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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/221891 | 3 | This is closely related to [this question.](https://mathoverflow.net/questions/221739/irreducibility-of-discriminant) Suppose I have the resultant $\mathcal{R}$ of two (or more polynomials) over $\mathbb{Q},$ and suppose $\mathcal{R}$ is *not* irreducible. What is the significance of the factors. In particular, what do... | https://mathoverflow.net/users/11142 | Reducibility of resultants | All such resultants and/or discriminants are geometrically irreducible in characteristic zero, and a power of an irreducible in general. This is actually covered by the geometric argument I quoted in my answer to the former question: [irreducibility of discriminant](https://mathoverflow.net/questions/221739/irreducibil... | 5 | https://mathoverflow.net/users/26522 | 221899 | 103,997 |
https://mathoverflow.net/questions/221102 | 7 | Find an asymptotically tight estimate for the sum
$$
A\_n^{k}(\lambda)= \sum\_{
\substack{a\_i\geq \lambda\_i
\\
a\_1+a\_2+\dots a\_k=n
}} \prod\_{i=1}^k a\_i!
$$
Is the leading term going to be
$$|\textrm{Number of Maximal Lambda}|(j-\lambda\_1-\lambda\_2-\lambda\_3- \lambda\_4+ \lambda\_{max})!\frac{1}{\lambda\_{\m... | https://mathoverflow.net/users/12337 | Sum of inverse of multinomial coefficients | I add two hopefully useful remarks.
I consider the general situation. In the sequel $k\geq 1$ and $\lambda=(\lambda\_1,\ldots,\lambda\_{k+1})$ are fixed, $s:=\sum\_{i=1}^{k+1}\lambda\_i$ and $n\geq s$.
Let $A\_n^{(k+1)}(\lambda):=\sum\limits\_{\stackrel{a\_i\geq \lambda\_i, i=1,\ldots,k+1}{a\_1+\ldots +a\_{k+1}=n}}... | 2 | https://mathoverflow.net/users/48831 | 221900 | 103,998 |
https://mathoverflow.net/questions/206348 | 32 | This question assumes familiarity with [combinatorial cardinal
characteristics of the continuum](http://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum).
Identify an infinite set $a\subseteq\mathbb{N}$ with its increasing
enumeration. Thus, for each natural number $n$, $a(n)$ is the $n$-th
element of $... | https://mathoverflow.net/users/2415 | Bidi: A new cardinal characteristic of the continuum? | Using a characterisation of $\min\{\mathfrak{d},\mathfrak{r}\}$ that comes from dualizing a result in Kamburelis' and Węglorz's paper called "Splittings":
>
> A. Kamburelis, B. Węglorz, *Splittings*, Archive for Mathematical Logic **35**, Issue 4 (1996) 263-277, doi:[10.1007/s001530050044](http://dx.doi.org/10.1007... | 19 | https://mathoverflow.net/users/67193 | 221901 | 103,999 |
https://mathoverflow.net/questions/221664 | 5 | Are there standard or known weights/metrics on cyclic orders?
[Cyclic orderings](https://en.wikipedia.org/wiki/Cyclic_order) are different ways of listing elements from a finite set, where you call two lists the same if they differ only by a rotation. For example, the ways that 4 people can sit at a circular table, w... | https://mathoverflow.net/users/8639 | Weights on cyclic orderings | cyclic permutations, and a need to define distances on them, pops up in literature on graph crossing numbers, e.g. we used it in <http://arxiv.org/abs/math/0404142>
It probably goes back all the way to at least
D.J. Kleitman, The crossing number of $K\_5$. J. Combinatorial Theory 9 (1970), 315–323.
| 2 | https://mathoverflow.net/users/11100 | 221905 | 104,002 |
https://mathoverflow.net/questions/221889 | 1 | See [here](https://mathoverflow.net/questions/221823/poisson-kernel-ex-y-textexp-i-theta-x-t-theta-y-t-ei-theta-x).
>
> Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W\_t = (X\_t, Y\_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp... | https://mathoverflow.net/users/81970 | Poisson kernel, expectation, an absolute value comes in | Yes, if $\tau$ is defined as $\min \{t\ge 0: Y\_t=0 \}$. Condition on $\tau =t$ and use that $Ee^{i\theta X\_t}=e^{i\theta x-\theta^2 t/2}$, as explained in the answer to the linked question. Thus
$$
Ee^{i\theta X\_{\tau}} = e^{i\theta x} E e^{-\theta^2\tau/2} .
$$
The distribution of $\tau$ and $Ee^{-b\tau}$ are discu... | 0 | https://mathoverflow.net/users/48839 | 221913 | 104,007 |
https://mathoverflow.net/questions/180723 | 3 | There is a nice theorem that for every simply connected, closed, smooth, oriented manifold $M$, we have for some $m \in \mathbb{N}$:
$$M \# \left(\mathop{\#}^m \left(\mathbb{C}\mathbb{P}^2 \# \overline{\mathbb{C}\mathbb{P}^2} \right) \right) \cong \left(\mathop{\#}^{m+b^+\_2(M)} \mathbb{C}\mathbb{P}^2 \right) \# \left(... | https://mathoverflow.net/users/13767 | Simply-connected 4-manifolds can be blown up and down to complex projective planes. How about non-simply-connected ones? | Two smooth oriented 4-manifolds $M,N$ with fundamental group $\pi$ admit maps to $B\pi$, the classifying maps of their universal covers. The manifolds $M$ and $N$ become diffeomorphic after connect summing with copies of complex projective space if and only if the images of the fundamental classes $[M],[N]$ in $H\_4(B\... | 8 | https://mathoverflow.net/users/81986 | 221915 | 104,008 |
https://mathoverflow.net/questions/221921 | 7 | Let $X\subseteq \mathbb{C}^n$ be an irreducible variety defined over $\mathbb{Q}$. I would like to show that for all but finitely many prime $p$ the variety $X(\mathbb{F}\_p)$, defined over $\mathbb{F}\_p$, is geometrically irreducible, i.e., $X(\overline{\mathbb{F}}\_p)$ is irreducible. Can someone give me a hint or a... | https://mathoverflow.net/users/8419 | Geometrically irreducible variety | I agree with the OP that there is no need to use fancy and sophisticated language and tools. (Maybe what he calls *variety* is nowadays better called *algebraic set*.)
(a) The case that $X$ is a hypersurface is well known, some people call it the Bertini-Noether Theorem. Here one has to show that an absolutely irredu... | 13 | https://mathoverflow.net/users/18739 | 221924 | 104,010 |
https://mathoverflow.net/questions/221919 | 3 | I have the following problem. Let's say we have $x\_{jk}$ it is an expression value of gene $j$ in a sample $k$. It is the average of expression levels across the cell types $s\_{ij}$, weighted by respective proportions $a\_{ki}$ ($i = 1 \cdots N$, $N$ is the disease type):
$$
x\_{jk} = \sum\_{i=1}^{N} a\_{ki}s\_{ij}... | https://mathoverflow.net/users/81988 | Better alternative to solve quadratic programming for large matrices | Your problem is also convex. Hence, a whole bunch of methods for convex optimization are available. Since projecting onto the constraints is not too difficult (project each row of $A$ onto the simplex), you could use projected gradient descent (many variants are available: you could use over- or under-relaxation or Nes... | 3 | https://mathoverflow.net/users/9652 | 221927 | 104,013 |
https://mathoverflow.net/questions/221898 | 8 | Fix a Riemannian metric on a manifold $M$. Suppose that we fix two points $x,y \in M$. We start with the space
$C^{\infty}\_{\searrow}(x,y) = \left\{\gamma: \mathbb{R}\to M:\,\lim\_{t\to-\infty}\gamma(t) = x, \lim\_{t\to\infty}\gamma(t)=y, \exists\,C,\delta > 0 \text{ such that} \vert\gamma'(t)\vert \leq Ce^{-\delta... | https://mathoverflow.net/users/26069 | Banach manifold of paths with endpoints on submanifolds | For each point $s$ in a submanifold $S\subset M$, there exists
*some* Riemannian metric on $M$, which makes a neighborhood $U\subset S$ of $s\in S$ totally geodesic in $M$. The definition of (open sets in) these mapping spaces is local in $S\_1,S\_2$ (see below), so the existence of metrics which make $S\_\pm$ near giv... | 2 | https://mathoverflow.net/users/66777 | 221931 | 104,015 |
https://mathoverflow.net/questions/221843 | 13 | Consider the following generalization of residual finiteness to
topological groups.
A locally compact Hausdorff group $G$ is called *residually compact* if
for every compact $K \subseteq G$ there is a normal cocompact lattice
$\Lambda \subseteq G$ such that
the projection $G \to G/\Lambda$ is injective on $K$.
Not... | https://mathoverflow.net/users/36367 | A generalization of residual finiteness to topological groups | Here's an answer to the mathematical part of the question (namely Q2).
An example is the group of adeles. Start from $\mathbf{A'}$ defined as the inverse image of $\bigoplus\_p\mathbf{Q}\_p/\mathbf{Z}\_p$ in $\prod\mathbf{Q}\_p$, with $\prod\mathbf{Z}\_p$ prescribed to be a compact open subgroup (the sum being over a... | 7 | https://mathoverflow.net/users/14094 | 221934 | 104,017 |
https://mathoverflow.net/questions/221420 | 7 | Let $H$ be the quaternions algebra. An $H^{\*}$ algebra is a normed ring $A$ which is simultaneously a unital left $H$ module and has an involution $\*$ with the following properties:
$\forall \lambda \in H, a,b \in A$
1.$\;\lambda(ab)=(\lambda a)b$
2. $\; \parallel ab\parallel \leq \parallel a \parallel \paralle... | https://mathoverflow.net/users/36688 | $H^{*}$ algebras as a generalization of $C^{*}$ algebras | Let assume that you consider unital algebra only (one can still study non unital algebra by unitarizing them, but notion of spectrum is always a little annoying when one want to consider non unital algebra) and that a $H^\*$ algebra is a real $C^\*$-algebra with morphism of real $C^\*$-algebra $\mathbb{H} \rightarrow A... | 4 | https://mathoverflow.net/users/22131 | 221939 | 104,019 |
https://mathoverflow.net/questions/221944 | 2 | Suppose $S\subset{\mathbb R}^n$ is an infinite subset that is in *general position* which means that the intersection of $S$ with every affine subspace of dimension $d<n$ always contains at most $(d+1)$ points. Are there
two distinct points $a,b\in S$ such that $(a+b)/2$ is contained
in the interior of the convex hull ... | https://mathoverflow.net/users/42506 | Internal edges in Convex Polytopes | If $n\geq 4$, let $S$ be a moment curve $f(t)=(t,t^2,\dots,t^n),t\in \mathbb{R}$. Any hyperplane contains at most $n$ points from $S$, since polynomial of degree at most $n$ has at most $n$ roots. So, any $n+1$ points of $S$ are in general position, hence $S$ is in general position. For any two points $a=f(u),b=f(w)$ t... | 6 | https://mathoverflow.net/users/4312 | 221948 | 104,020 |
https://mathoverflow.net/questions/221943 | 8 | Let $A\_n$ be the following $n\times n$ matrix: $(A\_n)\_{i,j}= \frac{1}{\sqrt{n}}\omega\_n^{i\cdot j}$ for all $0 \le i,j <n$, where $\omega\_n=e^{\frac{2\pi i}{n}}$.
I want to understand how $A\_n$ acts on the set $\{0,1\}^n \subseteq \mathbb{C}^n$ in the following specific sense:
1. Let $f(n)= \max\_{v\in\{0,1\}... | https://mathoverflow.net/users/31469 | Maximal $L_1$ norm of Fourier Transform of a Subset | Regarding Q1, if one selects $v$ randomly and uses Khintchines's inequality one obtains a lower bound $f(n) \gg n$, which complements the trivial upper bound of $f(n) \leq n$ coming from Plancherel and Cauchy-Schwarz. It looks like the same argument should also show that $f(n,w)$ is comparable to $w^{1/2} n^{1/2}$ for ... | 6 | https://mathoverflow.net/users/766 | 221958 | 104,024 |
https://mathoverflow.net/questions/221953 | 4 | Good morning everybody.
I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is $\mathbb R^3$ endowed with coordinates $(x,y,z)$ and an horizontal distribution spanned by $X=\partial\_x-\frac y2\partial\_z$ and $Y=\partial\_y+\frac x2 \partial\_z$... | https://mathoverflow.net/users/57571 | dirichlet problem in the heisenberg group | I would look at:
>
> A. Bonfiglioli, E. Lanconelli, F. Uguzzoni. *Stratified Lie Groups
> and Potential Theory
> for their Sub-Laplacians*. Springer, 2007. [DOI 10.1007/978-3-540-71897-0](http://dx.doi.org/10.1007/978-3-540-71897-0)
>
>
>
This book is always my first reference for this sort of general theory... | 3 | https://mathoverflow.net/users/4832 | 221960 | 104,026 |
https://mathoverflow.net/questions/221957 | 84 | *This question was previously asked on [Math SE](https://math.stackexchange.com/q/1488512/39599).*
---
Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma \to \mathbb{CP}^3$. It follows from the degree-genus formula ... | https://mathoverflow.net/users/21564 | Is there a complex surface into which every Riemann surface embeds? | The answer is negative. Suppose for contradiction that $S$ is such a surface, and let me first assume that it is smooth and projective.
Fix $g\geq 24$. Then the coarse moduli space of genus $g$ curves $M\_g$ is of general type (this is due to Harris, Mumford and Eisenbud, see for instance [The Kodaira dimension of th... | 86 | https://mathoverflow.net/users/2868 | 221969 | 104,029 |
https://mathoverflow.net/questions/221961 | 2 | Given a locally free sheaf $E$ of rank $r$ on a (smooth, projective, algebraic) surface, I want to know the dimension of the scheme parametrizing the zero-dimensional (meaning they have zero dimensional support) quotients of $E$ of length $k$.
I believe the answer should be $(r+1)k$. Indeed, the generic thing seems t... | https://mathoverflow.net/users/19088 | Dimension of Quot scheme of zero dimensional quotients of a locally free sheaf | Your dimension is correct, this and more can be found in this [article.](http://arxiv.org/abs/alg-geom/9704016)
| 3 | https://mathoverflow.net/users/70593 | 221971 | 104,031 |
https://mathoverflow.net/questions/221973 | -1 | I am studying $AC$-groups, i.e. groups in which the centralizer of every
non-identity element is abelian. Now I need an example of a group in which
the centralizer of every non-identity element is non-abelian.
Where can I find and how can I classify them?
| https://mathoverflow.net/users/78228 | Example of a group in which centralizers of every element are non-abelian | The smallest groups in which the centralizer of every element is non-abelian
have order $32$. You can find them with [GAP](http://www.gap-system.org) as follows:
```
gap> IsExample := G -> ForAll(ConjugacyClasses(G),
> cl->not IsAbelian(Centralizer(G,Representative(cl))));;
gap> n := 1;;
gap... | 3 | https://mathoverflow.net/users/28104 | 221974 | 104,032 |
https://mathoverflow.net/questions/221697 | 3 | Let $M,N$ be manifolds whose dimensions may be different. Let $f: M\longrightarrow N$ be a smooth map. What **geometric conditions on $f$** can we impose such that the induced homomorphism
$$
f^\*: H^\*(N;\mathbb{Z}\_2)\longrightarrow H^\*(M;\mathbb{Z}\_2)
$$
is injective?
What **geometric conditions on $f$** can we ... | https://mathoverflow.net/users/65800 | geometric conditions on maps between manifolds inducing monomorphisms on cohomology | Are you looking for things like the following?
Suppose $M$ and $N$ are of the same dimension connected, closed and $\mathbb{Z}/p\mathbb{Z}$ orientable. Suppose that the $(\mathbb{Z}/p\mathbb{Z})$ degree of $f$ is non-zero. Then the map $f^\*:H^\*(N)\rightarrow H^\*(M)$ is injective. This basically follows from Poinc... | 4 | https://mathoverflow.net/users/12156 | 221975 | 104,033 |
https://mathoverflow.net/questions/221972 | 3 | Is there any hope to prove this conjecture (or a similar one)?
>
> **Conjecture** Let $\Omega\_k$ be a family of convex (smooth) domains, and let $u\_k$ be convex Alexandrov solution of $$ \begin{cases}
> det(D^2u\_k)=f\_k&\mbox{ in }\Omega\_k\\ u\_k=0 &\mbox{ on
> }\partial\Omega\_k \end{cases} $$ with $0<\lambd... | https://mathoverflow.net/users/39115 | stability of the Monge-Ampère equation | Yes, this follows from Schauder theory for the Monge-Ampere equation and for linear equations. Subtracting the equations $\det D^2u\_k = f\_k$ and $\det D^2u = f$ gives, for $v\_k = u\_k - u$, the equation
$$a\_{ij}(x) (v\_k)\_{ij} = f\_k - f$$
where $a\_{ij}$ are coefficients depending on $D^2u\_k$ and $D^2u$ (to see ... | 4 | https://mathoverflow.net/users/16659 | 221979 | 104,035 |
https://mathoverflow.net/questions/180718 | 19 | $$
143\,\sqrt {3}\;{\mbox{$\_2$F$\_1$}\left(\frac{1}{2},\frac{1}{2};\,1;\,{\frac {3087}{8000}}\right)}=
40\,\sqrt {5}\;
{\mbox{$\_2$F$\_1$}\left(\frac{1}{3},\frac{2}{3};\,1;\,{\frac {2923235}{2924207}}\right)}
$$
While working on [another problem](https://mathoverflow.net/q/179108/454), I found two different ways of ex... | https://mathoverflow.net/users/454 | A hypergeometric puzzle | These formulas originate from modular parametrizations of the underlying hypergeometric functions and there are many such appearances in the literature (a collection of references can be found in <http://arxiv.org/abs/1302.0548> where such algebraic transformations are used in a related context). But the most classical... | 8 | https://mathoverflow.net/users/4953 | 221981 | 104,036 |
https://mathoverflow.net/questions/221920 | 52 | Have there been any studies of publication rates in Mathematics?
We are trying to construct a workload model for the Faculty of Science and Engineering at my institution. Part of this involves assigning a fixed number of "points" for each published paper. It seems that our colleagues in some of the sciences publish ... | https://mathoverflow.net/users/3684 | Publication rates in Mathematics | Here is the [AMS culture statement](http://www.ams.org/profession/leaders/culture/CultureStatement06.pdf) on publication rates in mathematics. Even the best young mathematicians publish average of two or fewer articles per year.
| 79 | https://mathoverflow.net/users/6269 | 221982 | 104,037 |
https://mathoverflow.net/questions/221980 | 3 | Let $L=[a,b]\cap\mathbb{N}$ with $a,b\in\mathbb{N}$, let $D\in\mathbb{N}$, and let $C=L^D$. Then I would like to know how many points are there in $C$ with the same given norm-2 $d$. I.e., I'm looking for $|A|$, where
$A = \{p\in C,\ ||p||\_2=d\}$
| https://mathoverflow.net/users/81978 | How many points are in such set with the same norm-2 | The answer is the coefficient of $t^{d^2}$ in the generating function
$\left( \sum\_{j=a}^b t^{j^2}\right)^D$.
| 2 | https://mathoverflow.net/users/13650 | 221983 | 104,038 |
https://mathoverflow.net/questions/221856 | 32 | At least 99% of books on functional analysis state and prove the Hahn-Banach theorem in the following form: *Let $p:X\to \mathbb R$ be sublinear on a real vector space, $L$ a subspace of $X$, and $f:L\to \mathbb R$ linear with $f\le p|\_L$. Then there is a linear $F:X\to\mathbb R$ with $F\le p$ and $F|\_L=f$*.
Howeve... | https://mathoverflow.net/users/21051 | Hahn-Banach theorem with convex majorant | If $p$ is convex, then $P(x)=\inf\_{t>0}t^{-1}p(tx)$ is sublinear, isn't it? Also, if a *linear* functional is dominated by $p$, it is also dominated by $P$. Finally, $P\le p$. So there is no non-trivial gain in generality whatsoever *unless* you start talking about extending non-linear functionals but then you should ... | 34 | https://mathoverflow.net/users/1131 | 221988 | 104,043 |
https://mathoverflow.net/questions/221990 | 0 | I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, specifically for $\theta(n)$, seem to approach the bound of $\theta(n) \leq n$ for all $n \in \mathbb{N}$.
1. Is this boun... | https://mathoverflow.net/users/74600 | Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa? | Although [Dusart](http://arxiv.org/pdf/1002.0442.pdf) has shown that $\vartheta(x)<x$ for all $x<8\cdot 10^{11}$, it is not true in general that $\vartheta(x)<x$. Indeed, Littlewood’s oscillation theorem implies that there is a positive constant $c$ and infinitely many $x$ for which $$\vartheta(x)>x+c\sqrt{x}\log\log\l... | 12 | https://mathoverflow.net/users/nan | 221994 | 104,045 |
https://mathoverflow.net/questions/221887 | 3 | The motivation of the question is that I try to test when a real number is not an cyclotomic integers. Or more specifically, when a positive real number is not a quantum dimension of a unitary fusion category?
We know that when $1\leq d<2$, $d$ is not a quantum dimension of a unitary fusion category if $d \neq 2\cos(... | https://mathoverflow.net/users/17787 | What are the necessary conditions for a real number to be a cyclotomic integers？ | I suspect you might be looking for the following fact:
>
> If $x$ is a cyclotomic integer, and $p$ a prime does not divide the discriminant, then the minimal polynomial of $x$ factors modulo $p$ into irreducible components *all of the same degree*.
>
>
>
See for example Theorem 4.6 in [**Elementary and analyti... | 7 | https://mathoverflow.net/users/3 | 221995 | 104,046 |
https://mathoverflow.net/questions/221968 | 5 | Is there a simple function that connects $k$ in $k$-planar graphs and genus of such graphs?
If there is no simple function is there any non-trivial upper and lower bound?
| https://mathoverflow.net/users/nan | $k$-planar graphs and genus | For fixed $k$ (even $k=1$) the genus can be arbitrarily large (linear in the number of vertices). Graphs with genus $g$ have $3n+O(g)$ edges by Euler's formula, while 1-planar graphs can have as many as $4n-O(1)$ edges. So dense 1-planar graphs (e.g. take a planar graph with all faces quadrilaterals and add two crossin... | 6 | https://mathoverflow.net/users/440 | 222002 | 104,048 |
https://mathoverflow.net/questions/222005 | 0 | *I know classical strong maximum principles for supersolutions of Laplacian operator, which says:*
>
> **Suppose $ u \in C^2(\Omega)\cap C(\overline{\Omega}) $ satisfies $ -\Delta u \geq 0 $ in $ \Omega$.**
>
>
> **If $ u \geq 0 $ on $ \partial \Omega $ then $ u \geq 0 $ in $ \Omega $. In fact, either $u > 0$ in ... | https://mathoverflow.net/users/76453 | Strong Maximum Principle for very weak supersolutions of Laplacian operator | As such the statement does not make sense: if $u \in L^1\_\mathrm{loc} (\Omega)$, its restriction on $\partial \Omega$ is not well-defined.
If $u \ge 0$ on $\Omega \setminus K$, where $K \subset \Omega$ is a *compact* subset and $-\Delta u \ge 0$ in the sense of distributions and if $\Omega$ is connected, then either... | 1 | https://mathoverflow.net/users/42047 | 222014 | 104,052 |
https://mathoverflow.net/questions/222012 | 9 | I have a very simple question regarding geometric morphisms of $\infty$ topoi, but have been unable to find the answer in Lurie's HTT (although it seems likely that its there somewhere and I just can't find it).
Classically, given two sites $C$ and $D$, a morphism of sites $f:C\to D$ induces a geometric morphism of ... | https://mathoverflow.net/users/43687 | Geometric morphism of $\infty$ topos | I never found any discussion about this in HTT either, but it turns out to work exactly the same way as in SGA 4.
That is, let $f^\* : P(D) \to P(C)$ denote the restriction functor on presheaves, and $f\_!$ its left adjoint (given by left Kan extension).
Since $f$ preserves covering families, $f\_!$ preserves local e... | 12 | https://mathoverflow.net/users/2503 | 222019 | 104,053 |
https://mathoverflow.net/questions/222024 | 5 | Suppose I have a smooth projective surface $S$ and a curve $C\subset S$. The self-intersection of $C$ is by definition the degree of the restriction to $C$ of the normal bundle of $C$ inside $S$.
By the very definition, if I deform $C$ inside $S$ then the self intersection does not change. Now, I wonder if this stays... | https://mathoverflow.net/users/4096 | Self intersection and deformations | I think that, *if the curve does not disappear*, then the answer is positive.
In fact, let $\pi \colon \mathcal{X} \to \Delta$ your deformation of surfaces over a disk, with $S\_1= \pi^{-1}(t\_1)$ and $S\_2 = \pi^{-1}(t\_2)$. The assumption that $C\_1 \subset S\_1$ deforms to $C\_2 \subset S\_2$ along $\pi$ means tha... | 5 | https://mathoverflow.net/users/7460 | 222025 | 104,055 |
https://mathoverflow.net/questions/222015 | 0 | What are the typical case we need to use [Non-negative least squares NNLS](https://en.wikipedia.org/wiki/Non-negative_least_squares)
$$
||Ax - B||^2
$$
instead of least-square $$ Ax-B$$ (or vice versa)?
And is there any drawback in applying them on large $A$ matrix (e.g. dimension 1000 row x 10 columns)
| https://mathoverflow.net/users/81988 | When to use non-negative-least square and least-square | You would want to constrain $x$ to non-negative values whenever a negative $x$ makes no physical sense, say because $x$ represents the intensity of a pixel, or the price of an object, or a frequency count, or a chemical concentration, etc. An overview of typical applications is given in [Nonnegativity Constraints in Nu... | 2 | https://mathoverflow.net/users/11260 | 222029 | 104,057 |
https://mathoverflow.net/questions/222028 | 4 | Is finitely generated projective module M of rank one over regular commutative notherian ring free?
Bass (Illinois Math J, 1963) showed that in case M is nonfinitely generated, it is free. I am wondering about finitely generated case.
Thanks!
| https://mathoverflow.net/users/82045 | projective module of rank one over notherian ring | **No**. Take the case of a Dedekind domain $A$ that it is not a principal ideal domain. It is a regular commutative notherian ring (of Krull dimension 1). The group of classes $\mathrm{Pic}(A)$ is not trivial, otherwise it would be a p.i.d. Therefore, there is a rank one projective module, in this case a fractional ide... | 7 | https://mathoverflow.net/users/6348 | 222032 | 104,059 |
https://mathoverflow.net/questions/222030 | 3 | Consider the Jordan function $J\_2(n)$ defined by
$$
J\_2(n) = \#\{x \in (\mathbb{Z}/n)^2 \mid ord(x) = n\}
$$
(this is OEIS [A007434](http://oeis.org/search?q=1%2C3%2C8%2C12%2C24%2C24%2C48%2C48&language=english&go=Search)). One can prove the following identity pretty easily:
$$
\sum\_{d \mid n} J\_2(d) \sigma\_1(n/d) ... | https://mathoverflow.net/users/1703 | Is there a nice generating function proof of the following identity? | The identity $G(s)=\frac{\zeta(s-2)}{\zeta(s)}$ is a corollary of the formula $J\_2(p^e) = p^{2e} - p^{2e-2}$. The last identity from the question is simple because $F(s)=\zeta(s)\zeta(s−1).$
| 8 | https://mathoverflow.net/users/5712 | 222037 | 104,062 |
https://mathoverflow.net/questions/210068 | 3 | In Fulton's Intersection theory Example 6.1.2，one considers two divisors on $\mathbf{P}^2$ given by $D\_1=A+2B,D\_2=2A+B$, where $A,B$ are lines meeting at a point.
Let $X=D\_1\times D\_2,Y=\mathbf{P}^2\times\mathbf{P}^2$, $V=\mathbf{P}^2$, $f\colon V\to Y$ is diagonal map. Consider the fiber product of closed immer... | https://mathoverflow.net/users/nan | Calculating the distinguished varieties of intersection product | Let's fix some notation for making things explicit. (The notation in your last 3 paragraphs is slightly confusing, so I'm making up my own). I take $A=V(z)$ and $B=V(y)$, hence $P=[1,0,0]$. Then $D\_1=V(y^2z)$ and $D\_2=V(yz^2)$, and hence
$$W=D\_1 \cap D\_2=V(y^2z,yz^2).$$
Since $D\_i$ are divisors, we can calculate t... | 2 | https://mathoverflow.net/users/54541 | 222039 | 104,064 |
https://mathoverflow.net/questions/222043 | 18 | For a finite group $G$, take $m$ as the largest integer such that $G$ has a subgroup $H\cong S\_m$ and $n$ as the smallest integer such that $G$ is itself isomorphic to a subgroup of $S\_n$. We then define the ***"squeezing number"*** of $G$ as $s(G):=\dfrac nm$. This is probably not a new idea, so pointers are welcome... | https://mathoverflow.net/users/29783 | "Squeezing" a finite group between symmetric groups | Here is an attempt assuming Goldbach's Conjecture. Express the required ratio as $n/m$ with $n-m \ge 8$ even, and take $G = S\_m \times C\_{pq}$, where $p$ and $q$ are distinct primes with $p+q=n-m$.
In fact it appears to have been proved that every sufficiently large even integer is the sum of four distinct primes, ... | 32 | https://mathoverflow.net/users/35840 | 222045 | 104,065 |
https://mathoverflow.net/questions/222049 | 4 | Let $C$ be a general genus $g$ curve, how can we describe the Neron-Severi group of its $n$-th self product $C^n=C\times \dots \times C$?
It is a lattice in $H^2(C^n,\mathbb{Z})\cong \mathbb{Z}^{n+2n(n-1)g^2}$, do we know its rank and basis if $C$ is general?
| https://mathoverflow.net/users/nan | Neron-Severi group for product of curves | Let me give the answer for $n=2$.
Fix a point $p \in C$ and call $x\_1, \, x\_2, \, \Delta$ the divisor classes
$\begin{equation\*}
\begin{split}
x\_1 & := \{(p, \, x) \, | \, x \in C\} \\
x\_2 & := \{(x, \, p) \, | \, x \in C \} \\
\Delta & := \{(x, \, x) \, | \, x \in C \}.
\end{split}
\end{equation\*}
$
Le... | 5 | https://mathoverflow.net/users/7460 | 222051 | 104,068 |
https://mathoverflow.net/questions/222047 | 3 | I saw this in [here](http://www.researchgate.net/publication/225152216_Biflatness_of_semigroup_algebras). Let $A$ be a Banach algebra, and let $\Lambda$ be a non-empty set. We denote by
$M\_\Lambda(A)$ be the set of $\Lambda\times\Lambda$ matrices $(a\_{ij})\_{i,j\in\Lambda}$ with entries in $A$ such that
$\|(a\_{ij})\... | https://mathoverflow.net/users/76115 | $M_Λ(A) → A ⊗ M_Λ(C)$ | The norm on $M\_\Lambda(A)$ is the one obtained by identifying it as a vector space (not an algebra) with $\ell^1(\Lambda\times\Lambda, A)$.
Similarly, the norm on $M\_\Lambda({\mathbb C})$ is the one obtained by identifying it as a vector space with $\ell^1(\Lambda\times\Lambda)$.
*Fact.* Given any index set $I$ a... | 4 | https://mathoverflow.net/users/763 | 222060 | 104,069 |
https://mathoverflow.net/questions/222063 | 6 | **In the following, an algebra will always mean a finite-dimensional associative commutative unital algebra (over some field $k$).**
Let $A$ be a $\mathbb{C}$-algebra. I am trying to understand how its group of units $A^{\times}$ sits topologically in $A$ relative to the locus of non-invertible elements $A\setminus A... | https://mathoverflow.net/users/1849 | Connectedness of units in finite-dimensional commutative complex algebras | For (Q1): A finite dimensional $\mathbb{C}$-algebra $A$ is Artinian, so $A$ is a product of Artin local algebras. The units of a product of algebras is the product of the units, so we may assume $A$ is local, with maximal ideal $\mathfrak{m}$. The complement of a subspace in a complex vector space is connected, so $A^\... | 9 | https://mathoverflow.net/users/5263 | 222066 | 104,072 |
https://mathoverflow.net/questions/222057 | 4 | Let $X$ be an abelian surface over $\mathbb{C}$. Consider the Kummer surface $K$ associated to $X$, that is the quotient of $X$ by the action of involution on $X$, $x\mapsto -x$. Kummer surface is a singular surface with 16 nodes. Does this mean that every curve through the 16 nodes is a singular curve? If we start wit... | https://mathoverflow.net/users/70211 | Curve through the 16 singular points of a Kummer surface | Regarding your first question, the answer is *no*.
Following J. Silverman's suggestion, let me provide an example of an abelian surface $A$ and a smooth curve $C$, fixed by the involution $(-1)\_A$, and containing all the $2$-torsion points of $A$. Therefore, the image $C'$ of $C$ in $K := \textrm{Kum}(A)$ will be a... | 3 | https://mathoverflow.net/users/7460 | 222068 | 104,074 |
https://mathoverflow.net/questions/222054 | 5 | **Please note edits after original post changing the specific form of the setup**
Let's say we have a stochastic differential equation:
$$
\mathrm{d}S\_t = |S^\beta| {(\mu \mathrm{d}t + \sigma\mathrm{d}W\_t)}
$$
where $W\_t$ is a standard brownian motion.
When $\beta=0$, the whole thing continues to be a brownian ... | https://mathoverflow.net/users/82057 | Between arithmetic and geometric Brownian motions: when are negative values possible? | Assuming that $\mu$ and $S\_0$ are positive, the process stays almost surely non-negative. This is easily seen as when $S$ hits zero, it has a deterministic drift upwards. However, the process does not necessarily stay strictly positive, the hitting probability of zero can be positive.
This model is well studied in t... | 3 | https://mathoverflow.net/users/20026 | 222071 | 104,076 |
https://mathoverflow.net/questions/222053 | 3 | Let us consider a countable set of sequences of positive numbers $\{(x\_n^{(1)}),(x\_n^{(2)}),\dots\}$ for which we have
* $(\forall k\in\mathbb{N}) \ \lim\_n x\_n^{(k)} = +\infty$, and
* $(\forall k\in\mathbb{N}) \ \lim\_n\frac{x\_n^{(k+1)}}{x\_n^{(k)}}=0$ (the growth rate of the sequences is in the descending order... | https://mathoverflow.net/users/82056 | The minimal growth rate of the countable family of sequences | Joel's answer gives the history of the result, and Fan Zheng's comment gives a one-word proof, "diagonalization", but it might be worthwhile to make the argument more explicit. The strategy for defining the sequence $(y\_n)$ is to make it a sequence of integers that is monotone non-decreasing but grows very slowly. Tha... | 8 | https://mathoverflow.net/users/6794 | 222072 | 104,077 |
https://mathoverflow.net/questions/222080 | 2 | Let $S$ be a density zero set rational primes, in the concrete situation
$\#\{p<X,p\in S\}=\mathcal{O}(x/(\log x)^{3/2-\delta})$ for all $\delta>0$.
Then can $\prod\_{p\in S}(1-\frac{1}{p+1})$ be zero? If nonzero, do we have a lower bound?
| https://mathoverflow.net/users/42690 | infinite product of (1-1/(p+1)) over a density 0 set of primes | Use $1-x \ge e^{-\frac{x}{1-x}}$ for $x \in (0,1)$ to establish
$$\ln \prod\_{p \in S}\bigg(1-\frac{1}{p+1}\bigg) \ge -\sum\_{p \in S} \frac{1}{p}.$$
This is tight since $1-x=e^{-x + O(x^2)}$ in a neighborhood of $x=0$ and $\sum\_p \frac{1}{p^2}$ converges.
So one needs an upper bound on $\sum\_{p \in S} \frac{1}{... | 11 | https://mathoverflow.net/users/31469 | 222083 | 104,083 |
https://mathoverflow.net/questions/222092 | 3 | Let $G$ be an algebraic group over a perfect field $k$. Then it is know that it can be written as an extension of an affine algebraic group and a proper algebraic group.
Is there a similar result for a group over a $k$-scheme of finite type $S$?
| https://mathoverflow.net/users/27398 | Chevalley devissage | The most obviously analogous statement over general bases is false. Here are two different counterexamples:
Let $S$ be a curve over $k$ and let $E$ be a Neron model elliptic surface. Over a point of good reduction, $E$ is proper, and over a point of bad reduction, $E$ is affine. There is no way to write $E$ as an ext... | 6 | https://mathoverflow.net/users/18060 | 222095 | 104,086 |
https://mathoverflow.net/questions/222096 | 7 | Let $\mathcal{C}$ be a pivotal tensor category. Feel free to assume finiteness, semisimplicity, fusion, sphericality, unitarity or whatever makes things interesting. Which of the following structures come for free?
1. Does the pivotal structure canonically define a *twist*: a natural transformation $\theta$ from the ... | https://mathoverflow.net/users/51107 | Twists, balances, and ribbons in pivotal braided tensor categories | Question 2: Given a pivotal braided category $\mathcal{C}$, there are 2 ways to endow $\mathcal{C}$ with twists under which $\mathcal{C}$ is a rigid balanced category. Conversely, given a rigid balanced category $\mathcal{C}$, there are 2 ways to endow $\mathcal{C}$ with a pivotal structure under which $\mathcal{C}$ is... | 11 | https://mathoverflow.net/users/351 | 222098 | 104,087 |
https://mathoverflow.net/questions/222097 | 8 | Let S be a subset of [1..N] containing no three-term arithmetic progression, and let h(S) be the size of the largest gap between two consecutive elements of S. By Roth's theorem, h(S) has to grow with N. But how fast? For instance, are there arbitrarily large N admitting a subset of [1..N] with no three-term AP and no ... | https://mathoverflow.net/users/431 | Subsets of [1..N] with no three-term arithmetic progressions and no large gaps | For the sake of having a reference, Ron Graham shows in ["On the growth of a van der Waerden-like function"](http://www.emis.ams.org/journals/INTEGERS/papers/g29/g29.pdf) that for a fixed $k$, there exists a 3AP-free subset of $\{1,2,\dots,N\}$ with gaps bounded by $k$ and $N\geq k^{c\log k}$, where $c$ is some absolut... | 8 | https://mathoverflow.net/users/2384 | 222102 | 104,088 |
https://mathoverflow.net/questions/222093 | 2 | This is related to a question that I already asked [Curve through the 16 singular points of a Kummer surface](https://mathoverflow.net/questions/222057/curve-through-the-16-singular-points-of-a-kummer-surface/222068#222068). I am new to Abelian varieties and I am trying to understand more about them.
Let $X=J(C)$ be... | https://mathoverflow.net/users/70211 | A curve in an abelian surface and its image in the Kummer surface | Let $b:\hat{X}\rightarrow X$ the blowing up of the 16 2-torsion points, and $E\_1,\ldots ,E\_{16}$ the exceptional $(-1)$ curves on $\hat{X}$. The involution $x\mapsto -x$ lifts to an involution $\sigma $ of $\hat{X}$.
You are looking at the linear system $|nb^\*C|-\sum E\_i$. The involution $\sigma $ acts on $H=H^0... | 3 | https://mathoverflow.net/users/40297 | 222108 | 104,089 |
https://mathoverflow.net/questions/222099 | 8 | Brauer's classic theorem states that any character of a finite group can be expressed as a linear combination with integer coefficients of characters induced by linear characters of p-elementary subgroups.
This is an improvement over Artin's theorem, in which the linear combination is with *rational* coefficients of ... | https://mathoverflow.net/users/43108 | Beyond Brauer's theorem | Jeremy Rickard's answer is perfectly correct, of course. It is also easy to see (by an argument close to Taketa's) that if $G$ is a finite simple group, and $\chi$ is a faithful (not assumed irreducible, though in fact it will be irreducible) complex character of minimal degree of $G$, then $\chi$ is not a non-negative... | 11 | https://mathoverflow.net/users/14450 | 222115 | 104,091 |
https://mathoverflow.net/questions/222114 | 7 | It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change at the limit. For example, one can perturb a sphere $S^2$ slightly so that it has no symmetry. As long as the Ricci flow... | https://mathoverflow.net/users/82084 | Ricci flow and isometry group | I think the phenomenon is much more general: If a sequence of metrics $d\_i$ on a compact metric space $X$ converges (pointwise on $X\times X$) to a metric $d$, and if $h$ lies in the intersection of the isometry groups of $(X, d\_i)$, then clearly $h$ is an isometry of $(X,d)$.
| 12 | https://mathoverflow.net/users/1573 | 222122 | 104,094 |
https://mathoverflow.net/questions/222125 | 2 | $ST$ is the weak set theory built upon identity theory and containing
1. the axiom for *empty set*,
2. the axiom for *adjunction* and
3. the axiom for *extensionality*.
It is known that $ST$ interprets Robinson Arithmetic, and so $ST$ is incomplete.
Is there a very weak set theory $ST^\*$ which is like $ST$ minu... | https://mathoverflow.net/users/37385 | Is Extensionality needed for the incompleteness of very weak set theories? | $\newcommand\ST{\text{ST}}$Here is an example of a set theory $\ST^\*$ with your properties. Let $\ST^\*$ be the theory that is just like $\ST$, but it asserts extensionality only for nonempty sets. So $\ST^\*$ is consistent with the existence of multiple empty sets, and therefore it does not prove extensionality. But ... | 3 | https://mathoverflow.net/users/1946 | 222128 | 104,096 |
https://mathoverflow.net/questions/222127 | 12 | In the notes of Vector Bundles and K-theory by Prof Allen Hatcher, on page 12 he proved a Proposition that for each vector bundle $E\to B$ with $B$ compact hausdorff there exists a vector bundle $E'\to B$ such that $E\oplus E'$ is the trivial bundle.
Later in example 3.6 he showed that the compactness of $B$ is an impo... | https://mathoverflow.net/users/33064 | Is every vector bundle over a noncompact finite-dimensional manifold a summand of a trivial bundle? | The proposition you refer to holds for any space homotopy equivalent to a finite dimensional CW complex. Here are the main points.
1. The property that any vector bundle $E$ over a space has a complementary bundle $E^\prime$ is preserved by homotopy equivalences.
2. Any finite dimensional CW complex is homotopy equiv... | 15 | https://mathoverflow.net/users/1573 | 222134 | 104,098 |
https://mathoverflow.net/questions/222129 | 8 | This is a reference request.
Do you know where I can find the dimensions of the faithful projective representations of $S\_n$ and $A\_n$ for $n\ge 5$?
Thank you in advance.
| https://mathoverflow.net/users/69558 | Faithful projective representations of symmetric groups | Maybe it's worth adding a reference to a modern treatment of Schur's work in book form, which might be easier to read than the original paper of Schur (note that I'm unrelated to the authors):
P.N. Hoffman and J.F. Humphreys, *Projective representations of the symmetric groups.*
Q-functions and shifted tableaux.
Oxfo... | 11 | https://mathoverflow.net/users/4231 | 222138 | 104,100 |
https://mathoverflow.net/questions/222139 | 3 | A quartic surface in $\mathbb{P}^3$ is said to be a "symmetroid" if its equation is obtained as the determinant of a 4x4 symmetric matrix of linear forms. It is well known that the general symmetroid has 10 nodes and the family of such surfaces seem to have the same dimension as the moduli space of 10 points in $\mathb... | https://mathoverflow.net/users/4096 | Quartic symmetroids and 10-points sets | The answer is *no*.
In fact, let $\Gamma \subset \mathbb{P}^3$ be the set of $10$ nodes of a general quartic symmetroid. Then the Gale transform $\Gamma' \subset \mathbb{P}^5$ of $\Gamma$ must be a zero dimensional scheme of length $10$ given by the simple intersection points of two Veronese surfaces, and this condi... | 2 | https://mathoverflow.net/users/7460 | 222145 | 104,103 |
https://mathoverflow.net/questions/221113 | 10 | For a topology $\mathcal{T}$ on a set $S$, where $\mathcal{T}$ does not have a finite base, I define the *grasp* $g(\mathcal{T})$ to be the least infinite cardinal $\kappa$ such that $\mathcal{T}$ has a base $\mathcal{B}$ with $|\mathcal{B}|= w(\mathcal{T})$ , ($w$ is the usual *weight* function), satisfying $$\mathcal... | https://mathoverflow.net/users/81583 | Two questions about the "grasp" cardinal function | **Proposition** (with Z. Szentmiklóssy): *It is consistent that
$\omega\_1=cf (2^{\omega})<g(D(2^{\omega}))=2^{\omega}$.*
Proof:
Assume GCH in the ground model.
For ${\alpha}<{\omega}\_1$ let
$$
P({\alpha})=Fn({\omega}\_{{\alpha}+1}\times {\omega}\_{{\omega}\_1+1},2;{\omega}\_{{\alpha}+1}),
$$
and
$$
P=\prod\_{{\... | 2 | https://mathoverflow.net/users/71011 | 222148 | 104,106 |
https://mathoverflow.net/questions/222058 | 2 | By Bessaga-Pelczynski Selection Principle, it is easy to check that both $l\_{p}(1\leq p<2)$ and $l\_{r}(1<r<p^{\*})$ contains no normalized weakly $p$-summable sequences. I do not know if it is the case for $L\_{r}[0,1]$. My question: Does
$L\_{r}[0,1](1<r\leq 2)$ contain normalized weakly $p$-summable sequences$(1<p<... | https://mathoverflow.net/users/41619 | Weakly $p$-summable sequences in $L_{r}$ | No, because $L\_r$ has cotype 2 for $r\le 2$. In fact, every bounded linear operator from $\ell\_q$ into $L\_r$ is compact when $r \le 2 < q$.
| 1 | https://mathoverflow.net/users/2554 | 222150 | 104,107 |
https://mathoverflow.net/questions/201275 | 10 | I recently noticed that there are *two* senses in which colimits are functorial, and I'm curious about their interplay.
Let $C$ be a cocomplete category. Then, on the one hand, for any diagram category $I$ we have a functor $\mathrm{colim} : \mathrm{Fun}(I,C) \to C$ (left adjoint to the "constant $I$-shaped diagram" ... | https://mathoverflow.net/users/303 | The bifunctoriality of co/limits | This observation has now been codified (in the $\infty$-categorical setting) in section 3 here: <http://arxiv.org/pdf/1510.03525v1.pdf>
| 1 | https://mathoverflow.net/users/303 | 222152 | 104,108 |
https://mathoverflow.net/questions/156363 | 3 | Let $A$ denote a symmetric matrix of non-negative entries, whose rows (and columns) sum up to the all-positive vector $d := A\mathbf{1}$ with $\mathbf{1}$ the all-one-vector. Denote $D := diag(d)$ by a capital letter, similar for any other vector.
Let $b$ denote any all-positive vector.
Assume there exists a positive... | https://mathoverflow.net/users/27164 | Convergence of Symmetric Iterative Proportional Fitting | Now, one year later, I found a recent publication by S. Kurras that answers my above question in all aspects. See here for the paper plus its supplement:
[Proceedings of AISTATS 2015](http://www.jmlr.org/proceedings/papers/v38/)
Indeed he proves that SIPF converges to the same limit as IPF, for any all-positive vecto... | 0 | https://mathoverflow.net/users/27164 | 222158 | 104,110 |
https://mathoverflow.net/questions/219236 | 3 | In standard convex programming, the objective function and each of the constraint inequalities are convex. in such case, if the KKT condition hold for a point, and Slater condition is also hold for the solution space, that point is global optimum.
However what if one of the constraint isn't convex but the solution spa... | https://mathoverflow.net/users/38361 | Generalization of standard convex problem | Let me sketch a proof that the KKT conditions imply global optimality in the case that the objective $f$ and $S$ is convex. No constraint qualification is needed.
Let us assume that the KKT conditions hold at $x$. For simplicity, let both inequality constraints be active at $x$, i.e., $g\_1(x) = g\_2(x) = 0$.
First... | 2 | https://mathoverflow.net/users/32507 | 222164 | 104,112 |
https://mathoverflow.net/questions/222157 | 12 | Often number theorists can bound the number of solutions to a diophantine equation based on the size of the points and the size of the coefficients. But this, as I understand it, can be a bit of a red herring - the genus of a curve is what is really controlling the number of points.
What I would like is some intuiti... | https://mathoverflow.net/users/82104 | Why does genus control the number of points | Even for curves, the situation becomes clearer if you also allow affine curves and look at integral points, or more generally, $S$-integral points in number field. If the curve is projective, that's the same as looking at rational points. Also, it's better to look at Euler characteristic $\chi(C)$ (so as to get integer... | 12 | https://mathoverflow.net/users/11926 | 222165 | 104,113 |
https://mathoverflow.net/questions/222136 | 0 | As we know, if the equation
$$a(x)y''+b(x)y'+c(x)=0 \ \ \ \ \ \ \ \ \ (1)$$
has a regular singular point at $x=x\_0$ then we seek solution of the equation as
$$y(x)=\sum\_{n=0}^{\infty}\beta\_n (x-x\_0)^{n+\lambda} \ \ \ \ \ \ \ (2)$$
but what about the case when the equation $(1)$ has several singular points? A... | https://mathoverflow.net/users/81920 | Frobenius method for multiple singular points | You may expand at every point, singular or non-singular. At a non-singular point
you will obtain 2 series with integer powers, with radius of convergence at least the distance to the closest singular point. At a regular singular point you obtain two series with radius of convergence at least the distance to the other s... | 1 | https://mathoverflow.net/users/25510 | 222166 | 104,114 |
https://mathoverflow.net/questions/222154 | 13 | Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have non-trivial isometry group?
| https://mathoverflow.net/users/82102 | Isometry group of a compact hyperbolic surface | In genus 2, every surface has a symmetry, namely a hyperelliptic involution. In higher genus, generic surfaces will not have any symmetries. If a surface has a non-trivial symmetry group, then the quotient by the symmetry group will be an orbifold, and the moduli space of hyperbolic structures on this orbifold will hav... | 23 | https://mathoverflow.net/users/1345 | 222167 | 104,115 |
https://mathoverflow.net/questions/222153 | 5 | I have been poking around the internet and nlab looking at the concept of large and small categories. My original focus was locally presentable categories of categories and I was thinking of finding categories like Hilb and Group and Set as colimits over diagrams of compact objects. This lead to trying to define "large... | https://mathoverflow.net/users/10007 | internalization of the concept of large and small category | I am not sure exactly what sort of thing you are looking for, but you would probably find the work on [algebraic set theory](http://www.phil.cmu.edu/projects/ast/) relevant and possibly interesting.
Algebraic set theory studies elementary set theory through category-theoretic methods. A central idea is an abstract no... | 7 | https://mathoverflow.net/users/1176 | 222179 | 104,118 |
https://mathoverflow.net/questions/222171 | 6 | My understanding is that the principal graphs are a pair of undirected bipartite graphs, $\Gamma\_+,\Gamma\_-$. They can be calculated from a fusion ring with a given simple object $X$. How to calculate such pair of graphs?
There is related dicussion [An embedding theorem for a fusion ring planar algebra?](https://ma... | https://mathoverflow.net/users/17787 | How to calculate the principal graphs of a fusion ring with a given simple object? | Probably what you really want is the fusion graph $\Gamma\_X$ with respect to your simple object $X$. It is a directed graph with vertices labelled by the simple objects of your category. Between the vertices labelled by simples $Y$ and $Z$, there are $N^{Y,X}\_Z=\operatorname{dim}(\operatorname{Hom}(Y\otimes X, Z))$ d... | 5 | https://mathoverflow.net/users/351 | 222183 | 104,120 |
https://mathoverflow.net/questions/222190 | 4 | In [Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces](https://www.math.stonybrook.edu/~mlyubich/Archive/Geometry/Teichmuller%20Space/Mirz3.pdf), Mirzakhani gave a recursive formula for WP volumes of moduli spaces $\mathcal{M}\_{g,n}(L)$ of bordered Riemann surfaces, and the cons... | https://mathoverflow.net/users/42690 | Explicit constant terms of volumes of moduli spaces | Norman Do calculates somes $V\_{g,n}(L)$ polynomials for small $g$ and $n$ in the appendix A of his thesis:
* Norman Nam Van Do, [Intersection theory on moduli spaces of curves via hyperbolic geometry](http://users.monash.edu/~normd/documents/Do-Phd-Thesis.pdf) (2008)
I'll copy constant terms here for future refere... | 4 | https://mathoverflow.net/users/43108 | 222207 | 104,130 |
https://mathoverflow.net/questions/222202 | 4 | About the Hardy-Littlewood conjecture by [Terence Tao](https://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-461113):
**Conjecture 2** (Prime tuples conjecture, quantitative form) Let ${k\_0 \geq 1}$ be a fixed nat... | https://mathoverflow.net/users/31356 | Upper bound for the first Hardy-Littlewood conjecture | This is explained for example in Iwaniec & Kowalski's "Analytic Number Theory", as an standard application of Selberg's $\Lambda^2$ sieve. See chapter 6, Elementary sieve methods. In particular you get $C\_{k\_0}=2^{k\_0}k\_0!$, for an upper bound, using your notation:
$${(2^{k\_0}k\_0! {\mathfrak G} + o(1)) \frac{x}... | 9 | https://mathoverflow.net/users/43108 | 222209 | 104,132 |
https://mathoverflow.net/questions/214947 | -1 | we know that every maximal ideal in $C(X)$ is in this form:
$$M^p=\left\{\,f \in C^\*(x):\ p\in cl\_{\beta X} Z\left(f\right)\,\right\}$$
and every maximal ideal in $C^\*(X)$ is
$$M^{\*p}=\left\{\,f\in C^\*(X):\ f^{\beta}\left(p\right)=0\,\right\}$$
and it is not necessary that
$$ M^p \cap C^\*(X) = M^{\*p}... | https://mathoverflow.net/users/78262 | Corresponding between prime ideals in $C(X)$ and $C^*(X)$ | In general, the prime ideals of $C(X)$ and $C^\*(X)$ is not in a one to one corresponding. In fact, The prime ideals of $C(X)$ contained in $M^p$ is in one to one corresponding with that of $C^\*(X)$ contained in $M^{\*p}$ if and only if $p\in \upsilon X$, where $\upsilon X$ is the Hewitt-Nachbin space (real compactifi... | 2 | https://mathoverflow.net/users/78262 | 222214 | 104,134 |
https://mathoverflow.net/questions/222219 | 4 | What I am going to ask is probably simple and maybe trivial. But I want to be sure that I am not missing any point. Let ${\bf G}\subseteq \mathrm{GL}\_n(\mathbb{C})$ be a simple and simply connected group. Then this group is defined over $\mathbb{Z}$. Hence I can talk about ${\bf G}(\mathbb{F}\_q)$ where the characteri... | https://mathoverflow.net/users/8419 | Simple groups of Lie type | I'll just make my comment an answer so as to close this question. By Theorem 24.17 of the book "Linear Algebraic Groups and Finite Groups of Lie Type" by Malle and Testerman we have the answer is yes unless $\mathbf{G}(\mathbb{F}\_q)$ is one of the groups
$\mathrm{SL}\_2(\mathbb{F}\_2)$, $\mathrm{SL}\_2(\mathbb{F}\_3... | 5 | https://mathoverflow.net/users/22846 | 222222 | 104,135 |
https://mathoverflow.net/questions/222218 | 10 | Maybe the question does not fit here.
Yesterday in my logic course, I presented a nice example about an application of model theory to group theory. The example is due to Hodges and as following: For any infinite simple group $G$, there is a countable simple group $H\subseteq G$.
Then a logician asked whether the u... | https://mathoverflow.net/users/14340 | Extending an infinite simple group | I will make my comment into an answer. It is a standard result (see for example Chapter 8 of Dixon & Mortimer's book on Permutation Groups) that, for an infinite cardinal $\kappa$, the only normal subgroups of the symmetric group ${\rm Sym}(\Omega)$ on a set $\Omega$ of cardinality $\kappa$ are the subgroups
${\rm Sym}... | 11 | https://mathoverflow.net/users/35840 | 222227 | 104,136 |
https://mathoverflow.net/questions/219621 | 12 | Let $c(n)$ in $\mathbb{Z}/2\mathbb{Z}[x]$ be defined by the recursion $$c(n+4)=c(n+3)+(x^4+x^3+x^2+x)c(n)+x^n\cdot(x+x^2),$$ and the initial conditions
$$c(0)=0,\quad c(1)=1,\quad c(2)=x,\quad c(3)=x^2.$$
**Question**: If 4 divides $n$, is $c(n)$ a sum of $c(k)$ with $k$ less than $n$?
**Remarks**:
1. I've check... | https://mathoverflow.net/users/6214 | A characteristic 2 polynomial recursion | EDIT (11/25/16)
The earlier version of this answer is sketchy, and as I've put up a complete version on arXiv (1603.03910 [math.NT], "A characteristic 2 recurrence with a Hecke algebra application"), I'm replacing my answer with references to this preprint. The argument, culminating in Theorem 2.10, involves nothing ... | 3 | https://mathoverflow.net/users/6214 | 222238 | 104,139 |
https://mathoverflow.net/questions/222228 | 5 | Ramanathan has defined the semistability of a principal $G-$bundle $E$ over a curve $X$ as follows:
>
> $E$ is semistable iff for any parabolic subgroup $P\subset G$, for any reduction of the structure group of $E$ to $P$: $\sigma:X\rightarrow E(G/P)$, and for any dominant caracter $\chi:P\rightarrow \mathbb C^\*$... | https://mathoverflow.net/users/66528 | Semistability of principal bundle vs vector bundle | The following paper contains a proof (Corollary 1) of the identification of Ramanathan-semistability for principal $GL\_n$-bundles and Mumford-semistability for the associated vector bundles:
* D. Hyeon and D. Murphy. Note on the stability for principal bundles. Proc. Amer. Math. Soc. 132 (2004), 2205-2213.
The ba... | 4 | https://mathoverflow.net/users/50846 | 222239 | 104,140 |
https://mathoverflow.net/questions/222242 | 3 | For a CM elliptic curve $E$ and its Grössencharakter, their conductors are both supported on bad primes of $E$. Moreover, by comparing their functional equation, there should be some obvious relations. I think this is how to deduce the functional equation for a CM elliptic curve from the one for its Grössencharakter.
... | https://mathoverflow.net/users/42690 | Conductor of a CM elliptic curve and its Grössencharacter | Here is a completely overkill explanation that nevertheless answers the question.
The question of the relation between the conductor of a CM elliptic curve and its associated Grössencharakter is related to something much stronger, namely automorphic induction.
Given a quadratic extension of number fields $E/F$ and ... | 7 | https://mathoverflow.net/users/3803 | 222245 | 104,144 |
https://mathoverflow.net/questions/222254 | 4 | I recently came across a construction that, in abstraction, leads to the following family of abelian groups: Fix $1<q<p$ with $q$ and $p$ relatively prime. The group $G\_{(p,q)}$ is given by the presentation
$$<g\_0, g\_1, g\_2, \dots \mid g\_i^p=g\_{i+1}^q,\ g\_ig\_j=g\_jg\_i>.$$
In retrospect, I quickly realized that... | https://mathoverflow.net/users/3400 | Recognize this countably generated abelian group? | There is an isomorphism
$$
\begin{align\*}
\varphi:G\_{(p,q)}&\to\mathbb{Z}[1/q],\\
g\_i&\mapsto\frac{p^i}{q^i}.
\end{align\*}
$$
To check surjectivity: for any $\frac{a}{q^n}\in\mathbb{Z}[1/q]$, we can find integers $b,c$ such that $b p^n+c q^n = a$ because $p^n$ and $q^n$ are relatively prime, and we will have $\va... | 10 | https://mathoverflow.net/users/5263 | 222256 | 104,146 |
https://mathoverflow.net/questions/144778 | 2 | Suppose we look at the $\mathbb {C}$ module $\mathbb{C}[x\_1,\dots,x\_n]=\oplus{S\_i}$ where $S\_i$ are the polynomials of degree $i$. Then we look at a subring $R$ (also a $\mathbb {C} $ module) generated by polynomials $p\_1,\dots,p\_m$ which have some algebraic relations among them. We grade $R=\oplus{R\_i}$ where $... | https://mathoverflow.net/users/41283 | Hilbert Regularity in relation to degree of generators | Let $I=(x^{10},y^{10})$. The maximum degree of a generator is 10. The Hilbert polynomial is 0, but the function does not attain this value until degree 19, so your guess isn't correct. It *is* related to the degree of the generators of the Gröbner basis; see for instance *Using Algebraic Geometry* (Cox, Little, O'Shea)... | 1 | https://mathoverflow.net/users/78010 | 222262 | 104,149 |
https://mathoverflow.net/questions/184853 | 3 | Let $\Phi$ be an irreducible root system of rank $\ell$. The fundamental invariants of $\Phi$ is a set of $\ell$ integers $d\_1, \cdots, d\_\ell$ canonically attached to $\Phi$.
Now suppose $\Psi$ is a closed subsystem of $\Phi$ and suppose $\Psi$ is generated by a subset of simple roots of $\Phi$. (In the language ... | https://mathoverflow.net/users/41301 | Fundamental invariants for root subsystems | I will give the following positive answer:
Let H be a split reductive subgroup of the split reductive group G of the same rank. Let WH and WG be their Weyl groups. Then every fundamental invariant of WH divides a fundamental invariant of WG.
Unsatisfactionally, this proof will not give a bijection between fundament... | 3 | https://mathoverflow.net/users/425 | 222265 | 104,151 |
https://mathoverflow.net/questions/222263 | 6 | Consider the following two symplectic matrices
$$
A \ = \
\left(\begin{array}{rrrr}%
1&0&0&0\\%
0&1&0&0\\%
0&0&-1&1\\%
0&0&-1&0\\%
\end{array}\right), \ \ \
B \ = \
\left(\begin{array}{rrrr}%
-1&0&0&-1\\%
0&0&-1&0\\%
0&1&-1&0\\%
1&0&0&0\\%
\end{array}\right).
$$
Is it true that the (Zariski-dense) group $\langle A,B \r... | https://mathoverflow.net/users/1568 | Structure of the group generated by two specific symplectic matrices | Your representation $p$ is *not* faithful, since we have
$$
(ABA^{-1}BA^{-1}BAB^{-1})^3 \ = \ 1.
$$
In particular, this means that
$$
(aba^{-1}ba^{-1}bab^{-1})^3 \ = \
\left(\begin{array}{rr}%
-24587&42408\\%
15048&-25955\\%
\end{array}\right)
$$
lies in the kernel of $p$.
| 7 | https://mathoverflow.net/users/28104 | 222267 | 104,152 |
https://mathoverflow.net/questions/222253 | 2 | Let given ring $R$ without zero divizors, where adittive group of $R$ with zero torsion. Let given subring $R\_0\leq R$, and $p$ is prime number, such that $\forall r\in R, \exists i>0 : p^ir\in R\_0$. Is it true that if $Nil(R\_0/pR\_0)=\{0\}$, then $Nil(R/pR) =\{0\}$?
Second question:
Is previous problem true in ... | https://mathoverflow.net/users/82143 | Subring of ring | No, this isn't true in general:
$$R\_0:= \mathbb{Z}[X] \le \mathbb{Z}[X,Y]/(X^2-pY)=: R$$
is a counter-example because
$R/pR = \mathbb{F}\_p[X,Y]/(X^2)$ has non-trivial radical.
$R$ is a domain since $X^2-pY$ is irreducible in $\mathbb{Z}[X,Y]$. To see that $R\_0$ embedds into $R$ suppose that $f\in \mathbb{Z}[X]$... | 3 | https://mathoverflow.net/users/17734 | 222277 | 104,156 |
https://mathoverflow.net/questions/222273 | 5 | I am working on a problem where the following sum appears:
$$F(s, t)=\frac{1}{\Gamma(1+2\alpha)}\sum\_{n=0}^{\infty}{\frac{s^{n} t^{n}}{\left[(s+1)(t+1)\right]^{n+1+\alpha}}\frac{\Gamma(n+1+2\alpha)}{\Gamma(n+1)}}$$
where $s, t$ are positive real numbers and $\alpha>-1/2$. If I let Wolfram Mathematica calculate this se... | https://mathoverflow.net/users/39163 | On a Sum of Gamma Functions | Expanding my comment.
Notice that $\frac{\Gamma(n+2\alpha+1)}{\Gamma(n+1)\Gamma(2\alpha+1)}=\binom{n+2\alpha}{n}=(-1)^n\binom{-2\alpha-1}{n}$ and thus the sum can be rewritten as
$$\frac{1}{[(s+1)(t+1)]^{\alpha+1}}\sum\_{n=0}^{\infty} \binom{-2\alpha-1}{n} \left(-\frac{st}{(s+1)(t+1)}\right)^n$$
$$ = \frac{1}{[(s+1)(... | 10 | https://mathoverflow.net/users/7076 | 222279 | 104,157 |
https://mathoverflow.net/questions/221992 | 5 | Let$\newcommand{\mM}{\mathcal{M}}$ $\mM\_{1,1}$ be the moduli stack of elliptic curves. Let $R$ be a Dedekind domain, say $\mathbb{Z}[1/N]$ for simplicity, and suppose we have a finite etale cover:
$$\mM\rightarrow\mM\_{1,1}[1/N]$$
Must the coarse moduli scheme $M$ of $\mM$ be smooth over $\mathbb{Z}[1/N]$? This is c... | https://mathoverflow.net/users/15242 | are the coarse moduli schemes of finite etale covers of $\mathcal{M}_{1,1}$ smooth? | Yes, $M$ is smooth.
In proving this we may focus on a fixed residue characteristic, so we loose no generality by assuming that $N$ is divisible by a prime $\ell \ge 3$. The morphism $\mathcal{Y}(\ell)[1/N] \rightarrow \mathcal{M}\_{1, 1}[1/N]$ is a $\mathrm{GL}\_2(\mathbb{Z}/\ell\mathbb{Z})$-torsor, and so is its ba... | 5 | https://mathoverflow.net/users/5498 | 222280 | 104,158 |
https://mathoverflow.net/questions/222272 | 13 | Forgive me for my ignorance, but I'm very surprised to learn that there are [two](https://en.wikipedia.org/wiki/Ivan_Matveyevich_Vinogradov) [Vinogradovs](https://en.wikipedia.org/wiki/Askold_Ivanovich_Vinogradov), both famous in the field of analytic number theory. Guessing from their names and [the Russian naming con... | https://mathoverflow.net/users/37103 | Two Vinogradovs? Is one the son of the other? | The answer seems to be no. It is hard to find direct confirmation, but every time their names are mentioned together it is made clear that there's no relationship.
>
> The Russian mathematician Askold Ivanovich Vinogradov is not to be
> confused with the other Russian mathematician (the mathematical
> great-grand... | 14 | https://mathoverflow.net/users/43108 | 222289 | 104,161 |
https://mathoverflow.net/questions/222225 | 2 | Let $X$ be a projective surface over $\mathbb{C}$, let $x\in X$ be the only singular point of $X$. Let $L$ be an ample line bundle on $X$. Consider the blow up $Y$ of $X$ along $x$, $f:Y\longrightarrow X$. Let $L'$ be the pull back of $L$ to $Y$ and let $E$ be the exceptional divisor.
We have the following short exa... | https://mathoverflow.net/users/70211 | Relation between curves in a complete linear system contained in another | In the following I suppose that by "curve" in a linear system you mean "effective divisor" vithout any claim about being irreducible and/or reduced.
1) Curves in |L'| are exactly the pull-back of curves in |L|. So "curves" in |L'-E| are obtained by pulling-back curves in L through x, and then remove $E$ with multipli... | 1 | https://mathoverflow.net/users/46104 | 222296 | 104,163 |
https://mathoverflow.net/questions/222299 | 10 | As the title says. Can we determine all the integral points on elliptic curves of the form
$$y^2=x^3+px$$
for a prime $p$? If yes, can someone explain me how? A good reference would also be sufficient.
| https://mathoverflow.net/users/76011 | Integral points on elliptic curves of the form $y^2=x^3+px$ | This is completely worked out in Walsh's paper:
* P. G. Walsh, [Integer Solutions to the Equation $y^2=x(x^2\pm p^k)$](http://projecteuclid.org/download/pdf_1/euclid.rmjm/1214947612) (2008)
In particular, for your elliptic curve $y^2=x^3+px$, there are at most 2 primitive integer solutions if $p>3$, and 4 if $p=3$.... | 13 | https://mathoverflow.net/users/43108 | 222300 | 104,164 |
https://mathoverflow.net/questions/222298 | 0 | Let $G = (V,E,W)$ be a weighted graph, where each edge $e = (v\_i,v\_j)$ has weight $w\_{ij} \in \mathbb Z^+ \cup \{0\}$. By replacing $e$ with $w\_{ij}$ copies of unweighted multiedges, a weighted graph $G$ is transferred as an unweighted multigraph. For example, a $2$-walk $v\_0 v\_1 v\_2$ in the weighted graph versi... | https://mathoverflow.net/users/23202 | Number of $k$-walks containing a vertex in an unweighted multigraph | Let $A = (w\_{ij})$ be the (weighted) adjacency matrix of $G$. Then:
(i) the number of closed $k$-walks containing vertex $v\_1$ equals $(A^k)\_{1,1}$;
(ii) the number of non-closed $k$-walks containing $v\_1$ as an end point equals $\sum\_{i\ne 1} (A^k)\_{i,1} + (A^k)\_{1,i}$;
(iii) the number of non-closed $k$-... | 1 | https://mathoverflow.net/users/7076 | 222307 | 104,168 |
https://mathoverflow.net/questions/222292 | 4 | Take $k$ consecutive composite integers from a prime gap. What is known about the largest prime divisor of their product?
It seems to me that except for the triplet $(8,9,10)$ and the pair $(8,9)$ , this largest prime divisor is always larger than $2k$, but I could not find an elementary (my level) proof.
As $k$ g... | https://mathoverflow.net/users/64384 | What is known about the largest prime divisor of the product of $k$ consecutive integers? | There have been several investigations into the largest prime factor of a product of consecutive integers; the Sylvester--Schur theorem is an early example. Here is a survey by Shorey and Tijdeman: <https://www.math.leidenuniv.nl/~tijdeman/shoretij.pdf>
In
Laishram, Shanta(6-TIFR-SM); Shorey, T. N.(6-TIFR-SM)
The ... | 8 | https://mathoverflow.net/users/16510 | 222309 | 104,169 |
https://mathoverflow.net/questions/222216 | 9 | Given a rational bivariate power series $F(x,y)=\sum{a\_{n,m}x^ny^m}$, the diagonal function $G(t):=\sum{a\_{n,n}t^n}$ is known to be algebraic, although not rational in general. I was wondering if there were some known conditions for it to be rational. The setting I'm working with has $a\_{n,m}$ non-negative integers;... | https://mathoverflow.net/users/41283 | When is the diagonal of a rational bivariate power series again rational | Assume for convenience that each $a\_{n,m}\in\mathbb{C}$. Consider the
proof in *Enumerative Combinatorics*, vol. 2, Theorem 6.3.3, that
$G(t)$ is algebraic. One computes the constant term (with respect to
$s$) of $F(s,t/s)$ (being careful of the meaning of $F(s,t/s)$). The
zeros of the denominator of $F(s,t/s)$ (with ... | 6 | https://mathoverflow.net/users/2807 | 222311 | 104,171 |
https://mathoverflow.net/questions/79371 | 11 | [This](http://www.math.tifr.res.in/putabstract.php?date=2011-02-03) talk by Jinhyun Park connects a lot of interesting themes, making me [curious](http://www.psychologicalscience.org/index.php/news/releases/curiosity-doesnt-kill-the-student.html) to read more about that. Do you know where?
| https://mathoverflow.net/users/451 | Hilbert's 3rd problem,number theory, motives, cyclic homology,... | This circle of topics is certainly one of my favourite surprising connections in mathematics. I will try to outline what little I understand of the big picture. Apologies for the length.
**Hilbert's 3rd problem and Dehn complexes:** As is well-known, Hilbert's 3rd problem asked for examples of tetrahedra of equal vol... | 26 | https://mathoverflow.net/users/50846 | 222320 | 104,174 |
https://mathoverflow.net/questions/221384 | 0 | Let $M$ be a type $II\_1$ factor von Neumann algebra, and let $G$ be a discrete group acting on $M$ which is free and ergodic. Is the crossed product von Neumann algebra $M \rtimes G$ type $II\_1$ factor?
| https://mathoverflow.net/users/73660 | types of crossed product von Neumann algebras | One assumption that would guarantee that the crossed product is a factor is that the action be properly outer. Incidentally, I am not sure what you mean by ``ergodic'' in this setting.
The crossed product is automatically a finite von Neumann algebra (if $M$ is any von Neumann algebra and $G$ is a discrete group, the... | 4 | https://mathoverflow.net/users/75274 | 222323 | 104,175 |
https://mathoverflow.net/questions/221935 | 4 | Given two subsets $A,B$ of the Cantor cube $2^\omega$ we write $A\le\_W B$ (resp. $A\le\_H B$) if there is a continuous (and injective) function $f:2^\omega\to 2^\omega$ such that $f^{-1}(B)=A$. The relation $\le\_W $ is the well-known (and well-studied) Wadge reducibility relation. The relation $\le\_H$ is called *the... | https://mathoverflow.net/users/61536 | Hurewicz versus Wadge hierarchy of zero-dimensional Borel sets? | After some thoughts I realized that the answers to Problems 1 and 3 are negative. Namely, each limit Wadge class contains infinitely many Hurewicz non-equivalent spaces. Indeed, take a sequence $(U\_n)\_{n\in\omega}$ of pairwise disjoint clopen subsets of the Cantor cube $2^\omega$ that converge to some point $x\_\inft... | 3 | https://mathoverflow.net/users/61536 | 222336 | 104,177 |
https://mathoverflow.net/questions/222330 | 2 | Let $\bf{G}$ be a simple and simply connected algebraic group over $\mathbb{Q}$. Is it true that the base extension of $\bf{G}$ over $\bar{\mathbb{F}}\_p$, i.e., ${\bf G}\times\_{\mathrm{Spec(\mathbb{Z})}}\mathrm{Spec}(\bar{\mathbb{F}}\_p)$, is simple and simply connected for all but finitely many p?
| https://mathoverflow.net/users/8419 | base extension of algebraic groups | The answer to the "reasonable" interpretation of your question is "yes". The real point is that the property of being "simply connected" in the sense of connected semisimple groups is characterized in terms of the root datum of the geometric fiber, and there is a "local constancy" property for root data in the setting ... | 3 | https://mathoverflow.net/users/81332 | 222337 | 104,178 |
https://mathoverflow.net/questions/222333 | 3 | I have an array of integers T[N] indexed from 1. For example T[1] = 1, T[2] = 4, T[3] = 5, T[4] = 9. Let's enumerate all subsets of this set in a certain order: increasing sum of the elements. In case of a tie, we compare the sorted lists of indexes of numbers in subsets.
For example:
>
> 1. {} = 0 indexes: 0
> 2... | https://mathoverflow.net/users/82181 | $k$-th subset in order of increasing sum | Since this is MathOverflow I'll give you the mathematical answer rather than the practical answer. I'll assume all numbers are positive; otherwise minor modifications would be needed to handle negative numbers. Also, I'm taking literally your request for the $k$th subset alone, rather than for all subsets up to that on... | 6 | https://mathoverflow.net/users/440 | 222341 | 104,181 |
https://mathoverflow.net/questions/222314 | 3 | I'm starting to study geodesic laminations on hyperbolic surfaces and in particular I'm focusing my attention on $\mathcal{PML}\_0(S)$, the space of projective classes of measured geodesic laminations with compact support on the surface $S$.
This space is compact (it is stated for example in the article "On Teichmuelle... | https://mathoverflow.net/users/nan | Why is $\mathcal{PML}_0(S)$ compact? | The original proof is given in Theorem 8.10.5 of ["The Geometry and Topology of 3-manifolds"](http://library.msri.org/books/gt3m/) by W. P. Thurston.
| 2 | https://mathoverflow.net/users/20787 | 222342 | 104,182 |
https://mathoverflow.net/questions/220975 | 8 | I'm a first year phd student in Germany. I've started my phd study one year ago and I'm currently confused about the topic I've chosen. The program is in the area of PDEs, and actually I didn't learn much about PDEs in my master study, only for instance the solution of the four fundamental types of PDEs and some Sobole... | https://mathoverflow.net/users/76318 | Asking for Advices for Choosing a Ph.D thesis problem (in PDE area) | Well, I'm not sure if I got your question correctly. If I understood apropriately, you are asking for real life models where the test functions have restrictions as being divergence/curl free. Fluid dynamics is one of the fields where this restriction appear, not only on the solution, but also on the test function. Che... | 2 | https://mathoverflow.net/users/33135 | 222348 | 104,184 |
https://mathoverflow.net/questions/222352 | 0 | What is the most commonly used treatment method of the moving interface in the classical two phase Stefan problems with the finite element method. Here I mean the water-ice two phase problem under freeze-thaw cycles. Can anyone give a somewhat detailed explanation?
| https://mathoverflow.net/users/82192 | The classical two phase Stefan problems | there is a very large literature, you could start for example from a [textbook](https://books.google.nl/books/about/The_Stefan_Problem.html?id=l0-t-sjVCqYC&redir_esc=y); an overview of numerical methods is given [here](http://www.sciencedirect.com/science/article/pii/S0377042705003730):
>
> In this paper, we presen... | 0 | https://mathoverflow.net/users/11260 | 222357 | 104,188 |
https://mathoverflow.net/questions/221640 | 6 | In contrast to classic results for arithmetic progressions of arbitrary length in one set at least of any finite partition of $\mathbb N$, it is easy to construct a partition in two sets of integers $A$ and $B$ such that neither $A$ nor $B$ contains an infinite arithmetic sequence. Actually (with choice, of course), on... | https://mathoverflow.net/users/17164 | A kind of anti-Ramsey result | According to [P. Erdõs, A. Hajnal: On a property of families of sets, Acta Math. Acad. Sci. Hungar. 12 (1961), 87--123](http://www.renyi.hu/~p_erdos/1961-11.pdf) (see page 90) the stronger result you formulated is a theorem of Bernstein from *F. Bernstein, "[Zur Theorie der trigonometrischen Reihen](https://doi.org/10.... | 5 | https://mathoverflow.net/users/71011 | 222364 | 104,192 |
https://mathoverflow.net/questions/219888 | 8 | Consider a sequence of complex polynomials $f \in \mathbb{C}[z]$, $f(0) \neq 0$, that are composed of a negligible fraction $o(\deg{f})$ of monomials. Are the zeros of such polynomials necessarily equidistributed in angle, for the uniform measure $d\theta/2\pi$ on $S^1 = \mathbb{C}^{\times} / \mathbb{R}^{> 0}$?
Certa... | https://mathoverflow.net/users/26522 | Angular distribution of zero sets of sparse polynomials | A lot depends on how exactly to understand this question. The crudest form (for an $m$-nomial of degree $n$ with non-zero free term, the number of all zeroes in a sector of aperture $2\pi\theta$ is $\theta n$ with an error at most $m$) is an exercise in elementary complex analysis. Indeed, consider a sector of aperture... | 4 | https://mathoverflow.net/users/1131 | 222373 | 104,197 |
https://mathoverflow.net/questions/222353 | 15 | I might direct this question to Urs Schreiber directly, but just in case someone else has some interesting examples, I'll make the question public.
The formulation of differential cohomology in certain cohesive $\infty$ topos seems, to me, to be an incredibly powerful generalization of the concepts surrounding classi... | https://mathoverflow.net/users/43687 | Examples of differential cohomology in cohesive $\infty$ topos | If we consider just shape modality "$\Pi$" (or "ʃ") and flat modality $\flat$ (which are sufficient for the [differential cohomology hexagon](http://ncatlab.org/nlab/show/differential+cohomology+diagram), then the traditional [arithmetic fracture squares](http://ncatlab.org/nlab/show/fracture+theorem) are the left half... | 10 | https://mathoverflow.net/users/381 | 222377 | 104,198 |
https://mathoverflow.net/questions/222331 | 18 | **Question:** Is there a linear recurrence sequence $(u\_n)\_{n\geq0}$ (on the rationals, but I would also be interested by reals) for which $\text{Pos}(u) = \{i \mid u\_i > 0\}$ is precisely the set of Fibonacci numbers?
*Additional infos:*
* It is well known that the set $\{i \mid u\_i = 0\}$ is quite easy to des... | https://mathoverflow.net/users/7687 | For a linear recurrence sequence $(u_n)_{n\geq 0}$, can $\{i \mid u_i > 0\}$ be the set of Fibonacci numbers? | Solutions to (complex) linear recurrences are of the form
$$\sum\_i c\_i n^{e\_i} \alpha\_i^n.$$
To build such a function that is positive only on Fibonacci numbers, take $n^2(\alpha^n + \bar{\alpha}^n-2)+c$ where $\alpha = \exp(2\pi i / \phi), \phi = \frac{\sqrt{5} + 1}{2}$. For this to be positive, $\Re(\alpha^n... | 27 | https://mathoverflow.net/users/2954 | 222379 | 104,199 |
https://mathoverflow.net/questions/222349 | 7 | I'm studying root systems and coming up with an observation:
Let $\Phi$ be an irreducible root system and $\Phi^+$ be a positive root system. Denote by $\Delta=\{\alpha\_1, \alpha\_2,\ldots,\alpha\_n\}$ be the corresponding base. For $\beta, \gamma \in \Phi^+$, $\alpha\_1, \alpha\_2 \in \Delta$ with the properties th... | https://mathoverflow.net/users/50437 | A technical question about root systems | Consider the root system $\mathsf{B}\_n$ with the standard numbering of the fundamental roots (that is, $\alpha\_n$ is short). Take $\alpha\_{n-2}$ and $\alpha\_n$ as a pair of orthogonal roots (indeed, $\alpha\_{n-2}+\alpha\_n$ is not a root), and take $\gamma=\alpha\_{n-1}+\alpha\_n$. Then $\beta\_1=\alpha\_{n-2}+\al... | 4 | https://mathoverflow.net/users/5018 | 222383 | 104,201 |
https://mathoverflow.net/questions/206346 | 3 | Suppose that for each field $F$ a linear map $X(F): H\_M^{p,q}(F, \mathbb{Q}) \longrightarrow H\_M^{p,q}(F,\mathbb{Q})$ is given, such that $X$ commutes with inclusions of fields and transfers for finite field extensions. Is it proved, conjectured or doubted that each such map is just a multiplication on a rational num... | https://mathoverflow.net/users/21620 | Do there exist nontrivial motivic cohomology operations preserving weights? | I agree with Mikhail Bondarko's comment, the condition in the question is certainly too weak. However, looking at cohomology operations (as the title suggests) something can be said. The basic issues involved in the question seem to be
1. What can we say about stable operations for rational motivic cohomology?
2. Ho... | 7 | https://mathoverflow.net/users/50846 | 222389 | 104,204 |
https://mathoverflow.net/questions/212706 | 16 | The sequence $(a\_n)\_{n \ge 0}$ satisfies, $a\_0 = a\_1 = 1$ and the recursion relation:
$$a\_n = \sum\limits\_{k=0}^{[n/2]} \frac{a\_k}{(n-2k)!}$$
where, $[x]$ is the nearest integer to $x$ not exceeding it.
Alternatively define $a\_n$'s as:
$$\sum\limits\_{n=1}^{\infty} a\_nx^n = \exp\left(\sum\limits\_{n=0}... | https://mathoverflow.net/users/62680 | Asymptotics of a recurrence relation | It is not hard prove the bounds you want by purely real variable techniques. First note that the $a\_n$ are non-negative for all $n$. For a general non-negative sequence $a\_n$, and real numbers $N>0$, put
$$
F(N) = \sum\_{n=0}^{\infty} a\_n e^{-n/N},
$$
and assume that there are constants $\alpha >1$, and positive... | 12 | https://mathoverflow.net/users/38624 | 222394 | 104,206 |
https://mathoverflow.net/questions/222393 | 0 | I know that for a noetherian ring, it's hilbert series can be written as $$HS(t)=\frac{P(t)}{\prod\_{i=1}^d{(1-t^{d\_i})}}$$ where $P(t)$ is polynomial, and there are $d$ generators of degrees $d\_1,\dots,d\_k$.
I want a partial converse, namely if I'm given a Hilbert series and I know the orders of the poles, when ... | https://mathoverflow.net/users/41283 | When can one infer degrees of generators of a ring from its hilbert series | Your second paragraph is not accurate as stated. You need the
generators to be algebraically independent. At any rate, the ring
$\mathbb{C}[x^2, xy, y^2, z^4]$ has Hilbert series $1/(1-t^2)^3$, so
the generator of degree four cannot be gleaned from the Hilbert
series.
| 7 | https://mathoverflow.net/users/2807 | 222397 | 104,207 |
https://mathoverflow.net/questions/222403 | 13 | I'm very keen to deepen my understanding of arithmetic and diophantine problems. In the past I studied some algebraic, analytic and sieve based number theory. Recently I've been reading *Weil - Basic Number Theory* which covers some early results of Fermat and Euler in their original forms and then discusses modern int... | https://mathoverflow.net/users/82207 | Arithmetical results to help study arithmetic geometry? | That will depend on your more specific interests. One point of entrance into arithmetic geometry could be to have a good understanding of Szpiro's discriminant-conductor inequality for non-isotrivial elliptic curves over a function field, particularly the proof embodied by the Kodaira-Spencer class, and to see how this... | 13 | https://mathoverflow.net/users/26522 | 222409 | 104,212 |
https://mathoverflow.net/questions/222434 | 2 | I have another question about laminations on surfaces. As usual let $\mathcal{S}$ be the set of homotopy classes of simple closed curves in $S$ and $\mathcal{PML}\_0(S)$ be the set of projective classes of measured geodesic laminations on a surface $S$. I know there is an injective map
$i:\mathcal{S}\rightarrow \math... | https://mathoverflow.net/users/nan | Why simple closed curves are dense in $\mathcal{PML}_0(S)$? | See [McMullen's 2005 note](http://www.math.harvard.edu/~ctm/home/text/class/harvard/275/05/html/home/course/course.pdf)s, Cor 2.4 Another proof can be found in Thurston's notes (chapter 8).
| 2 | https://mathoverflow.net/users/11142 | 222436 | 104,224 |
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