does that imply? (or
: why c... | https://mathoverflow.net/users/105615 | Categorical mapping class group action | [This is an elaboration of parts of Mark Penney's answer]
A natural source of categorical actions of the mapping class group is the category assigned by any 4d TFT to a surface. Such categories are often written as Fukaya categories - eg Donaldson theory and Seiberg-Witten theory will attach Fukaya categories of modu... | 8 | https://mathoverflow.net/users/582 | 263674 | 118,598 |
https://mathoverflow.net/questions/263631 | 3 | Let $f: \mathbb{R}^n \rightarrow \mathbb{C}$ be in $L^2\cap L^1,$ then the Fourier transform is in $L^2 \cap L^\infty.$
Does this imply that we can take common norms in the sense that we can estimate
$$\sup\_{x\_i \in \mathbb{R}} \sqrt{\int\_{\mathbb{R}^{n-1}} \left\lvert \hat{f}(x\_1,...,x\_i ,...,x\_n) \right\rv... | https://mathoverflow.net/users/105648 | Boundedness of different Fourier transforms | Yes, such bounds follow by considering partial Fourier transforms and then using the norm bounds you quoted. Let me take $n=2$ for ease of notation. Write $g(x,t)=\int f(x,k) e^{2\pi i kt}\, dk$. Then, for example (and as desired),
$$
\sup\_y \|\widehat{f}(x,y)\|\_{L^2(dx)} =\sup\_y \|g(x,y)\|\_{L^2(dx)}\le \left\| \in... | 7 | https://mathoverflow.net/users/48839 | 263682 | 118,600 |
https://mathoverflow.net/questions/263679 | 1 | For an elliptic curve $y^2=4x^3-g\_2x-g\_3$ over $\mathbb{Q}$, it is known that the local parameter at $O$ (the identity point) could be written by $-\frac{x}{y}$. Is it possible to write down the local parameter at $P\in E[\ell]$ for some prime $\ell\geq3$ with $P\neq O$ in the similar way (maybe, in terms of $x$ and ... | https://mathoverflow.net/users/44005 | Local parameter at torsion points of elliptic curve | Let $P=(a,b)$. Then $x-a$ is a local parameter at $P$, since it has a zero of order one at P. (It's two zeros are $P$ and $-P$.) Or if you want a local parameter that is defined over the field of definition of $E$, you can take the $\ell$-division polynomial $$\psi\_\ell(x):=\prod\_{P\in E[\ell]/\{\pm1\}} \bigl(x-x(P)\... | 5 | https://mathoverflow.net/users/11926 | 263684 | 118,601 |
https://mathoverflow.net/questions/263657 | 3 | Let $T=(\mathbb{C}^\*)^n$ act on $\mathbb{P}^n$ torically by
$$t.[x\_0:\dots:x\_n]=[x\_0\;:\;t\_1x\_1\;:\;\ldots \;:\;t\_nx\_n]$$
I would like to know an expression for
1. the equivariant Chern character $\mathrm{ch}^T(\chi\cdot\mathcal{O}(d))$, where $\chi\cdot\mathcal{O}(d)$ is the equivariant line bundle isomorp... | https://mathoverflow.net/users/4721 | Equivariant characteristic classes on $\mathbb{P}^n$ | I am just posting my comments as an answer. Denote by $A$ the equivariant Chow ring $\text{CH}^\*\_T(\text{Spec}(\mathbb{C}))$. For $i=1,\dots,n$, denote by $\lambda\_i\in A^1$ the first Chern class of the character $\chi\_i$ with $\chi\_i(t) = t\_i$. Denote by $\lambda\_0$ the class $0$, i.e., the first Chern class of... | 3 | https://mathoverflow.net/users/13265 | 263700 | 118,608 |
https://mathoverflow.net/questions/263702 | 7 | As I understand things, one of the classical reasons to care about modular forms was their relation to interesting arithmetic functions/counting questions, i.e. on sums of squares and partitions. When I read Diamond and Shurman’s book([Diamond&Shurman]A First Course in Modular Forms), this point of view was briefly men... | https://mathoverflow.net/users/105675 | The importance of relations between automorphic forms and arithmetic functions | Let me just highlight one aspect that I find particularly interesting.
Without any doubt, one the most basic arithmetic functions is $\tau\_k(n)$, which counts the number of ways $n$ can be written as a product of $k$ factors. It is the $k$-fold convolution of the constant $1$ function with itself, so it is multiplic... | 7 | https://mathoverflow.net/users/11919 | 263705 | 118,610 |
https://mathoverflow.net/questions/263692 | 9 | Let $A$ be a tiling of $\mathbb{R}^{2}$ using regular polygons. Assume that the tiling is edge-to-edge. Assume also that there are two directions of periodicity, so that $\mathbf{u},\mathbf{v}\in \mathbb{R}^{2}$ are linearly independent vectors, and $A+\mathbf{u}=A+\mathbf{v}=A$.
Question: Must there always exist ort... | https://mathoverflow.net/users/90186 | Periodic tilings of the plane by regular polygons | I claim that the tiling <https://upload.wikimedia.org/wikipedia/commons/6/66/5-uniform_310.svg> does not admit a orthogonal period.
The basis for the period lattice is given by the two vectors $$ v\_1 = \begin{pmatrix} 3 + \sqrt{3} \\ -1 \end{pmatrix}, v\_2 = \begin{pmatrix} 1/2 \\ (2+\sqrt{3})/2 \end{pmatrix}. $$ S... | 10 | https://mathoverflow.net/users/36579 | 263709 | 118,612 |
https://mathoverflow.net/questions/263703 | 11 | Consider the sequence $a\_n=2^{2n}\binom{2n}n^{-1}$. [Stirling's approximation](https://en.wikipedia.org/wiki/Stirling's_approximation) shows that $a\_n\sim \sqrt{\pi n}$, thus
$$\sum\_{n\geq0}\frac{\pi}{2a\_n}\qquad \text{and} \qquad
\sum\_{n\geq0}\frac{a\_n}{2n+1}$$
are both divergent series. However, their differenc... | https://mathoverflow.net/users/66131 | Two divergent series conspiring? | We have
$$ f(x):=\sum\_{n\geq 0}\frac{x^{2n}}{a\_n} =
\frac{1}{\sqrt{1-x^2}} $$
and
$$ g(x):=\sum\_{n\geq 0} \frac{a\_n}{2n+1}x^{2n} =
\frac{\sin^{-1}x} {x\sqrt{1-x^2}}. $$
It is routine to compute that
$$ \lim\_{x\to 1-}\left(\frac 12\pi f(x)-g(x)\right)=1 $$
and then apply [Abel's theorem](https://en.wikipedia.o... | 29 | https://mathoverflow.net/users/2807 | 263710 | 118,613 |
https://mathoverflow.net/questions/263084 | 10 | I have submitted a paper on applied probability in one of SIAM journals. The paper is under review for 9 months. I asked the editor 1 month ago about it, I was told that one review report has come and they are waiting for the other. Will it be ok if I politely inquire about it now (I have inquired 3 times till now afte... | https://mathoverflow.net/users/94452 | Is it fine to inquire about a paper that's been under review for around 9 months? | My experience is in Pure Mathematics, not in Applied Probability. In my area, at least, it is unusual to have a full report recommending acceptance (pending revisions) in under a year, although it can happen much faster. It has happened to me within a few weeks, but that is the exception rather than the rule. My first ... | 3 | https://mathoverflow.net/users/3651 | 263714 | 118,614 |
https://mathoverflow.net/questions/263715 | 5 | In their paper [**Sign changes of Hecke eigenvalues**](https://arxiv.org/pdf/1405.7671.pdf), Matomaki and Radziwill showed that (**Theorem 1.2** of the paper) for a large enough $x$ , the number of sign changes of sign changes of in the non-vanishing sequence of Hecke eigenvalues $(\lambda\_f(n))\_{n\leq x}$ is
$\asym... | https://mathoverflow.net/users/76102 | Proof of a theorem about the size of the number of sign changes of Hecke eigenvalues | The proof of Theorem 1.2 relies on Propositions 3.4 and 3.5. In particular, if $h$ is sufficiently large but fixed, the bound you quote is true for a positive proportion of $x\in\mathbb{N}\cap[X,2X]$. Call such an $x$ nice.
Consider a maximal family $F$ of nice points $x\in[X,2X]$ such that any two of them differ by ... | 5 | https://mathoverflow.net/users/11919 | 263727 | 118,617 |
https://mathoverflow.net/questions/263725 | 4 | Let $p\geq 5$ be a prime number, let $E= \{(a,b,c,n) \in \mathbb{N}^4 ~|~ a^p+b^p+c^p=3^n\}$.
I know for all $k \geq 0$, $(3^k,0,0,kp), (0,3^k,0,kp), (0,0,3^k,kp) \in E$ and $(3^k,3^k,3^k,kp+1) \in E$.
Let $F\_1=\{(3^k,0,0,kp) ~| ~k \in \mathbb{N}\}$, $F\_2= \{(0,3^k,0,kp) ~| ~k \in \mathbb{N}\}$, $F\_3= \{(0,0,3^k... | https://mathoverflow.net/users/105688 | Non trivial solutions of $a^p+b^p+c^p=3^n$ | Partial answer. The $n$-conjecture implies at most finitely many
counterexamples over the *integers* for $p \ge 5$ besides yours
and some additional negative.
The [n-conjecture](http://cr.yp.to/bib/1994/browkin.pdf).
is a generalization of abc and basically says that the if
$a\_1 + \ldots + a\_n=0$, no proper subsum... | 3 | https://mathoverflow.net/users/12481 | 263731 | 118,620 |
https://mathoverflow.net/questions/263716 | 2 | I have a huge sparse linear system $Ax = b$ where I know that an eigenvalue/eigenvector pair is $1$ and a vector of all $1$'s. Can this knowledge help me in solving the linear system at all? It seems like it should.
| https://mathoverflow.net/users/105684 | Solving linear system when one eigenvalue is known | Yes and no.
1. If you modify slightly the implementation of a Krylov subspace method such as GMRES, you can construct a method with a projection subspace that contains the known eigenvector. This essentially lets you always include the known eigenvector in the search space, for free, saving one matvec and ensuring th... | 4 | https://mathoverflow.net/users/1898 | 263735 | 118,624 |
https://mathoverflow.net/questions/263719 | 1 | Given $B>0$ and $n\in\Bbb N$ what is the probability that a given $n\times n$ integer matrix with all entries bound by absolute value $<B$ is non-singular? I am looking for precise scaling.
| https://mathoverflow.net/users/10035 | On non-singularity of integer matrices with bounded entries | If I understand the question correctly, this is a known hard problem if $B$ is fixed, but $n$ is growing. See, for example, this paper, written by some fairly smart people:
On the singularity probability of discrete random matrices
Jean Bourgain, Van Vu, P. M. Wood (sorry, insert citation is failing), JFA 2010.
How... | 4 | https://mathoverflow.net/users/11142 | 263742 | 118,626 |
https://mathoverflow.net/questions/263741 | 3 | The following question arises while I am reading a paper of [B. N. Cooperstein.](https://link.springer.com/article/10.1007/BF02761072)
In the Table 1 of his paper, two groups have smaller permutation degree compared to other members in their family: $\mathrm{PSp}\_4(3)$ with permutation degree 27 and $\mathrm{PSU}\_3... | https://mathoverflow.net/users/18286 | On some classical groups with small permutation degree | For me the article by B. N. Cooperstein is behind a paywall.
But I think the object you are looking for is the Hoffman–Singleton graph. There is a wikipedia page about that graph, see
<https://en.wikipedia.org/wiki/Hoffman%E2%80%93Singleton_graph>
| 4 | https://mathoverflow.net/users/105705 | 263744 | 118,627 |
https://mathoverflow.net/questions/263368 | 5 | Let $p$ be an arbitrary distribution over $\mathbb{N}$, and $m\geq 1$ be an integer. Given an infinite sequence of i.i.d. draws $(X\_i)\_{i\geq 1}$ from $p$, define a *collision* as a pair $(i,j)$ with $i<j$ and $X\_i=X\_j$.
Let $M\_m$ be the minimum integer $\ell$ such that $m$ collisions happen in $(X\_i)\_{1\leq i... | https://mathoverflow.net/users/37266 | Birthday problem with unequal probability: expected number of draws before the $m$-th collision? | (1) Simple bounds:
For $m=1$ (see [here](https://mathoverflow.net/questions/257027)
(or [here](http://eprint.iacr.org/2005/318) and [here](https://arxiv.org/abs/1402.5547))) the inequalities
\begin{align\*}
\sqrt{\frac{\pi}{2}}{1\over \lVert p\rVert\_2}&\leq \mathbb{E}(M\_1)\leq \sqrt{\frac{\pi}{2}}{1 \over \lVer... | 7 | https://mathoverflow.net/users/48831 | 263749 | 118,628 |
https://mathoverflow.net/questions/263745 | 10 | Imagine a drug, a pill that you swallow, which is designed to dissolve in your
stomach at a constant rate. It must be shaped such that the surface area
remains constant when the volume is "eroded" uniformly over its surface.
Two dimensions
--------------
Define "erosion" of a distance $\delta$ from a shape as remov... | https://mathoverflow.net/users/105703 | Solids with constant surface area during "erosion" | It seems that such pill exists.
Take a ball and drill a hole through it, so you get a solid torus;
we assume it has smooth boundary $\Sigma$.
By Gauss--Bonnet formula, we gave
$$\int\limits\_\Sigma G=0,$$
where $G$ denotes Gauss curvature.
Denote by $H$ the mean curvature of $\Sigma$;
it is mostly very negative i... | 10 | https://mathoverflow.net/users/1441 | 263753 | 118,629 |
https://mathoverflow.net/questions/263752 | 3 | In studying singular spaces, it is often important to pick an appropriate stratification which encodes the singularity structure. One class of such stratifications are called "Whitney stratifications" after the work of H. Whitney (with many following contributions by Thom, Mather..) . A Whitney stratification obeys cer... | https://mathoverflow.net/users/26208 | Whitney Conditions vs Equisingularity | I am not sure what you mean by "equisingular" stratification. But I guess you would like to say that *$X$ is equisingular along $Y$* if the local rings $\mathcal{O}\_{X,x}$ have constant multiplicity for all $x \in Y$. The *equisingular stratification* would be a stratification with respect to the condition being *equi... | 3 | https://mathoverflow.net/users/37214 | 263758 | 118,631 |
https://mathoverflow.net/questions/263761 | 2 | How do we compute the sheaf cohomology of the universal subbundle and universal quotient bundle of the Grassmannian $G(k,V)$?
| https://mathoverflow.net/users/16356 | Sheaf cohomology of the universal sub and quotient bundles of the Grassmannian | See them as pushforwards of line bundles from the flag manifold by using
Borel-Weil fiberwise. Then use Borel-Weil up on the flag manifold.
Jerzy Weyman's book is the source I know of for these techniques.
| 5 | https://mathoverflow.net/users/391 | 263762 | 118,633 |
https://mathoverflow.net/questions/263750 | 8 | In his paper *Index for subfactors* [Invent. Math., vol. 72 (1983), pp. 1-26], Vaughan Jones proved his remarkable **index rigidity theorem**, i.e., the fact that the possible index values for a (type II$\_1$) subfactor are precisely those in the set
$$
\{4\cos(\pi/n)^{2}\,:\,n\geq 3\}\cup [4,+\infty].
$$
In particular... | https://mathoverflow.net/users/46541 | In what sense do Jones' original subfactors come from quantum SU(2) | You can construct a subfactor (under very mild assumptions) from an object $X$ in a rigid C\*-tensor category, by taking the limit of inclusions
$$
{\rm End}(X^{\otimes n}) \simeq \{ \iota\_X \otimes T \mid T \in {\rm End}(X^{\otimes n})\} \subset {\rm End}(X^{\otimes n+1})
$$
as $n \to \infty$. One good entry point i... | 8 | https://mathoverflow.net/users/9942 | 263768 | 118,635 |
https://mathoverflow.net/questions/263638 | 9 | The following set theoretical question is inspired by a question from recursion theory:
>
> **Question**: Is there an $L$-random real $r$ which is a minimal cover over another real $x$?
>
>
>
Where a minimal cover $r$ over $x$ means that $x\in L[r] \wedge r\not\in L[x]$ but there is not real $z$ so that $x\in ... | https://mathoverflow.net/users/14340 | Minimal cover v.s random reals | Suppose $r$ is random over $L$, $x \in 2^{\omega} \cap L[r]$ and $r \notin L[x]$. For $y \in 2^{\omega}$, let $y\_0, y\_1 \in 2^{\omega}$ denote the even and odd parts of $y$ - So $y = y\_0 \oplus y\_1$. Let $B$ denote the random algebra and $A$, the complete subalgebra of $B$ generated by $x$. Let $G\_A$ be an $A$-gen... | 6 | https://mathoverflow.net/users/2689 | 263774 | 118,637 |
https://mathoverflow.net/questions/263771 | 1 | In [a recent MO post](https://mathoverflow.net/questions/263708/is-there-a-trace-inequality-for-the-product-of-a-sequence-of-hermitian-postive-d), pallab1234 ask for trace inequalities for which counterexample were given. I wish to probe in a different direction.
Suppose $A, B$ are $n\times n$ symmetric matrices (wit... | https://mathoverflow.net/users/66131 | a follow up question on traces of matrices | I realized that you asked a different question after writing the answer, but thought it might be helpful anyway.
There are no equalities that depend only on the "string" of As and Bs.
We first check the base case, $k = 2$, with the rearrangement $ABAB$. This fails for a basic check of 2 by 2 matrices.
Now we can... | 3 | https://mathoverflow.net/users/44191 | 263785 | 118,639 |
https://mathoverflow.net/questions/263792 | 7 | Let $\Gamma$ be a discrete subgroup of $SL\_2(\mathbb{R})$ which has a cusp at $\infty.$ suppose that $\mu(\Gamma\setminus\mathbb{H})<\infty,$ consider the Eisenstein series :$$E(z,s,\Gamma)=\sum\_{\gamma\in\Gamma\_\infty\setminus\Gamma}\dfrac{y^s}{|cz+d|^{2s}}$$
what is the analytic properties of $E(z,s,\Gamma)$ ?
| https://mathoverflow.net/users/44319 | Analytic properties of Eisenstein series | The main properties of these Eisenstein series (meromorphic continuation, functional equation, poles and residues) are discussed and derived in Chapter 6 of Iwaniec: Spectral methods of automorphic forms (2nd edition, AMS, 2002). For their role in the spectral decomposition of $L^2(\Gamma\backslash\mathbb{H})$ see Chap... | 6 | https://mathoverflow.net/users/11919 | 263796 | 118,644 |
https://mathoverflow.net/questions/263769 | 8 | Let $H\subseteq\mathbb C^n$ be a smooth co-oriented real codimension one hypersurface. If $H$ is **weakly pseudo-convex**, then holomorphic maps $u:\Delta\to\mathbb C^n$ ($\Delta$ denotes the unit disk) satisfy the following **maximum principle** with respect to $H$:
>
> If $u(0)\in H$ and $u$ maps a neighborhood o... | https://mathoverflow.net/users/35353 | Most general maximum principle for non-integrable almost complex structures | You may already know this, and this is not the most general statement, probably, but it does work for all almost complex structures (integrable or not):
First, some notation: Using the almost complex structure $J$ on $X$, split the exterior derivative into graded pieces $d^{s,t}:\Omega^{p,q}(X)\to\Omega^{p+s,q+t}(X)$... | 5 | https://mathoverflow.net/users/13972 | 263802 | 118,645 |
https://mathoverflow.net/questions/263797 | 13 | Let $\operatorname{Inv}(\mathfrak{S}\_n):=\{\pi\in\mathfrak{S}\_n: \pi^2=1\}$ be the set of [*involutions*](http://mathworld.wolfram.com/PermutationInvolution.html) in the symmetric group $\mathfrak{S}\_n$. Denote $I\_n:=\#\operatorname{Inv}(\mathfrak{S}\_n)$. Let $\operatorname{tr}(\pi)$ be the number of fixed points ... | https://mathoverflow.net/users/66131 | trace and involution permutations: Part I | Let us identify an involution $\sigma$ in $\mathfrak{S}\_n$ with a set partition $\Pi\_\sigma$ of $[n] := \{1,2,...,n\}$ into nonempty blocks of size at most two in the obvious way: we have $\{a,b\} \in \Pi\_\sigma$ if and only if $\sigma(a)=b$. This is obviously a bijection between involutions and such set partitions;... | 20 | https://mathoverflow.net/users/25028 | 263803 | 118,646 |
https://mathoverflow.net/questions/263751 | 2 | In here Lemma $4$ using pigeonhole says:
For $T\_1,\dots,T\_s\in\Bbb R$ with $1\leq T\_1,\dots,T\_s<p$ and $\prod\_{i=1}^sT\_i > p^{s−1}$ and any integers $a\_1,\dots,a\_s$ there is an integer $t$ coprime to $p$ such that
$$\min\_{
k\in\Bbb Z}|ta\_i − kp| \ll T\_i,\quad\quad i = 1,\dots,s$$ holds.
**First of all is... | https://mathoverflow.net/users/nan | Essential clarifications on application of pigeonhole principle | First, we prove the Lemma 4. Considering the linked paper (that was in the previous version of this question), $p$ must be a prime number.
We apply the pigeon-hole principle in the following way:
Take $T\_i'$ to satisfy $T\_i\leq T\_i'$ and $p/T\_i' \in \mathbb{N}$. This is possible without changing $T\_i$ too mu... | 1 | https://mathoverflow.net/users/21090 | 263810 | 118,648 |
https://mathoverflow.net/questions/263399 | 5 | The standard rearrangement theorem for conditionally convergent series says that the terms in a conditionally convergent series can be rearranged so that the new sum is any desired number, or $\pm\infty$.
Let me set up some notation to investigate this a little further. My coefficients will be determined by a functio... | https://mathoverflow.net/users/3634 | The stabilizer of the conditionally convergent series | [Making comment into answer.] If there is a constant $C$ such that $|\sigma(n)−n| \leq C$ for all $n$, then $\sigma$ is sum-preserving. And this "boundedness" property is preserved by compositions and inverses. So one might guess that perhaps these are all of the sum-preserving permutations. But it is not the case. The... | 3 | https://mathoverflow.net/users/88133 | 263815 | 118,652 |
https://mathoverflow.net/questions/263654 | 10 | *I asked this on MathStackExchange and was instructed it would be better here.*
I've recently been learning about moduli spaces of instantons on $\mathbb{C}^{2}=\mathbb{R}^{4}$. From what I can gather, one can consider the framed moduli space of torsion-free sheaves on $\mathbb{P}^{2}$ of rank $N$ and second Chern cl... | https://mathoverflow.net/users/105661 | Instanton Moduli Space on ALE Spaces | For every $k$, $M$ and $N$ positive integers, one can consider the moduli space $\mathcal{M}\_M(k,N)$ of $U(N)$ instantons of instanton number (=second Chern class) $k$ on the resolution of the $A\_{M-1}$ surface singularity. The case $M=1$ corresponds to instantons on $\mathbb{C}^2$. In particular we have at least thr... | 5 | https://mathoverflow.net/users/25309 | 263816 | 118,653 |
https://mathoverflow.net/questions/263798 | 4 | Does anyone know a good description of homotopy classes of continuous functions $\Sigma\_g \longrightarrow \mathbb{R}P^2$, where $\Sigma\_g$ is the closed oriented surface of genus $g > 1$.
Thanks for the attention!
| https://mathoverflow.net/users/25511 | Homotopy classes of continuous functions $\Sigma_g \longrightarrow \mathbb{R}P^2$ | This [mathstackexchange answer](https://math.stackexchange.com/questions/36488/how-to-compute-homotopy-classes-of-maps-on-the-2-torus) gives a nice description of the set of homotopy class of maps from $T^2=\Sigma\_1$ to any space $X$, and includes a particular mention of the example $X={\mathbb R}P^2$. Answers to th... | 7 | https://mathoverflow.net/users/6668 | 263817 | 118,654 |
https://mathoverflow.net/questions/263764 | 14 | I wonder whether there are problems whose statement do not mention toric varieties (nor simple polytopes, vanishing sets of binomials, etc.), but whose proof nicely and essentially uses them?
To give an idea, two simple random examples -- not for toric varieties, but for other concepts:
* a one-line computation of ... | https://mathoverflow.net/users/43639 | Application of toric varieties for problems that do not mention them | There are lots of applications of toric varieties to singularities, e.g., the [proof](https://www.math.ubc.ca/~karu/papers/semist.pdf) of the weak factorization theorem in characteristic zero. (Indeed, the name of the linked paper is "Torification and factorization of birational maps.") The weak factorization theorem s... | 7 | https://mathoverflow.net/users/51424 | 263828 | 118,657 |
https://mathoverflow.net/questions/257268 | 7 | The [Eulerian numbers](https://en.wikipedia.org/wiki/Eulerian_number) enjoy many different presentations among which I write the two-variable recursive definition: $A(n,0)=1$ and $A(n,k)=0$ for $k<0$ so that
$$A(n,k)=(k+1)A(n-1,k)+(n-k)A(n-1,k-1).$$
However, my curiosity is regarding a certain "vanishing-variables" for... | https://mathoverflow.net/users/66131 | a new representation for Eulerian numbers? | Quiet a short algebraic proof may go as follows (I cite my own [manuscript](https://arxiv.org/abs/1512.07136)). For a polynomial $f(x\_1,\dots,x\_{n+1})$ of degree at most $n$ we define a linear operator $$\Phi[f]=\text{Sym}\, \frac{f(x\_1,\dots,x\_{n+1})}{(x\_1-x\_2)(x\_2-x\_3)\dots (x\_n-x\_{n+1})},$$
where $\text{Sy... | 5 | https://mathoverflow.net/users/4312 | 263844 | 118,666 |
https://mathoverflow.net/questions/263596 | 8 | Given a vector bundle $E \to M$ with connection $\nabla$, we get a twisted de Rham sequence using the exterior covariant derivative:
$$\mathcal{E} \xrightarrow{d^\nabla=\nabla} \Omega^1\_M \otimes\_{\Omega^0\_M} \mathcal{E} \xrightarrow{d^\nabla} \Omega^2\_M \otimes\_{\Omega^0\_M} \mathcal{E} \xrightarrow{d^\nabla} \cd... | https://mathoverflow.net/users/56938 | Relative version of de Rham cohomology with local coefficients | I had a look in Ramanan's Global Calculus, and sure enough he gives the necessary details:
Let $\mathbb{R}\_M$ denote the constant sheaf corresponding to $\mathbb{R}$ on $M$.
Note that $\Omega^k\_M\otimes\_{\mathbb{R}\_M} \mathcal{L} = \Omega^k\_M\otimes\_{\Omega^0\_M} \mathcal{E},$ and hence the connection is reco... | 2 | https://mathoverflow.net/users/56938 | 263847 | 118,667 |
https://mathoverflow.net/questions/263848 | 2 | Actually, I will be asking two, but related, questions.
>
> **Question 1.** Is there some theory about recurrent relations with *several* indices. For example, If we have a relation on $A\_{i;j}$: $$A\_{i+\alpha; j+\beta} = F(A\_{i+\alpha-1;j+\beta -1}, \ldots, A\_{i;j}).$$
>
>
>
And related
>
> **Question... | https://mathoverflow.net/users/104044 | Recurrent relation with several indices ( How many $m$-dim cubes in $n$-dim cube ) | This was an answer before the typo was edited by the OP on $Q\_m^{n+1}=2Q\_m^n+Q\_{m-1}^{n-1}$.
Let $F(x,y)=\sum\_{n,m\geq0}Q\_n^mx^my^n$ be a generating function. Based on the recurrence relation alone, you should be getting
$$F(x,y)=\frac{P(x,y)}{1-2y-xy^2};$$
for some polynomial $P(x,y)$ which depends on the initi... | 4 | https://mathoverflow.net/users/66131 | 263853 | 118,669 |
https://mathoverflow.net/questions/263855 | 11 |
>
> Is it true that for every finitelty generated subgroup $H$ of infinite index in a free
> group $F$ on the two letters $\{x,y\}$, there exists a finite index
> subgroup $K$ of $H$, such that the normal subgroup $N$ of $F$
> generated by $K$ is of infinite index in $F$ ?
>
>
>
The normal subgroup $N$ of $F$... | https://mathoverflow.net/users/38889 | Normal closures of finitely generated subgroups of a free group | This is true, though the proof uses some heavy machinery! Theorem A.1 of Agol--Groves--Manning's appendix to Agol's proof of the [Virtual Haken conjecture](https://www.math.uni-bielefeld.de/documenta/vol-18/33.pdf) states:
>
> Let $G$ be a hyperbolic group, let $H\leq G$ be a quasi-convex virtually special subgroup... | 12 | https://mathoverflow.net/users/1463 | 263862 | 118,671 |
https://mathoverflow.net/questions/263827 | 3 | Consider two densely defined, strictly positive, self-adjoint operators $A$ and $B$ with the following property
$$\|A^k x\| \simeq \|B^k x\|, \quad\forall x \in D(A^k)= D(B^k),$$
for $k=1,2,\cdots, M$, where $M$ can be $\infty$.
Do we have for fixed $t>0$,
$$\|T(t) x \| \simeq \|S(t) x\|, $$
where $T(t),S(t)$ are the... | https://mathoverflow.net/users/41105 | A sufficient condition for two semigroups to be norm equivalent? | The condition $\|A^k x\| \simeq \|B^k x\|$ can be rewritten, by squaring both sides, as $\langle x, A^{2k} x \rangle \simeq \langle x, B^{2k} x \rangle$, i.e. $A^{2k} \leqslant C B^{2k}$ and $B^{2k} \leqslant C A^{2k}$. Recall now that the function $s \mapsto s^{\alpha}$ is operator monotone for $\alpha \in (0,1]$. It ... | 1 | https://mathoverflow.net/users/24953 | 263865 | 118,672 |
https://mathoverflow.net/questions/263858 | 1 | Let $E$ be an elliptic curve over $\mathbb{Q}$. Let us fix a rational prime $\ell$ (odd, if you want) and an $\ell$-torsion point $P\in E[\ell]$. For a prime $\mathfrak{p}$ of $K=\mathbb{Q}(E[\ell])$, let us say that $P$ has *nonsingular reduction* at $\mathfrak{p}$, if $\tilde{P}\neq\tilde{O}$ in $\tilde{E}\_\mathfrak... | https://mathoverflow.net/users/44005 | Torsion points with exactly one singular prime (on elliptic curves) | The status of your "implicit assumption" isn't quite clear -- do you want all primes where $E$ has bad reduction to be in $S\_P$, or only a subset of them?
If you take E to be Cremona's 27a1, $y^2 + y = x^3 - 7$, then $E$ has a rational 3-torsion point $(3, -5)$, and for all rational primes $p \ne 3$ of E, the mod $p... | 0 | https://mathoverflow.net/users/2481 | 263867 | 118,674 |
https://mathoverflow.net/questions/263656 | 7 | In my graduate discrete math course I talked about real-rootedness of some combinatorial polynomials, and as a homework asked for a proof of the real-rootedness of the matching polynomial (matching generating function) of the complete graph,
$$
p\_n(x) = \sum\_{k \geq 0} m\_{n,k} x^k
$$
where $m\_{n,k}$ is the number... | https://mathoverflow.net/users/21690 | Matching polynomial of complete graph | I am not sure about how to use this recurrence to show that the roots are real and distinct, but I think if we assume we know this via other methods, I can show that the roots of $p\_n$ and $p\_{n-2}$ are interlaced.
Indeed, let $r\_1,\dots,r\_k$ be the roots of $p\_{n-2}$, and consider the intervals
$(\infty,r\_1],(... | 1 | https://mathoverflow.net/users/105771 | 263870 | 118,675 |
https://mathoverflow.net/questions/263523 | 3 | Let me be more specific: If $A=BC$, where $A$ and $C$ are given [**Laplacian matrices**](https://en.wikipedia.org/wiki/Laplacian_matrix), how to calculate $B$? The graph corresponding to $A$ is a **directed ring**, which is strongly connected and $1\_n$ and $1\_n^T$ are right and left eigenvectors respectively. The gra... | https://mathoverflow.net/users/105594 | If $A=BC$, where $A$ and $C$ are given Laplacian matrices, how to calculate $B$? | If the system of linear equations is consistent, its solution is $B=AC^+$, where $C^+$ is the [pseudoinverse](https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse) of $C$. Depending on the strictness of your definition of "closed formula", this may already fit the requirements. Otherwise, there are more ex... | 2 | https://mathoverflow.net/users/1898 | 263872 | 118,676 |
https://mathoverflow.net/questions/263845 | 6 | In a discussion on a youtube video on the hydra game I jokingly mentioned how everyone was assuming that $\varepsilon\_0$ was well-ordered. This lead to a bit of disagreement (in a nice way!) about the existence of the well-ordering of $\varepsilon\_0$, assuming that $\varepsilon\_0$ exists in whatever weak theory one ... | https://mathoverflow.net/users/4177 | Existence of well-ordering of epsilon_0 in weak theories | The *existence* of $\epsilon\_0$ and its order is not a problem, its *well-foundedness* is.
Cantor normal forms (recursively expanded) of ordinals below $\epsilon\_0$ can be written as strings over a finite alphabet, and in that form can be manipulated in weak fragments of arithmetic, say in $I\Delta\_0+\mathrm{EXP}$... | 10 | https://mathoverflow.net/users/12705 | 263888 | 118,678 |
https://mathoverflow.net/questions/263686 | 1 | I am, struggling to see whether the first moment when two processes are different (in terms of their finite dimensional distributions) can be defined in terms of their filtrations and would appreciate any suggestions / clarifications.
Here is an example of what I am trying to do: suppose that $X$ is a standard Browni... | https://mathoverflow.net/users/42754 | Whether the first moment two stochastic processes differ can be formulated in terms of filtrations? | I think I have found an answer to this question and will post it here for the future reference.
Suppose that we start with two filtrations $\mathbb{H}=(\mathcal{H}\_t)\_{t\ge0}$ and $\mathbb{F}=(\mathcal{F}\_t)\_{t\ge0}$. Consider $\tau\_1$ and $\tau\_2$ - stopping times with respect to both filtrations such that
$$
... | 0 | https://mathoverflow.net/users/42754 | 263895 | 118,679 |
https://mathoverflow.net/questions/263879 | 7 | We know that a smooth Riemannian manifold with nonnegative curvature is an Alexandrov space (with induced metric) of nonnegative curvature.
What about the converse? That is, given a smooth metric ***d*** on a smooth manifold ***M*** such that ***M*** is an Alexandrov space with nonnegative curvature, can we find a sm... | https://mathoverflow.net/users/90512 | Can we realize the smooth metric of an Alexandrov space with nonnegative curvature by a Riemannian structure? | Yes, smooth distance functions plus Alexandrov means Riemannian,
but you should make all the definitions precise.
After Otsu and Shioya, there was a paper of Perelman ["DC structure on Alexandrov space with curvature bounded below"](http://www.math.psu.edu/petrunin/papers/alexandrov/Cstructure.pdf). The key "new" ing... | 7 | https://mathoverflow.net/users/1441 | 263911 | 118,683 |
https://mathoverflow.net/questions/263137 | 8 | This is a special case of [this question](https://mathoverflow.net/questions/262829/linear-combinations-of-low-degree-polynomials-with-agreement-guarantee).
Let $\mathbb{F}$ be a finite field and $\mathbb{F}\_{\leq d}[x,y]$ the set of bivariate polynomials over $\mathbb{F}$ of degree at most $d\ll|\mathbb{F}|$. Do th... | https://mathoverflow.net/users/90531 | Question about polynomials over finite fields | The condition that $A$ and $B$ are nonempty disjoint can be replaced with the condition that they are distinct, as we may just replace $A$ and $B$ with $A \cap (A - B)$ and $B \cap (A- B)$ respectively.
Let $q = |\mathbb F|$.
Let $A$ and $B$ be two random linear subspaces of the space of polynomials of degree $\leq... | 2 | https://mathoverflow.net/users/18060 | 263914 | 118,685 |
https://mathoverflow.net/questions/263882 | 3 | Is there a way\* to prove that a Bonnet Surface $S$ in isothermal coordinates in $R^3$ with mean curvature ($H$) and Gaussian curvature ($K$) both non-constant and where $(H^2-K)=c$, (with c positive constant), must have negative Gaussian curvature?
Thanks in advance for any help!
Alex
* \*so do not use formulas $A... | https://mathoverflow.net/users/90594 | A kind of surfaces | It is not possible to prove that any surface $S\subset\mathbb{R}^3$ that has both $H$ and $K$ nonconstant but $H^2-K=c^2$ for some constant $c>0$ must have $K$ be negative. In fact, one cannot conclude anything about the sign of $K$ from only these hypotheses, beyond the fact that $K\ge -c^2$.
In fact, it's not diff... | 5 | https://mathoverflow.net/users/13972 | 263924 | 118,691 |
https://mathoverflow.net/questions/263901 | 6 | ... and really *without even the possibility of having objects*, so it's not a matter of just finding the "correct" flavour of Fukaya category to use.
**Question:** Does there exist interesting symplectic manifolds $(M,\omega)$ without:
* Any unobstructed Lagrangians (and therefore, with a $Ob(Fuk(M,\omega)) = \var... | https://mathoverflow.net/users/105615 | Does there exists a Fukaya category with no objects | Let's assume that $M$ is an $n$-dimensional exact symplectic manifold and we are only interested in Fukaya categories of closed exact Lagrangian submanifolds. Then for any subcritical Weinstein manifold, $\mathcal{F}(M)$ is trivial. This follows from the fact that there is a well-defined open-closed string map
$\math... | 3 | https://mathoverflow.net/users/43423 | 263926 | 118,693 |
https://mathoverflow.net/questions/263912 | 6 | It is conjectured that for a discrete, finitely presented group $G$ such that $BG$ satisfies Poincaré duality, there actually exists a closed manifold $M$ which is homotopy equivalent to $BG$.
This is somehow pointing in the opposite direction as Borel's conjecture, which implies that the homeomorphism type of such ... | https://mathoverflow.net/users/14233 | Every PD group is $\pi_1$ of an aspherical manifold | The reference for the first appearance of the conjecture (still without the condition that the PD group has to be a priori finitely presented) seems to be <http://www.worldcat.org/title/homological-group-theory-proceedings-of-a-symposium-held-at-durham-in-september-1977-on-homological-and-combinatorial-techniques-in-gr... | 6 | https://mathoverflow.net/users/39082 | 263929 | 118,694 |
https://mathoverflow.net/questions/263921 | 3 | This is a follow up on [my earlier MO question](https://mathoverflow.net/questions/263797/trace-and-involution-permutations).
Let $\operatorname{Inv}(\mathfrak{S}\_n):=\{\pi\in\mathfrak{S}\_n: \pi^2=1\}$ be the set of [*involutions*](http://mathworld.wolfram.com/PermutationInvolution.html) in the symmetric group $\m... | https://mathoverflow.net/users/66131 | trace and involution permutations: Part II | Let us use the convention that $\mathfrak{S}\_0$ has a unique element $\sigma$ which is the identity (hence an involution) and has $\mathrm{tr}(\sigma)=0$. We also use $[n] := \{1,2,\ldots,n\}$.
The claimed identity is that for any $k\geq 0$,
$$ \sum\_{n=0}^{\infty} \frac{z^n}{n!} \sum\_{\sigma \in \mathrm{Inv}(\math... | 5 | https://mathoverflow.net/users/25028 | 263930 | 118,695 |
https://mathoverflow.net/questions/263903 | 6 | I apologise in advance if my question is too basic.
Some notation:
1. $(X,\cal{X})$ denotes a measurable metric space
where $X$ is a metric space and
$\cal{X}$ is the associated Borel sigma algebra.
2. $B(X)$ is the space of all bounded continuous
functions defined on $X$.
Let $\{\mu\_n\}$ and $\{\nu\_n\}$ be se... | https://mathoverflow.net/users/98969 | Convergence of Radon Nikodym derivatives | I'm going to assume that your space is locally compact (as well as $\sigma$-compact), so that $X$ is the union of a sequence of compact sets where each lies in the interior of the next.
In this case, the answer to your question is **yes**.
Fix any $g \in B(X)$ with $g \geq 0$. We need $\int\_X g \, d\mu\_n \to \int... | 3 | https://mathoverflow.net/users/15570 | 263937 | 118,699 |
https://mathoverflow.net/questions/263904 | 3 | Let $G$ be a weakly amenable group, in the sense that it has a net of finitely supported functions $\varphi:G\to \mathbb{C}$ which converge point wise to 1 and their cb norm is bounded uniformly by some constant $C$.
It is known that this is equivalent to completely bounded approximation property (CBAP) for the redu... | https://mathoverflow.net/users/104535 | CBAP for the full group $C^*$-algebra | The answer to the second question is also negative. Let $(u\_g)\_{g\in G}$ be the generating unitaries of $C^{\ast}(G)$. Suppose that $(\varphi\_i)\_{i\in I}$ is a net a functions whose associated multipliers $m\_{\varphi\_{i}}: C^{\ast}\_{r}(G) \to C^{\ast}\_r(G)$ give CBAP of $C^{\ast}\_{r}(G)$. Suppose now that the ... | 4 | https://mathoverflow.net/users/24953 | 263943 | 118,701 |
https://mathoverflow.net/questions/263927 | 9 | I'm looking for a reference explaining under what conditions the internal Yoneda lemma holds; in particular, I am wondering if it is known what properties of the ($2$-)category of categories are responsible for the following.
Take a (small) category $\mathcal C$, i.e. a category internal to the category of sets. Prom... | https://mathoverflow.net/users/75650 | Yoneda Lemma for internal presheaves | The Yoneda lemma holds in any finitely complete $2$-category $\mathscr{K}$, such as the $2$-category $\text{Cat}(\mathscr{C})$ of internal categories in a finitely complete category $\mathscr{C}$.
The statement is that for any object $B$ of $\mathscr{K}$, any morphism $b \colon 1 \to B$, and any discrete fibration [... | 11 | https://mathoverflow.net/users/57405 | 263949 | 118,703 |
https://mathoverflow.net/questions/263836 | 3 | Let $f : C\rightarrow S$ be a proper smooth morphism of relative dimension 1 over a connected scheme $S$.
Let $G$ be a finite group of order invertible on $S$ which acts faithfully and $\mathcal{O}\_S$-linearly on $C/S$. Let $\omega\_{C/S} = \Omega^1\_{C/S}$ denote the dualizing sheaf of $C/S$, then the $G$-action on... | https://mathoverflow.net/users/88840 | For an action of a finite group $G$ on a curve $C/S$, is the induced action on $H^0(C,\omega_{C/X}^{\otimes m})$ "independent of the fiber"? | Based on the comments, it is clear that this question is as much about modular representation theory (in the tame case) as about curves and pluricanonical sections.
Let $G$ be a finite group of order $\ell$. Let $\mathbb{Z}[1/\ell]\to R$ be a ring homomorphism. Let $M$ be an $R$-module. Let $G\to \text{Aut}\_{R-\tex... | 3 | https://mathoverflow.net/users/13265 | 263956 | 118,705 |
https://mathoverflow.net/questions/263953 | 0 | Let $f$ be the distribution of a normal variable $\mathcal{N}(0,1)$, ie
$$ f(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$
I have to solve the equation:
$$x+y = f(x) - f(y)$$
I was working with mathematica and it gives me that the solution is $x=-y$. I would very happy if it's true, but I have no idea to prove that it is ... | https://mathoverflow.net/users/105812 | A simple equation with a normal distribution | Since $f$ is even, your question is equivalent to solving $x-y=f(x)-f(y)$, for $x,y\in\mathbb{R}$. Assume $x\neq y$. Then by mean value theorem, there exists some $t$ in the interval $]x,y[$, such that $$\frac{f(x)-f(y)}{x-y}=\frac{-te^{\frac{-t^2}{2}}}{\sqrt{2\pi}}.$$
We are hence reduced to prove that the function $$... | 2 | https://mathoverflow.net/users/105480 | 263960 | 118,707 |
https://mathoverflow.net/questions/263980 | 6 | If $H$, $H\_0$ are two separable Hilbert spaces and $H$ is continuously and densly embedded in $H\_0$, it is possible to construct a sequence of linear operators
$$ P\_n : H\_0 \to H $$
such that for all $x \in H\_0$ one has convergence $P\_n x \to x$ in the $H\_0$-norm.
The motivation is to generalize the idea of sm... | https://mathoverflow.net/users/13970 | Approximating dense subspaces of Fréchet spaces | Here is a very simple method for separable Hilbert spaces (which easily generalizes to Frechet spaces with Schauder bases): Take an orthonormal basis $(e\_k)\_k$ in $H\_0$ and choose $f\_{n,k}\in H\_1$ such that $\|e\_k-f\_{n,k}\|\_0 \le 1/(n^2+k^2)$. Then define $P\_n(x)=\sum\_{k=1}^n \langle e\_k,x\rangle\_0 f\_{n,k}... | 6 | https://mathoverflow.net/users/21051 | 263985 | 118,714 |
https://mathoverflow.net/questions/263957 | 1 | Recall that $(a;q)\_0:=1,\,(a;q)\_n=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})$ and
$(a;q)\_{\infty}=(1-a)(1-aq)(1-aq^2)\cdots$. Let's introduce the following (generalized) concept.
A *colored overpartition* (COP) is a partition where the last occurrence of each distinct number may receive any one of $c$ colors. The numbe... | https://mathoverflow.net/users/66131 | partition theory: meet the COP | Rather than coloring the last occurrence of each distinct number, we
can equivalently for each part $i$ color all the $i$'s with the same
color. Thus
$$ \sum\_{n\geq 0}\bar{p}\_c(n)q^n = \prod\_{i\geq
1}(1+c(q^i+q^{2i}+q^{3i}+\cdots)) $$
$$ = \prod\_{i\geq 1}\frac{1+(c-1)q^i}{1-q^i}. $$
| 5 | https://mathoverflow.net/users/2807 | 263988 | 118,715 |
https://mathoverflow.net/questions/263646 | 6 | I'm working on some research problems relating to random matrix products, and this is taking me into areas of mathematics I've not previously studied: Lie groups, representation theory, and real algebraic groups. In this area I encounter a lot of questions which I don't have the tools for, but which are probably easy f... | https://mathoverflow.net/users/1840 | Simultaneous triangularisation of an exterior power of a set of matrices | Here's a simple counterexample to Question 1: Let $d=4$ and $k=2$. Let $X\subset\mathrm{GL}\_4(\mathbb{R})$ consist of a single element $J$ where $J^2=-I$. Then $J$ is not conjugate to any upper triangular matrix (over $\mathbb{R}$) since it does not have real eigenvalues. Meanwhile, since
$$
\bigl(\Lambda^2(J)\bigr)^... | 8 | https://mathoverflow.net/users/13972 | 263991 | 118,716 |
https://mathoverflow.net/questions/263863 | 14 | This is [cross-posted](https://math.stackexchange.com/questions/2173639/does-the-doob-dynkin-lemma-hold-for-any-separable-measure-space) from math.se, where I got no responses.
Let's say that a measurable space $(Z, \mathcal Z)$ has the "[Doob-Dynkin](https://en.wikipedia.org/wiki/Doob%E2%80%93Dynkin_lemma) property"... | https://mathoverflow.net/users/41669 | Does the Doob-Dynkin lemma hold for any measurable space that separates points? | A characterization of spaces with the Doob-Dynkin property was given by N. Pintacuda in his paper *Sul lemma di misurabilità di Doob* (1989). Unfortunately, the paper is in Italian and it doesn't show up on Google.
L. Pratelli gave another proof of Pintacuda's result in his paper *Sur le lemme de mesurabilité de Doo... | 10 | https://mathoverflow.net/users/90906 | 263994 | 118,718 |
https://mathoverflow.net/questions/263996 | 3 | I am trying to evaluate an indefinite integral of the form
$\int \frac{dz}{A u\_1^2 + Bu\_2^2 + Cu\_1u\_2}$
where $u\_1$ and $u\_2$ are two independent solutions to the ODE
$u'' + F(z)u = 0$
This integral has arisen because the quantity $\sqrt{A u\_1^2 + Bu\_2^2 + Cu\_1u\_2}$ is a solution to the Ermakov-Pinney... | https://mathoverflow.net/users/105826 | An indefinite integral containing functions that are solutions to a 2nd order linear ODE | Assuming $4AB-C^2 \ne 0$, let $$ J(z) =\frac{2}{W \sqrt{4AB-C^2}}\;\arctan \left(\frac{2B}{\sqrt{4AB-C^2}} \frac{u\_2(z)}{u\_1(z)} + \frac{C}{\sqrt{4AB-C^2}}\right)$$
where $W$ is the (constant) Wronskian of $u\_1$ and $u\_2$.
Then I get
$$ \dfrac{dJ}{dz} = \frac{1}{A u\_1^2 + B u\_2^2 + C u\_1 u\_2}$$
| 4 | https://mathoverflow.net/users/13650 | 264007 | 118,722 |
https://mathoverflow.net/questions/262774 | 7 | Consider a normal form game with $n$ players (and finitely many options per player) defined by finite option sets $A\_1,\ldots,A\_n$ and payoff matrices $u\_1,\ldots,u\_n: \prod\_{j=1}^n A\_j \to \mathbb{R}$. Let $N$ be the set of Nash equilibria, which is a subset of the set $S := \prod\_{j=1}^n S\_j$ of mixed strateg... | https://mathoverflow.net/users/17064 | Topology of the set of Nash equilibria of a normal form game | Nash-equilibria satisfy a universality theorem. This is due to Ruchira Datta's paper "Universality Of Nash Equilibria", <https://arxiv.org/pdf/math/0301001.pdf>. From her abstract:
>
> Every real algebraic variety is isomorphic to the set of totally mixed Nash equilibria of some three-person game, and also to the ... | 4 | https://mathoverflow.net/users/21684 | 264012 | 118,724 |
https://mathoverflow.net/questions/264013 | 22 | What is the origin of the term "exterior" in "exterior calculus"? How does this term relate to "interior products" and "inner products", if it does at all?
| https://mathoverflow.net/users/2082 | Etymology of "exterior" in "exterior calculus" | I think it's well known to have been introduced by **Grassmann**. He explains the word choice in *Die lineale Ausdehnungslehre* [(1844, pp. x-xi)](https://archive.org/stream/dielinealeausde00grasgoog#page/n15):
>
> I have shown how one can understand as product of two segments the parallelogram (...); this product ... | 35 | https://mathoverflow.net/users/19276 | 264026 | 118,726 |
https://mathoverflow.net/questions/263966 | 26 | This question has its origin in Morse theory ([see this paper](http://www3.nd.edu/~lnicolae/Morse-count.pdf)) but it can be given an entirely elementary and amusing formulation.
The game of plates and olives starts with an empty table and ends with an empty table and it consists of a succession of the following eleme... | https://mathoverflow.net/users/20302 | A game of plates and olives | The answer to Question 1 is that yes, the belief is justified. By Stirling's approximation, $(2n+1)!!=n^n(2/e)^{n(1+o(1))}$ or $\log (2n+1)!! = n \log n +n(2/e)(1+o(1))$, so for an affirmative answer to Question 1 it is enough to get an upper bound on $G\_n$ of the form $G\_n \leq n^nC^n$ for some constant $C$.
Here'... | 10 | https://mathoverflow.net/users/21690 | 264034 | 118,729 |
https://mathoverflow.net/questions/263885 | 36 | Can you tell me an algebraic integer, with all archimedean absolute values less than 2, which is not an eigenvalue of $\pi\_1 + \pi\_2$ for any two permutation matrices $\pi\_1,\pi\_2$?
Is it conceivable that *every* algebraic integer satisfying the archimedean condition can be so expressed?
| https://mathoverflow.net/users/431 | Tell me an algebraic integer that isn't an eigenvalue of the sum of two permutations | Yes: if $\alpha$ is an algebraic integer which obeys $|\alpha| < 2$ for all archimedean norms $|\ |$ then $\alpha$ is an eigenvalue of a sum of two permutations matrices.
I remark that this is not true when $|\alpha|=2$, for example, $(1+\sqrt{-15})/2$ is an algebraic integer with absolute value $2$ which is not twic... | 44 | https://mathoverflow.net/users/297 | 264035 | 118,730 |
https://mathoverflow.net/questions/264016 | 7 | The class of walk-regular graphs contains the vertex-transitive graphs and the distance-regular graphs. However, there are walk-regular graphs that are neither vertex-transitive nor distance-regular. In particular, I believe that I found a first example of a cubic walk-regular graph that is neither vertex-transitive no... | https://mathoverflow.net/users/75248 | Are there only finitely many distinct cubic walk-regular graphs that are neither vertex-transitive nor distance-regular? | A graph is called semisymmetric if it is regular, edge-transitive but not vertex-transitive.
Semisymmetric graphs are walk-regular hence they provide example of graphs that are regular and walk-regular but not vertex-transitive.
This answers your question, as it is known that there are infinitely many semisymmetr... | 8 | https://mathoverflow.net/users/22377 | 264044 | 118,731 |
https://mathoverflow.net/questions/264003 | 4 | Let $md\_r^k(n)$ be the minimum diameter over all $r$-regular, $k$-connected graphs on at least $n$ vertices. (Let us assume $r, k \geq 2$).
>
> **Problem:** Find lower and upper asymptotic bounds on $md\_r^k(n)$.
>
>
>
There are fewer than $r^{(d+1)}$ vertices within distance $d$ of any given vertex in an $r$... | https://mathoverflow.net/users/99278 | What is the minimum diameter of $r$-regular, $k$-connected graphs? | When $r\ge 3$ it's not too hard to prove that a random $r$-regular graph on $n$ vertices has diameter on the order $\log n$ and is $r$-connected with positive probability. This is in Bollobas' Random Graphs book: there is a section on random regular graphs. If you want specified smaller connectivity, add a (fixed size)... | 1 | https://mathoverflow.net/users/36212 | 264045 | 118,732 |
https://mathoverflow.net/questions/264042 | 5 | Let $(G, +)$ be a commutative group. The endomorphism set $\text{End}(G)$ of all group endomorphisms $f:G\to G$ is a ring, where $+$ is taken pointwise and the multiplication is the composition of endomorphisms.
Suppose $f\in \text{End}(G)$ is not bijective (i.e. not a unit in the ring) and not a zero-divisor. Can it... | https://mathoverflow.net/users/8628 | Irreducible elements in endomorphism rings | If $R$ is a commutative unital ring, then $R \cong\_{\sf Ring} {\rm End}\_{{\sf Mod}\_R}(R\_R)$, with the elements of $R$ acting on $R$ by left multiplication. Now, let $R$ be any non-atomic integral domain and note that the multiplicative monoid of ${\rm End}\_{{\sf Mod}\_R}(R\_R)$ is a divisor-closed submonoid of the... | 7 | https://mathoverflow.net/users/16537 | 264048 | 118,734 |
https://mathoverflow.net/questions/264049 | 0 | Let $f\in \mathbb{Z}[x]$ and suppose $f(x) = a\_nx^n + a\_{n-1}x^{n-1} + \ldots + a\_0$ where $a\_n > 0$ and $a\_0 \in \{-1, +1\}$. Is there for every $n\in \mathbb{N}$ an integer $m> n$ such that $f(m)$ is (positive and) composite?
| https://mathoverflow.net/users/8628 | Does every integer polynomial evaluate to infinitely many composite numbers? | Yes.
Choose $n\_0$ large, so the polynomial is always positive, and let $k = f(n\_0)$. Then choose $n = n\_0 + ik$ for $i$ large; we have that $f(n)$ is congruent to $f(n\_0)$ mod $k$, so as long as $i$ is large enough, $f(n)$ is composite.
| 4 | https://mathoverflow.net/users/44191 | 264050 | 118,735 |
https://mathoverflow.net/questions/264047 | 4 | The Milnor-Wolf theorem states that any *finitely generated* solvable group has either polynomial or exponential growth.
>
> Is there an analogous result for locally compact compactly generated groups?
> (or rather, for some smaller class of group? connected ones? connected Lie?)
>
>
>
| https://mathoverflow.net/users/74799 | Milnor-Wolf theorem for topological groups | I found the following paper:
Yves Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques, Bulletin de la Société Mathématique de France (1973) Volume: 101, page 333-379
The analogous result (see Corollaire III.3) is proved for all compactly generated soluble locally compact groups as well as some o... | 4 | https://mathoverflow.net/users/4053 | 264057 | 118,736 |
https://mathoverflow.net/questions/263989 | 6 | Let $G\_{1}$ and $G\_{2}$ be two groups. Suppose that we have a morphism $\mathbb{Z}[G\_{1}]\rightarrow \mathbb{Z}[G\_{2}] $ of bialgebras is it true that this morphism comes from a morphism of groups $G\_{1}\rightarrow G\_{2}$ ?
In case when the answer is "no", is it true that if $\mathbb{Z}[G\_{1}]\rightarrow \mathbb... | https://mathoverflow.net/users/82229 | groupring morphisms and bialgebra | I will show that any bialgebra homomorphism $\mathbb QM\_1\to \mathbb QM\_2$ of monoid algebras is induced by a monoid homomorphism $M\_1\to M\_2$. This will imply what the OP wants.
An element $g$ of a bialgebra is called group-like if $\Delta(g)=g\otimes g$ and $\eta(g)=1$ where $\eta$ is the counit. It is well kno... | 5 | https://mathoverflow.net/users/15934 | 264070 | 118,738 |
https://mathoverflow.net/questions/264056 | 1 | This question comes from physics...
We have the following functional (or function dose not matter):
$$S\left[x\right]:=\intop\_{-\frac{t\_{0}}{2}}^{+\frac{t\_{0}}{2}}dt\,\mathcal{L}\left[x\right]\::\:{\displaystyle \mathcal{L}}:={\displaystyle \frac{m}{2}\left(\frac{dx}{dt}\right)^{2}-V\left[x\right]}$$
Religiously... | https://mathoverflow.net/users/36676 | Analytic continuation of definite integral? | This image should explain the Wick rotation, I hope:
The segment of the contour along the real $t$ axis may need to be deformed so that it avoids any poles, but when it encloses no poles the contour integral is zero. The contributions from the two quarter circles c... | 3 | https://mathoverflow.net/users/11260 | 264076 | 118,740 |
https://mathoverflow.net/questions/263982 | 15 | For a directed acyclic graph (DAG) $G$, denote by $G^\star$ the undirected graph obtained from $G$ by ignoring direction of its arcs. Let $e(G)=e(G^\star)$ be the number of arcs in $G$ (or edges in $G^\star$).
An arc $(u,v)$ in $G$ is called a *shortcut* if there exists a directed path from $u$ to $v$ in $G$ differen... | https://mathoverflow.net/users/7076 | Maximum matching in a graph with no "shortcuts" | Yes, your conjecture is true, even without the assumption that $G$ does not contain shortcuts. The following proof is due to Sam Fiorini.
*Proof.* Let $P \subseteq \mathbb{R}^{E(G^\star)}$ be the matching polytope of $G^\star$. That is, $P$ is the convex hull of the set of characteristic vectors of matchings of $G^\... | 13 | https://mathoverflow.net/users/2233 | 264099 | 118,746 |
https://mathoverflow.net/questions/264103 | 2 | Let $f$ be a real-valued function. Suppose I want to find a local maximum of $f$, but I decide to work with an ''approximation'' to $f$ --let us call it $g$. What is a suitable notion of ''approximation'' that gives conditions so that the critical points of $g$ approximate those of $f$, and such that the critical point... | https://mathoverflow.net/users/11674 | Question about optimizing a given function by optimizing an approximation | In general, small perturbations of the objective may change the set of local maxima drastically. Just think of a flat local maximum and adding a small wiggling (so also uniform approximation does not really help).
However, there is a notion of convergence of functions, that is build in a way to ensure convergence of ... | 1 | https://mathoverflow.net/users/9652 | 264105 | 118,748 |
https://mathoverflow.net/questions/264094 | 5 | In theorem 1.2 of Brian Conrad's handout [Operations with Pseudo-Riemannian metrics](http://math.stanford.edu/%7Econrad/diffgeomPage/handouts/metricops.pdf), the author writes
>
> **Theorem 1.2.** Every $C^p$ vector bundle $E\to M$ over a $C^p$ manifold with corners $0\leq p\leq \infty$ admits a Riemannian metric. ... | https://mathoverflow.net/users/69037 | Morrey & Grauert - real analytic vector bundles admits analytic Riemannian metric | I'm not sure this should be an answer, but here is a proof that might be more conceptual resting on one non-trivial fact: if $E\rightarrow M$ is a real analytic fiber bundle over a real analytic manifold $M$ which admits a continuous section, then $E\rightarrow M$ admits a real analytic section.
If you take this one... | 3 | https://mathoverflow.net/users/49247 | 264109 | 118,749 |
https://mathoverflow.net/questions/259046 | -1 | For sets and functions, I think the following data are equivalent:
1. A function $g:A\times B\to B$ such that $(\pi\_1,g):A\times B\to A\times B$ is a bijection;
2. a function $A\to \mathrm{Aut}B$.
*Proof.* The first condition is equivalent to $(\pi\_1,g)$ having singleton fibers. The fiber of $a,b$ is the intersec... | https://mathoverflow.net/users/69037 | General description of transition arrows of covering morphisms in family fibrations | I believe this is true. Note that $\mathrm{Aut}(A)$ is a triple equalizer of three maps $\mathrm{End}(A) \times \mathrm{End}(A) \to \mathrm{End}(A)$, the composites in both orders and the constant map at $\mathrm{id}\_A$. Since $H$ is a right adjoint, it preserves limits, and in particular this triple equalizer. Moreov... | 1 | https://mathoverflow.net/users/49 | 264111 | 118,751 |
https://mathoverflow.net/questions/262893 | 0 | [This post](https://mathoverflow.net/questions/151033/asymptotics-of-the-maximum-of-binomial-random-variables) derived the tail bound for the maximum of independent and identically distributed binomial r.v.'s based on normal approximation. Is there a similar result in the literature for finding the bounding constant (i... | https://mathoverflow.net/users/65953 | Tail bound for maximum of independent (but not identical) binomial random variables | You can derive some conditions applying the Bernstein's trick. For any positive $t$, we have
$$
P\left( \max\_{i\le m}X\_i > c\_{n,m}\right) = P\left( \exp\{t\max\_{i\le m}X\_i\} > e^{t c\_{n,m}}\right) \le e^{-t c\_{n,m}}E\left[ \max\_{i\le m} e^{tX\_i}\right]
$$
Since $X\_i$ follows a binomial distribution
$$
E\lef... | 1 | https://mathoverflow.net/users/27234 | 264124 | 118,755 |
https://mathoverflow.net/questions/264120 | 11 | I faced a hard question in kernel methods theory, which I can't answer for about one week. Initially it was formulated in terms of positive valued functions, but it could be reformulated easier:
Let $\{x\_1, \dotsc, x\_n\}$ and $\{y\_1, \dotsc, y\_n\}$ be two sets of real positive numbers. Prove that matrix $A$ is po... | https://mathoverflow.net/users/nan | Prove that matrix is positive definite | **Update:** I originally claimed to prove that $A$ is strictly positive definite, but there was a bug in the strictness part. I have revised the proof to show that $A$ is positive semidefinite. For an example to see that $A$ need not be strictly positive definite let $x\_i=y\_i$ for all $i$. Then $A = xx^T$ is rank one... | 19 | https://mathoverflow.net/users/5963 | 264125 | 118,756 |
https://mathoverflow.net/questions/264112 | 2 | In Ivey and Landsberg's book *Cartan for Beginners*, the end paragraph of example 5.8.2 claims that linear Weingarten surfaces can be constructed by a space curve. They cite an older book from 1945 that is in French (*Les Systemes Exterieurs et leurs Applications Geometriques*) which explicitly carries out this constru... | https://mathoverflow.net/users/103158 | Construction of a linear Weingarten surface from a space curve | You'll find a discussion of the analysis of linear Weingarten surfaces via exterior differential systems in [these lecture notes of mine](https://services.math.duke.edu/~bryant/MSRI_Lectures.pdf). Particularly look at Section 5.1, where it is discussed at length. If you've already read Ivey and Landsberg, this should b... | 7 | https://mathoverflow.net/users/13972 | 264129 | 118,758 |
https://mathoverflow.net/questions/263500 | 5 | Recall that an almost complex structure $J$ on a manifold $M^{2n}$ is called *tamed* if there exists a symplectic form $\omega$ on $M^{2n}$ such that $\omega(v,Jv)>0$ for any non-zero tangent vector $v$.
**Question.** Is there an almost complex structure $J$ on a closed ball $B^4$, such that any $C^{\infty}$-small pe... | https://mathoverflow.net/users/13441 | Almost complex structures on a 4-ball that are not tamed | The following construction provides plenty of examples of non-tamed almost complex structures:
Consider an almost complex structure $J$ on $B^4$ for which the contact hyperplanes of the *overtwisted* contact structure on $S^3=\partial B^4$ (in the same homotopy class as the standard tight contact structure) become $J... | 6 | https://mathoverflow.net/users/48067 | 264139 | 118,761 |
https://mathoverflow.net/questions/264113 | 0 | Let $(R,m)$ and $(S,n)$ be local Noetherian rings such that $S$ is a faithfully flat extension of $R$. Let $J\subsetneq I $ ideals of $R$.
**Can we relate $l\_R(I/J)$ and $l\_S(IS/JS)$?**
PS: Here $l(-)$ denotes the length.
| https://mathoverflow.net/users/9485 | Behaviour of length function under faithfully flat extension | **Remark.** Recall that any finitely generated module $M$ has a filtration
$$0 = M\_0 \subsetneq M\_1 \subsetneq \ldots \subsetneq M\_n = M$$
whose successive subquotients $M\_i/M\_{i-1}$ are isomorphic to $R/\mathfrak p\_i$ for some prime ideal $\mathfrak p\_i \subseteq R$. If all the $\mathfrak p\_i$ are maximal, the... | 2 | https://mathoverflow.net/users/82179 | 264145 | 118,762 |
https://mathoverflow.net/questions/264137 | 6 | Let $k$ be a ring (resp. profinite ring), $G$ a group (resp. profinite group), and $k[G]$ the group algebra (resp. completed group algebra).
For any such $G$, we may associate to it the group of units $k[G]^\times$ of $k[G]$, and this association is clearly functorial. Has this functor been studied at all?
For exam... | https://mathoverflow.net/users/15242 | What is known about the functor $G\mapsto k[G]^\times/k^\times$? | Forget about exactness; this functor isn't *additive*, which pretty much tanks any hope of doing homological algebra to it even if you restrict to abelian groups.
Here's a special case that's easy to understand. Suppose $G$ is finite and $k$ has characteristic not dividing $|G|$. Then $k[G]$ is semisimple, so we hav... | 6 | https://mathoverflow.net/users/290 | 264151 | 118,763 |
https://mathoverflow.net/questions/264090 | 3 | Let $a$ and $b$ be distinct positive real numbers. Let $(a\_n)$ and $(b\_n)$ be sequences of natural numbers such that $a\_n\sim an$ and $b\_n\sim bn$. All the limit relations here are for $n\to\infty$. Let $p\_n$ and $q\_n$ be the **coprime** natural numbers such
\begin{equation\*}
\frac{p\_n}{q\_n}=\frac{a\_n^{(n)}... | https://mathoverflow.net/users/36721 | Asymptotics for the number of digits of the ratio of binomial coefficients | So here is an answer, edited first from an unproved guess at the correct $f(a,b)$ and then a terse proof. The following version has a few more details added.
I claim that
$$
f(a,b)=\int\_0^{b+1} \left(\left\lfloor \frac{a+1}x\right\rfloor
-\left\lfloor\frac ax\right\rfloor
-\left\lfloor\frac{b+1}x\right\rfloor
+\left... | 3 | https://mathoverflow.net/users/11054 | 264152 | 118,764 |
https://mathoverflow.net/questions/264100 | 12 | It seems that people often talk of "doing Morse theory" on loop spaces in two quite different contexts.
Case 1: When one does Morse theory on a loop space $\Omega(M; p,q)$ using the energy functional as the Morse function, then critical points are geodesics and the index is the index is computed via the second varia... | https://mathoverflow.net/users/59235 | Morse theory in infinite dimensions | The first case has finite indices and parabolic gradient flow; the second infinite (co)indices and elliptic gradient flow.
In more detail, the Morse theory of the energy functional $E$ on $X:=\Omega(M;p,q)$ has the following behavior:
1) For generic metrics it's a Morse function (and for other special metrics of in... | 18 | https://mathoverflow.net/users/2356 | 264156 | 118,767 |
https://mathoverflow.net/questions/264168 | 0 | Let $E^c\_n$ be the expected number of connected components of a simple undirected graph on the vertex set $\{1,\ldots,n\}$. (Every possible edge in $\big\{\{a, b\}: a, b\in \{1,\ldots,n\} \land a \neq b\big\}$ is picked with probability $1/2$.)
Is $\{E^c\_n: n\in\mathbb{N}\}$ bounded? If yes, what is the least upper... | https://mathoverflow.net/users/8628 | Expected number of connected components as $V(G)$ grows large | $E\_{n+1}^c \leq E\_n^c + \frac{1}{2^n}$, so as $E\_0^c = 0$, we have that $E\_n^c \leq 2$.
Reason: For a graph $G$, define $|G^c|$ as the number of components of $G$. Then for any labeled graph $G$, there are $2^n$ graphs with $n + 1$ vertices such that the subgraph on the first $n$ vertices is $G$. The one such tha... | 4 | https://mathoverflow.net/users/44191 | 264169 | 118,770 |
https://mathoverflow.net/questions/264162 | 27 | I am wondering what is the etiquette of publishing a "folklore" result?
Though special cases of the result are well-known, the proof is not readily available in any reference text or paper I've seen. I will include it in the background of my thesis, and I could stick it onto another paper which I am working on (in w... | https://mathoverflow.net/users/56938 | Etiquette of publishing folklore results | You ask for the "etiquette", which may differ from field to field. For the research community in computer science, the fate of one particular folk theorem has been documented in loving detail by David Harel in [On Folk Theorems](http://www.wisdom.weizmann.ac.il/~dharel/SCANNED.PAPERS/OnFolkTheorems.pdf) (1980).
One t... | 10 | https://mathoverflow.net/users/11260 | 264170 | 118,771 |
https://mathoverflow.net/questions/179989 | 3 | I am looking for explicit formulas for the weight multiplicities of some particular irreducible representations of $SO(2m)$.
It is possible that they have been already computed; in this case I will appreciate a reference. Otherwise, How could I compute them?
These *particular* representations are as follows: let $\p... | https://mathoverflow.net/users/20052 | Weight multiplicities for some particular representations of SO(2m). | For $\mu$ a weight, let $||\mu||\_1$ denote the one-norm of $\mu$ (the sum of the absolute values of its entries) and let $Z(\mu)$ be the number of zero coordinates of $\mu$.
Let $k\geq0$ and $1\leq p\leq n$.
Write $r(\mu)=(k+p-||\mu||\_1)/2$.
If $r(\mu)$ is a non-negative integer, then
\begin{align\*}
m\_{\pi\_{\Lamb... | 2 | https://mathoverflow.net/users/20052 | 264178 | 118,773 |
https://mathoverflow.net/questions/264180 | 7 | Each automorphism of the quaternion algebra is inner and it is an orthogonal mapping with the determinant 1.
Let $f: \mathbb H \rightarrow \mathbb H$ be in $SO(4)$. Does there exists a quaternion $q$ with $\|q\|=1$ and an automorphism $g$ of $\mathbb H$ such that $f(x)=qg(x)$ for $x\in \mathbb H$ ?
| https://mathoverflow.net/users/105159 | Automorphisms and isometries of the quaternions | In other words, what you are asking is whether every $f\colon\mathbb{H}\to\mathbb{H}$ in $\mathit{SO}\_4$ takes the form $x\mapsto \bar u x v$ where $u,v$ are unit quaternions (the connection with your notation is that then $f$ is the composition of the inner automorphism $g\colon x\mapsto \bar v x v$ with left-multipl... | 15 | https://mathoverflow.net/users/17064 | 264188 | 118,777 |
https://mathoverflow.net/questions/264189 | 3 |
>
> Let $\pi:\mathbb{N}\to\mathbb{N}$ be a bijection. Then does there exist another bijection $\nu:\mathbb{N}\to\mathbb{N}$ and a constant $C$ such that
> $$
> \frac{1}{n} + \frac{1}{\pi(n)} \leq \frac{C}{\nu(n)}
> $$
> for all $n$? If so, can the constant be chosen independent of $\pi$?
>
>
>
While the harmon... | https://mathoverflow.net/users/27013 | Adding the harmonic sequence and a permutation of it | Yes, we may achieve even $\nu(n)\leqslant 2\min(n,\pi(n))$. Indeed, define $\rho(n)=\min(n,\pi(n))$ and let $t\_1,t\_2,\dots$ be an enumeration of the positive integers such that $\rho(t\_1)\leqslant \rho (t\_2)\leqslant \rho(t\_3)\leqslant \dots$. Then $\rho(t\_k)\geqslant k/2$, otherwise we may find three different n... | 4 | https://mathoverflow.net/users/4312 | 264196 | 118,780 |
https://mathoverflow.net/questions/264183 | 2 | Let $k$ be an infinite non-algebraically closed field, $X$ a smooth projective curve on $k$ and $E$ a locally-free sheaf on $X$ of rank at least $2$. Denote by $\bar{k}$ the algebraic closure of $k$, $X\_{\bar{k}}$ the base-change of $X$ to $\bar{k}$ and $E\_{\bar{k}}$ the pull-back of $E$ to $X\_{\bar{k}}$.
Is the ... | https://mathoverflow.net/users/43198 | Confusion regarding Riemann-Roch for vector bundles | Riemann--Roch is "the same" over any field. But in any event, this isn't so helpful for you since it only shows that the Euler characteristic $h^0(X,E)-h^1(X,E)$ is the same over $\bar k$ as it is for $k$.
The key phrase in your setting is "cohomology and base change". Cohomology and base change results tell you when... | 6 | https://mathoverflow.net/users/35353 | 264200 | 118,782 |
https://mathoverflow.net/questions/264197 | 2 | Let $k$ be an infinite field. Assume that the index of the algebraic closure $\bar{k}$ over $k$ is strictly greater than $2$. Let $U$ be a non-empty open subset of some affine space over $k$. Is it true that $U$ always contain a $k$-rational point?
| https://mathoverflow.net/users/43198 | Rational points on open subsets of affine space | Here is a short proof that, for an infinite field $k$, and all non-zero polynomials $F \in k[x\_1,\ldots,x\_n]$ in $n$ variables, there exists an $n$-tuple $a\_1,\ldots,a\_n \in k$ such that
$$
F(a\_1,\ldots,a\_n) \neq 0.
$$
We do induction on $n$. For $n=1$, the assertion follows from the fact that $F$ has finitely ma... | 8 | https://mathoverflow.net/users/17907 | 264212 | 118,783 |
https://mathoverflow.net/questions/264204 | 14 | Let $f:X\to Y$ be a morphism of algebraic varieties over $\mathbb C$. Assume that
a) $f$ is bijective on $\mathbb C$-points
b) $X$ is connected
c) $Y$ is normal.
Does it imply that $f$ is an isomorphism? If not, are there stronger reasonable conditions under which it is true?
| https://mathoverflow.net/users/3891 | Bijection implies isomorphism for algebraic varieties | I am just expanding the comment above. The statement is local on $Y$, so assume that $Y$ is connected. Since $Y$ is normal, this is equivalent to assuming that $Y$ is irreducible. Grothendieck's formulation of Zariski's Main Theorem is EGA $\textrm{III}\_2$, Théorème 4.4.3, p. 136. According to that theorem, there exis... | 13 | https://mathoverflow.net/users/13265 | 264216 | 118,786 |
https://mathoverflow.net/questions/263945 | 5 | I start with some known preliminaries on the problem:
**Classical result.** The one-dimensional Cauchy functional equation
$$
\forall x,y \in \mathbb{R}, \,\,\,f(x+y)=f(x)+f(y)
$$
with $f:\mathbb{R}\to \mathbb{R}$ is only solved by the trivial solutions $f(x)=cx$, for some $c \in \mathbb{R}$, if $f$ satisfies for som... | https://mathoverflow.net/users/32898 | On a generalization of the classical Cauchy's functional equation | As I have just learned from Janusz Matkowski, this follows from Theorems 5.5.2 and 18.2.1 in Kuczma's book (the same mentioned in my comments to the OP), after ruling out the trivial case when the cone $C$ is a line or a half-line. In particular, Theorem 5.5.2 is about the (unrestricted) Cauchy functional equation in $... | 3 | https://mathoverflow.net/users/16537 | 264224 | 118,788 |
https://mathoverflow.net/questions/264226 | 3 | Let $\sigma\_{m, r}$ be the degree-$r$ elementary symmetric polynomial in $m$ variables. Let $X\_{m, r}$ be the zero set of $\sigma\_{m, r}$ and $S\_{m, r}$ its singular locus. I.e.,
$S\_{m,r}$ is the set of common zeroes of $\sigma\_{m, r}$ and its partials $\frac{\partial}{\partial x\_1} \sigma\_{m,r}, \cdots, \frac{... | https://mathoverflow.net/users/97414 | Singular locus of zero set of elementary symmetric polynomial | Notice that the vanishing of $\frac{\partial}{\partial x\_1} \sigma\_{m,r}, \cdots, \frac{\partial}{\partial x\_m} \sigma\_{m,r}$ implies the vanishing of $\sigma\_{m,r}$ since
$$\sigma\_{m,r}=\frac{1}{r}\left( x\_1\frac{\partial}{\partial x\_1} \sigma\_{m,r}+ \cdots + x\_m\frac{\partial}{\partial x\_m} \sigma\_{m,r}\... | 5 | https://mathoverflow.net/users/2384 | 264229 | 118,790 |
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