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https://mathoverflow.net/questions/292283 | 1 | Let $(\mathcal{G}, \mathcal{G}^\*, \delta)$ be a Lie bialgebra. Suppose that the structure constant on $\mathcal{G}^\*$ and $\mathcal{G}$ are
\begin{align}
& [t^a, t^b]\_\* = C\_c^{ab} t\_c, \\
& [t\_a, t\_b] = f\_{ab}^c t\_c,
\end{align}
respectively.
Let $r = r^{ab} t\_a \otimes t\_b \in \mathcal{G} \otimes \mathca... | https://mathoverflow.net/users/11877 | How to show that the structure constant on $\mathcal{G}^*$ is $C_{c}^{ab} = f_{cd}^b r^{ad} + f_{cd}^a r^{db}$? | It is important to note that the literature often uses the shorthand $$[X\otimes 1+1\otimes X, Y\otimes Z]=(ad\_X\otimes 1+1\otimes ad\_X)(Y\otimes Z)=[X,Y]\otimes Z+X\otimes [Y,Z]$$ where $1$ is the identity map. We compute $C^{ab}\_c=\langle [t^a,t^b]\_\*,t\_c\rangle$ as follows. Recall that there is implied summatio... | 2 | https://mathoverflow.net/users/31556 | 295806 | 130,034 |
https://mathoverflow.net/questions/295810 | 3 | Is there a geometric interpretation of a surface subgroup being Zariski dense? Or, conversely, given a $\Pi\_1$ injective surface in a 3-manifold, is there a geometric or topological requirement on the surface so that the corresponding group is Zariski dense?
The theorems I've seen that show existance of Zariski den... | https://mathoverflow.net/users/122234 | Is there a geometric interpretation of a Zariski dense surface subgroup? | I am interpreting the question as meaning that the OP asks for a geometric interpretation of a Zariski dense surface subgroup in $SL(n, \mathbb{Z})$ for $n=3, 4.$ The existence of such has been shown by Long, Reid, and Thistlethwaite, in a series of papers, using methods of $3$-dimensional topology (and a healthy dose ... | 1 | https://mathoverflow.net/users/11142 | 295817 | 130,038 |
https://mathoverflow.net/questions/295451 | 5 | Is there an example of a finite group $A$ that acts on a finite group $C$ irreducibly (that is, $C$ has no proper nontrivial $A$-invariant subgroup)
such that there exists an epimorphism $$\tau \colon A \ltimes C \to A$$ which does not split?
| https://mathoverflow.net/users/38889 | A finite group that splits and does not split | I don't believe this is possible. Let $G$ be a minimal counterexample. Then $G = AC$ with $C \unlhd G$ and $A$ a complement of $C$, and there exists $D \unlhd G$ with $G/D \cong A$, but $D$ has no complement if $G$. Note that $G/D \cong G/C \Rightarrow |C|=|D|$. Since $A$ acts irreducibly on $C$, we have $C \cap D = 1$... | 5 | https://mathoverflow.net/users/35840 | 295833 | 130,043 |
https://mathoverflow.net/questions/236277 | 13 | Let $N\subseteq M$ be an inclusion of semi-finite factors with normal faithful semi-finite traces $\operatorname{Tr}\_N$ and $\operatorname{Tr}\_M$ respectively. Let $T: M^+\to \widehat{N^+}$ be the unique trace-preserving normal faithful semi-finite operator valued weight.
As for normal weights, we define
\begin{ali... | https://mathoverflow.net/users/351 | Is the domain of an operator valued weight closed under Hahn-Jordan decomposition? | Here is a counter-example. Which is sadly rather long. I use [Haagerup's original paper](https://www.sciencedirect.com/science/article/pii/0022123679900533) for the definition of $T$ below.
Let $M$ be the $\ell^\infty$ direct sum of the matrix algebras $\mathbb M\_n$ with $\tau\_M$ being the sum of the un-normalised ... | 3 | https://mathoverflow.net/users/406 | 295842 | 130,045 |
https://mathoverflow.net/questions/232043 | 6 | Because I still have no idea how it is possible for me to write down seemingly important equations ... that don't make any sense (at least for me) and because I haven't got any helpful comment so far, I'll be happy to offer a +100 bounty (that is, almost all my reputation!) not for a definitive anwser to this weird que... | https://mathoverflow.net/users/88057 | What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory) | Classical Bayesian analysis rests on first chosing a *prior measure* $m$, either finite (proper) or infinite (improper), then deriving the posterior probability of an event $A$ conditional on observed $B$ as $Pr(A|B)=m(A\cap B)/m(B)$.
As I said in my first answer, this is possible for $[0,1]^{[0,1]}$ with the product... | 2 | https://mathoverflow.net/users/75422 | 295845 | 130,048 |
https://mathoverflow.net/questions/295832 | 8 | Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ we set $N\_0(v) = N(v) = $ $\{w\in V: \{v,w\} \in E\}$ and for $k\in \omega$ let $$N\_{k+1}(v) = N\_k(v) \cup \bigcup\big\{N(z): z\in N\_k(v)\big\}.$$
We define the *neighborhood fingerprint* of $G$ as $F\_G: V\times \omega \to \omega$ defined by $(v,k) ... | https://mathoverflow.net/users/8628 | Neighborhood fingerprint of a graph | The answer is no.
Among [Andries E. Brouwer's web pages at TU Eindhoven](http://www.win.tue.nl/~aeb/), the [Cages](http://www.win.tue.nl/~aeb/graphs/cages/cages.html) page lists three non-isomorphic graphs on $70$ vertices (they are $(3,10)$-cages, found by O'Keefe & Wong 1980) with the same distance distribution $1+... | 7 | https://mathoverflow.net/users/49003 | 295847 | 130,049 |
https://mathoverflow.net/questions/295682 | 3 | **I am specifically interested in computing:
$$\mathbb{E}[S\_p S\_q S\_s S\_t]$$**
where $S\_t=\frac{dN\_t}{dt}$ and $N\_t$ is a Poisson process (so $S\_t$ is a "Poisson pulse train"):
$$\mathbb{P}(N\_t=n)=\frac{(\lambda t)^n}{n!}e^{-\lambda t} \; , \; t>0$$
**Here is my attempt:**
Assuming $t>s>p>q$, I tried to co... | https://mathoverflow.net/users/111000 | General result for the N-point correlation of the Poisson process (and its derivative)? | This question (and its generalization) is conventiently addressed by considering the moment-generating functional. Let $N\_t$ be a counting process with (possibly stochastic and time-dependent) intensity $\lambda\_t$ (you recover your case by setting $\lambda\_t=\lambda$, where $\lambda$ is a constant).
Suppose that ... | 4 | https://mathoverflow.net/users/69603 | 295848 | 130,050 |
https://mathoverflow.net/questions/295854 | 1 | In 1974, W. B. Johnson and E. Odell observed that there are subspaces $X$ of $L\_{1}$ with the Schur property. In 1980, J. Bourgain and H. P. Rosenthal constructed a subspace $X$ of $L\_{1}$ such that $X$ has the Schur property, but $X$ is not isomorphic to a subspace of $l\_{1}$. Hence, I have the first question as fo... | https://mathoverflow.net/users/41619 | Can every Banach space with the Schur property embed into $L_{1}(\mu)$ for some $\mu$? | Absolutely not. Take the the $\ell\_1$-sum of $\ell\_\infty^n$ ($n\in \mathbb N$). If that embedded into $L\_1(\mu)$, then you would have found $c\_0$ in some ultrapower of $L\_1(\mu)$, which is impossible.
| 3 | https://mathoverflow.net/users/15129 | 295855 | 130,051 |
https://mathoverflow.net/questions/295841 | 1 | I would like to know if there exists a Liouville theorem for solutions $u : \mathbb{R}^n \to \mathbb{R}$ of uniformly elliptic equations of the kind
$$
D\_i \left( a\_{ij} D\_j u \right) + b\_i D\_i u = 0.
$$
I assume the coefficients $a\_{ij},b\_i \in C^{\infty}(\mathbb R^n) \cap L^{\infty}(\mathbb{R}^n)$.
Any hint... | https://mathoverflow.net/users/86341 | A Liouville theorem for a uniformly elliptic equation in divergence form | What do you mean by the Liouville theorem? If the absence of bounded or positive harmonic functions, then the answer is "no" due to the presence of a vector field $b$. The corresponding counterexample can be constructed already for $n=1$. Take the diffusion coefficient $a=a\_{11}$ to be equal identically 1, and let $b=... | 2 | https://mathoverflow.net/users/8588 | 295857 | 130,052 |
https://mathoverflow.net/questions/295814 | 5 | Suppose that a discrete group $\Gamma$ acts on a compact Hausdorff space $X$ via homeomorphisms. This action induces an action on $C(X)$, the space of all continuous functions from $X$ to $\mathbb{C}$, by $s.f(x)=f(s^{-1}x)$. The action is said to be 'prime' if $C(X)$ doesn't admit any invariant unital $C^\*$-subalgebr... | https://mathoverflow.net/users/40212 | Example of a prime action on a compact Hausdorff Space | Thompson's group $V$, which is a finitely presented infinite simple group, consists of all homeomorphisms of the Cantor set $\{0,1\}^\mathbb N$ that can be described the following way. A *prefix code* is a collection of finite words none of which is a prefix of another. A finite prefix code $C$ is maximal if it is not ... | 2 | https://mathoverflow.net/users/15934 | 295860 | 130,053 |
https://mathoverflow.net/questions/295862 | 2 | To what extend do the various different *Sobolev inequalites* hold if I replace the usual **target space** $\mathbb R$ by an arbitrary Banach space, the notion of derivative by **Frechét derivative** and the usual real-valued integral by the **Bochner integral** (correspondingly adapting the definition of *weak derivat... | https://mathoverflow.net/users/78554 | Sobolev inequalities for Banach-valued functions | The question you are asking (and a lot of related material) is discussed in great detail in the article "[W. Arendt and M. Kreuter: Mapping theorems for Sobolev spaces of vector-valued functions](https://dx.doi.org/10.4064/sm8757-4-2017)" (to appear in Studia Mathematica; [preprint](https://arxiv.org/abs/1611.06161) av... | 3 | https://mathoverflow.net/users/102946 | 295870 | 130,056 |
https://mathoverflow.net/questions/295764 | 3 | Apologies if this question is a bit simplistic/vague for MO:
I'm looking for an all-purpose definition in the literature of when a sufficiently generic filter "canonically codes" a generic real. Examples of what I mean come from everyone's favorite notions of forcing to add certain reals "on purpose":
In Cohen forc... | https://mathoverflow.net/users/16107 | A definition of the generic real coded by a generic filter? | In the examples that you gave, the question whether you take a "union of stems" or "intersection of conditions" is really just a matter of notation.
For example, there are several ways to define conditions in Silver forcing: as partial functions from $\omega$ to $2$ with coinfinite domains, or as sufficiently unifor... | 8 | https://mathoverflow.net/users/14915 | 295885 | 130,059 |
https://mathoverflow.net/questions/295875 | 42 | **Question** Is there a connection between Abel and Galois theories of polynomial equations?
Recall that for every polynomial $p(x)\in \mathbb{Q}[x]$ (say, without the free coefficient), Abel considered the monodromy group of the Riemann surface of the analytic function $w(z)$ defined by $p(w(z))+z=0$.. There is an e... | https://mathoverflow.net/users/nan | Abel and Galois (and Arnold) | The action of the monodromy group of $w(z)$ on the fiber $p^{-1}(a)$ for a non-critical value $a$ of $p$ (that is $|p^{-1}(a)|=\deg p$) is the same as the action of the Galois group of $p(x)+z$ over $\mathbb C(z)$ on the roots of $p(x)+z$ in some splitting field. One can see this by comparing each of these groups with ... | 26 | https://mathoverflow.net/users/18739 | 295894 | 130,062 |
https://mathoverflow.net/questions/293724 | 3 | Given a domain $\Omega \subset \Bbb R^n$ and $\Delta\varphi=f$ where $\varphi:\Bbb R^n \to \Bbb R$ is unknown and $f:\Omega\to \Bbb R$ is a blackbox function (for each $\bf x$ it provides $f({\bf x})$, but we don't know what $f$ actually is), and in addition we may know **one of the following**,
1. discrete data poin... | https://mathoverflow.net/users/98958 | Numerical iterative methods for Poisson equation | In the first case, when you are given a finite set of points, your problem is not well defined. There are in general arbitrarily many solutions if you are just given a finite set of function values.
In the second case you have to perform two steps. First you discretize your problem, then you solve the resulting linea... | 3 | https://mathoverflow.net/users/112614 | 295903 | 130,065 |
https://mathoverflow.net/questions/295912 | 20 | The notion of beauty has historically led many mathematicians to fruitful work. Yet, I have yet to find a mathematical text which has attempted to elucidate what exactly makes certain geometric figures aesthetically pleasing and others less so. Naturally, some would mention the properties of elegance, symmetry and surp... | https://mathoverflow.net/users/56328 | Mathematical theory of aesthetics | George D Birkhoff, [*Aesthetic Measure*](https://rads.stackoverflow.com/amzn/click/0674730224), 1933
>
> An attempt to bring the basic formal side of art within the purview of simple mathematical formula defining aesthetic measure. Contents: the basic formula; polygonal forms; ornaments and tilings; vases; diatonic... | 16 | https://mathoverflow.net/users/454 | 295920 | 130,074 |
https://mathoverflow.net/questions/295909 | 2 | Let's say we have category with one object $N$ and infinite number of arrows, which are named as natural numbers, with the same law of composition, where $id$ arrow 0.
I try to understand 1) if categorical product/coproduct could be defined in such category and 2) if it could have properties of usual binary operations ... | https://mathoverflow.net/users/122288 | Is product/coproduct in category with only one object possible? | The category you describe lacks any interesting products and coproducts. But there is a category with only one object that has very interesting products and coproducts. Namely, let $A$ be a [$III\_1$ factor](https://en.wikipedia.org/wiki/Von_Neumann_algebra#Type_III_factors), and consider the category with only one obj... | 5 | https://mathoverflow.net/users/78 | 295926 | 130,078 |
https://mathoverflow.net/questions/295889 | 4 | I have asked this in math.SE (<https://math.stackexchange.com/questions/2698772/factorizations-in-terms-of-characters>) but it was barely viewed.
I have seen mention in different places that the number of solutions to the factorization equation $\pi\_1\cdots \pi\_r=1$ in $S\_n$, is given by
$$ \frac{|C\_{\lambda\_1}|... | https://mathoverflow.net/users/83671 | Factorizations in terms of characters | Your formula is a simple consequence of the fact that for any finite group $G$, the elements $\frac{\chi(1)}{|G|}\sum\_{w\in G}\chi(w^{-1})w$ form a set of orthogonal idempotents for the center of the group algebra $\mathbb{C}G$, where $\chi$ ranges over all irreducible characters of $G$. The proof uses only the orthog... | 5 | https://mathoverflow.net/users/2807 | 295930 | 130,079 |
https://mathoverflow.net/questions/295915 | 5 | I am looking at page 32 (beginning of Chapter 5) [here](https://web.math.princeton.edu/~chang/zur.pdf). We are given a formally self-adjoint, metrically defined differential operator $A$ on $(M^n,g)$ of order $2l$ with positive definite leading symbol, such that $A\_{\tilde{g}}=c^{-2l}A\_g$ whenever $\bar{g}=c^2g$ is a... | https://mathoverflow.net/users/122291 | Reference for Weyl's law for higher order operators on closed Riemannian manifolds | One possible reference is Seeley's paper on [Complex powers of Elliptic Operators](http://inspirehep.net/record/51406?ln=en), where Seeley did it for the Laplacian (page 6). But the discussion carries over to all elliptic $\Psi DO$s without much difficulty. For a "modern" expository article, see [this](http://www.uni-m... | 2 | https://mathoverflow.net/users/18850 | 295933 | 130,081 |
https://mathoverflow.net/questions/295891 | 3 | Let $G$ be a finite $p$-group of nilpotency class $c$ and of derived length $d$.
As is well known, we have $d\leq \lfloor\log\_2 c\rfloor+1$,
(<https://groupprops.subwiki.org/wiki/Derived_length_is_logarithmically_bounded_by_nilpotency_class>).
Does there exist any information or any classification of finite $p$-g... | https://mathoverflow.net/users/27962 | The nilpotency class and the derived length of a $p$-group | My guess is "no", at least as regards a classification. A check with GAP shows that 30,591 of the 34,297 groups of order $5^7$ satisfy this equality; that 7,882 of the 9,310 groups of order $3^7$ satisfy the equality; and that 53,499 of the 56,092 groups of order $2^8$ satisfy the equality.
Here's an easy bit of GAP ... | 3 | https://mathoverflow.net/users/41862 | 295943 | 130,085 |
https://mathoverflow.net/questions/295897 | 3 | Let $G$ be a reductive group over a nonarchimedean local field $F$. Let $\pi$ be an irreducible, cuspidal representation of $G$, with contragredient $\tilde{\pi}$. Then $\tilde{\pi}$ is cuspidal.
A character of $G$ is unramified if it is trivial on all compact subgroups of $G$. The group of unramified characters of $... | https://mathoverflow.net/users/77909 | Contragredient of a cuspidal representation | This is already false for $G={\rm GL}(1,F)$. In that case a cuspidal irreducible representation is a smooth character $\chi$ of $F$. The contragredient is $\chi^{-1}$. We have $\chi \sim \chi^{-1}$ iff $\chi^2$ is unramified. There are easy counter-examples.
Let us give another counter-example in higher rank. Take $... | 6 | https://mathoverflow.net/users/4767 | 295954 | 130,087 |
https://mathoverflow.net/questions/295757 | 0 | My understanding of [Lecture #33, 34: The Characteristic Function for a Diffusion](http://stat.math.uregina.ca/%7Ekozdron/Teaching/Regina/441Fall14/Notes/L33-34-Nov24.pdf):
---
As an alternative to directly computing the characteristic function of a random variable $X\_t$ in a stochastic process $\{X\_t\}\_{t \in... | https://mathoverflow.net/users/69696 | Expected properties for a PDE whose solution is supposed to be something that doesn't exist | 1. Lognormal does have a characteristic function
2. It has no closed form
3. The value problem solution doesn't have an Ansatz directly related to its terminal condition as with the value problems for ABM and OU
4. The value problem solution rather has a series solution.
5. Such series solution gives the characteristic... | 0 | https://mathoverflow.net/users/69696 | 295962 | 130,091 |
https://mathoverflow.net/questions/295944 | 2 | While it is easy to see that $H^1(\mathbb{R})$ are Hölder $1/2$-continuous, I started wondering whether this implies that $\delta\_x(\varphi)=\varphi(x)$ is continuous as a functional
$$\delta\_x:H^1(\mathbb{R}) \rightarrow \mathbb{R}?$$
I believe it is false, but do not know a counterexample.
Since it was asked ... | https://mathoverflow.net/users/122309 | Is the Delta distribution a continuous functional on $H^1(\mathbb{R})$? | Let $f:=\varphi$, $a:=\|f\|\_2$, $b:=\|f'\|\_2$, so that $\|f\|\_{H^1}=a+b$; see e.g. [Wikipedia](https://en.wikipedia.org/wiki/Sobolev_space) for the definition of $H^k$. Without loss of generality, $x=0$. For all $y\in[0,1]$, we have
$$|f(y)-f(0)|\le\int\_0^y|f'(t)|dt\le\int\_0^1|f'(t)|dt\le\sqrt{\int\_0^1|f'(t)|^2 ... | 2 | https://mathoverflow.net/users/36721 | 295963 | 130,092 |
https://mathoverflow.net/questions/295852 | 6 | Let $R$ be a regular, local $\mathbb{Q}$-algebra with a regular system of parameters $x\_1, \dotsc, x\_n$, and let
$$f \colon \mathbb{Q}[X\_1, \dotsc, X\_n]\_{(X\_1, \dotsc, X\_n)} \rightarrow R$$
be the map given by $X\_i \mapsto x\_i$. Then $f$ is flat (for instance, by Bourbaki, cf. EGA III, 0.10.2.2).
Is $f$ a ... | https://mathoverflow.net/users/70964 | Is this morphism of regular local rings regular? | You do not need $R$ excellent. Since $\mathbb{Q}[X\_1, \dotsc, X\_n]\_{(X\_1, \dotsc, X\_n)}$ is excellent it follows from [M. André, Localisation de la lissité formelle.](https://link.springer.com/article/10.1007/BF01168230)
| 3 | https://mathoverflow.net/users/92322 | 295971 | 130,096 |
https://mathoverflow.net/questions/295955 | 5 | I have troubles with the theory of existence of quasi-conformal homeomorphisms realizing Beltrami coefficients. Let $X$ be a (compact) Riemann surface and $f \colon X \rightarrow \mathbb{C}$ be smooth. Then on a coordinate chart $(U,z)$ of $X$, $x \in X$, the quotient $\left( \frac{\overline{\partial} f}{\partial f} \r... | https://mathoverflow.net/users/117619 | Clarification on Beltrami Differentials | I am not really familiar with Imayoshi and Taniguchi's book on Teichmüller theory, but here is my understanding of Beltrami differential, which I learned from Hubbard's book *Theichmüller Theory and Applications to Geometry, Topology, and Dynamics Volume 1*. He defines Beltrami differentials (forms) in Chapter $4$.
... | 5 | https://mathoverflow.net/users/74772 | 295978 | 130,099 |
https://mathoverflow.net/questions/234480 | 20 | Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, endowed with the usual topology of semi-norms
$$|f|\_{K}=\sup\_{|p|\leq k}~\sup\_{x\in K}|(\partial/\partial x)^p f(x)|,$$
where ... | https://mathoverflow.net/users/89429 | Density of polynomials in $C^k(\overline\Omega)$ | **No**, the polynomials will not be dense in general.
The following example is essentially one-dimensional. Let $C\subset[0,1]$ be the usual ternary Cantor set and $g\colon[0,1]\to[0,1]$ the Cantor function (a.k.a. Devil's staircase). Then $g$ is continuous and locally constant on the open set $U := (0,1)\setminus C$... | 9 | https://mathoverflow.net/users/14849 | 295985 | 130,103 |
https://mathoverflow.net/questions/295976 | 7 | I would like to start by saying that any comment or idea is highly appreciated.
Let us observe that for Hilbert-Schmidt operators $H\_1,H\_2$ on an infinite-dimensional separable complex Hilbert space $H$ and a bounded self-adjoint operator $T: H \rightarrow H$
$$\operatorname{Tr} \left(H\_1 ([T,H\_2])^\*\right) = \o... | https://mathoverflow.net/users/122309 | Existence of spectral gap | The answer to question (i) is negative. First of all, I'd like to remark that the shouldn't be an $i$ in the definition of $S\_T$ therefore for me $S\_T^2(H)$ will be $[T,[T,H]]$.
Let us first consider the case in which $T=\textrm{diag}(c\_i)$ is a diagonal operator. Then $\textrm{Tr}(S\_T^2(H) H^{\ast}) = \sum\_{i,j... | 5 | https://mathoverflow.net/users/24953 | 295988 | 130,104 |
https://mathoverflow.net/questions/295929 | 8 | Recall that a *Type III code* of rank $r$ is a linear subspace $C \subset \mathbb F\_3^r$ which is self-dual for the standard inner product. (These occur only when $r$ is divisible by $4$.) Elements of $C$ are called *code words*. The *Hamming weight* of a code word is its number of non-zero entries. I will call a code... | https://mathoverflow.net/users/78 | Are there Type III codes with small but nonzero "index"? | Index $24$ isn't hard for length $36$.
For example, the Type III code with generator matrix
```
+ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 0 + - + 0 - - + 0 + + - + + - 0 -
0 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 0 - + - - 0 0 - 0 0 0 0 - + - - -
0 0 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + 0 - + 0 + - - - + + - + 0 - 0 - +
0 0 0... | 4 | https://mathoverflow.net/users/14830 | 295996 | 130,107 |
https://mathoverflow.net/questions/295873 | 5 | Let $K\_n$ denote the complete graph on $n$ vertices. Let us denote the vertices simply by $1,\ldots,n$. Suppose that, for each edge $ij$, with $1\leq i<j \leq n$, we assign an ordered basis $(p^+\_{ij}, p^-\_{ij})$ of $\mathbb{C}^2$, where we think of the latter as the space of complex polynomials of degree less than ... | https://mathoverflow.net/users/81645 | A combinatorial question on complete graphs and polynomials | This is a nice question! Thanks for contacting me by email; I want to summarize here some of the email discussion that we've had.
Your problem bears a family resemblance to [Rota's Basis Conjecture](http://michaelnielsen.org/polymath1/index.php?title=Rota%27s_conjecture). The well-known connection between the Alon–Ta... | 4 | https://mathoverflow.net/users/3106 | 295998 | 130,108 |
https://mathoverflow.net/questions/295867 | 5 | Consider a pair of holomorphic functions $f,g \in \mathcal{O}(\Delta)$ on the complex unit disk $\Delta = \{|z| < 1\}$ that both satisfy $f(0) = g(0) = 0$ and $f'(0) = g'(0) = 1$. *Does the domain
$$
f(\Delta) \cdot g(\Delta) := \{f(z)g(w) \mid z, w \in \Delta \} \subset \mathbb{C}
$$
contain the unit disk $\Delta$, wi... | https://mathoverflow.net/users/26522 | The largest disk contained by a 'product' of two simply connected plane regions with unit conformal radii | Let's start with the simple reduction. Notice that $f(\Delta)$ and $g(\Delta)$ are connected open sets containing small disks near the origin, so if one of them is unbounded, $f(\Delta)g(\Delta)=\mathbb C$.
Let $a\in\Delta\setminus\{0\}$ (we certainly have $0=f(0)g(0)$, so the origin is never problematic) be not in ... | 5 | https://mathoverflow.net/users/1131 | 296002 | 130,110 |
https://mathoverflow.net/questions/296011 | 7 | I've noticed a couple of conference titles which reference something called
"topology in dimension 3.5," such as [this one](http://math.rice.edu/NewsEvents/Conferences/TopologyConference2016/index.html) and [this one](https://sites.google.com/sjsu.edu/thompscharbyfest/home). This subject seems quite mysterious to me —... | https://mathoverflow.net/users/97265 | What is "topology in dimension 3.5"? | cw answer: As mentioned in the comments, this just refers to the relations between 3-dimensional and 4-dimensional topology.
| 4 | https://mathoverflow.net/users/14094 | 296015 | 130,112 |
https://mathoverflow.net/questions/296012 | 3 | This question attempts to generalize Hall's theorem to taking "triangles" instead of edges.
Say we are given $3$ disjoint sets over vertices $A,B,C$. Consider a graph who's edges only connect vertices from those $3$ different sets.
We say this arrangment is $A$-good if there exists $|A|$ disjoint triangles in this g... | https://mathoverflow.net/users/104594 | Generlization of Halls theorem to triangles | Determining whether a $k$-partite $k$-uniform hypergraph has a perfect matching is NP-complete for $k\geq 3$, so any condition equivalent to being $A$-good is bound to be somewhat complicated. Finding a neccessary and sufficient conditions for hypergraphs is considered a difficult open problem for which a lot of partia... | 6 | https://mathoverflow.net/users/2384 | 296021 | 130,115 |
https://mathoverflow.net/questions/296004 | 5 | To describe homotopy invariant algebraic structures on spaces, there are different approaches.
* The Stasheff / Boardman–Vogt / May approach, where operations and equations are replaced by spaces of operations, witnessing higher homotopy coherence.
* The Segal approach, where the structure maps that would be isomorph... | https://mathoverflow.net/users/16109 | Homotopy invariant structure: Stasheff versus Segal | As requested, my comments in the form of an answer:
At the end of *Categories and Cohomology Theories* Segal gives a fairly detailed sketch of how to compare these theories in the harder, $\mathbb{E}\_{\infty}$ case. The same sketch works in the $\mathbb{A}\_{\infty}$-case. Later, May-Thomason elaborated on Segal's r... | 7 | https://mathoverflow.net/users/6936 | 296024 | 130,116 |
https://mathoverflow.net/questions/295989 | 2 | Suppose we have a deterministic complete [finite automaton](https://en.wikipedia.org/wiki/Deterministic_finite_automaton) which is synchronized, meaning we have a [reset word](https://en.wikipedia.org/wiki/Synchronizing_word), i.e. a word which resets the automaton to a definite state, regardless from which state we st... | https://mathoverflow.net/users/37580 | For synchronizing eulerian finite state machines every proper subset of states has some larger state set leads to this subset | This is proved in Section 4 of Kari's paper [here.](https://pdfs.semanticscholar.org/3b73/3fae84ed37f9d1bd061ccdabb6af73a4f9f4.pdf)
Essentially the same proof is in chapter 15 of my book the Representation Theory of Finite Monoids done from a more representation theoretic viewpoint. The main trick is to use ascending... | 2 | https://mathoverflow.net/users/15934 | 296025 | 130,117 |
https://mathoverflow.net/questions/296008 | 8 | Let $x, y \in \mathbb{R}^{n}\_{\geq 0}$ satisfy $\sum\_i x\_i = \sum\_i y\_i = 1$ and let $c \geq 1$. I am trying to find out whether the following inequality holds:
\begin{equation}
\left(\sum\_i x\_i y\_i\right)^{2c-1} \leq \sum\_i(x\_i y\_i)^c
\end{equation}
It seems like there should be a simple proof using sta... | https://mathoverflow.net/users/122335 | Inequality for the inner product in the probabilistic simplex | *It seems like there should be a simple proof using standard inequalities*
Most certainly, most certainly. Nate essentially had it though he was doing Jensen on the wrong side.
We have $z\mapsto z^{2c-1}$ convex, so by Jensen
$$
\sum(x\_iy\_i)^c=\sum x\_i(x\_i^{\frac{c-1}{2c-1}}y\_i^{\frac{c}{2c-1}})^{2c-1}\ge\left... | 10 | https://mathoverflow.net/users/1131 | 296027 | 130,119 |
https://mathoverflow.net/questions/296038 | 0 | Let $X=\{x\_1,x\_2,...,x\_n\}$ and $Y=\{y\_1,y\_2,...,y\_n\}$ be sets over a finite field $F$ with $p=char(F)>2$. Assume
$$x\_1^k+x\_2^k+...+x\_n^k=y\_1^k+y\_2^k+...+y\_n^k,\ 1\leq k\leq n$$
I wanna show that $X=Y$.
| https://mathoverflow.net/users/84871 | Show that sets are equal | Use the standard notations $e\_k=\sum\_{A\subset \{1,\dots,n\}, |A|=k} \prod\_{i\in A} x\_i$, with the conventions $e\_0=1$ and $e\_m=0$ for $m>n$; $p\_k=\sum\_{i=1}^n x\_i^k$.
If $n=p$, the statement is true if you require your conditions for all $k$, not just $k\le n$.
Indeed, Newton's identities say that
$$
ke... | 5 | https://mathoverflow.net/users/1306 | 296043 | 130,127 |
https://mathoverflow.net/questions/296054 | 5 | I ask a question on math stack exchange about Hodge conjecture and have not got any reply or comment.
<https://math.stackexchange.com/questions/2704988/clarify-hodge-conjecture>
so I decide to post this question on MO. Let's focus on smooth projective varieties defined over $\mathbb{Q}$. The Hodge conjecture could ... | https://mathoverflow.net/users/87910 | Misunderstanding of Hodge conjecture | The Hodge conjecture implies that the functor
$R\colon M\_{num}(k,\mathbb{Q})→HS(\mathbb{Q})$ is fully faithful when $k$ is the algebraic closure of $\mathbb{Q}$, not $\mathbb{Q}$ itself, as your example illustrates.
| 10 | https://mathoverflow.net/users/122351 | 296055 | 130,130 |
https://mathoverflow.net/questions/296045 | 2 | Let $(X,\Sigma,\mu)$ be a measure space.
The semi-finite version of $\mu$ on $(X,\Sigma)$ is denoted $\mu\_{\rm sf}$ and given by
$$
\mu\_{\rm sf}(E) = \sup\{\mu(A) \mid A \subseteq E \text{ measurable, } \mu(A) \lt \infty\}, \quad \text{for}E \in \Sigma .
$$
>
> Consider the map $i: L^2(\mu)\longrightarrow L^2(\... | https://mathoverflow.net/users/113054 | Identity map between $L^2(\mu)$ and $L^2(\mu_{\rm sf})$ | Yes, it is bijective.
p4sch has already shown the map is well-defined and injective (in fact, an isometry). I claim it is also surjective. (I assume throughout that $\mu$ is a positive measure.)
Note it is an exercise to show that $\newcommand{musf}{\mu\_{\rm sf}} \musf$ is indeed a countably additive measure.
**... | 4 | https://mathoverflow.net/users/4832 | 296072 | 130,134 |
https://mathoverflow.net/questions/296068 | 14 | Let $X$ be a co-finite topological space. If $|X| \ge 2^{\aleph\_0}=\mathfrak c$, then $X$ is contractible (<https://en.wikipedia.org/wiki/Contractible_space>) . Indeed, there is a bijection $f: X \times (0,1) \to X$; fix a point $a \in X$; define $H: X \times [0,1] \to X$ as $H(x,0)=x, \forall x \in X; H(x,1)=a,\foral... | https://mathoverflow.net/users/nan | Continuum Hypothesis and the fact that every co-finite topological space, with uncountable underlying set , is contractible | Nice question!
I claim that this property does not necessarily imply CH. As Todd
guessed in his comment, the answer is related to certain cardinal
characteristics of the continuum.
Specifically, let us define the *closed-partition number* to be the size $\kappa$ of the smallest nontrivial partition of the unit inte... | 14 | https://mathoverflow.net/users/1946 | 296084 | 130,140 |
https://mathoverflow.net/questions/295809 | 7 | The answer to the question is almost surely negative (as almost always in Banach space theory) but I cannot find a relevant example.
Is there an example of an infinite-dimensional Banach space $X$ such that $X^{\*\*}$ does not contain infinite-dimensional reflexive subspaces?
Note that such a space must be HI-satur... | https://mathoverflow.net/users/121555 | Reflexive subspaces of bidual Banach spaces | The answer is that there is indeed an example of such space. This is established in Theorem 6.27 of:
Argyros, Spiros A.; Arvanitakis, Alexander D.; Tolias, Andreas G. Saturated extensions, the attractors method and hereditarily James tree spaces. Methods in Banach space theory, 1–90, London Math. Soc. Lecture Note Se... | 7 | https://mathoverflow.net/users/848 | 296089 | 130,141 |
https://mathoverflow.net/questions/296088 | 20 | Thanks to the fibrations
\begin{align\*}
SO(n) \to SO(n+1) &\to S^n\\
SU(n) \to SU(n+1) &\to S^{2n+1}\\
Sp(n) \to Sp(n+1) &\to S^{4n+3}
\end{align\*}
we know that
\begin{align\*}
\pi\_i(SO(n)) \cong \pi\_i(SO(n+1)) \cong \pi\_i(SO), \quad i &\leq n-2\\
\pi\_i(SU(n)) \cong \pi\_i(SU(n+1)) \cong \pi\_i(SU), \quad ... | https://mathoverflow.net/users/21564 | The first unstable homotopy group of $Sp(n)$ | The answer appears to be in the paper [Homotopy groups of symplectic groups](https://doi.org/10.1215/kjm/1250524819) by Mimura and Toda. They claim the calculation was already in a paper of [Harris](https://www.ams.org/journals/tran/1963-106-01/S0002-9947-1963-0143216-6/S0002-9947-1963-0143216-6.pdf), but that was stat... | 27 | https://mathoverflow.net/users/18060 | 296091 | 130,143 |
https://mathoverflow.net/questions/295965 | 6 | Let $G = G(K)$ be a Chevalley group over an algebraically closed field $K$ of characteristic $p > 0$. Consider the finite group $G(q) = G(\mathbb{F}\_q)$. (For example, if $G = \operatorname{SL}\_n(K)$ then $G(q) = \operatorname{SL}\_n(\mathbb{F}\_q)$).
It was proven by Steinberg that every irreducible $\mathbb{F}\_q... | https://mathoverflow.net/users/38068 | Are indecomposable representations of a finite group of Lie type absolutely indecomposable? | This is rarely the case for all $V$. $KG(q)$ would need to have finite representation type, so $G(q)$ would need to have a cyclic Sylow $p$-subgroup. Otherwise a counterexample can be found by taking an indecomposable module over some field extension of $\mathbb{F}\_q$, that is not defined over $\mathbb{F}\_q$ and rega... | 9 | https://mathoverflow.net/users/22989 | 296095 | 130,145 |
https://mathoverflow.net/questions/296070 | 10 | It is a 20th century result that there exists only one algebraically closed field for a given characteristic $p$ and cardinality $\kappa>\aleph\_0$, up to isomorphism. Is there a better way to imagine such fields, other than adjoining $\kappa$ transcendental elements to $\mathbb Q$ or $\mathbb F\_p$ and taking the alge... | https://mathoverflow.net/users/114143 | Algebraically closed field of cardinality greater than $\mathfrak c$ | Here is a different way to think about how these fields arise.
In characteristic zero, all such fields in any uncountable cardinality $\kappa$ can be viewed as arising as a nonstandard version of the algebraic closure the rationals.
That is, take any nonstandard model of arithmetic $M$ of uncountable size $\kappa$... | 11 | https://mathoverflow.net/users/1946 | 296100 | 130,148 |
https://mathoverflow.net/questions/296105 | 3 | Looking at the connection between modular forms as sections and automorphic representations it is to me somewhat clear why automorphic representations are (demanded to be) admissible $G(\mathbb{A}^\infty)$ modules.
Is there a similar reason for the $(\mathfrak{g},K)$-module structure(like some nice analytic structur... | https://mathoverflow.net/users/94298 | What role do $(\mathfrak{g},K)$- modules play in the construction of automorphic vector bundles | Given any admissible representation of $G(\mathbb{R})$ one can construct a $(\mathfrak{g}, K)$ module from it. Isomorphism of the admissible representations is not the same as isomorphism of the $(\mathfrak{g}, K)$ modules (called infinitesimal isomorphism), but these notions agree for unitary representations. Therefor... | 2 | https://mathoverflow.net/users/6084 | 296119 | 130,154 |
https://mathoverflow.net/questions/295395 | 9 | This question is meant to be viewed under moderate large cardinal hypotheses, e.g., enough to ensure $\aleph\_1^{L[x]}<\aleph\_1$ for all reals $x$.
In analogy with the (well-developed) theory of countable Borel equivalence relations, what can be said about *countable* $\mathbf\Sigma^1\_2$ (or $\mathbf\Delta^1\_3$, w... | https://mathoverflow.net/users/16107 | Countable $\mathbf\Sigma^1_2$ equivalence relations | We will show the generalization of the Feldman-Moore theorem on countable equivalence relations to the context of thin $\kappa$-Suslin equivalence relations where $\kappa$ is any infinite cardinal. Under further hypotheses, that is determinacy axioms, then every $\Sigma^1\_{2n+2}$ set of reals is $\delta^1\_{2n+1}$-Sus... | 2 | https://mathoverflow.net/users/3859 | 296125 | 130,155 |
https://mathoverflow.net/questions/296034 | 0 | Denote by $\precsim$ the order comes from "Murray-von Neumann" equivalence in the projection lattice of a von Numann algebra. Let e and f be two projections in a **properly infinit**e von Numann algebra M. Do $e\precsim f$ and $1-e\precsim 1-f$ imply $e\sim f$?
| https://mathoverflow.net/users/84700 | $e\precsim f$ and $1-e\precsim 1-f$ imply $e\sim f$? | No. Take e=0 and 0 < f < 1 such that both f and 1−f are infinite, with (1−f)~1.
Then e≾f because 0≾f for any projection f.
Also 1−e≾1−f because 1≾1−f, which holds by definition of f.
| 3 | https://mathoverflow.net/users/402 | 296134 | 130,158 |
https://mathoverflow.net/questions/292274 | 14 | I'm working with different definitions of proper action (Cartan, Bourbaki and Palais) and the relation between them. All the spaces I'm working with are $T\_{3.5}$, the definitions are:
If $U$ and $V$ are subsets of a $G$-space $X$ then we say that $U$ is **thin relative** to $V$ if $\{g \in G \; : \; gU \cap V \neq ... | https://mathoverflow.net/users/112651 | Action that is Bourbaki proper but not Palais proper | Let $M$ and $N$ be smooth finite dimensional manifolds with $M$ compact, and let $\dim(N)\ge \dim(M)$. Let $\text{Imm}(M,N)$ be the space of smooth immersions $N\to N$, which is a smooth manifold modelled on spaces $\Gamma(f^\*TN)$ of smooth sections alonf immersions. Consider the regular Frechet Lie group $\text{Diff}... | 3 | https://mathoverflow.net/users/26935 | 296139 | 130,162 |
https://mathoverflow.net/questions/294095 | 2 | Consider a dynamical system described by the following coupled non-linear differential equation
\begin{align}
\dot{x}\_1(t) &= v + a\_{12}\sin(x\_2(t)-x\_1(t)) + a\_{13}\sin(x\_3(t)-x\_1(t))\\
\dot{x}\_2(t) &= w + a\_{21}\sin(x\_1(t)-x\_2(t)) + a\_{23}\sin(x\_3(t)-x\_2(t))\\
\dot{x}\_3(t) &= w + a\_{31}\sin(x\_1(t)-x\_... | https://mathoverflow.net/users/62673 | On local attractivity of a coupled non-linear differential equation | Since the RHS is $2 \pi$-periodic in all variables, one can consider it on the three-dimensional torus $(\mathbb{R}/2 \pi \mathbb{Z})^3$.
Assume $a\_{21} = a\_{31}$. Then the two-dimensional torus
$$
T := \{\, (x\_1, x\_2, x\_2): x\_1, x\_2 \in \mathbb{R}/2 \pi \mathbb{Z} \,\}
$$
is an invariant submanifold. To inves... | 2 | https://mathoverflow.net/users/121784 | 296143 | 130,163 |
https://mathoverflow.net/questions/296076 | 1 | Let $L$ be a two-dimensional subspace in the space of all linear operators from $\mathbb{E}$ to $\mathbb{F}$ ($\mathbb{E}$ and $\mathbb{F}$ are linear spaces, possibly finite-dimensional). Let $A,B,C$ be three non-proportional (i. e. every two of them form a basis for $L$) linear operators from $L$. It is easy to prove... | https://mathoverflow.net/users/85336 | Rank of Operators in a Two-Dimensional Operator Subspace | A generalization should be like this.
**Proposition.** Let $L$ be a two-dimensional subspace in the space of all linear operators from $\mathbb{E}$ to $\mathbb{F}$ ($\mathbb{E}$ and $\mathbb{F}$ are finite-dimensional linear spaces). Suppose there are $k+2$ pairwise non-proportional operators $A\_{1},\ldots,A\_{k+2}$... | 1 | https://mathoverflow.net/users/85336 | 296149 | 130,167 |
https://mathoverflow.net/questions/296148 | 1 | I am trying to read [this paper](http://arxiv.org/abs/math/0611317) by Lawrence Breen.
It starts with the definition of a torsor.
>
> Let $G$ be a bundle of groups on a space $X$. The following definition of a principal space is standard, but note the occurrence of structural bundle of groups,rather than simple a... | https://mathoverflow.net/users/118688 | Notion of Torsors |
>
> I do not understand what does it mean to say bundle of groups on a space? Does it mean that G as a set is disjoint union on groups indexed bybelements of X?
>
>
>
Presumably there's also a condition that the groups "vary continuously" in some sense, just like a vector bundle isn't any disjoint union of arbit... | 5 | https://mathoverflow.net/users/6427 | 296151 | 130,168 |
https://mathoverflow.net/questions/296082 | 3 | Let $G$ be a primitive group acting on a set $\Omega$ with $n$ elements. By Cameron/Liebeck (essentially a consequence of the Classification + O'Nan-Scott), there are two possibilities:
(a) $G$ has a subgroup of index $\leq n$ isomorphic to an alternating group,
(b) $G$ is of size $\leq n^{O(\log n)}$.
In case ... | https://mathoverflow.net/users/398 | Length of composition series in a primitive group | To answer your new question, Theorem 1.3 of [this arXiv paper](https://arxiv.org/abs/1712.05520) proves that the composition length $c(G)$ of primitive $G \le S\_n$ satisfies
$$c(G) \le \frac{8}{3} \log\_2 n - \frac{4}{3}$$
and also identifies the examples in which equality holds.
| 6 | https://mathoverflow.net/users/35840 | 296152 | 130,169 |
https://mathoverflow.net/questions/296154 | 3 | An operator $T\colon X\rightarrow Y$ is said to be strictly cosingular provided that for no infinite-dimensional Banach space $Z$ there exist surjective operators $R\colon X\rightarrow Z$ and $S\colon Y\rightarrow Z$ such that $R=ST$; Equivalently, there is no infinite-codimensional subspace $V$ of $Y$ such that $Q\_{V... | https://mathoverflow.net/users/41619 | Strictly cosingular operators and $l_{1}$-strictly cosingular operators into $L_{1}[0,1]$ | Yes, they are the same. Pełczyński proved that strictly singular, strictly $\ell\_1$-singular, strictly cosingular, and weakly compact operators on $L\_1$ are all the same. This is Theorem 1 in
>
> A. Pełczyński, On strictly singular and strictly cosingular operators. II. Strictly singular and strictly cosingular o... | 2 | https://mathoverflow.net/users/15129 | 296155 | 130,170 |
https://mathoverflow.net/questions/296159 | 11 | We investigate the Hilbert space $\ell^2(\mathbb{N}\_0)$ with standard orthonormal basis vectors $e\_n:=(0,...,0,1,0,...).$
Consider the family of self-adjoint rank $1$ projections $P\_n\bullet:= \langle \bullet,e\_n \rangle e\_n.$
Take any $n\in\mathbb{N}\_0$. My question is this: Does there exist a bounded linear... | https://mathoverflow.net/users/119875 | Operator that commutes with projections | Maybe a quicker way to see this is, if $TP\_m = P\_mT$ for all $m \neq n$ then $TP = PT$ where $P =\sum\_{m\neq n} P\_m = I - P\_n$. Since $T$ commutes with $I$, it must therefore commute with $P\_n$.
| 17 | https://mathoverflow.net/users/23141 | 296168 | 130,176 |
https://mathoverflow.net/questions/296099 | 3 | I am looking for an example of a group ring $\mathbb{Z}[G]$ of a finite group $G$ along with a lattice $I$ (in the case at hand the word 'lattice' means: a $\mathbb{Z}[G]$-submodule which is additionally a finitely generated free $\mathbb{Z}$-subalgebra [corrected]) such that
\begin{equation}
M:=\mathbb{Q}[G] / I
\end{... | https://mathoverflow.net/users/122368 | Looking for example of quotient of group algebra by ideal of group ring which fails to be injective | So long as $G$ is nontrivial, the augmentation ideal of $\mathbb{Z}[G]$ still works.
If $I$ is any submodule of $\mathbb{Z}[G]$ then there is a short exact sequence of $\mathbb{Z}[G]$-modules
$$0\to\mathbb{Z}[G]/I\to\mathbb{Q}[G]/I\to\mathbb{Q}[G]/\mathbb{Z}[G]\to0.$$
The last term is always injective, so if the midd... | 2 | https://mathoverflow.net/users/22989 | 296169 | 130,177 |
https://mathoverflow.net/questions/296162 | 14 | Let $X$ be a compact complex smooth manifold with holomorphically trivial canonical class.
>
> It is true that any (sufficiently small?) deformation of the complex structure of $X$ also has holomorphically trivial canonical class?
>
>
>
| https://mathoverflow.net/users/16183 | Deformations of Calabi-Yau manifolds | The answer in general is no. Nakamura has constructed [here](http://projecteuclid.org/euclid.jdg/1214432677) (pp.90, 96-99, solvmanifolds of type III-(3b)) an example of a compact complex (non-Kähler) manifold $M$ with $TM$ holomorphically trivial (so in particular $K\_M$ is holomorphically trivial) which has arbitrari... | 23 | https://mathoverflow.net/users/13168 | 296172 | 130,178 |
https://mathoverflow.net/questions/200242 | 9 | What is known about the decidability of (first-order formulas in) the structure $(\mathcal{L}(H),\leq)$, where $\mathcal{L}(H)$ is the collection of all closed linear subspaces of a (separable) Hilbert space $H$, and $X\leq Y$ means $X\subseteq Y$? (Clearly meets and joins always exist and are first-order definable, so... | https://mathoverflow.net/users/16107 | Decidability of the Hilbert lattice and quantum logic | A year after you posted the question, Fritz showed the common theory of all such lattices is undecidable:
<https://arxiv.org/abs/1607.05870>
---
In reponse to @MattF's query I'll post an example of how infinite dimension differs from finite. Namely, the lattice of closed subspaces is modular only in the finite-di... | 7 | https://mathoverflow.net/users/4600 | 296175 | 130,179 |
https://mathoverflow.net/questions/296171 | 13 | **Do you know a good reference for the existence and uniqueness of a smooth structure on $3$-manifolds?**
As far as I understand topological $3$-manifolds admit a unique smooth structure.
I could find the following references for this result:
It follows from Hauptvermutung for $3$-manifolds (Theorems 3 and 4 in [2]... | https://mathoverflow.net/users/121665 | Unique smooth structure on 3-manifolds | An alternative to Moise's paper for the existence and uniqueness of piecewise linear (PL) structures on topological 3-manifolds is the paper "The triangulation of 3-manifolds" by A.J.S. Hamilton in Quart. J. Math. Oxford (2), 27 (1976), 63-70. The result is stated as Theorem 2 there and proved in the rest of the paper ... | 17 | https://mathoverflow.net/users/23571 | 296190 | 130,184 |
https://mathoverflow.net/questions/296009 | 2 | For a Lie group $G$ with compact Lie subgroup $K$, we say that $(G,K)$ is a pair of Gelfand type if the representation $L^2(G/K)$ of $G$ is multiplicity free, that is, if it is a direct integral of distinct irreducible representations.
Can there exist a pair of dual representations in $L^2(G/K)$ for a Gelfand pair $... | https://mathoverflow.net/users/90430 | Gelfand pairs and (self)-dual representations | I gave in the comments the example of the Gelfand pair $(G,K) = (\mathbf R/\mathbf Z,0)$ for which every non-trivial irreducible representation arises together with its (distinct) dual representation. But actually this is completely general, at least when $G$ is compact: the representation $L^2(G/K)$ is self-dual. So b... | 3 | https://mathoverflow.net/users/10265 | 296199 | 130,187 |
https://mathoverflow.net/questions/296111 | 1 | If $f(z)$ is an entire function of exponential type $\tau$ and $p$ a positive number such that that $$\int\_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$
then it can be proven that $$\int\_{-\infty}^{+\infty}|f(x+iy)|^pdx\leqslant e^{\tau p|y|}\int\_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$ for all $y$.
I was wondering if such... | https://mathoverflow.net/users/39180 | Plancharel-Pólya inequality for functions of exponential type | Yes, this is true. This is Theorem 11 on chapter 2 (page 82) in R.Young's book (An introduction to nonharmonic Fourier analysis). The proof is based on an application of the Phragmén–Lindelöf principle.
| 1 | https://mathoverflow.net/users/16040 | 296214 | 130,191 |
https://mathoverflow.net/questions/293967 | 3 | Let
* $U$, $H$, $\tilde H$ be infinite-dimensional separable $\mathbb R$-Hilbert spaces
* $Q$ be a self-adjoint and nonnegative nuclear linear operator on $U$
* $\Psi$ be a Hilbert-Schmidt operator from$^1$ $Q^{1/2}U$ to $H$
* $\tilde Q:=\left(\Psi Q^{1/2}\right)\left(\Psi Q^{1/2}\right)^\ast$
* $\Phi$ be a Hilbert-S... | https://mathoverflow.net/users/91890 | Estimate for the composition of two Hilbert-Schmidt operators | We'll need the following fact: Let $U\_i,H$ be $\mathbb R$-Hilbert spaces and $A\_i\in\mathfrak L(U\_i,H)$ with $$A\_1A\_1^\ast=A\_2A\_2^\ast.\tag4$$ Then, $$\iota\_{12}:=A\_2^{-1}A\_1\operatorname P\_{(\ker A\_1)^\perp}$$ and $$\iota\_{21}:=A\_1^{-1}A\_2\operatorname P\_{(\ker A\_2)^\perp}$$ are well-defined partial i... | 0 | https://mathoverflow.net/users/91890 | 296223 | 130,195 |
https://mathoverflow.net/questions/296048 | 3 | It is known that under $MA+ \neg CH$, every Corson compact space with the countable chain condition (ccc) is merizable. It is also known that, under $CH$, there exist nonmetrizable Corson compact spaces with ccc. Is it consistent with $ZFC+ \neg CH$ the existence of a nonmetrizable Corson compact space
with ccc?
| https://mathoverflow.net/users/122189 | Nonmetrizable Corson compacta with ccc | Yes, take any model of $\neg CH$ where a Suslin Line exists (for example, start with a ground model where $CH$ does not hold and use Tennenbaum's original forcing for adding a Suslin Tree).
We can assume that our Suslin Line is compact. Indeed, let $Y$ be its Dedekind completion. Then $Y$ is a compact ccc linearly o... | 5 | https://mathoverflow.net/users/11647 | 296228 | 130,196 |
https://mathoverflow.net/questions/292405 | 7 | I am reading Brinon, Conrad "Notes on $p$-adic Hodge theory" and I can't find any reference for the proof of Theorem 9.1.8, namely the injectivity of the Frobenius endomorphism of $A\_{cris}$. Does anyone know where to find it or how to prove it?
| https://mathoverflow.net/users/118305 | Injectivity of Frobenius on $A_{cris}$ | $\newcommand{\Z}{\mathbb{Z}}$
$\def\cO{\mathcal{O}}$There seems to be a suspiciously straightforward proof by analyzing the Witt coordinates of the elements of $A\_{cris}$ so there might well be a mistake here.
The ring $A\_{cris}$ is the $p$-adic completion of the divided power envelope of the ideal $(\xi)$ in the r... | 2 | https://mathoverflow.net/users/39304 | 296230 | 130,197 |
https://mathoverflow.net/questions/296222 | 3 | The following inverse semigroup associated to a directed graph came up in my research. I've read that from an inverse semigroup one may derive a $C^\*\!$-algebra whose generators are partial isometries (but I don't yet understand how to do it). I am aware of the existence of the well-known $C^\*\!$-algebras derived fro... | https://mathoverflow.net/users/7227 | an inverse semigroup (and perhaps a $C^*\!$-algebra) associated with a directed graph | Your inverse monoid without the partial sum relations seems like a variation of McAlister's monoid [here](https://ac.els-cdn.com/S0021869397973014/1-s2.0-S0021869397973014-main.pdf?_tid=2c958ec1-ed8e-4e03-b119-ed7302662dbd&acdnat=1522084399_8160c3d8c4cc28c11587b91f24bcf4c8) but for paths in a graph instead of words ove... | 1 | https://mathoverflow.net/users/15934 | 296234 | 130,198 |
https://mathoverflow.net/questions/296229 | 0 | Let $f(z)$ be a holomorphic function defined on the disk $|z|\le 2$. Suppose $|f(z)|<1$ for $|z|\le 2$. It looks like there is a constant $c>0$ such that $|f(z)'|<c$ on the disk $|z|\le 1$ (for example, $c=1$?). I wonder if this is true. If yes, is an optimal such constant can be found?
| https://mathoverflow.net/users/13441 | Bounding the derivative of a holomorphic function on a disk by its absolute value | Yes, this is true. The simple reason is that bounded functions form a normal family. Therefore their derivatives are uniformly bounded on every compact.
To obtain the estimate $|f'(z)|<1$, apply Cauchy theorem:
$$|f'(z)|=\left|\frac{1}{2\pi}\int\_{|\zeta-z|=1}\frac{f(\zeta) d\zeta}{(\zeta-z)^2}\right|\leq 1.$$
Equali... | 2 | https://mathoverflow.net/users/25510 | 296242 | 130,199 |
https://mathoverflow.net/questions/296245 | 2 | Let $p\_1,p\_2,...,p\_n$ are given probabilities. ($\sum\_{i=1}^n p\_i =1, p\_i \geq 0 $). Is there any distribution, which picks $k\leq n$ distinct elements from $1,2,...,n$ such that $P(i \in S) = k p\_i$ and $(i\neq j)$ $P(i \in S, j \in S) = c p\_i p\_j$, where S is our distribution realization and $c$ is some cons... | https://mathoverflow.net/users/119108 | Sampling with non-uniform probabilities | Usually not. Denote $f\_i=\mathbb{1}\_{i\in S}$, then $\mathbb{E} f\_i=\mathbb{E} f\_i^2=kp\_i$, $\sum f\_i\equiv k$, $\mathbb{E} f\_if\_j=cp\_ip\_j$ if $i\ne j$. Thus
$$k^2p\_i=k\mathbb{E} f\_i=\mathbb{E} \sum\_j f\_if\_j=cp\_i(1-p\_i)+kp\_i.$$
If $p\_i>0$, this gives $c(1-p\_i)=k^2-k$, and if $p\_i\ne p\_j$ and $k>1$... | 1 | https://mathoverflow.net/users/4312 | 296249 | 130,201 |
https://mathoverflow.net/questions/296261 | 9 | Is there an example of a rational smooth projective variety over a perfect field of characteristic $p$, that is not liftable to characteristic zero?
| https://mathoverflow.net/users/nan | Liftable rational varieties | Two such examples were given by Achinger and Zdanowicz [AZ17], both of which satisfy a whole bunch of other good properties (e.g. their classes in the Grothendieck ring of varieties are polynomials in the Lefschetz motive $\mathbb L = [\mathbb A^1]$). The easiest one to state is probably the following:
**Example.** (... | 11 | https://mathoverflow.net/users/82179 | 296264 | 130,207 |
https://mathoverflow.net/questions/296212 | 2 | Let $A$ be a generalized Cartan matrix and let $\mathfrak{g}$ be the Kac-Moody Lie algebra associated to $A$. There is an associated graph of $\mathfrak{g}$ which is known as the Dynkin diagram of $\mathfrak{g}$ (At least in the affine case I know this is true).
My question is which Lie super algebras has an associa... | https://mathoverflow.net/users/33047 | Graph of a Lie super algebra | See §15 in [Dictionary on Lie Superalgebras](https://arxiv.org/abs/hep-th/9607161), by L. Frappat, A. Sciarrino, P. Sorba. They define the Dynkin diagram of basic Lie superalgebras (i.e., those with an even nondegenerate invariant bilinear form, and whose even part is reductive).
| 3 | https://mathoverflow.net/users/106114 | 296277 | 130,209 |
https://mathoverflow.net/questions/296268 | 2 | Apologies if this question might be trivial or has been asked already (haven't found an equivalent post), but I am trying to figure out whether the following is true:
Given two convex sets $\mathcal{X} \subseteq \mathbb{R}^n$ and $\mathcal{Y} \subseteq \mathbb{R}^n$, is $\mathcal{Z} := \{x \odot y ~|~ x \in \mathcal{... | https://mathoverflow.net/users/122451 | "Minkowski Multiplication" of Convex Sets? | No. Consider $X=Y=\{(u,u+1): u\in R^{\ge 0}\}$. Then $(0^2,1^2)$ and $(2^2,3^2)$ are both in $Z$. If $Z$ is convex then their average $(2,5)$ must also be in $Z$. But $uv=2, (u+1)(v+1)=5$ has no real solutions, so $(2,5)$ is not in $Z$ and $Z$ is not convex.
UPDATE: Even if $X$ and $Y$ are required to contain the ori... | 4 | https://mathoverflow.net/users/nan | 296278 | 130,210 |
https://mathoverflow.net/questions/296051 | 5 | In Nekovar's introductory paper "Beilinson's Conjecture"
<http://math.stanford.edu/~conrad/BSDseminar/refs/BeilinsonintroII.pdf>
The conjecture is formulated for smooth projective varieties over $\mathbb{Q}$. However all the statements and proofs seem to equally work even for smooth projective varieties defined ove... | https://mathoverflow.net/users/87910 | Does Beilinson's conjecture on values L-functions work for smooth projective varieties over a number field | Yes, you can formulate Beilinson's conjectures for smooth projective varieties over a number field. I would recommend the following survey paper by Dinakar Ramakrishnan: *Regulators, algebraic cycles and values of $L$-functions*. Other references, with an emphasis on the equivariant version of the conjecture, include: ... | 6 | https://mathoverflow.net/users/6506 | 296288 | 130,213 |
https://mathoverflow.net/questions/296289 | 4 | Given a non-trivial group $G$ and $g\in G\setminus \{e\_G\}$ where $e\_G$ is the neutral element, it is easy to show using Zorn's Lemma, that there is a subgroup not containing $g$ that is maximal amongst the subgroups contained in $G\setminus\{g\}$.
What is an example of infinite groups $G, H$ with $G\not \cong H$ a... | https://mathoverflow.net/users/8628 | Maximal subgroups not containing a specific element | Olshanskii showed that for $p>10^{75}$, there are continuumly many nonisomorphic Tarski monsters. These are countably infinite groups whose proper nontrivial subgroups are cyclic of order $p$.
On any group $\Gamma$ one can define an equivalence relation $E\_{\Gamma}$ which relates two elements if they generate the sa... | 13 | https://mathoverflow.net/users/75735 | 296291 | 130,214 |
https://mathoverflow.net/questions/296295 | 0 | Let $G$ be an infinite vertex-transitive graph (this means that for every $u,w \in V(G)$ there exists an automorphism $\tau$ of $G$ such that $\tau(u) = v$).
We assume that $G$ is undirected, and does not have loops.
Suppose that there exists some $B \in \mathbb{N}$ such that $G$ does not contain a clique of size $B$... | https://mathoverflow.net/users/38889 | coloring infinite vertex transitive graph without large cliques | Apart of the specific Mycielski-like construction from the article in Dominic's answer, we may take the **universal triangle free-graph**, which satisfies the following conditions:
(i) $G$ has countable number of vertices;
(ii) $G$ does not contain triangles;
(iii) for any finite set $V\_0$ of vertices of $G$ and... | 6 | https://mathoverflow.net/users/4312 | 296299 | 130,217 |
https://mathoverflow.net/questions/296225 | 9 | If $\kappa$ is a singular cardinal, a *scale for $\kappa$* consists of an increasing sequence $\langle \kappa\_i : i < \mathrm{cf}(\kappa) \rangle$ converging to $\kappa$ and a sequence of functions $\langle f\_\alpha : \alpha < \kappa^+ \rangle$ that is linearly ordered and dominating in the partial order of $\prod\_{... | https://mathoverflow.net/users/11145 | PCF theory and good points in scales | Jing is correct in stating that the result follows from the referenced results in the Abraham-Magidor handbook chapter.
A general theorem, which can be proven in the same way, is the following result:
**Theorem:** Suppose that $\kappa$ is a singular cardinal and $\vec{f} = \langle f\_\alpha \mid \alpha < \lambda \r... | 8 | https://mathoverflow.net/users/26002 | 296303 | 130,219 |
https://mathoverflow.net/questions/296292 | 5 | I want to interpret the degree of the field of rationality of an automorphic form as a notion of size, analogously to the conductor, and this question is about the possible obstructions to do so. The results I know are limited to the case of Hecke cusp forms, I hense recall them and raise the natural questions and resu... | https://mathoverflow.net/users/43737 | Fields of rationality as a notion of automorphic size | I haven't thought about the case of Maass forms, but I can tell you what is expected about the weight.
*Maeda's conjecture* implies that all cuspidal eigenforms in $S\_k(1)$ are conjugate, so $\bar d(\pi)$ should be $\dim S\_k(1)$, where $\bar d$ denotes the degree of the Galois closure. In particular, the answer to ... | 3 | https://mathoverflow.net/users/6518 | 296308 | 130,221 |
https://mathoverflow.net/questions/296003 | 3 | Let $T$ be a complete theory (say in a countable language, though this may not be required). Assume that $T$ is complete with infinite models.
Question 1) When is the theory of $T$ an almost sure theory? What are the references for these results.
Note that I have left the term "almost sure" undefined. This is beca... | https://mathoverflow.net/users/nan | When are pseudofinite theories almost sure theories? | In this answer, let's only think about theories in countable relational languages.
Most of the time, when people say that $T$ is an "almost-sure theory", they mean it arises as the limit theory for some sequence of probabilistic constructions which admits a zero-one law. We could formalize this by asking for a seque... | 1 | https://mathoverflow.net/users/2126 | 296317 | 130,225 |
https://mathoverflow.net/questions/296324 | 2 | Let $\eta$ be a continuous bounded function on $(0, \infty)^{2}$ so that $\eta(0,0)=1$. Let $A$ be a bounded operator on $\ell^{2}(\mathbb{Z}\_{\geq 0})=\ell^{2}$ (by bounded operator I will always mean such an object) and let $A\_{j,k}$ be its "matrix entries" with respect to the standard basis of $\ell^{2}$.
Suppos... | https://mathoverflow.net/users/122479 | Convergence of sequence of images of Schur multipliers | By the uniform bound on $\|A^{(N)}\|$ and linearity, SOT convergence follows from the $\ell^2$-norm convergence, for every $i$, of the $i$-th column $A^{(N)} e\_i$ to the $i$-th column $A e\_i$. This convergence is straightforward (say by the dominated convergence theorem).
| 4 | https://mathoverflow.net/users/10265 | 296326 | 130,228 |
https://mathoverflow.net/questions/296325 | 0 | Let $F$ be a finite Galois extension of the rational function field $\mathbb Q(x)$. Let $k$ be the field of constants of $F$, i.e., the algebraic closure of $\mathbb Q$ in $F$. Is $k$ necessarily a Galois extension of $\mathbb Q$?
| https://mathoverflow.net/users/46987 | Field of constants of a Galois extension of function fields | Fix an embedding of $F$ in $\overline{\mathbb Q(x)}$. Then $F$ is Galois over $\mathbb Q(x)$ if and only if every automorphism $\sigma$ of $\overline{\mathbb Q(x)}$ over $\mathbb Q(x)$ satisfies $\sigma(F)=F$. Since $\sigma$ sends $\overline{\mathbb Q}$ to $\overline{\mathbb Q}$, doesn't that immediately imply that $\s... | 3 | https://mathoverflow.net/users/11926 | 296327 | 130,229 |
https://mathoverflow.net/questions/296302 | 0 | Let $n,k\in\mathbb N$, $x\in(0,1/2)$.
You start $n$ empty bins; each can accommodate at most $k$ balls.
At each iteration, you choose an $x$ fraction of the *non-full* bins and add one ball to each. (if this number is not an integer, pick a subset that is larger than an $x$ fraction.)
>
> **How to maximize the ... | https://mathoverflow.net/users/47499 | How to play the following game? | I don't have a proof at the moment that your strategy is the optimum one--I suspect that it is though. For now at least though, We can show your bound is at least, asymptotically optimum.
On the one hand, if $m$ is the number of non-full bins at the start of round $i$, then you will be adding $xm$ balls. But at most ... | 1 | https://mathoverflow.net/users/122188 | 296335 | 130,231 |
https://mathoverflow.net/questions/296353 | 2 | So apparently the [Krylov-Bogoliubov theorem](https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogolyubov_theorem) says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$.
Of course, if $X$ is just a single point then there's only one such $\mu$. Also, i... | https://mathoverflow.net/users/4600 | Non-uniqueness in Krylov-Bogoliubov theorem | Not at all. Take $f(x)=x^2$ on the unit interval. For examples of uniquely ergodic homeomorphisms of the Cantor set see [The prevalence of uniquely ergodic systems](https://mathscinet.ams.org/mathscinet-getitem?mr=252604) by Jewett. A simple explicit example is provided by the boundary action of any hyperbolic automorp... | 4 | https://mathoverflow.net/users/8588 | 296356 | 130,235 |
https://mathoverflow.net/questions/296351 | 3 | This question was raised in the comment by Todd Trimble at [how to proof there is a natural number n, the first four digits of n! Is 2018?](https://mathoverflow.net/questions/296321/how-to-proof-there-is-a-natural-number-n-the-first-four-digits-of-n-is-2018). I thought the question may be posted separately, as even par... | https://mathoverflow.net/users/36721 | Is the sequence $(\log(n!)\mod1)_{n\in\mathbb N}$ dense in the interval $[0,1]$? | Let $a$ be the base of logarithm here. Consider the function $$f(x)=\frac{(x+1/2)\ln x-x}{\ln a}.$$
As noted in the question, it is enough to prove that the sequence $f(n)$ is dense modulo $1$. In fact, this sequence is equidistributed. By Weyl's criterion, it is enough to show that for any nonzero integer $k$ we hav... | 6 | https://mathoverflow.net/users/101078 | 296357 | 130,236 |
https://mathoverflow.net/questions/296350 | 14 | Recall that we say that a bounded measurable set $S\subset\mathbb R^n$ is said to be *Caccioppoli* if the indicator function $1\_S$ is BV, and we set
$$
\operatorname{perim}(S)=\| \nabla 1\_S\|\_{TV}
$$
where $\|\cdot\|\_{TV}$ denotes the total variation. So, if $S$ and $T$ are Caccioppoli sets, is it known whether $S\... | https://mathoverflow.net/users/94022 | Is the intersection of two Caccioppoli (i.e. finite perimeter) sets Caccioppoli? | **That is true.** Caccioppoli sets are also known as sets of finite perimeter.
>
> **Theorem.** *Suppose $f\in L^1(\mathbb{R}^n)$ vanishes outside the unit cube $[0,1]^n$. For $i=1,2,\ldots,n$ consider the function
> $V\_if(x\_1,\ldots,x\_{i-1},x\_{i+1},\ldots,x\_n)=
> V\_0^1f(x\_1,\ldots,x\_{i-1},\cdot,x\_{i+1}... | 14 | https://mathoverflow.net/users/121665 | 296360 | 130,239 |
https://mathoverflow.net/questions/296273 | 5 | Let $n>1$ be an integer. Consider the set $C\_n := \{0,1, \dots , n-1\}$.
An *Eulerian ordering* of $C\_n$ is an ordering $r\_1, \dots, r\_n$ of its elements such that:
$$\forall i \le n \ \forall j<i \ \exists k < i \text{ with } \frac{n}{gcd(n,r\_k-r\_i)} \text{ prime and } \frac{gcd(n,r\_k-r\_i)}{gcd(n,r\_j-r\... | https://mathoverflow.net/users/34538 | Eulerian ordering of the integers modulo n | The basic idea is to separate out the action of each prime, to the maximal extent possible. In the example given in the above comments, the idea was to ignore (the remainder modulo) $2$ as long as possible, ignore (the remainder modulo) $3$ as long as possible given that $2$ was being ignored, etc. I was inspired by th... | 4 | https://mathoverflow.net/users/44191 | 296366 | 130,242 |
https://mathoverflow.net/questions/296271 | 7 | What is currently known about lifts of tropical varieties to varieties over $\mathbb{Q}\_p$ or its extensions? Starting with an appropriate rational polyhedral cone complex what are the obstructions to its deformation to a (toric or log-smooth) variety over a $p$-adic field classified by? In which major cases are they ... | https://mathoverflow.net/users/nan | $p$-adic lifts of tropical varieties | There are moduli spaces of tropical lifts. They satisfy "Murphy's law": Any behavior which can happen on a scheme of finite type can happen on them. See Katz and Payne "[Realization spaces for tropical fans](https://arxiv.org/abs/0909.4582)" for the details.
| 3 | https://mathoverflow.net/users/297 | 296367 | 130,243 |
https://mathoverflow.net/questions/296382 | 8 | I heard about this problem an year ago, but I just can't remember the name.
The problem goes like this: study the sets
>
> $\{a\_1,a\_2,\dotsc,a\_m\}\subseteq\mathbb{N}$ such that if $1\leq i<j\leq m$, then $a\_i a\_j+1$ is a perfect square.
>
>
>
Is there a technical term for such sets?
| https://mathoverflow.net/users/122510 | How are such sets of natural numbers called? | Sets of $m$ integers with this property are called *integer Diophantine $m$-tuples*. A good starting point to learn about them (and about *rational* Diophantine $m$-tuples) is [this paper](https://arxiv.org/pdf/1507.00569.pdf) by
Dujella, Kazalicki, Mikic, and Szikszai.
| 15 | https://mathoverflow.net/users/9924 | 296385 | 130,245 |
https://mathoverflow.net/questions/44737 | 28 | I have been thinking about this question for quite some time but now [this](https://mathoverflow.net/questions/44680/norm-of-commutators-bis) question by Denis Serre revived some hope.
**Question.** Let $x,y$ be invertible matrices (say, over $\mathbb C$) and $[x,y,y]=x$ where $[a,b]=a^{-1}b^{-1}ab$, $[a,b,c]=[[... | https://mathoverflow.net/users/nan | Invertible matrices satisfying $[x,y,y]=x$ | The answer is "No". Indeed, consider the 1-related group $G=\langle x,y \mid [x,y,y]=x\rangle$ That group has a presentation $\langle a,b,t \mid a^t=ab, b^t=ba\rangle$ (easy to check). Thus it is an ascending HNN extension of the free group. The group $G$ is hyperbolic (proved by Minasyan using the Bestvina-Feighn comb... | 13 | https://mathoverflow.net/users/nan | 296388 | 130,248 |
https://mathoverflow.net/questions/295904 | 3 | For $0<\alpha<n$ and $n\geq 2$ we define the *Riesz potential* by
$$
(I\_\alpha f)(x) = \frac{1}{\gamma(\alpha)}
\int\_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\alpha}}\, dy\, ,
\quad
\text{where}
\quad
\gamma(\alpha)=
\frac{\pi^{\frac{n}{2}}\, 2^\alpha\,\Gamma\left(\frac{\alpha}{2}\right)}
{\Gamma\left(\frac{n-\alpha}{2}\r... | https://mathoverflow.net/users/121665 | Composition of Riesz potentials | This is an extended version of my comment.
---
Let $g\_t$ be the Gauss–Weierstrass kernel,
$$
g\_t(x) = \frac{1}{(4 \pi t)^{n/2}} \, e^{-|x|^2/(4t)} .
$$
If $\alpha \in (0, n)$, we have
$$
\begin{aligned}
\frac{1}{\Gamma(\tfrac{\alpha}{2})} \int\_0^\infty g\_t(x) t^{\alpha/2 - 1} dt & = \frac{1}{2^n \pi^{n/2} \... | 3 | https://mathoverflow.net/users/108637 | 296396 | 130,251 |
https://mathoverflow.net/questions/296399 | 3 | Is there an established name for graphs, that can be decomposed into
* a tree with at least three leaf nodes and
* a connected two-regular graph with the tree's leaf nodes as vertices?
examples of those graphs are the edge-graphs of polyhedra with one facet, that is edge-adjacent to all other facets.
| https://mathoverflow.net/users/31310 | Name for Biconnected Tree+Cycle Graph | Ok, since it is close enough for the OP (as evidenced by the comments) I will transfer my comment to an answer so that the question can be neatly wrapped up.
So a **Halin graph** (named after Rudolf Halin) is built from a tree with no vertices of degree 2 that is embedded in the plane and whose leaves are then connec... | 4 | https://mathoverflow.net/users/1492 | 296414 | 130,259 |
https://mathoverflow.net/questions/296425 | 16 | Let $K$ be a number field. Is it necessarily true that $\mathbb{Q}$ is a first-order definable subset of $K$? Equivalently (since in any number field, its ring of integers is a definable subset), is $\mathbb{Z}$ necessarily a definable subset of $K$, or of $\cal{O}\_K$?
Edit: And if not, is $\mathbb{Q}$ always interp... | https://mathoverflow.net/users/83073 | Are the rationals definable in any number field? | According to R. S. Rumely, Undecidability and Definability for the theory of global fields, AMS Trans., 262, pp. 195-217, prime subfield is always definable in global field, and in number case, you can define $\Bbb N$.
| 12 | https://mathoverflow.net/users/81055 | 296427 | 130,263 |
https://mathoverflow.net/questions/296430 | 2 | Let $D(A)$ be the derived $(\infty,1)$-category of some abelian category $A$. For which $A$ is $D(A)$ locally cartesian closed?
Replace $D$ with $D^b$ or similar if appropriate.
I essentially want to show that for any $f:x\to y$ in $D(A)$, the pullback functor $f^\*: D(A)/y\to D(A)/x$ has a right adjoint $f\_\*$. H... | https://mathoverflow.net/users/122538 | When is the derived category $D(A)$ locally cartesian closed? | This will almost never happen. Since $D(A)$ has a terminal object 0, if it's locally cartesian closed, then it's also cartesian clsoed. To be cartesian closed means that $x \oplus (-) : D(A) \to D(A)$ is a left adjoint, and in particular preserves colimits. In particular, it preserves the initial object: $x \oplus 0 = ... | 5 | https://mathoverflow.net/users/2362 | 296432 | 130,264 |
https://mathoverflow.net/questions/296426 | 2 | Suppose there is a random variable, $X$, with finite variance, and c.d.f. $F(x)$. Does this imply that the upper Matuszewska index of $\bar F(x)$ exists and is strictly smaller than $-2$?
The upper Matuszewska index of $f(x)$ is defined as the infimum of $\alpha$ such that there exists a $C$ such that for each $\Lamb... | https://mathoverflow.net/users/60836 | Matuszewska Index and finite variance | The answer is no: $\bar F$ does not have to have any negative Matuszewska index (any tail function $\bar F$ trivially has any nonnegative Matuszewska index).
Indeed, take any negative real $\alpha$.
Suppose that $P(X=n^n)=c/n^{3n}$ for natural $n$, where $c:=1/\sum\_1^\infty1/n^{3n}$. Then $EX^2<\infty$. However,
... | 1 | https://mathoverflow.net/users/36721 | 296435 | 130,267 |
https://mathoverflow.net/questions/296440 | 3 | Consider Propositional Lax Logic ($PLL$)
* <https://www.uni-bamberg.de/fileadmin/uni/fakultaeten/wiai_professuren/grundlagen_informatik/papersMM/pll.pdf>
The Hilbert system of $PLL$ takes as axiom schemata all theorems of (or a complete set of axioms for) the Intuitionistic propositional calculus plus the modal axi... | https://mathoverflow.net/users/122435 | Modal collapse upon addition of the law of the excluded middle to an Intuitionistic modal logic | I'll write $\to$ instead of $\supset$, and $\bot$ instead of false, below.
Since Law of Excluded Middle is given, I'll argue using classical propositional logic.
Since $M\to\bigcirc M$ is already given as Axiom $\bigcirc$R, let's prove $$\bigcirc M\to M.$$
We are given
$$\neg\bigcirc\bot.\tag{\*}$$
First, by Axiom $\... | 2 | https://mathoverflow.net/users/4600 | 296446 | 130,271 |
https://mathoverflow.net/questions/278441 | 4 | First, some background: I wanted to prove that, if $f$ is a measurable function such that $\nabla f\in L^p\_\text{loc}(\mathbb R^n)$, then $f\in L^p\_\text{loc}(\mathbb R^n)$, $p\in(1,\infty)$. This is proven, for instance, in the book Sobolev Spaces by Vladimir Maz'ya, but I don't like the proof there. I was thinking ... | https://mathoverflow.net/users/96932 | What is the dual space of $L^p$(conservative vector fields on a bounded set)? | **Here is a short, elementary, and self-contained proof of the result you wanted to prove.** It is similar to the one given in Maz'ya's book, but simpler.
For a related post see: <https://mathoverflow.net/a/297392/121665>
>
> If $f\in L^1\_{\rm loc}$ or even if $f$ is a distribution and $\nabla
> f\in L^p$, then $f... | 3 | https://mathoverflow.net/users/121665 | 296464 | 130,278 |
https://mathoverflow.net/questions/283397 | 3 | The law of the hitting time of a 1-dimensional Brownian motion $W$ is well known, but I can't find any information on the density of the hitting time of $|W|$.
I define $T=\inf \{t>0,|W|(t)= 1\}$. One can find some useful informations about this random variables like its expectation 1, variance $\frac{2}{3}$ or Lapla... | https://mathoverflow.net/users/114223 | Does the hitting time of +1/-1 of a Brownian motion posess a density? | The law of $T$ (and more generally the law of hitting times of Bessel processes) has been studied extensively in the literature, in particular by Marc Yor. In the survey
[PROBABILITY LAWS RELATED TO THE JACOBI THETA AND
RIEMANN ZETA FUNCTIONS, AND BROWNIAN EXCURSIONS](http://www.ams.org/journals/bull/2001-38-04/S0273... | 4 | https://mathoverflow.net/users/48356 | 296466 | 130,279 |
https://mathoverflow.net/questions/296216 | 2 | I need a formula for the 2-adic valuation of the number of proper equivalence classes of primitive positive definite binary quadratic forms of discriminant $-D$, call it $h\_0(-D)$. I'm sure the answer is well-known, and in fact I have found it on slides for a talk somewhere (see [here](http://chengshantian.weebly.com/... | https://mathoverflow.net/users/49340 | 2-parts of class numbers of binary quadratic forms for non-fundamental discriminants | Pretty sure what you want is D. A. Buell, *Binary Quadratic Forms*. Chapter 9 is called The 2-Sylow Subgroup. Back in chapter 7, pages 117-118, he compares form class numbers $h(\Delta) $ and $h(\Delta p^2),$ also $h(4 \Delta)$
As indicated in comments, the number of genera is predictable (and is a power of 2). Howev... | 1 | https://mathoverflow.net/users/3324 | 296471 | 130,282 |
https://mathoverflow.net/questions/296394 | 4 | For integer $n>1$ define $q(n)=\frac{\log(\rm{rad}(n))}{\log(n)}$
where $\rm{rad}(n)$ is the radical of $n$, the product of the disctinct
prime factors.
For real $A$ and integer $N$ define $S\_{N,A}=\#\{n : 1 < n <N,q(n)\le A\}$.
$s(N,A)=\frac{S\_{N,A}}{N}$ and $s^\*(N,A)=\frac{\log(S\_{N,A})}{\log(N)}$.
Q1. Are ... | https://mathoverflow.net/users/12481 | Numbers up to $N$ with small radical | It may be more natural to consider
$$
N(x,y) = \# \{ n\le x: \text{rad}(n) \le y\},
$$
and the question is essentially about $N(x,x^{\alpha})$ (with $\alpha =1/2$ in your question 2). The quantity $N(x,y)$ has been analyzed in detail by [Robert and Tenenbaum](https://perso.univ-st-etienne.fr/rool6510/Nxy.pdf), and t... | 5 | https://mathoverflow.net/users/38624 | 296473 | 130,284 |
https://mathoverflow.net/questions/296460 | 10 | I would like to know if following is correct.
**Statement.** Suppose we have a smooth (i.e., $C^\infty$) almost complex structure on $\mathbb R^4$ and $C\_1, C\_2$ are two $J$-holomorphic curves passing through $(0,0)$, tangent at $(0,0)$ and regular at $(0,0)$. Then there exist $C^{\infty}$ smooth complex coordinate... | https://mathoverflow.net/users/13441 | Two smooth tangent almost complex curves in a $4$-manifold | This follows from theorem 6.2 (and the first sentence in the proof) of Mario J. Micallef and Brian White, *The structure of branch points in minimal surfaces and in pseudoholomorphic curves*, **Ann. of Math.** (2) 141 (1995), no. 1, 35–85.
| 5 | https://mathoverflow.net/users/13268 | 296474 | 130,285 |
https://mathoverflow.net/questions/296408 | 5 | A minimal ideal of a commutative ring $R$ is a nonzero ideal which contains no other nonzero ideal.
Let $X $ be a completely regular topological space and $C (X) $ the ring of all real valued continuous functions over $X $. Is there any characterization for minimal ideals of $C (X) $?
| https://mathoverflow.net/users/122527 | Minimal ideals of the ring of continuous functions | Let us first observe that an ideal $I$ of a commutative ring $R$ with identity is *minimal* in OP's sense if and only if $I$ is a non-zero [simple module](https://en.wikipedia.org/wiki/Simple_module) over $R$. From now on, we will favour this terminology.
Note also that a topological space $X$ with trivial topology, ... | 7 | https://mathoverflow.net/users/84349 | 296476 | 130,286 |
https://mathoverflow.net/questions/295661 | 2 | Let $R$ be a Prufer domain. If $0 \ne a \in R$ is such that $Ra \cap Rb$ is principal ideal for every $b \in R$, then is it true that $Ra+Rb$ is also principal for every $b\in R$ ?
Over Prufer domains, torsion-free modules are flat , so if $K$ is the fraction field of $R$ then any subring $S$ of $K$ containing $R$, i... | https://mathoverflow.net/users/nan | GCD and LCM of elements in Prufer domain | In fact, for any given nonzero $a$ and $b$, if $Ra\cap Rb$ is principal so is $Ra+Rb$. Here is one way to see it (surely there must be a more down-to-earth proof). Without assuming $Ra\cap Rb$ principal, we have an exact sequence of $R$-modules
$$\begin{array}{ccccccccc}
0&\longrightarrow&Ra\cap Rb&\longrightarrow& Ra\... | 4 | https://mathoverflow.net/users/7666 | 296486 | 130,289 |
https://mathoverflow.net/questions/296483 | 2 | It is easy to see that [vertex-transitive](https://en.wikipedia.org/wiki/Vertex-transitive_graph) graphs must be [regular](https://en.wikipedia.org/wiki/Regular_graph).
This question looks for regular graphs that are "the opposite" of vertex-transitive.
**Question.** Is there an integer $N\in\mathbb{N}$ such that g... | https://mathoverflow.net/users/8628 | Strongly rigid connected $k$-regular graphs | I will use the blocks of Steiner triple systems. Suppose $\mathcal{S}$ is a Steiner triple system on $v$ points. Then $v\cong1,3$ mod 6 and there are $v(v-1)/2$ blocks. The block graph has the blocks of the triple system as its vertices, two are adjacent if they have a point in common. It is
strongly regular.
If a tr... | 4 | https://mathoverflow.net/users/1266 | 296501 | 130,294 |
https://mathoverflow.net/questions/296489 | 5 | Let $f:X\rightarrow Y$ be a regular map of smooth connected algebraic varieties (say over an algebraically closed field). I know that the image $f(X)$ is only a constructible set, in general, but I am interested in conditions that ensure $f(X)$ being an algebraic variety.
**A precise question:** suppose the different... | https://mathoverflow.net/users/5301 | When is the image of a regular map an algebraic variety? | I am correcting the first sentence of the comment.
Even if the rank of $df$ is constant, the image may be only **constructible**. Let $X$ be the complement of the $s$-axis in the $(s,t)$-affine plane, $X=\{(s,t): t\neq 0\}.$ Let $Y$ be the affine $3$-space with coordinates $(u,v,w).$ Let $f$ be the function $f(s,t)=(... | 6 | https://mathoverflow.net/users/13265 | 296502 | 130,295 |
https://mathoverflow.net/questions/296328 | 7 | Let $T:\mathbb Q(x)\to \mathbb Q(x)$ be the operator of inverse logarithmic derivative, i.e. $$Tf=\frac{f}{f'}.$$ Define $$p\_n(x)=T^n\left(x-\frac{x^2}{2}\right).$$ Let $f\_n(x) \in \mathbb Z[x]$ be the numerator of $p\_n(x)$, taken with positive leading term. The sequence $D\_n$ of discriminants of our polynomials st... | https://mathoverflow.net/users/101078 | Discriminant of numerator of inverse logarithmic derivative operator iteration | First, let's make the substitution $x\to 1-x$, so the function $x-\frac12x^2$
becomes $\frac12(1-x^2)$. The key here is that it is an even function, and the
discriminant of even and odd polynomials are essentially squares. More precisely,
$$\begin{aligned}
\text{Disc}\bigl( A(x^2) \bigr) &= \pm\text{Resultant}\bigl(A(... | 10 | https://mathoverflow.net/users/11926 | 296503 | 130,296 |
https://mathoverflow.net/questions/296516 | -1 | Consider the following statement (P):
>
> For every subset $\emptyset \not = A \subset P(\omega)$, there exists a subset $B\subset A$ such that $B \subset A$ such that :
>
>
> 1) $B$ is countable (i.e. is either finite or has cardinal $\aleph\_0$)
>
>
> 2) $\bigcap B = \bigcap A$
>
>
> 3) $\bigcup B = \bigcup... | https://mathoverflow.net/users/95470 | countable and uncountable subset of $P(\omega)$ | It is relatively consistent with ZF that this is impossible.
Suppose that there is an infinite Dedekind finite set $A\subset P(\omega)$. Since ZF proves that the continuum is bijective with the set of branches through the binary tree $2^{<\omega}$, we may assume by labeling the nodes of this tree that
no finite sub... | 4 | https://mathoverflow.net/users/1946 | 296517 | 130,301 |
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