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https://mathoverflow.net/questions/163212 | 1 | How to show?
$${}\_2F\_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$
It numerically is very close, came up when evaluating:
$$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 \times 3}{1 \times 3 \times 5} + \frac{1 \times 2 \times 3 \times 4}{1 \times 3 \times 5 \times 7} + \cdots = 1 + \fr... | https://mathoverflow.net/users/36221 | Hypergeometric sum specific value | This is equivalent to showing that $\displaystyle\sum\_{n=2}^\infty\frac{n!}{(2n-1)!!}=\frac\pi2$ , which, after multiplying both the
numerator and the denominator with $(2n)!!=2^n\,n!$, and taking into account that $\dfrac{(2n)!}{n!^2}=$
$=\displaystyle{2n\choose n}$, can be rewritten as $\displaystyle\sum\_{n=2... | 3 | https://mathoverflow.net/users/39602 | 163244 | 85,501 |
https://mathoverflow.net/questions/163200 | 5 | In order to prove non-uniqueness of singular vectors when a repeated singular value is present, the book (Trefethen-Bau, considered the most authotitative book on the subject), argues as follows: Let $\sigma$ be the first singular value of A, and $v\_{1}$ the corresponding singular vector. Let $w$ be another linearly i... | https://mathoverflow.net/users/49490 | About partial uniqueness of SVD | We have $\|Av\_1\|=\sigma$. Assume, $\|Av\_2\| < \sigma$. Now, $w= av\_1+bv\_2$, so \begin{equation}
\|Aw\|^2=|a|^2 \|Av\_1\|^2 + |b|^2 \|Av\_2\|^2 + 2ab\langle Av\_1,Av\_2\rangle < \sigma^2 + 2ab\langle Av\_1,Av\_2\rangle.\quad (\*)
\end{equation}
Let $u\_1$ be the first left-singular vector of $A$, so that $Av\_1 =... | 3 | https://mathoverflow.net/users/8430 | 163267 | 85,509 |
https://mathoverflow.net/questions/163255 | 3 | In differential geometry, tangent vectors are considered operators.
At point p, the local tangent space is defined as
$$
T\_p(M)=\{X^i\partial\_i|X\in R^n\}
$$
This is quite easy to understand for me.
However, I study information geometry recently and get stuck with the tangent vectors defined on statistical manifol... | https://mathoverflow.net/users/49550 | Why tangent vector of statistical manifold is a function? | There are situations where you think of a function space as an infinite-dimensional manifold. Then the tangent space is also infinite-dimensional, and also naturally a function space. The tangent space to a product manifold is the direct sum of the tangent spaces, and a function space is like a (contiuum) product, so i... | 4 | https://mathoverflow.net/users/327 | 163271 | 85,511 |
https://mathoverflow.net/questions/162987 | 2 | One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F\_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for the discrete logarithm problem. A very powerful tool for the study of those abelian varieties is the theory of complex m... | https://mathoverflow.net/users/48499 | Canonical lifts from $\mathbb F_q$ and CM-theory | As pointed out by Stankewicz (this is just a repeat of their answer) there is an equivalence of categories between ordinary abelian varieties over $\mathbf{F}\_q$ and lattices $\Lambda$ with an F; the precise form is Theorem 7 of the paper "Vari\'et\'es ab\'e1iennes ordinaires sur un corps fini", Inventiones mathematic... | 3 | https://mathoverflow.net/users/44817 | 163273 | 85,512 |
https://mathoverflow.net/questions/163274 | 52 | Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes.
Let $\mathrm{Li}(x)$ be the offset [logarithmic integral](http://en.wikipedia.org/wiki/Logarithmic_integral), let $\pi(x)$ be the [prime counting function](http://en.wi... | https://mathoverflow.net/users/6514 | Are the primes normally distributed? Or is this the Riemann hypothesis? | These questions on the spacings between primes are expected to be true, but are far from being proved. They are not directly related to RH, but seem to encode other relations among zeros. The conjecture (1) follows from the Hardy-Littlewood prime $k$-tuples conjectures; this was established by Gallagher. More precise v... | 68 | https://mathoverflow.net/users/38624 | 163275 | 85,513 |
https://mathoverflow.net/questions/162913 | 0 | Recently I was reading an interesting proof for Bohr's lemma by the tool of hyperbolic metric, however I have a following question:
Given a holomorphic map $f$ on $D$, $f(0)=0$, and $|f|<1$ on $D\_{r}=\{|z|<r\}$ and $|f|=1$ somewhere on $\partial D\_{r}$. Denote $\rho\_{U}$ as the hyperbolic metric in $U$
We define... | https://mathoverflow.net/users/11966 | A question for hyperbolic metric in the proof for Bohr's lemma | Beardon and Pommerenke give an optimal estimate (up to a fixed multiplicative error) on the density of the hyperbolic metric on a multiply-connected plane domain $W$. It is of the form
$$ \rho\_W(z) \approx \frac{1}{\operatorname{dist}(z,\partial W)\cdot \max(M,1)},$$
where $M$ is the maximal modulus of an essential ro... | 1 | https://mathoverflow.net/users/3651 | 163283 | 85,515 |
https://mathoverflow.net/questions/163227 | 2 | I am wondering if we can minimize a strictly convex quadratic function in finite time, subject to linearly equality and nonnegativity constraints.
Thanks!
| https://mathoverflow.net/users/49425 | Minimize a strictly convex quadratic function subject to linearly equality and nonnegativity constraints in finite time? | Although complexity analysis can give you some insight on the difficulty of your problem, it is unlikely that will settle your question in full-generality.
For example: in the oracle model, a strongly convex function can be minimized in time $O(\ln(1/\varepsilon))$. However, since your domain is a general polyhedron,... | 4 | https://mathoverflow.net/users/39129 | 163289 | 85,517 |
https://mathoverflow.net/questions/152909 | 2 | Consider the lienard vector field $\cases{
x'=y -F(x) \\
y'=-x }
$ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined on $R^{2}$ such that $[X,Y]=0$.
can we conclude that $Y$ is tangent to solutions of lienard vector field?
Namely we ask:
... | https://mathoverflow.net/users/36688 | The centralizer of Lienard equation | This is not a complete answer, but it should give some elements towards it (especially regarding the link with PDEs). First, the Liénard system has a non-trivial centralizer in the case $F''=0$, *i.e.* $F(x)=cx$ for a real constant $c$, as the radial vector field belongs to it. This trival case put aside, write $Y=A\pa... | 3 | https://mathoverflow.net/users/24309 | 163291 | 85,518 |
https://mathoverflow.net/questions/160691 | 8 | Raphael Douady's [thesis](http://citeseer.uark.edu:8080/citeseerx/showciting;jsessionid=82A0592660D37BA6C373CABF62CF034A?cid=2510347), *Applications du théorème des tores invariants*, has been cited in numerous papers by many experts.
According [Wikipedia](http://en.wikipedia.org/wiki/Raphael_Douady), *he proves of ... | https://mathoverflow.net/users/11028 | Raphael Douady's thesis: Applications du théorème des tores invariants | I'm sorry that my PhD thesis wasn't published indeed and is hard to find (up to my knowledge it is only available at university of Paris 7). As I have a scanned copy of it, I asked people at my software company Riskdata to include it in the list of posted papers, althogh it is not really related to math finance, which ... | 23 | https://mathoverflow.net/users/49570 | 163293 | 85,519 |
https://mathoverflow.net/questions/162938 | 3 | If $\gamma$ and $\eta$ are two smooth curves in a smooth manifold $M$, is it possible to find a diffeomorphism of $M$ such that $f \circ \gamma = \eta$? What if one removes the assumption of smoothness, working with topological manifolds (or even plain topological spaces) and homeomorphisms?
(Later edit: Argghh! I ap... | https://mathoverflow.net/users/54780 | Diffeomorphism with prescribed behaviour | If your definition of "curve" is embeddings of say $[0,1]$ in a manifold, then the result follows in dimensions $n\geq 2$ by the isotopy extension theorem, together with the observation that the unit tangent bundle of a connected manifold is connected. In dimension $n=1$ the diffeomorphism would not always be isotopic ... | 2 | https://mathoverflow.net/users/1465 | 163295 | 85,520 |
https://mathoverflow.net/questions/163294 | 11 |
>
> What is the limit, as $n \to \infty$, of the expected distance between two
> points chosen uniformly at random within a unit edge-length hypercube
> in $\mathbb{R}^n$?
>
>
>
For $n=1$, the average distance is $\frac{1}{3}$.
For $n=2$, it is approximately $0.52$.
For $n=3$, approximately $0.66$ (Robbins' co... | https://mathoverflow.net/users/6094 | Limit of distance between two random points in a unit $n$-cube | The limit is $\infty$. See equation (2) of <http://mathworld.wolfram.com/HypercubeLinePicking.html>
| 9 | https://mathoverflow.net/users/1847 | 163296 | 85,521 |
https://mathoverflow.net/questions/163286 | 3 | I have some -possibly- simple but broad questions: Where to begin the study of von Neumann Algebras? Which are the important questions in the field that guide current research? I'm interested in learning the basics to start working at research level in the field.
My backgroud is: some basic Functional and Real Analys... | https://mathoverflow.net/users/34679 | References for von Neumann Algebras | For a rapid introduction see <http://www.math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf> (Von Neumann Algebras, by Vaughan F.R. Jones -- "The purpose of these notes is to provide a rapid introduction to von Neumann algebras which gets to the examples and active topics with a minimum of technical baggage. ")
A su... | 8 | https://mathoverflow.net/users/32389 | 163299 | 85,523 |
https://mathoverflow.net/questions/163301 | 21 | Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a $3$-dimensional topological quantum field theory (going down to surfaces), Chern-Simons theory.
Later other authors (Reshe... | https://mathoverflow.net/users/290 | What is Chern-Simons theory expected to assign to a point? | I have a proposal for what Chern-Simons should assign to a point:
>
> The $\otimes$-category of (certain) representations of $\widetilde{\Omega G}$.
>
>
>
Here, $\Omega G$ is the based loop group, and the tilde indicates that one should take the central extension inherited from the level $k$ central extensi... | 18 | https://mathoverflow.net/users/5690 | 163305 | 85,525 |
https://mathoverflow.net/questions/163292 | 3 | Consider the following setup: three bounded, 'nice' convex sets $A \subseteq B \subsetneq C \subset \mathbb{R}^2$, and three points $x,y,z\in \partial A\cap \partial B\cap \partial C$ (see edit below). Imagine a triangle inscribed in a circle, with some convex curves between the edges of the triangle and the circle, 'c... | https://mathoverflow.net/users/48403 | Perimeter of a 'trapped' convex set | Let me add the required modification (as a separate "*answer*" to keep each text clean). I could simply replace $\ f\ $ by $\ f(x\ y):= (a\!\cdot\!x\ \,\frac ya)$, where $\ 0<a<3\ $ and $\ a\ $ is very close to $\ 3$. But I prefer to provide a specific example.
Let me repeat that the answer to the **Question** in gen... | 5 | https://mathoverflow.net/users/8385 | 163311 | 85,527 |
https://mathoverflow.net/questions/105942 | 14 | Let $K$ be an imaginary quadratic field with Hilbert class field $H$, and let $E$ be an elliptic curve defined over $H$ with complex multiplication by the ring of integers $O\_K$ of $K$. It is known that for an integral ideal $\mathfrak{m}$ of $O\_K$, $K(j(E),h(E[\mathfrak{m}]))$ is the ray class field of K modulo $\ma... | https://mathoverflow.net/users/16120 | Elliptic Curves with CM and Class Field Theory | In fact Shimura handled the case of an abelian variety $A$ with complex multiplication by an order $O$ inside the maximal order $O\_K$ of the CM field $K$.
A very good modern reference is the following article by Marco Streng:
[An explicit version of Shimura's reciprocity law for Siegel modular functions.](http://pub.m... | 10 | https://mathoverflow.net/users/26737 | 163328 | 85,533 |
https://mathoverflow.net/questions/163329 | 27 | Let me start by acknowledging the existence of this thread: [Mathematics and cancer research](https://mathoverflow.net/q/87575)
It is well-known that mathematical modeling and computational biology are effective tools in cancer research. When I started college and declared the math major, this was the direction I env... | https://mathoverflow.net/users/10547 | Algebra and cancer research | You might like to look at this paper:
* Monica Nicolau, Arnold J. Levine, and Gunnar Carlsson, [Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival](https://doi.org/10.1073/pnas.1102826108), Proceedings of the National Academy of Sciences, Febru... | 28 | https://mathoverflow.net/users/10366 | 163334 | 85,535 |
https://mathoverflow.net/questions/67335 | 9 | Suppose given a noetherian ring $R$. On the stable category $R\text{-}\underline{\text{mod}} := R\text{-mod}/R\text{-proj}$, we have the Heller operator
$$
\Omega : R\text{-}\underline{\text{mod}} \to R\text{-}\underline{\text{mod}} \; ,
$$
defined by a choice of short exact sequences $\Omega X \to P \to X$ with $P$ ... | https://mathoverflow.net/users/9300 | Heller operator without left adjoint? | This is answered by Proposition 1.7 in
[Auslander, Maurice; Reiten, Idun. [Homologically finite subcategories](http://ebooks.cambridge.org/chapter.jsf?bid=CBO9780511661853&cid=CBO9780511661853A007&tabName=Chapter). Representations of algebras and related topics (Kyoto, 1990), 1--42, London Math. Soc. Lecture Note Ser.,... | 5 | https://mathoverflow.net/users/18756 | 163335 | 85,536 |
https://mathoverflow.net/questions/163316 | 4 | A finitely generated group G is said to satisfy Schreier's index formula if for every subgroup H of index k in G we have: d(H) - 1 = k(d(G) - 1). For example, a finitely generated free group satisfies Schreier's index formula.
For a group G we denote by G' its commutator subgroup.
Let F be a free group on a finite ... | https://mathoverflow.net/users/38889 | Schreier's index formula | I claim that a residually finite, finitely generated group $G$ satisfies Schreier's formula iff it is free.
Here is the idea. Let $X$ be the generating set of $G$ of size $d(G)$. Suppose $G$ is not free on that generating set. Suppose $K$ is the presentation complex for $G$ wrt generators $X$ and some set of relatio... | 8 | https://mathoverflow.net/users/15934 | 163352 | 85,545 |
https://mathoverflow.net/questions/163342 | 2 | Consider multisets of the form $A = \{a\_1,\dots,a\_n\}$ of integers. Let $q = P(a\_i = a\_j)$ when $i$ and $j$ are chosen independently and uniformly from $\{1,\dots, n\}$. Let $B$ be the set of integers in $A$. We know that $|B| \leq n$. Finally let $p\_b = P(a\_i = b)$ when $i$ is chosen uniformly from $\{1,\dots, n... | https://mathoverflow.net/users/45564 | How to minimize $-\sum p_b \ln{p_b}$? | First note that $q = \sum\_b p\_b^2$. We're going to use an instance of the AM-GM inequality:
$$
\sum\_b p\_b\cdot p\_b \geq \prod\_b p\_b^{p\_b}.
$$
This gives
$$
-\sum\_b p\_b \log p\_b
=
\log \Bigl( \prod\_b p\_b^{-p\_b} \Bigr)
\geq
\log \Bigl( 1\Big/\sum\_b p\_b^2\Bigr)
=
-\log q.
$$
This gives a lower bound.
... | 3 | https://mathoverflow.net/users/586 | 163360 | 85,548 |
https://mathoverflow.net/questions/163366 | 5 | Let $G$ be a finite abelian group (of odd order if it's significant) acting on sphere $S^2\subset\mathbb{R}^3$. So my question: is it true that $G$ has a fixed point?
| https://mathoverflow.net/users/43459 | Finite group acting on sphere | See
Fixed points of abelian actions on S2
JOHN FRANKSa1, MICHAEL HANDELa2 and KAMLESH PARWANIa3
(Ergodic Theory and Dynamical Systems, 2007)
| 4 | https://mathoverflow.net/users/11142 | 163369 | 85,552 |
https://mathoverflow.net/questions/163357 | 6 | If $Q:\mathbb{Z}^{2k}\to \mathbb{Z}$ is any positive definite integer -valued quadratic form in $2k$ variables, then it is well known, that the $\textbf{theta series}$ $\theta\_Q(z):=\sum\_{m\in\mathbb{Z}^{2k}}q^{Q(m)}\ (q=e^{2\pi i z})$ is a modular form of weight $k$ on the congruence group $\Gamma\_0(N)$ ( for some ... | https://mathoverflow.net/users/45300 | eisenstein part of the theta function | I believe the formula you're looking for is a formula of Siegel that expresses the $n$th Fourier coefficient of an Eisenstein series as a product of local densities. There are lots of papers giving formulas for these local densities in terms of the arithmetic of $Q$ in the literature. My two favorite sources are
$\bu... | 8 | https://mathoverflow.net/users/48142 | 163370 | 85,553 |
https://mathoverflow.net/questions/163363 | 3 | $A \in S(K)$ iff $A$ is a subalgebra of some member of $K$
$A \in H(K)$ iff $A$ is a homomorphic image of some member of $K$
It is trivial to see the containment $SH \leq HS$. Taking a simple example of rings also proves $HS \neq SH$. Currently I am stuck with proving $SH(K) = HS(K)$ where K is a monounary algebra ... | https://mathoverflow.net/users/46237 | H S class operator and its equality | I'm assuming that $(Z,successor)$ represents the set of integers with the successor function $s\colon Z\to Z$, $s(a)=a+1$.
I think the following works: we want to show that any homomorphic image of a subalgebra of $(Z,s)$ is a subalgebra of a homomorphic image of $(Z,s)$. A subalgebra of $(Z,s)$ is either the whole ... | 1 | https://mathoverflow.net/users/3959 | 163378 | 85,557 |
https://mathoverflow.net/questions/163384 | 6 | Let $R$ be a commutative ring and let $A\_1$ and $A\_2$ be (not necessarily commutative) $R$-algebras. Under which conditions on $A\_1$ and $A\_2$ is the following true:
For every projective $A\_1$-module $P\_1$ and every projective $A\_2$-module $P\_2$ we have that $P\_1\otimes\_R P\_2$ is projective as a $A\_1\otimes... | https://mathoverflow.net/users/49605 | is the tensor product of projective modules again projective? | Since $P\_1$ is projective there exists $Q\_1$ $A\_1$-module and an isomorphism
$$ P\_1\oplus Q\_1 = A\_1^{\oplus I\_1}$$
for some index set $I\_2$. Analogously there exists $Q\_2$ $A\_2$-module and an index set $I\_2$ such that
$$ P\_2\oplus Q\_2 = A\_2^{\oplus I\_2} $$
Tensoring the two previous relations
$$P\_1 \oti... | 13 | https://mathoverflow.net/users/43054 | 163389 | 85,560 |
https://mathoverflow.net/questions/163377 | 14 | **Background**
I am including this information about real higher order derivatives because it does not seem to be common knowledge. I also include a review of the complex Hessian.
If $f:\mathbb{R}^n \to \mathbb{R}^k$ is a function, then its derivative $Df:\mathbb{R}^n \to \textrm{Hom}(\mathbb{R}^n,\mathbb{R}^k)$. $... | https://mathoverflow.net/users/1106 | What is the "complex third derivative"? | The reason the complex Hessian (actually, it ought to be called the 'Hermitian Hessian', since it defines an Hermitian form at every point, but 'the complex Hessian' is entrenched in the literature) is so important is that it is the unique (up to constant multiples) linear second order differential operator from real f... | 23 | https://mathoverflow.net/users/13972 | 163390 | 85,561 |
https://mathoverflow.net/questions/163387 | 12 | This is a companion contrast to [the earlier analogous question for unit $n$-cubes](https://mathoverflow.net/q/163294/6094),
where the answer (provided by several respondents) is $\infty$ .
>
> What is the limit, as $n \to \infty$, of the expected distance between two
> points chosen uniformly at random within a u... | https://mathoverflow.net/users/6094 | Limit of distance between two random points in a unit-radius $n$-sphere | Both points will be very close to (let's pretend: on) the surface with prob almost 1. Call the first point the north pole. By concentration of measure for the sphere, a randomly chosen second point is almost guaranteed to be almost on the equator, so the limit should be $\sqrt{2}$.
| 15 | https://mathoverflow.net/users/48839 | 163393 | 85,564 |
https://mathoverflow.net/questions/157415 | 7 | For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits\_{n=1}^{\infty} (1-q^{n})$, let for brevity $e\_k:=\eta(q^k)$.
An **eta product identity** (or **eta identity** for short) is then defined as a homogenous polynomial in the $e\_k$ with integer coefficients such that its Taylor s... | https://mathoverflow.net/users/29783 | Under which constraints are there only finite numbers of irreducible eta product identities? | I feel like you had a little bit of trouble coming up with a good notion of "irreducible". A lot of the things you say suggest you want identities that "cannot be built from simpler ones." Here's a way of thinking about it that may be helpful.
Given a level $N$, consider the polynomial ring $R$ over $\mathbb{C}$ on $... | 4 | https://mathoverflow.net/users/48142 | 163398 | 85,566 |
https://mathoverflow.net/questions/163404 | 4 | Let $A=\{z\in\mathbf{C}:1/2<|z|<1\}$ be an open annulus. Let us cover $A$ by 3 open sets:
$U\_0,U\_1$ and $U\_2$ which we assume to be all homeomorphic to a 2 dimensional open disc. Moreover, we assume that $U\_{ij}:=U\_{i}\cap U\_j$ are also homeomorphic to two dimensional open discs and that $U\_0\cap U\_1\cap U\_2=\... | https://mathoverflow.net/users/11765 | Why Cech cohomology does not compute sheaf cohomology on an open annulus | Functions $(f\_0,f\_1,f\_2)$ as you requested do in fact exist. Your confusion comes from the fact that you are trying to impose the cocycle condition on the intersection $U\_0\cap U\_1\cap U\_2$, which is empty. That is the $f\_i$'s are such that
$$ f\_2(x) = f\_1(x)+1\ \forall x\in U\_1\cap U\_2$$
$$ f\_2(x) = f\_0(x... | 19 | https://mathoverflow.net/users/43054 | 163410 | 85,567 |
https://mathoverflow.net/questions/163407 | 3 | Let $B \subset \mathbb{N}$ be a set of natural numbers such that $|B \cap [1,N]| \sim N^\gamma$, for some $\gamma > 0$ with the following property:
For any pair of positive integers $k,n$ we define $B\_{k,n} = \{b \in B: b \equiv k \pmod{n}\}$. Then $|B\_{k,n} \cap [1,N]| \sim \frac{1}{n} |B \cap [1,N]|$ as $N \right... | https://mathoverflow.net/users/10898 | Is every sufficiently dense well mixed set an additive basis? | Fix an $h\in\mathbb N$. Choose your favourite irrational, $\alpha$. Let $B=\{n\in\mathbb N\colon \langle\alpha n\rangle\in [0,\frac{1}{2h})\}$. (Here $\langle x\rangle$ denotes the fractional part of $x$. Then if $n\_1,\ldots,n\_h\in B$, and $m=n\_1+\ldots+n\_h$, then $\langle m\alpha\rangle=\langle n\_1\alpha\rangle +... | 6 | https://mathoverflow.net/users/11054 | 163413 | 85,569 |
https://mathoverflow.net/questions/163317 | 4 | Let S be a finite simple nonabelian group, w a word in a finite number of variables which is not a power of another word. Must there be a substitution of elements of S in w such that the resulting element is not 1?
Equivalently, let S be a finite simple group and F a free group on a finite number of variables. Let w ... | https://mathoverflow.net/users/38889 | Simple groups and words | The answer is 'no' if you fix $S$ and consider all (primitive) words $w$ (see Derek Holt's answer). However, if you fix the word $w$, then $w$ takes a non-trivial value on $S$ (equivalently, $S$ is generated by values of the word $w$) for all but finitely many non-abelian finite simple groups $S$. This is due to the fa... | 5 | https://mathoverflow.net/users/4053 | 163416 | 85,571 |
https://mathoverflow.net/questions/163394 | 5 | If $X$ is a set and $I$ is an ideal on $X$. Let $\mathbb{P}$ be the forcing poset consisting of $I^+$ subsets of $X$ with the subset partial ordering. Let $G$ be $\mathbb{P}$-generic filter over $M$, where $M$ is the ground model.
As in Jech Lemma 22.13, it can be shown that $G$ is a $M$-ultrafilter on $X$. I presum... | https://mathoverflow.net/users/43354 | Generic Ultrapower as a Class | I think it is historically important to see that the earlier results involving generic ultrapowers, such as those of Solovay in the late 1960s, did not depend on the far-future realizations of Laver and Woodin. The Laver/Woodin result is important and interesting, but it seems that the known applications of generic ult... | 7 | https://mathoverflow.net/users/11145 | 163421 | 85,574 |
https://mathoverflow.net/questions/163396 | 1 | I'm looking at a M/M/1 queue system and trying to show that $\{M\_t\}\_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda\_i = \lambda$ and $\mu\_i = \mu \forall i$. In the current situation we're adding the non-trivial probability that a new clie... | https://mathoverflow.net/users/49607 | M/M/1 Queue with probability of new customer leaving | You are interested in a model with "balking". See e.g.
>
> János Sztrik, Basic Queueing Theory, Chapter 2.2., p. 25-29, [link](http://irh.inf.unideb.hu/~jsztrik/education/16/SOR_Main_Angol.pdf)
>
>
>
I assume that the probability you gave was not meant to be a probability of leaving but rather of staying.
| 1 | https://mathoverflow.net/users/34050 | 163424 | 85,576 |
https://mathoverflow.net/questions/163400 | 0 | let $b\_{n,k}$ be the numbers defined formally by $$X^n=\sum\_{k=0}^n b\_{n,k}\binom{X}{k}$$ where $\binom{X}{n}=\frac{1}{n!}\prod\_{k=0}^{n-1}(X-k)$.
I am looking for an equivalent of $b\_{n,k}$ when $k$ is fixed and $n\to+\infty$.
Thanks in advance for any answer.
| https://mathoverflow.net/users/33128 | Equivalent of Stirling-like numbers | You are looking for a scaled version of the [Stirling numbers of the second kind](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind):
Note that $\binom{X}{n} = \frac{(x)\_n}{n!}$ where $(x\_n)$ is the falling factorial (or Pochhammer symbol).
| 2 | https://mathoverflow.net/users/1056 | 163425 | 85,577 |
https://mathoverflow.net/questions/162829 | 4 | I have recently been interested in the incompressible Euler equation, but since I am new to the topic, I would like to inquire what are the standard sources/references (for self-study) regarding the existing theory. For instance, are there some good survey papers which also include the classical theory for weak solutio... | https://mathoverflow.net/users/46298 | Classical theory for the incompressible Euler equation (reference request) | Classical existence and regularity results can be found in the survey <http://arxiv.org/abs/math/0703406> (Euler Equations of Incompressible Ideal Fluids, by C. Bardos and E. S. Titi -- <http://www.mccme.ru/~ansobol/otarie/slides/Russ-Math-Surveys-Euler-Bardos.pdf> is the published version).
P.S. This paper <http://a... | 4 | https://mathoverflow.net/users/32389 | 163428 | 85,578 |
https://mathoverflow.net/questions/163430 | 16 | I'm stuck in some algorithms in hyperbolic groups, which may be rather simple.
Let $G$ be a hyperbolic group given by a finite presentation. It is known that the hyperbolicity constant $\delta$ can be computable from the given presentation (by papasoglu, but I could not find/read the related reference).
We also kn... | https://mathoverflow.net/users/37302 | Algorithms in hyperbolic groups | An algorithm for computing the hyperbolic "thinness" constant $\delta$ is described in the paper:
Epstein, David B. A.; Holt, Derek F
Computation in word-hyperbolic groups.
Internat. J. Algebra Comput. 11 (2001), no. 4, 467–487.
and I did implement it. It works OK on reasonably straightforward examples like surface... | 16 | https://mathoverflow.net/users/35840 | 163433 | 85,579 |
https://mathoverflow.net/questions/163412 | -1 | V is a variety of commutative semi group satisfying the identity $x^2 = x^3$.
I need to prove that:
$|F\_V(\{x\_1\dots,x\_n\})|$ = $3^n -1$.
Any hints on this ?
$F\_V$ is V-free algebra.
| https://mathoverflow.net/users/46237 | Variety of commutative semi group | Let $S\_n = F\_V(\{x\_1, ..., x\_n\})$. Since $S\_n$ is commutative, its elements can be written in the form $x\_1^{r\_1} \cdots x\_n^{r\_n}$ where $r\_1 + \dots + r\_n > 0$. Morevover, since the semigroup $S\_n$ satisfies the identity $x^2 = x^3$, you may assume that each $r\_i$ is equal to $0$, $1$ or $2$. It follows... | 1 | https://mathoverflow.net/users/38236 | 163434 | 85,580 |
https://mathoverflow.net/questions/163409 | 11 | If $M$ is a Riemannian manifold with $\Delta $ its Laplacian, how can we define $(1-\Delta)^{1/2}$?
The book I am reading says that $(1-\Delta)^{1/2}$ is an invertible first-order pseudo-differential operator with the inverse $(1-\Delta)^{-1/2}$. Naively, I try to construct $(1+|\xi|^2)^{1/2}$ as its principal symbol... | https://mathoverflow.net/users/16323 | How to define the square root of $1-\Delta $? | There is a general result of Seeley which states that if $A$ is an elliptic, selfadjoint positive scalar pseudo-differential operator of order $k$ on a compact Riemann manifold, then for any $\newcommand{\bR}{\mathbb{R}}$$s\in\bR$ the operator $A^s$ defined by functional calculus is in fact described by an elliptic sel... | 19 | https://mathoverflow.net/users/20302 | 163442 | 85,583 |
https://mathoverflow.net/questions/163308 | 4 | This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector bundle on a non compact Riemann surface is trivial. In fact I think it holds with vector bundle replaced with principal $G... | https://mathoverflow.net/users/7 | Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$ | $\def\ZZ{\mathbb{Z}}\def\CC{\mathbb{C}}$ Let $E$ be an elliptic curve with affine chart $\{ (x,y) : y^2 = x+ax^2+bx^3 \}$ . We will write $p$ for the point $(0,0)$ and $\infty$ for the puncture. We want a holomorphic function on $E$ with a simple zero at $p$. We'll write $\omega$ for the nonvanishing holomorphic form
$... | 6 | https://mathoverflow.net/users/297 | 163449 | 85,584 |
https://mathoverflow.net/questions/163453 | 2 | There is an army of interesting constructions in AT, and Understanding them are usually very helpful for appreciate the theory underneath. So I would like to invite you to share those examples that you think are beautiful and illuminating to the study of homotopy theory.
Following is what comes to my mind at first pl... | https://mathoverflow.net/users/40345 | Beautiful constructions in algebraic topology that facilitate one's understanding of homotopy theory | Links of singularities of complex algebraic varieties are fascinating geometrical and topological objects.
For example Brieskorn manifolfs. A Brieskorn manifold is the intersection of a small sphere around the origin with the singular complex hypersurface
$$z\_0^{k\_1}+\cdots+z\_n^{k\_n}=0$$
they are closed smooth... | 5 | https://mathoverflow.net/users/27816 | 163466 | 85,589 |
https://mathoverflow.net/questions/163467 | 4 | I have seen references to the "tilting method" in the theory of large deviations. Is there a simple explanation of what this is, exactly?
| https://mathoverflow.net/users/11142 | What is "tilting" in the context of large deviations? | Tilting refers to a change of measure of the form $e^{\lambda \cdot x}/C(\lambda)$ where $C(\lambda)=E(e^{\lambda\cdot X})$.
(The setup is that the measure for which you are proving large deviations is a topological vector space $X$, and $\lambda$ is in the dual space to $X$). The name tilting comes because the effect ... | 13 | https://mathoverflow.net/users/35520 | 163471 | 85,593 |
https://mathoverflow.net/questions/163464 | -3 | I am trying to find the -1 moments of sum of N geometric random variable, i.e. $E[\frac{1}{\sum\_{i=1}^N X\_i}]$
Suppose the probability mass function is $f\_X(x) = (1 - p)^{x - 1} p$
The moment generating function of $\sum\_{i = 1}^N X\_i$ will be
$E[e^{t \sum X\_i}] = ( \frac{pe^t}{1 - (1-p)e^t})^N$
Let $h(t)... | https://mathoverflow.net/users/49474 | Suppose I know $\int h(t) dt = H(t)$, is there a way to find $\int h(t)^N dt$? | If you set $u=e^t$, we get
$$
\int \left(\frac{pu}{1-(1-p)u}\right)^N\frac{du}{u}
$$
According to Wolfram Alpha, with $p=1/2$,
$$
\int \left(\frac{e^t}{2 (1-e^t/2)}\right)^N dt = \left(\frac{e^t}{2-e^t}\right)^N (1-e^t/2)^n \frac{{}\_2F\_1(n, n, n+1, e^t/2))}{n}+\text{constant}
$$
where ${}\_2F\_1$ is the [hypergeometr... | 1 | https://mathoverflow.net/users/4600 | 163476 | 85,595 |
https://mathoverflow.net/questions/163445 | 3 | I am looking for advice concerning a specific situation related to centre manifold theory (compare Perko 2001).
The part which is known
-----------------------
Let's consider a differential equation in higher-dimensional Euclidean space which admits equilibrium points. Centre manifold theory yields the following ni... | https://mathoverflow.net/users/42006 | Centre manifold theory for a curve of equilibrium points | You will want to look for *Normally Hyperbolic Invariant Manifolds* or NHIMs. Briefly said, these are invariant submanifolds (i.e. a curve of equilibria is a special case) with hyperbolic dynamics in the normal directions: the spectrum (in this case: eigenvalues) in the normal directions must be bounded away from zero ... | 4 | https://mathoverflow.net/users/3928 | 163477 | 85,596 |
https://mathoverflow.net/questions/163483 | 3 |
>
> what are notable/ prominent inductive proofs relating to fractals?
>
>
>
the motivation for this question is:
* fractals are very difficult mathematical objects to work with, and many problems/questions about them lie at the boundary of decidable/undecidable (and many are undecidable).
* so it would be use... | https://mathoverflow.net/users/20793 | notable inductive proofs relating to fractals | Spectral decimation is an inductive process where the eigenvalues of a natural Laplacian on "nice" fractals is computed inductively. The idea is that by using a sequence of finite graphs to approximate the graph and graph Laplacians on these graphs one can inductively determine the eigenvalues of the Laplacian on the n... | 2 | https://mathoverflow.net/users/11332 | 163486 | 85,598 |
https://mathoverflow.net/questions/163462 | 1 | I want to ask some question related with the sieve of Eratosthenes.
The sieve of Eratosthenes: write it as $E\_1(x) (=\pi(x)-\pi(\sqrt x)+1)$.
Then we have an obvious result
$$E\_1(x)/x\ln^{-1}x = 1,$$
as $x\rightarrow \infty$ by PNT.
The question comes, we can think weight $a$ (positive integer) to each summatio... | https://mathoverflow.net/users/49625 | Question on an arithmetic function with the sieve of Eratosthenes | Just as the original sieve of Eratosthenes formula is a rewriting of
$$
E\_1(x) = \sum\_{n\le x} \prod\_{\substack{p\le\sqrt x \\ p\mid n}} (1-1) = \sum\_{\substack{n\le x \\ p\mid n\Rightarrow p>\sqrt x}} 1,
$$
this modification is the same as
$$
E\_a(x) = \sum\_{n\le x} \prod\_{\substack{p\le\sqrt x \\ p\mid n}} (1-a... | 2 | https://mathoverflow.net/users/5091 | 163488 | 85,599 |
https://mathoverflow.net/questions/163408 | -2 | Let $A\_n = \{(1,\ldots,n) , f \}$ where $f(i) = (i+1)$ if $i \neq n $ otherwise $f(n) = 1$.
This describes a mono unary algebra.
The proof for $HPS \neq SPHS$ I know uses metabelian groups and was published by
George Bergman in Algebra Universalis.
Here I need to prove $SHPS \neq SPHS$ for the class of mono unary al... | https://mathoverflow.net/users/46237 | SHPS and SPHS inequality using monounary algebra | The only subalgebras of $A\_p$ are $\varnothing$ and $A\_p$. I claim that the only homomorphic images of $A\_p$ are $A\_p$ and the one-element algebra.
**Note:** there was an arithmetical error in the following paragraph. Fixed.
Indeed, let $\Phi$ be a congruence on $A\_p$ that contains a pair $(a,b)$ with $a\neq b... | 1 | https://mathoverflow.net/users/3959 | 163499 | 85,604 |
https://mathoverflow.net/questions/163451 | 21 | It seems that in spite of the Bailey–Borwein–Plouffe formula it is still unknown whether $\pi$ is normal in base 16. What are the difficulties in using it for this purpose?
In a comment to his answer to [a question about decimal normality of $\pi$](https://mathoverflow.net/questions/51853/what-is-the-state-of-our-ign... | https://mathoverflow.net/users/8588 | Normality of $\pi$ in base 16 | The usual suspects in this subject (Borwein, Bailey, etc.) have written good expositions of this topic. What follows is shamelessly stolen from them. An easier example to explain, that illustrates the main idea, is log 2. We have the formula
$$\log 2 = \sum\_{n=1}^\infty {1 \over n 2^n}.$$
You can, if you like, call th... | 22 | https://mathoverflow.net/users/3106 | 163501 | 85,605 |
https://mathoverflow.net/questions/163500 | 5 | Suppose $f:\mathbb{Z}\_{\geq 0}\to\mathbb{Z}\_{\geq 0}$ is an arithmetic function that grows slower than the identity map. Has it already been conjectured that, under this general hypotheses, $\pi(n+f(n))-\pi(n-f(n))\sim \dfrac{2f(n)}{\log n}$? Or are there any known counterexamples?
Thanks in advance.
| https://mathoverflow.net/users/13625 | Has this strengthening of the PNT already been conjectured? | [This paper](http://www.math.boun.edu.tr/instructors/yildirim/paper/paper9.pdf) by Yildirim gives a good survey on this. Clearly, the question is equivalent to determining functions $\Phi$ such that $\pi(x+\Phi(x))-\pi(x)\sim \frac{\Phi(x)}{\log x}$. Quoting the paper, this was proved by Heath-Brown for $\Phi(x)=x^{\fr... | 16 | https://mathoverflow.net/users/23008 | 163506 | 85,607 |
https://mathoverflow.net/questions/163493 | 4 | Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl\_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of $\mathbb{P}^4$ and $E$ the exceptional divisor. I would like to compute te intersection numbers of these divisors.
... | https://mathoverflow.net/users/nan | Blow-up of $\mathbb{P}^4$ along a quadric surface | The intersection numbers $H^4 =1,\; H^3\cdot E = 0,\; H^{2}\cdot E^2 = -2,\; H\cdot E^3 = -6$ are correct. The last one $E^4 = H^0\cdot E^4 = -18$ is wrong.
A way to compute $E^4$ is the following. The blow-up $X$ of a smooth quadric surface $Q\subset\mathbb{P}^4$ is isomorphic to the blow-up of a smooth $4$-dimensio... | 2 | https://mathoverflow.net/users/14514 | 163508 | 85,608 |
https://mathoverflow.net/questions/163478 | 1 | I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form
$$R(z):= \sum\_{j=1}^{n} \frac{a\_j}{z-p\_j},$$
where the $a\_j$'s are nonzero complex numbers and the $p\_j$'s are distinct complex numbers.
Any relevant reference is welcome.
Note : The [Gauss-Lucas... | https://mathoverflow.net/users/1162 | On the geometry of roots of a sum of complex linear fractions | Section 21 in Chapter V of "Geometry of Polynomials" by Morris Marden, AMS, Mathematical Surveys 3, 1966 may contain what you are looking for, Lemma 21.1.on page 98 is as follows:
Let $C\_j(z) := |z - c\_j|^2 - r\_j^2$ and $C\_j$ be the circular region defined by $\pm C\_j(z) \leq 0$, let $p\_j$ be in $C\_j$ and let ... | 1 | https://mathoverflow.net/users/37855 | 163512 | 85,610 |
https://mathoverflow.net/questions/163463 | 3 | [Here](https://mathoverflow.net/questions/162633/is-continuity-of-a-functor-stable-under-pullback), I asked wether taking lax pullback preserves continuity, but got no precise answer.
However, I have found [this](http://arxiv.org/abs/1401.6247) recent article by Riehl and Verity which proves something very similar, b... | https://mathoverflow.net/users/4477 | Follow up: Is continuity preserved when taking comma object? | Our paper specializes to the case of 1-categories with (co)limits and (co)limit-preserving functors, not to 2-categories.
But surely this is true. I'm defining the comma object in the 2-category of (strict) 2-categories, 2-functors, and 2-natural transformations. The defining universal property of $F \downarrow G$ c... | 10 | https://mathoverflow.net/users/2181 | 163516 | 85,611 |
https://mathoverflow.net/questions/163515 | 20 | Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can this be proved?
Motivation
----------
This infinite family of polynomials comes from a [recent paper of mine](http:... | https://mathoverflow.net/users/11598 | How to prove that every polynomial in an infinite family is irreducible over Q? | The same Puiseux-series tactic works (effectively) to list all $y$
for which $p(X,y)$ has a quadratic factor. For example, the linear
coefficient of such a factor is $a = -(r\_i+r\_j)$ for some distinct $i,j$
where $r\_1,\ldots,r\_5$ are the roots of $p(X,y)$. But for large $y$
there are three real and two imaginary ro... | 22 | https://mathoverflow.net/users/14830 | 163523 | 85,612 |
https://mathoverflow.net/questions/163526 | 2 | Let $G$ be any graph with at least one edge and let $e$ be any edge of $G$. Let $G-e$ denote the subgraph of $G$ obtained by deletion of the edge $e$. Assume that $G$ has $n$ vertices.
Suppose that $\lambda\_1(G)\geq\cdots\geq \lambda\_n(G)$ and $\lambda\_1(G-e)\geq\cdots\geq \lambda\_n(G-e)$ are eigenvalues of $G$ an... | https://mathoverflow.net/users/19075 | Eigenvalues of a graph and its one-edge-delation graph | I think this is false and there are counterexamples.
Here is a sage session.
```
sage: ed,e=([(0, 2), (0, 4), (0, 5), (1, 3), (1, 5), (2, 4), (3, 5)], (0, 2))
sage: G=Graph(ed);F=Graph(G);F.delete_edge(e)
sage: eg=G.adjacency_matrix().eigenvalues();eg.sort(reverse=1);eg
[2.414213562373095?, 1.732050807568878?, -0.... | 2 | https://mathoverflow.net/users/12481 | 163529 | 85,613 |
https://mathoverflow.net/questions/163549 | 3 | Let $\mathbf{f}$ be a Hida family of ordinary $p$-adic modular forms, and let $V(\mathbf{f})$ be the corresponding $\Lambda$-adic Galois representation (a quotient of the inverse limit
$$ \varprojlim\_r H^1\_{et}(X\_1(Np^r), \mathbf{Z}\_p) $$
with respect to the trace maps).
It follows from work of Wiles that if the ... | https://mathoverflow.net/users/2481 | Interpolation of periods for a Hida family of modular forms | Have a look at Ohta's paper "Ordinary p-adic ´etale cohomology groups
attached to towers of elliptic modular curves. II" (Mathematische Annalen); he strengthens there the results from his previous paper to include all congruence classes, using a result of Gross.
| 4 | https://mathoverflow.net/users/39954 | 163552 | 85,624 |
https://mathoverflow.net/questions/163558 | 2 | Let $G$ be a finite group and $G^{\prime}$ be its commutator subgroup. Let $\mathbb{Z}$ and $\mathbb{Q}$ denote the integers and rationals. $\mathbb{Z}$ and $\mathbb{Q/Z}$ treated as trivial $G$-module. We know that $H^1(G,\mathbb{Z})Hom(G,\mathbb{Q/Z}) \cong Hom(G/G^{\prime},\mathbb{Q/Z}) \cong G/G^{\prime} \cong H\_1... | https://mathoverflow.net/users/49668 | What are finite groups $H$ such that $H^n(H,\mathbb{Q/Z}) \cong H_n(G,\mathbb{Z})$? | It's true for any finite group (for $n>0$).
The long exact sequence of cohomology for the short exact sequence $0\to\mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}\to0$ gives $H^n(G,\mathbb{Q}/\mathbb{Z})\cong H^{n+1}(G,\mathbb{Z})$, and the Universal Coefficient Theorem gives $\operatorname{Ext}^1(H\_n(G,\mathbb{Z})... | 4 | https://mathoverflow.net/users/22989 | 163561 | 85,626 |
https://mathoverflow.net/questions/163527 | 11 | Let $M$ be a connected noncompact parallelisable smooth $n$-manifold. (By Hirsch-Smale theory, $M$ can be immersed into $\mathbb{R}^n$.) I am interested in the following two properties:
P1: $M$ can be smoothly embedded into $\mathbb{R}^{n+1}$.
P2: $M$ admits a smooth embedding into $\mathbb{R}^{n+1}$ which is trans... | https://mathoverflow.net/users/49655 | Immersions of $n$-manifolds in $\mathbb{R}^n$ versus embeddings in $\mathbb{R}^{n+1}$ | I was slightly confused by the wording of your question. I interpret it that you are asking first if an immersion into $R^n$ implies an embedding into $R^{n+1}$ (P1) and if so, if you get an embedding that is transverse to a coordinate vector field (P2). It wasn't clear to me whether you wanted to specify the immersion... | 7 | https://mathoverflow.net/users/3460 | 163570 | 85,630 |
https://mathoverflow.net/questions/163572 | 3 | Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some radius $\delta$ around $0 \in \mathbb{C}$ so that $0$ is the only critical value in $D$, and we can choose a ball $B$ of ... | https://mathoverflow.net/users/42325 | Simple example of isolated critical point with non-semisimple monodromy | Let us consider the polynomial $$f(x, \, y)= x^5+y^5+x^2y^2.$$ It defines a germ of isolated singularity at the origin of $\mathbb{C}^2$, whose Milnor number is $\mu(f) = 11$.
[This](http://lilypond.org/vc/texi2html.git/test/singular_manual/res/singular/sing_663.htm) computation with Singular shows that the correspon... | 4 | https://mathoverflow.net/users/7460 | 163576 | 85,632 |
https://mathoverflow.net/questions/163534 | 5 | Consider a set of $N$ points in $n$-dimensional space, i.e.
\begin{align\*}
\{x\_1, \dots, x\_N\} \subset \mathbb R^n.
\end{align\*}
Let us be given a finite family of non-injective matrices
\begin{align\*}
\{M\_j \in \mathbb R^{m \times n} : j = 1, \dots, J\},
\end{align\*}
e.g. $m<n$.
In a nutshell, the problem I... | https://mathoverflow.net/users/nan | Reconstructing set of points from one-dimensional images | Denote by $S$ your finite collection of $N$ points in $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$. Here is how you can recover $S$ from the knowledge of its images via a finite collections of linear maps of rank $<n$. More precisely one can use a universal family consisting of roughly $\frac{N^4}{2}$ matrices of type $(n-1)... | 2 | https://mathoverflow.net/users/20302 | 163580 | 85,634 |
https://mathoverflow.net/questions/163581 | 3 | Let $\mathcal{D}(\mathbb{R})$ be the continuous dual of $C^\infty\_c(\mathbb{R})$, the space of compactly-supported smooth functions. There is a nice characterization of distributions $a\in\mathcal{D}(\mathbb{R})$ that are localized at a point $r\in\mathbb{R}$; $a$ must be a linear combination of distributional derivat... | https://mathoverflow.net/users/27121 | Zero currents localized along a submanifold | A special case is this:
Let $u\in\mathcal{E}'(\mathbb{R}^{n+m})$. Then $supp(u)\subseteq\mathbb{R}^n\times\{0\}$ iff $u$ is a linear combination of the form $\sum\_{\substack{\alpha\in\{0\}\times\mathbb{N}^m \\ |\alpha|\leq k}} c\_\alpha \partial^\alpha \iota(v\_\alpha)$ for certain $v\_\alpha\in\mathcal{E}'(\mathbb{... | 4 | https://mathoverflow.net/users/3041 | 163587 | 85,635 |
https://mathoverflow.net/questions/163588 | 3 | Hechler forcing is described on page 278, Jech.
Does anyone know when Hechler forcing was first used in a publication?
| https://mathoverflow.net/users/nan | What year was Hechler forcing created? | The citation is given on page 283 of Jech (at least, in the 3rd Millenium edition); it is to Hechler's paper "On the existence of certain cofinal subsets of $^\omega\omega$," in the collection "Axiomatic Set Theory II" (see <http://www.ams.org/books/pspum/013.2/9987/pspum9987.pdf>), which is the only paper by Hechler i... | 4 | https://mathoverflow.net/users/8133 | 163591 | 85,636 |
https://mathoverflow.net/questions/163426 | 4 | Let $p \in [1,\infty)$. Does there exist $C>0$ such that for every $f \in W^{1,p}([0,1],\mathbb{R})$ we have
$$\|f\|\_{L^\infty}\leq C\|f\|\_{L^p}^{1-\frac{1}{p}}\|f\|\_{W^{1,p}}^{\frac{1}{p}}?$$
My Motivation is the following: For $m \in \mathbb{N}$ I have an approximation operator $Q^h$ depending on the meshwidth $h>... | https://mathoverflow.net/users/35593 | Estimate infinity norm with Lp and W1p norm | Yes. This is a known inequality, called e.g. "Gagliardo-Nirenberg inequality" in Brezis' book, Comment 1 to Chapter 8, page 233.
| 3 | https://mathoverflow.net/users/26039 | 163595 | 85,637 |
https://mathoverflow.net/questions/163540 | 5 | Let $F$ be a free group on $d$ generators.
Denote by $F\_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F\_k$. What is the normal subgroup growth of $G$?
Explicitly, for each natural number $n$, what is the number of normal subgroups of index not greater than $n$ in $G$? An asymptotic result will also be i... | https://mathoverflow.net/users/38889 | Normal Subgroup Growth | As is stated in Lubotzky-Segal's book on this topic, very little is known about normal subgroup growth for soluble groups. However, we can say that a rank d free $k$-step solvable group, $k \geq 2$, has normal subgroup growth that is very close to $n^{\log(n)}$. Indeed, this group maps onto any metabelian group generat... | 6 | https://mathoverflow.net/users/3970 | 163601 | 85,642 |
https://mathoverflow.net/questions/163619 | 4 | The circle method is often used to estimate the number of solutions to the equation $$x\_1 + x\_2 + ... x\_k = N$$ if for all $i$ $x\_i\in A\subseteq\mathbb{N}\_0$ and some subset of the nonnegative integers. When we try to use the circle method if $A$ isn't a subset of the nonnegative integers, one runs into issues, s... | https://mathoverflow.net/users/40983 | Circle method on things other than the integers | The circle method is a very powerful and versatile tool which can be made to work in a wide range of situations. It certainly works over number fields, see for example
Skinner - Forms over number fields and weak approximation.
The circle method can even be made to work over function fields, see e.g.
Lee - Forms i... | 2 | https://mathoverflow.net/users/5101 | 163635 | 85,656 |
https://mathoverflow.net/questions/163431 | 1 | Let $K=\mathbb{Q}[\alpha]$ where $\alpha$ is integral over $\mathbb{Z}$ such that the Galois hull of $K$ can be embedded in $\mathbb{R}$. Let $S=\mathbb{Z}[\alpha]$. Let $x\_1, \ldots , x\_n$ be a $\mathbb{Q}$-basis of $K$ and let $f: K \to \mathbb{Q}$ be a $\mathbb{Q}$-linear map such that $f(x\_i \cdot x\_j)=\delta\_... | https://mathoverflow.net/users/36563 | lattice in number field already a fractional ideal? | When $\alpha$ is fixed at the beginning, this is not always true, and it doesn't matter whether $K$ is or isn't totally real.
Note first that any full-rank lattice in any number field admits *some* order of the field as endomorphisms. (Proof: The action of any algebraic integer $y$ generating the field on the lattice... | 6 | https://mathoverflow.net/users/49003 | 163643 | 85,660 |
https://mathoverflow.net/questions/161497 | 7 | It is well known that [Ackermann's function](https://en.wikipedia.org/wiki/Ackermann_function) is not primitive recursive. Therefore, the theories of primitive recursive arithmetic (PRA) and of $\Sigma\_1$-induction ($I\Sigma\_1$) cannot prove the totality of Ackermann's function.
However $\Sigma\_2$-induction suffic... | https://mathoverflow.net/users/3365 | Does the totality of Ackermann's function prove the consistency of $\Sigma_1$-induction? | The answer is apparently yes.
Recall the Grzegorczk hierarchy $\mathcal{E}^n$. This a hierarchy which classifies computable functions based on their growth rate, see [Wikipedia: Grzegorczyk hierarchy](https://en.wikipedia.org/wiki/Grzegorczyk_hierarchy).
Now on this hierarchy can be extended to ordinal level, see [... | 6 | https://mathoverflow.net/users/3365 | 163644 | 85,661 |
https://mathoverflow.net/questions/161260 | 4 | I have a proof that if $n$ is an integer such that $n>1$ and $k$ is any field, then if $g$ is an element of $\mathrm{SL}(n,k)$ of infinite order then the set of all $h$ with the property that $g$ and $h$ generate a free group of rank two is Zariski dense.
<http://rupertmccallum.com/free_groups.pdf>
I am trying to ... | https://mathoverflow.net/users/15482 | ubiquity of free subgroups of special linear groups | I think this is an interesting result with a cool method. By the way, you might want to look at Theorem 2 on page 158 of Borel's "On free subgroups of semisimple groups" (Enseign. Math. (2), 29(1-2):151–164, 1983). Using this and the Ping-Pong lemma, you can prove results that are similar (but not identical) to yours. ... | 1 | https://mathoverflow.net/users/3970 | 163653 | 85,668 |
https://mathoverflow.net/questions/163641 | 3 | If $X\_1\supseteq X\_2\supseteq \ldots$ is a sequence of "nice" compact spaces, I would like to know whether the natural map from $H\_\*(\cap X\_i)$ to the inverse limit $\lim \, H\_\*(X\_i)$ is surjective. In particular, if there exist nonzero $\beta\_i\in H\_q(X\_i)$ such that the inclusion-induced homomorphism $i\_\... | https://mathoverflow.net/users/10072 | Homology of infinite intersection | To expand on my answer above, consider the sequence $\dots\subset Y\_2\subset Y\_1$
of spaces where $Y\_i=X\_i$ for $i\leq n$ and $Y\_i=X\_n$ for $i>n$.
Milnors proof proceeds by taking $M\_i$ to be the mapping cylinder of $X\_{i+1}\rightarrow X\_i$ and $M\_0$ the cone over $X\_0$ with vertex $t$. Then one takes the... | 3 | https://mathoverflow.net/users/4181 | 163677 | 85,680 |
https://mathoverflow.net/questions/163528 | 9 | Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let
$$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n\_k,$$
be a non-singular quadric hypersurface over $k$. Then it is well-known that $X\times\_k \bar k$ is birational to some projective space, ... | https://mathoverflow.net/users/5101 | Is any quadric birational to a product of Brauer-Severi varieties? | Consider the projective quadric $V$ given by
$$
2x^2+y^2+z^2+w^2=0
$$
over $\mathbb{Q}$. Inspired by Jason Starr's remark on splitting fields, I will prove that $V$ is not birational to a product of Severi-Brauer varieties over $\mathbb{Q}$.
Clearly $V$ has no points over $\mathbb{R}$, hence no rational points. So i... | 8 | https://mathoverflow.net/users/17907 | 163679 | 85,681 |
https://mathoverflow.net/questions/163684 | 3 | Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}\_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?
| https://mathoverflow.net/users/5259 | Intersection theory on M_{g,n} | Perhaps these papers could be useful:
* C. Faber, "*Maple program for calculating intersection numbers on moduli spaces of curves*", (<http://math.stanford.edu/~vakil/programs/index.html>).
* S. Yang, "*Intersection numbers on $\overline{M}\_{g,n}$*", (<http://msp.org/jsag/2010/2-1/jsag-v2-n1-p01-s.pdf>).
The algor... | 2 | https://mathoverflow.net/users/14514 | 163687 | 85,685 |
https://mathoverflow.net/questions/163654 | 9 | Do You know any successful applications of the geometric group theory in the number theory? GTG is my main field of interest and I would love to use it to prove new facts in the number theory.
| https://mathoverflow.net/users/42695 | Links between Geometric Group Theory and Number Theory | Another reference showing the connections between geometric group theory and number theory is the book of
A. Lubotzky on "Discrete Groups, Expanding Graphs and Invariant Measures."
There are many special topics where a connection between number theory and geometric group theory arises. As an example see <http://arxi... | 3 | https://mathoverflow.net/users/32332 | 163688 | 85,686 |
https://mathoverflow.net/questions/163669 | 1 | For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there is a similar formula to compute these dimensions when you consider a projective bundle instead of a projective space.
| https://mathoverflow.net/users/49710 | Bott formula for projective bundles | As Sasha says, the best analogy is probably the following. Let $B$ be a complex variety (or analytic space), $E$ a vector bundle on $B$ of rank $r+1$, $\pi:P=\mathbb{P}(E)\rightarrow B$ the corresponding projective bundle. Then $R^p\pi\_\*(\Omega ^q\_P)\cong \Omega \_B^{q-p}$ for $p\leq r$, $=0$ for $p>r$, and there ar... | 1 | https://mathoverflow.net/users/40297 | 163691 | 85,688 |
https://mathoverflow.net/questions/163668 | 4 | I was talking to a friend and the following set $S$ came up.
Let $f$ be some real valued function tending to infinity.
Let $S$ be a subset of natural numbers such that $|S \cap [1,N]| = N^{\delta}+ O(N^{\delta}/f(N))$ and further for any $r,m \in \mathbb{N}$ we have
$$
| \{ x \in S \cap [1,N] : x \equiv r (\text{mod... | https://mathoverflow.net/users/48408 | Existence of a certain subset of natural numbers equidistributed modulo $m$ for every $m$ | As mentioned by [Aaron Meyerowitz](https://mathoverflow.net/users/8008/aaron-meyerowitz), a set of the form $$S=\left\{\left[k^{1/\gamma}\right]:\ k\in\mathbb{N}\right\}$$ should work for any $1/2<\gamma<1$. Notice that for such a value of $\gamma$, $$\left[(k+1)^{\gamma}\right]-\left[k^{\gamma}\right]=\begin{cases}
1 ... | 6 | https://mathoverflow.net/users/12176 | 163692 | 85,689 |
https://mathoverflow.net/questions/163711 | 17 | **Does Peano's theorem apply to spaces with infinite dimension? Or is there a counterexample?**
Here, Peano's theorem is:
Let $E$ be a space with finite dimension. Consider a point $(t\_0,x\_0) \in \Re \times E$, constants $ a, b > $ 0 and a continuous function $$F: [t\_0 - a, t\_0 + a] \times B\_b[x\_0] \longright... | https://mathoverflow.net/users/49716 | I would like to have a counter example that Peano's theorem does not apply to spaces with infinite dimension | No, Peano's existence theorem fails completely in infinite-dimensional spaces: there are counterexamples in every infinite-dimensional Banach space. This is a theorem of Godunov (A. N. Godunov, *Peano's theorem in Banach spaces*, Functional Analysis and its Applications **9** (1975), 53-55, <http://dx.doi.org/10.1007/B... | 23 | https://mathoverflow.net/users/4720 | 163713 | 85,700 |
https://mathoverflow.net/questions/163562 | 3 | In his paper “Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche” I. Schur has stated that Frobenius has communicated to him a simple direct proof of his finite version of the Rogers – Ramanujan identities. I could not find a reference to this proof. Does anyone know where this direct proof has be... | https://mathoverflow.net/users/5585 | Reference for Frobenius’s proof of Schur’s finite version of the Rogers - Ramanujan identities | Early history of the Rogers-Ramanujan identities is discussed by Hardy in "Ramanujan: Twelve lectures suggested by his life and work", Chelsea,
third edition, 1978. But Hardy doesn't mention Frobenius. Neither is Frobenius mentioned in Andrew V. Sills' comprehensive review <http://www.combinatorics.org/ojs/index.php/el... | 3 | https://mathoverflow.net/users/32389 | 163716 | 85,702 |
https://mathoverflow.net/questions/41284 | 9 | What is known about explicit (not necessarily closed-form) solutions to the recurrence
$$R^n\_k= (\alpha n) R^{n-1}\_k + (\alpha' n + \beta' k) R^{n-1} \_{k-1},$$
with initial condition $R\_0^0 = 1$ and with $R^n\_k = 0$ for $n < 0$ or $k < 0$? Special cases of this are closely related to recurrences satisfied by some ... | https://mathoverflow.net/users/9716 | Solving a general two-term combinatorial recurrence relation | A general solution of the Graham-Knuth-Patashnik 6.94 problem
$$
R^{n}\_k=(\alpha \, n + \beta\, k + \gamma)\, R^{n−1}\_k+ (\alpha′\, n+ \beta′\, k +\gamma′)\, R^{n−1}\_{k−1} + \delta\_{n,0}\delta\_{k,0}\,,
$$
with $R\_{n}^k=0$ if $n<0$ or $k<0$, can be found in the paper ``Bivariate generating functions for a cl... | 5 | https://mathoverflow.net/users/49726 | 163721 | 85,705 |
https://mathoverflow.net/questions/163722 | 7 | Let $W\_t$ be a Wiener process and consider the time integral
$$ X\_T:= \int\_0^T W\_t dt $$
It is often mentionend in literature that $X\_T$ is a Gaussian
with mean 0 and variance $T^3/6$.
I am interested in learning more about the process $X\_T$ for $T>0$.
Except for the description of the individual random vari... | https://mathoverflow.net/users/3757 | Properties of the time integral of Wiener process | As an integral of a zero-mean Gaussian process, your $X\_T$ is a zero-mean Gaussian process as well. Its covariance function can be calculated via
$$ c(s, t) = \int^s\_0 \int^t\_0 \min(u, v) \; du \; dv \; , $$
which yields
$$ c(s, t) = \frac{\min(s, t)^2}{6} \left( 3 \max(s, t) - \min(s, t) \right) . $$
In ter... | 12 | https://mathoverflow.net/users/24274 | 163726 | 85,707 |
https://mathoverflow.net/questions/163743 | 8 | As is well-known, if we quotient $SU(2)$ by the action of $U\_1$, embedded in the diagonal as $(e^{i \theta}, e^{-i \theta})$, we get the $2$-sphere. As is also well-known, if we quotient $SU(3)$ on the diagonal by $U(1) \times U(1)$, embedded in the diagonal as $(e^{i \theta\_1}, e^{i \theta\_2}, e^{-i(\theta\_1 + \th... | https://mathoverflow.net/users/49349 | Quotienting $SU(3)$ by $U(1)$? | The various different ways that $\mathrm{U}(1)\simeq S^1$ can appear as a subgroup of $\mathrm{SU}(3)$ are indexed by a lattice of rank $2$, and the $7$-dimensional quotients are now known as Aloff-Wallach manifolds, after a paper by Simon Aloff and Nolan Wallach, *An infinite family of distinct 7-manifolds admitting p... | 16 | https://mathoverflow.net/users/13972 | 163746 | 85,711 |
https://mathoverflow.net/questions/163749 | 13 | I'm trying to motivate the notion of integrality in a ring extension. It seems that the following would be a good motivation, because it would show that the notion of algebraic elements over a ring is not useful.
Here's the thing I believe is true: Let $R\subseteq S$ be a ring extension. The set of elements of $S$ th... | https://mathoverflow.net/users/33757 | Algebraic Closure of a Ring is Not a Ring? | You definition of "algebraic" is very strange when $R$ is not a domain. Due to that strangeness, it is easy to find examples: Let $k$ be a field, let $R = k \times k$ and let $S = k[t] \times k[t]$. Then $(t,0)$ obeys $(0,1) \cdot (t,0)=(0,0)$ and $(0,t)$ obeys $(1,0) \cdot (0,t) = (0,0)$.
However, I claim that $(t,t... | 16 | https://mathoverflow.net/users/297 | 163752 | 85,713 |
https://mathoverflow.net/questions/163694 | 14 | My question stems from [Misha](https://mathoverflow.net/users/21684/misha)'s [answer](https://mathoverflow.net/questions/156022/finite-index-free-subgroups-of-mathrmsl3-mathbbz/156042#156042) of a MathOverflow question. Misha supplied the following question in his answer:
>
> **Open question**: Does there exist a f... | https://mathoverflow.net/users/3970 | Subgroups of $SL_3(\mathbb{Z})$ that are finitely generated, Zariski-dense, infinite index, and torsion-free | Here is what's known about this question:
1. The problem is hard and requires new ideas. The situation in the $SL(4,Z)$ case is very different and the analogy is misleading.
2. $\Gamma=SL(3,Z)$ contains no singular semisimple elements (of infinite order). This implies that there cannot be "nontrivial" semisimple RAAG... | 11 | https://mathoverflow.net/users/21684 | 163754 | 85,714 |
https://mathoverflow.net/questions/142051 | 9 | Euclid's first three postulates are the basis of compass and straight edge constructions which are as complex as arithmetic.
The constructions themselves may be expressed as a formula with each of the six operations of addition, subtraction, multiplication, division, complex conjugate, and square root corresponding t... | https://mathoverflow.net/users/34859 | Is compass and straight edge geometry complete? | Here is a precise formulation, which shows the problem to be open.
Let $L$ be the language of Tarski-style geometry, with the two relations $Collinear(p,q,r)$ and $Equidistant(p,q,r,s)$.
Let $E$ be the constructible numbers, and $R$ the real numbers. Then $E$ and $R$ are structures for $(+,\times)$, and $E^2$ and $... | 8 | https://mathoverflow.net/users/nan | 163758 | 85,717 |
https://mathoverflow.net/questions/120494 | 1 | Consider three $N \times N$ Hermitian matrices $A\_0$, $A\_1$, $A\_2$. Consider the function
\begin{align}
f(t\_1,t\_2)=\lambda\_{\text{min}}(A\_0+t\_1A\_1+t\_2A\_2)
\end{align}
where $\lambda\_{\text{min}}$ denotes the minimum eigenvalue. $f(t\_1,t\_2)$ is clearly a concave function. How do we find the sub-gradient of... | https://mathoverflow.net/users/27249 | Subgradient of Minimum Eigenvalue | Note that $f(t)=g(A(t))$, where $A(t\_1,t\_2)=A\_0+t\_1A\_1+t\_2A\_2$ is affine and $g(B)=\lambda\_{min}(B)$ which is concave on $B$. Now, to obtain the supergradient of this function you just need to use the chain rule of convex analysis.
First, it is easy to see that the Jacobian of $A$ is
$$ D A(t) = A^{\ast} = [A... | 1 | https://mathoverflow.net/users/39129 | 163761 | 85,719 |
https://mathoverflow.net/questions/163763 | 12 | Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group has a unique right-Haar measure (up to a multiplicative constant). Similarly one can define the left-Haar measure as bein... | https://mathoverflow.net/users/39129 | Entropy for Haar measure on $O(n)$ | This is a theorem of [K. Berg,](http://www.springerlink.com/index/J5X0728070568177.pdf) that the Haar measure is the measure of maximal entropy for automorphisms of compact groups. See, for example, these [lecture notes.](http://maths.dur.ac.uk/~tpcc68/entropy/Ch6November2011.pdf)
An information-theoretic approach ha... | 11 | https://mathoverflow.net/users/11260 | 163766 | 85,722 |
https://mathoverflow.net/questions/160440 | 6 | Let $A$ be an unbounded linear operator of domain $D(A)$ defined on a Banach space $X$. Suppose that $A$ generates a $C\_0$-semigroup $T(t)$ which is uniformly bounded. I would like to know if there are additional assumptions under which the following conjecture is true:
For every compact subset $K$ of $X$, there exi... | https://mathoverflow.net/users/24060 | A question about uniformly bounded semigroups | If I understand your question correctly, this would mean that the function $t\mapsto T(t)x$ is Lipschitz continuous, which is equivalent for $x$ to be in the Favard space $\text{Fav}(A)$$. See for example Defintion 8.2. in these [lecture notes](https://numerical-analysis.uibk.ac.at/isem-mathematik/images/1/10/ISEM15_Le... | 6 | https://mathoverflow.net/users/12898 | 163767 | 85,723 |
https://mathoverflow.net/questions/163732 | 7 | Questions:
* Is there some understanding of the reason, why the euclidean TSP allows a PTAS, whereas the metric TSP in general does not and, is the PTAS stable under sufficiently small perturbation of the edge weights of an euclidean TSP?
* If sufficiently small perturbations of the edge weights do not harm a PTAS, ... | https://mathoverflow.net/users/31310 | What is the Essential Reason that allows a PTAS for the EUCLIDEAN TSP? | One of the key ingredients in Arora's proof is the Patching Lemma, which allows you to reduce the number of crossings between subregions while expanding the length of a tour by a controlled amount. An analogue of the Patching Lemma can be proved for more general normed spaces, but not for general metric spaces.
Note,... | 4 | https://mathoverflow.net/users/3106 | 163773 | 85,726 |
https://mathoverflow.net/questions/163770 | 2 | This question might not be suitable for MO since i know nothing about Calabi-yau varieties aside the fact that they are used in string theory to compactify additional dimensions, but still, it makes want to explore a bit this subject. It seems, judging by a quick Google search, that one can associates a(n) (Hasse-Weil)... | https://mathoverflow.net/users/13625 | l-functions of calabi-yau varieties | The short answer is "in general, no". Here are some reasons:
1. The L-function is not even well-defined unless we fix a model over a number field (to so say, choose equations for your variety with coefficients in a number field). Most notions of mirror symmetry do not take this into account.
2. Ignore the previous po... | 9 | https://mathoverflow.net/users/26218 | 163777 | 85,728 |
https://mathoverflow.net/questions/163775 | 5 | Assume that $G$ is a finite $p$-group, $p$ odd, with a non-trivial elementary abelian Frattini subgroup. Then both $\Phi(G)$ and $G/ \Phi(G)$ are vector spaces over $\mathbb{F}\_p$. Is it possible to get a bound for $\dim \Phi(G)$ as a (polynomial) function of $\dim G / \Phi(G)$?
Edit: I should probably write a littl... | https://mathoverflow.net/users/49750 | Bound for the Frattini subgroup of a $p$-group | There can be no polynomial bound which is independent of $p$. Consider the group $G = C\_{p} \wr C\_{p},$ where $C\_{p}$ denotes the cyclic group of order $p.$ Then $\Phi(G) = G^{\prime}$ has order $p^{p-1},$ (and is elementary Abelian) yet $[G: \Phi(G)] = p^{2}.$
| 7 | https://mathoverflow.net/users/14450 | 163778 | 85,729 |
https://mathoverflow.net/questions/156678 | 9 | In Montgomery's [A Tour of Subriemannian Geometries, Their Geodesics and Applications](http://rads.stackoverflow.com/amzn/click/0821841653), problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?".
Can someone explain what are the major difficulties faced in proving the same ? Suggestions for suitabl... | https://mathoverflow.net/users/42015 | Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard? | The reason the problem is hard is that we do not have a good handle on what abnormal (=singular) geodesics can look like. See the chapter of my book that describes abnormal geodesics. Progress is being made in the Carnot case, but it is slow.
I would look at a fairly recent article
(EXTREMAL CURVES IN NILPOTENT LIE... | 16 | https://mathoverflow.net/users/2906 | 163782 | 85,730 |
https://mathoverflow.net/questions/163731 | 3 | I am searching for the correct term for the following, if it exists.
A set $X\subset \mathbb{R}^2$ is called $r$-convex if for any two points $x\_1, x\_2\in X$ such that there exists an arc of radius $r$ connecting these two points, at least one arc lies in the set $X$.
Note 1: Obviously, there are always two arcs ... | https://mathoverflow.net/users/39752 | Generalization of notion of convexity | I can not comment posts because of too small reputation :-) It will change I hope.
In connection with usul's comment:
Also strongly convex sets are considered. Roughly speaking, one could require that together with two points the whole lens is contained in a set. For a definition see for instance here: <http://arxi... | 2 | https://mathoverflow.net/users/47248 | 163791 | 85,733 |
https://mathoverflow.net/questions/163797 | 27 |
>
> Let $G$ be a finite subgroup of $\mathrm{GL}\_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
>
>
> Assume that the eigenvalues of $A$ are included in a circle arc of length $<\frac{\pi}3$, and the eigenvalues of $B$ are included in a circle arc of length $<\pi$.
>
>
> Then $AB=B... | https://mathoverflow.net/users/nan | Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$ | This is a Theorem of Frobenius, whose proof can be found in the book of Isaacs on Character theory (14.15).
There are several results of this type, including theorems of H. Blichfeldt. These are connected to proving Jordan's theorem about finite subgroups of ${\rm GL}(n,\mathbb{C}).$ There is an interesting section ... | 36 | https://mathoverflow.net/users/14450 | 163798 | 85,734 |
https://mathoverflow.net/questions/163796 | 3 | Let $R$ be a commutative Noetherian domain, $K$ its fraction field, and $J$ a fractional ideal (i.e. a finitely generated sub-$R$-module of $K$) such that $J^2=J$. Is it true that $J=0$ or $J=R$? If not in this generality, can someone provide a counterexample?
I can prove that the claim is true if we assume $R$ is lo... | https://mathoverflow.net/users/17988 | Idempotent fractional ideals of a Noetherian domain | This is not true. The ring $R = k[t^2, t^3]$ with $J = k[t]$ is a counter example.
| 6 | https://mathoverflow.net/users/49765 | 163801 | 85,737 |
https://mathoverflow.net/questions/163802 | 4 | Let $f:\tilde{X} \to X$ be a normalization of projective variety. Let $L$ be a very ample line bundle on $X$. Is $f^\*L$ a very ample line bundle on $\tilde{X}$? If not true in general, is there any dimension restriction on $X$ for which this is true?
| https://mathoverflow.net/users/46578 | Does normalization of projective varieties preserve very ampleness | 1) There are certainly many cases where $f^\*L$ is still very ample: e.g. if $X$ is an irreducible curve, any line bundle on $X$ of degree $\geq 2g(\tilde{X} )+1$ will have this property.
2) A typical situation where $f^\*L$ is not very ample is when $H^0(\tilde{X},f^\*L )\cong H^0(X,L)$. For instance, take for $X$ a... | 6 | https://mathoverflow.net/users/40297 | 163806 | 85,739 |
https://mathoverflow.net/questions/163809 | 3 | Any binary relation $R\subseteq X\times Y$ gives rise to a Galois connection between the powersets of $X$ and $Y$ in a well known way (on MO you can see it e. g. in [this answer](https://mathoverflow.net/a/71616/41291); in fact, such Galois connections are in one-to-one correspondence with binary relations).
It follo... | https://mathoverflow.net/users/41291 | Which complete lattices arise as images of the Galois connections induced by binary relations? | Yes. Any complete lattice arises from a Galois connection. For instance, if $(X,\leq)$ is a poset, then the complete lattice associated with the Galois connection induced by the relation $\leq$ is the Dedekind-MacNeille completion of $X$. In particular, if $X$ is a complete lattice, then $X$ is the complete latticed ob... | 6 | https://mathoverflow.net/users/22277 | 163816 | 85,743 |
https://mathoverflow.net/questions/163676 | 7 | Let $G(t,x)$ be the heat kernel
$$
G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.
$$
Here is one approximation to $G(t,x)$:
$$
G\_\epsilon(t,x)=e^{-t/\epsilon} \sum\_{k=1}^\infty \left(\frac{t}{\epsilon}\right)^k \frac{1}{k!} G(k\epsilon,x).
$$
The question is: can one show that f... | https://mathoverflow.net/users/36814 | About the convergence rate for an approximation to the heat kernel | One can prove a stronger estimate in fact. Suppose $u$ and $v$ are positive with $v>u$ say. Note that
$$
|G(u,x)-G(v,x)| \le \int\_{u}^v \Big| \frac{d}{dt} G(t,x)\Big| dt =\int\_{u}^{v}
\frac{e^{-x^2/2t}}{\sqrt{2\pi t}} \Big|\frac{x^2}{2t^2}-\frac{1}{2t}\Big| dt.
$$
Integrating this over $x\in {\Bbb R}$ we obtain
$... | 7 | https://mathoverflow.net/users/38624 | 163820 | 85,745 |
https://mathoverflow.net/questions/163815 | 7 | Since I have studied analysis as well as algebra recently, I am familiar to work on integrablities, and such concepts when I look at topologies. Currently, I am studying algebraic geometry, and I want to ask If it is meaningful to work on such concepts on the Zariski topology. I have searched quite a lot, and I have he... | https://mathoverflow.net/users/36867 | Is it meaningful to work on convergencies, integration, etc. on the Zariski topology? | I've decided to expand my comment into an answer.
The point is what do you think a topology is for. If you think that a topology is for talking about convergence of sequences, then **no** the Zariski topology doesn't do anything for you in that regard. However I contend that in geometry (even differential geometry) p... | 21 | https://mathoverflow.net/users/43054 | 163828 | 85,749 |
https://mathoverflow.net/questions/163833 | 5 | As in title - I'm looking for a proof of the existence of a countable recursively inaccessible or recursively Mahlo ordinals, especially the first one. When looking for it in all the papers I stumbled across their existence wasn't questioned, and no proofs were referenced. On Wikipedia (where I found the first mention ... | https://mathoverflow.net/users/30186 | Proof of existence of recursively inaccessible and Mahlo ordinals | An ordinal $\xi$ is *recursively inaccessible* when it is an admissible limit of admissible ordinals. To be admissible means that $L\_\alpha$ is a model of the Kripke-Platek axioms of set theory. It is not difficult to see that $\omega\_1$ itself is admissible, since $L\_{\omega\_1}\models\text{KP}$, and by taking elem... | 9 | https://mathoverflow.net/users/1946 | 163835 | 85,751 |
https://mathoverflow.net/questions/163808 | 3 | This is a soft-question, but I haven't found an answer anywhere: do the factors of the Iwasawa decomposition of the pseudo-orthogonal group SO(p, q) have a simple form, in the same way that the factors of the Iwasawa decomposition of SL(n) do? I've only ever seen SL(n) given as an example, and I wondered why.
| https://mathoverflow.net/users/34884 | Iwasawa decomposition of the pseudo-orthogonal group | Indeed, it is easy to write out the Iwasawa decomposition for orthogonal groups $SO(p,q)$, say with $p\ge q$. By a change of variables, we can make the quadratic form $\pmatrix{ 0\_q & 0 & 1\_q \\ 0 & -1\_{p-q} & 0 \\ 1\_q & 0 & 0\_q}$. The maximal compact $K=S(O(p)\times O(q))$, which was diagonal blocks with the quad... | 4 | https://mathoverflow.net/users/15629 | 163839 | 85,754 |
https://mathoverflow.net/questions/163837 | 11 | I'm trying to understand the paper
Arbarello, Enrico, Cornalba, Maurizio,
Calculating cohomology groups of moduli spaces of curves via algebraic geometry.
Inst. Hautes Études Sci. Publ. Math. No. 88 (1998), 97–127 (1999).
At the very top of page 103 (of the journal; this is the 7th page of the paper) they assert w... | https://mathoverflow.net/users/49778 | Why is the kth cohomology group of the DM-compactification of the moduli space of curves pure of weight k? | Let me elaborate on my comment, adding some details and references.
The space $\overline{\mathcal{M}}\_{g, \, p}$ is an *almost Kähler $V$-manifold*. This means that it has only quotient singularities and admits a bimeromorphic, proper modification which is a Kähler manifold.
Then the assertion follows by applying ... | 10 | https://mathoverflow.net/users/7460 | 163842 | 85,755 |
https://mathoverflow.net/questions/163847 | 31 | The fundamental group of the Hawaiian earring is very complicated, but since it's "1-dimensional" one might guess that the higher homotopy groups vanish. Do they? Since the Hawaiian earring does not have a universal cover, the standard approach to showing that higher homotopy groups of graphs vanish does not apply.
| https://mathoverflow.net/users/22 | Are the higher homotopy groups of the Hawaiian earring trivial? | Despite the non-existence of a universal covering space, there is still an object that acts like a universal covering space for the earring space. It is actually quite useful for understanding the structure of the fundamental group as a subgroup of the inverse limit $\varprojlim F\_n$ of free groups. The "generalized u... | 37 | https://mathoverflow.net/users/5801 | 163850 | 85,758 |
https://mathoverflow.net/questions/163638 | 3 | Are there necessary/sufficient conditions for a functor $f\colon \mathcal C\to \widehat{\mathcal A}$ to induce an equivalence $\text{Lan}\_yf=F\colon \widehat{\mathcal C}\leftrightarrows \widehat{\mathcal A}$, so that $\cal A,C$ are Morita equivalent categories?
| https://mathoverflow.net/users/7952 | Morita equivalence via Kan extension | Expanding on Zhen's comment into an answer, the Cauchy completion of $A$ can be identified with the small-projective objects in $\widehat{A}$ (i.e. those where mapping out of them preserves all small colimits). Thus, an equivalence of presheaf categories must restrict to an equivalence of Cauchy completions. It follows... | 6 | https://mathoverflow.net/users/49 | 163855 | 85,761 |
https://mathoverflow.net/questions/163584 | 6 | Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of [$T$-modules](http://ncatlab.org/nlab/show/Eilenberg-Moore+category) $\mathsf{Mod}(T)$ is cocomplete ([Linton](http://link.springer.com/chapter/10.1007/BFb008308... | https://mathoverflow.net/users/2841 | Universal property of module categories over monads | Here is a partial answer with some literature pointers.
It's a general fact that in any bicategory with local reflexive coequalizers (i.e. reflexive coequalizers in each hom-category, preserved by composition on either side), Eilenberg-Moore objects coincide with Kleisli objects. I believe this was first observed by ... | 6 | https://mathoverflow.net/users/49 | 163857 | 85,762 |
https://mathoverflow.net/questions/163856 | 3 | If there exists a nontrivial vector field $V\not=0$ in Riemannian manifold $M$ and an open set $U\subset M$ such that $\nabla\_{X}V=0$ in $U$ for any vector field $X$ in $M$, then dose $U$ have to be flat?
That is, if a Riemannian maniflod exists a vector field $V$ parallel transport along any vector field, then is t... | https://mathoverflow.net/users/43941 | If there exists a nontrivial vector field $V$ such that $\nabla_{X}V=0$ for any vector field $X$, the manifold must be flat? | If you take a manifold like $M = \mathbb R \times M'$ with the usual metric on $\mathbb R$ and where $M'$ is some Riemannian manifold, the vector field induced from $\mathbb R$ will satisfy this condition but $M$ will not be flat in general.
However, if you impose instead that you have $\dim M$ many independent para... | 12 | https://mathoverflow.net/users/4622 | 163858 | 85,763 |
https://mathoverflow.net/questions/163843 | 15 | Let $E$ be the elliptic curve $y^2=x^3+1$ and $p \equiv 1 \pmod{3}$ a prime. Computing the number of points mod $p$ of $E$ using the naive method gives:
$$ \#E(\mathbb F\_p) = 1+ \sum\_{x=0}^{p-1} \left( 1 + \left(\frac{x^3+1}{p}\right)\right) \equiv p+1 + \sum\_{x=0}^{p-1}(x^3+1)^{\frac{p-1}2}\pmod{p}$$
developing the... | https://mathoverflow.net/users/10670 | Congruence for the number of points in the elliptic curve $y^2 = x^3+b \pmod{p}$ | This is really just my comment above, but maybe it's a reasonable answer. Congruences between certain binomial coefficients and the representations of primes by some quadratic forms go back to Gauss and Jacobi. The two examples given in the question and a number of other examples are discussed in the paper [Binomial co... | 13 | https://mathoverflow.net/users/38624 | 163861 | 85,764 |
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