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https://mathoverflow.net/questions/59563 | 27 | The space of configurations of $k$ distinct points in the plane
$$F(\mathbb{R}^2,k)=\lbrace(z\_1,\ldots , z\_k)\mid z\_i\in \mathbb{R}^2, i\neq j\implies z\_i\neq z\_j\rbrace$$
is a well-studied object from several points of view. Paths in this space correspond to motions of a set of point particles moving around avoid... | https://mathoverflow.net/users/8103 | Configuration space of little disks inside a big disk | I have a number of results on hard disks in various types of regions, and preprints are in progress. The terminology "hard spheres" (or "hard disks" in dimension 2) comes from statistical mechanics, and I believe Fred Cohen is following my lead on this. (See for example the hard disks section of Persi Diaconis' [survey... | 32 | https://mathoverflow.net/users/4558 | 59611 | 37,023 |
https://mathoverflow.net/questions/59614 | 1 | If I have a function $f(z,\alpha)$ (let's keep it a polynomial of order $\geq 2$ in $z$, for simplicity), what would be a necessary condition for there to be branch points for this function? A friend mentioned that $f(z,\alpha)$ and $f\_{\alpha}(z,\alpha)$, where the second term is the partial derivative w.r.t. $\alpha... | https://mathoverflow.net/users/13916 | Necessary condition for a branch point | Your question is not very clear. However, I guess you are asking for the branch points of the cover of $\mathbb{C}$ defined by $f(z, \alpha)=0$.
In this case, let us assume for the sake of symplicity that $f(z, \alpha)$ is *monic* in $z$; then
$f(z, \alpha)=z^n + f\_{n-1}(\alpha)z^{n-1}+ \cdots + f\_0(\alpha)$.
... | 2 | https://mathoverflow.net/users/7460 | 59622 | 37,029 |
https://mathoverflow.net/questions/59495 | 3 | Suppose $K$ is an $n$-dimensional $C^2$ convex body in $\mathbb{R}^{n+1}$. We choose two distinct directions $z\_0, z\_1\in\mathbb{S^{n}}$. If $P\_1$ and $P\_2$ are the corresponding hyperplanes($z\_0\perp P\_1$ and $z\_1\perp P\_2$) and $K'$ is the projection of $K$ on $P\_1\cap P\_2$, what is the $Vol(K')$?
We know t... | https://mathoverflow.net/users/12145 | volume of the projected body | This is easier at least for me if we forget about the inner product.
Let $K$ be a convex body in the vector space $X$ and assume that the origin lies in the interior of $K$. If you know $K$, you know its support function $h: X^\* \rightarrow \mathbb{R}$. Now let $N \subset X$ be a linear subspace and $\pi: X \rightar... | 1 | https://mathoverflow.net/users/613 | 59624 | 37,031 |
https://mathoverflow.net/questions/38300 | 2 | Let $u(t) = \Sigma\_{k=1}^n c\_k e^{i \lambda\_k t} (c\_k \in \mathbb C, \lambda\_k \in \mathbb R) $ be an exponential polynomial of order $n$ with purely imaginary exponents. We can assume that the exponents are $\delta$-separated.
Define $E\_n$ to be the collection of all exponential polynomial of order $n$. i.e.,... | https://mathoverflow.net/users/6766 | How to find the almost period of an exponential polynomial | You are basically interested in what is called $\epsilon$-dual Characters.
For a set $\Lambda \subset \R^d$ we define
$$\Lambda^\epsilon := \{ t \in \R^d | \left| e^{2 \pi x \cdot t} -1 \right| < \epsilon \, \forall x \in \Lambda \}\,.$$
In your case $\Lambda := \{ \lambda\_1, .., \lambda\_n \}$ and for all $t \... | 1 | https://mathoverflow.net/users/11552 | 59629 | 37,032 |
https://mathoverflow.net/questions/59620 | 17 | I have been trying to learn the method of Chabauty and Coleman to find rational points on curves; I have been reading an exposition by McCallum and Poonen which was pointed out to me by Emerton in [this question](https://mathoverflow.net/questions/56011/why-should-i-believe-the-mordell-conjecture).
Let $X$ be a curv... | https://mathoverflow.net/users/5744 | Rational points à la Chabauty-Coleman | If $r=2$ then $r'>0$. For an example where $r'=1$, take a curve such that the jacobian has a nontrivial endomorphism $f$
and such that the group of points in the jacobian is generated by $P,f(P)$ for some point $P$
Now find a prime $p$ splitting in $\mathbb{Q}(f)$ so that $f(P)=\alpha P$ for some $p$-adic number $\alph... | 7 | https://mathoverflow.net/users/2290 | 59632 | 37,033 |
https://mathoverflow.net/questions/59635 | 4 | Let $n, d$ be positive integers. I am interested in the open subset
$\mathcal U\_{n,d} \subset \mathbb P H^0 ( \mathbb P^n\_{\mathbb R}, \mathcal O\_{\mathbb P^n\_{\mathbb R}}(d))$ corresponding to sections without real zeros. When $d$ is odd and $n$ arbitrary, the set
$\mathcal U\_{n,d}$ is empty since (homeogeneou... | https://mathoverflow.net/users/605 | Hypersurfaces without real points | If $s$ is a non-zero section whose image lies in $\mathcal U\_{n,d}$, then it has constant sign on $V^\ast:=\mathbb R^{n+1}\setminus\{0\}$ and after possibly multiplying by $-1$ we may assume that $s$ is strictly positive on $V^\ast$. The strictly positive $s$ form an open convex cone $C$ (we do not assume that $0$ bel... | 3 | https://mathoverflow.net/users/4008 | 59642 | 37,037 |
https://mathoverflow.net/questions/59645 | 3 | I can show that $\sum^\infty\_{k=1}{{1}\over{k^2}} = {{\pi^2}\over{6}}$ without to much hassle just using two representations of ${{sin(x)}\over{x}}$ but I cant find any proof nearly as simple when I try to determine what $\sum^\infty\_{k=1}{{1}\over{k^2+1}}$ equals.
Does there exist an elementary proof? or must I r... | https://mathoverflow.net/users/13927 | algebraic proof of an infinite sum | You can evaluate a series for cotangent at the square root of -1: see
<http://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions> .
| 7 | https://mathoverflow.net/users/6153 | 59646 | 37,039 |
https://mathoverflow.net/questions/59558 | 3 | So we all know already that next identities follow:
$3^2+4^2=5^2$
$3^3+4^3+5^3=6^3$
So it raises my question:
For $(\*)n^k+(n+1)^k+...+(n+m)^k=(n+m+1)^k$ are there infinite triples (n,m,k) s.t `(*)` is satisifed?
Any literature on this question or more general question from this follows?
Thanks.
| https://mathoverflow.net/users/13904 | Integer solutions of $n^k+(n+1)^k+\cdots+(n+m)^k=(n+m+1)^k$ | I do not have a definite answer, but in view of the discusion, and since it connects, as requested, the question to problems investigated in the literature, some remarks.
I claim that:
>
> If an answer to this question is *known*, then it is of the form that there are infinitely many solutions.
>
>
>
This m... | 4 | https://mathoverflow.net/users/nan | 59649 | 37,041 |
https://mathoverflow.net/questions/59580 | 16 | Assume you have a Poisson point process of constant intensity $\lambda$ in the Euclidean plane. From this point process we construct the Delaunay triangulation (or the Voronoi tessellation for that matter).
It is known [Stoyan et all] that the expected degree of a typical vertex has degree 6. Moreover, there are sev... | https://mathoverflow.net/users/13825 | Expected Degree of a vertex in Delaunay Triangulations | The expected degree for the Delaunay triangulation on hyperbolic space will depend on the density of the points. With low enough density points, you should get arbitrarily high degree. You should be able to get the expected degree as high as you want for a point distribution in the plane by taking the conformal represe... | 6 | https://mathoverflow.net/users/2294 | 59652 | 37,043 |
https://mathoverflow.net/questions/59661 | 5 | Hi,
given a discrete simple Random walk on a symmetric graph, what is known about the probability of the random walker to return to a starting site at step $n$? Specifically, I am interested in the probability that at step $n$ the random walk has returned to its starting vertex.
What I have found is e.g. in the Bo... | https://mathoverflow.net/users/13932 | Probability of return at step $n$ of a Random walk to its starting vertex | The simple random walk on the circle graph $\mathbb{Z}/N\mathbb{Z}$ of size $N$ can be realized as the class modulo $N$ of simple random walk on $\mathbb{Z}$, hence
$$
p\_{00}^{(n)}(\mathbb{Z}/N\mathbb{Z})=\sum\_{k\in\mathbb{Z}}p\_{0,kN}^{(n)}(\mathbb{Z}).
$$
Now, for every $k$,
$$
p\_{0,k}^{(n)}(\mathbb{Z})=2^{-n}{n\c... | 1 | https://mathoverflow.net/users/4661 | 59665 | 37,049 |
https://mathoverflow.net/questions/59643 | 8 | Let G be a finite subgroup of U(n), the unitary group acts on $\mathbb{C}^n$. If there is a unit vector $x$ in $\mathbb{C}^n$ such that g(x) is almost orthogonal to x, for all $g\in G$ except the identity, can we perturb x so that g(x) is exactly orthogonal to x, for all $g\in G$ except the identity? More precisely, ca... | https://mathoverflow.net/users/9858 | A Perturbation problem for U(n) | If gx is a bounded distance away from x (which in particular occurs when gx is nearly orthogonal to x), then g is a bounded distance away from the identity. Since U(n) is compact, this and the pigeonhole principle forces the group G to have bounded cardinality; in particular, the set of all such groups is compact (if o... | 10 | https://mathoverflow.net/users/766 | 59672 | 37,052 |
https://mathoverflow.net/questions/59410 | 7 | I am actually studying coset geometries (in the sense of Tits and Buekenhout) for the sporadic simple group of Suzuki. I came aware that Buekenhout found in 1979 a geometry over the following diagram
```
c 6
O----------O----------O
1 4 4
```
However, I couldn't find any information... | https://mathoverflow.net/users/12039 | A rank 3 geometry for the sporadic simple group of Suzuki | I finally found the maximal parabolic subgroups of this geometry. Let us first denote the types of the elements with 0,1 and 2 when reading the diagram from left to right, and let us denote with $G\_0$, $G\_1$ and $G\_2$ the stabilizer of an element of type 0, 1 and 2 respectively. Then we have:
$$
G\_0 = G\_2(4),\qu... | 7 | https://mathoverflow.net/users/12039 | 59676 | 37,055 |
https://mathoverflow.net/questions/59605 | 12 | Can any one recommend me a good introductory book in Riemann Surface?
| https://mathoverflow.net/users/13913 | Reference in Riemann Surfaces | It depends partly what you are more interested in, geometry or analysis. There are two relevant categories: compact complex manifolds of dimension one (and holomorphic maps), and algebraic complex curves (and rational maps). The approach in the wonderful book of Miranda is to consider the functor from algebraic curves ... | 14 | https://mathoverflow.net/users/9449 | 59688 | 37,062 |
https://mathoverflow.net/questions/59633 | 6 | I'm trying to compute an (s-t) maximum flow through a network which includes a number of arc pairs ((u,v), (v,u)) that have equal, negative capacities (weights). I'm not aware of any efficient algorithms that solve this problem directly, so I am trying to think of a way to transform the problem so that it can be passed... | https://mathoverflow.net/users/13922 | Maximum flow with negative capacities? | The correspondence between max flow with lower capacities and max flow with negative capacities as described in the linked homework problem is based on the convention that $f(u,v)=-f(v,u)$ for all arc pairs $\{(u,v),(v,u)\}$. <http://computingscience.nl/docs/vakken/an/an-maxflow.ppt> contains a reduction of max flow wi... | 4 | https://mathoverflow.net/users/12674 | 59689 | 37,063 |
https://mathoverflow.net/questions/59018 | 6 | The following question feels to me like a standard sort of 'fact' in birational geometry, but I can't seem to write down a correct set of details. Hopefully someone can point me to a reference and not a counter example!
Suppose $X$ is a variety (reduced and irreducible over an algebraically closed field, perhaps of c... | https://mathoverflow.net/users/5100 | equations defining a subvariety | *Credit*: This answer came out of trying to understand why auniket's answer (a.k.a. counterexample) works.
1) auniket is correct that for dimension reasons $T$ cannot surject onto $S$, so in particular my comment about $X$ being normal possibly helping is irrelevant. So is $T$.
2) It seems to me that there is a muc... | 2 | https://mathoverflow.net/users/10076 | 59691 | 37,064 |
https://mathoverflow.net/questions/59686 | 3 | I am reading Mandelbrot, and stubling upon his use of the limit ("almost a Hölder exponent")
$$\lim\_{\varepsilon\to0}\frac{\log(f(x+\varepsilon) - f(x))}{\log(\varepsilon)}.$$
To simplify, lets assume that $f$ is non-decreasing, and of course the limit of $\epsilon$
is from the right. Calling this (me, not Mandelbro... | https://mathoverflow.net/users/6494 | Mandelbrot and "log-derivative" | Suppose $f(x) = \int\_0^x d\mu(t)$ for some measure $\mu$. Then your log-derivative is (almost) equivalent to the quantity $\alpha(x) = \lim\_{r \to 0} \frac{\log \mu(]x-r,x+r[)}{\log r}$. This is known as the *local dimension* of $\mu$ at $x$, and it has been much studied in geometric measure theory. One particular to... | 5 | https://mathoverflow.net/users/13944 | 59704 | 37,072 |
https://mathoverflow.net/questions/59706 | 3 | Let $S$ be a base scheme. Let $X$ be a scheme over $S$ and let $G$ be a group scheme over $S$ acting on $X$ via $\sigma: G \times\_S X \to X$. Suppose that we have a scheme $Y$ over $S$ together with $\varphi: X \to Y$ such that $\varphi \circ \sigma = \varphi \circ p\_2$ (where $p\_2: G \times\_S X \to X$), $\varphi$ ... | https://mathoverflow.net/users/1107 | geometric quotient | As far as I recall from Mumford's book there are two more conditions an a "geometric quotient", namely that $\varphi$ is submersive and that ${\mathcal O}\_Y \to \varphi\_\*({\mathcal O}\_X)^G$ is an isomorphism. Moreover, the condition on geometric fibers is only equivalent to
(\*) surjectivity of $\varphi$ and the ... | 6 | https://mathoverflow.net/users/13302 | 59709 | 37,074 |
https://mathoverflow.net/questions/59708 | 2 | Given a funtion $f \in L^p([0,1])$ (take $p=\infty$ if you'd like), and also a measure preserving map $s:[0,1] \to [0,1]$ (meaning $s$ pushes Lebesgue measure forward to itself) I would like to know if there exists some $f^{\ast} \in L^p([0,1])$ such that $f^{\ast} \circ s = f$. If $s$ is invertible this is of course o... | https://mathoverflow.net/users/8755 | Existence of rearrangements of functions in $L^p([0,1])$ when given a measure preserving map | Consider $$s(x) = \begin{cases}2x, & x \in [0,1/2] \\\ 2x-1,&x \in ]1/2,1]\end{cases}.$$
Then for an arbitrary $f^{\ast}$, we have
$$(f^{\ast}\circ s)(x+1/2) = (f^{\ast}\circ s)(x)$$
for all $x \in ]0,1/2]$. Certainly not all functions $f \in L^p([0,1])$ possess this property.
Also, measure preserving maps need not... | 3 | https://mathoverflow.net/users/11716 | 59710 | 37,075 |
https://mathoverflow.net/questions/59671 | 1 | I want to learn something about Weil groups, so I want to read Deligne's paper"Les Constantes des Equations Fonctionnelles Des Fonctions L". But I feel not so comfortable to read a so long French paper. It is better if some one had translated it into English.
Does any one has the English translation of this paper?
| https://mathoverflow.net/users/13466 | English version of Deligne's paper"Les Constantes des Equations Fonctionnelles Des Fonctions L" | This is a community wiki answer collecting some of the comments:
While it seems unlikely that Deligne's article has been translated, there are several alternative sources in English that may be of use, including:
1. Tate's article *[Number-theoretic background](http://ams.org/online_bks/pspum332/pspum332-ptIII-1.pd... | 10 | https://mathoverflow.net/users/2874 | 59719 | 37,078 |
https://mathoverflow.net/questions/59713 | 4 | From a discussion with some friends, this *apparently easy* problem has come out; I decided to post it here, because I believe that the answer is non-trivial and the maths beneath interesting. Partial solutions, ideas or possible approaches are welcome too!
Suppose that there are $P$ boxes (of infinite capacity) and ... | https://mathoverflow.net/users/13388 | Probability estimates for "beans & boxes" | A classical way of tackling such kind of problems is via Poisson approximations.
For example, consider a Poisson point process in $(0,1) \times (0,\infty)$ with unit intensity. The number $N\_k(T)$ of points in $(\frac{k}{P},\frac{k+1}{P}) \times (0,T)$ is distributed as a Poisson random variable with mean $\frac{T}{P}... | 4 | https://mathoverflow.net/users/1590 | 59720 | 37,079 |
https://mathoverflow.net/questions/59317 | 9 |
>
> **Question.** Give a (possibly elementary) example of a probability measure preserving action $\rho\colon G \curvearrowright X$ of a finitely-generated discrete group $G$ on a standard borel space $X$ with a probability measure, such that
>
>
> 1. the equivalence relation generated by $\rho$ is ergodic and amen... | https://mathoverflow.net/users/2631 | amenable equivalence relation generated by an action of a non-amenable group | The answer is yes, such an action exists.
What is needed for the construction is the following very nice example of an action of a non-amenable group on $\mathbb Z$, which I just learned from Gabor Elek.
Consider a graph with vertices given by $\mathbb Z$ and unoriented edges between $n$ and $n+1$.
Pick a random ... | 7 | https://mathoverflow.net/users/8176 | 59721 | 37,080 |
https://mathoverflow.net/questions/59714 | 4 | Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every $m\in M$ and every non-negative integers $k$. Defininig relations between the generators are given by
$m^{(0)} = e$
$m^{... | https://mathoverflow.net/users/13086 | Is the functor of divided powers a weakly monoidal functor? | This is a well-known result and, apart from terminology, should be found in
Roby, Norbert
Lois polynômes multiplicatives universelles. (French. English summary)
C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 19, A869–A871.
Note that this becomes most natural if one interprets your $D\_r(M)$ (usually denoted $\Gamma... | 12 | https://mathoverflow.net/users/4008 | 59722 | 37,081 |
https://mathoverflow.net/questions/59732 | 1 | Let $F$ and $G$ be two primitive functions of the Selberg class, and let $\mathbb{A}$ be the set of values taken by the function which maps a prime number $p$ to $a\_{p}(F)\overline{a\_{p}(G)}$. $\mathbb{A}$ is finite or countable, so that there exists $I\subset\mathbb{N}^\*$ such that $\displaystyle{\mathbb{A}=\bigcup... | https://mathoverflow.net/users/13625 | Selberg's orthonormality conjecture and density | No. (This is very unlikely to hold for anything except $L$-functions with only finitely many distinct coefficients; the simplest counter-example is to take an elliptic curve $E$ and its associated $L$-function $L(E,s)$, normalized to have critical line at $1/2$, and $F=G=L(E,s)$; for any non-zero real number $x>0$, due... | 6 | https://mathoverflow.net/users/20038 | 59744 | 37,091 |
https://mathoverflow.net/questions/59727 | 4 | Hello,
Do we know if the axiom of choice is needed for Chevalley's valuation/place extension theorem (i.e. the theorem that states that for every valued field and a field extension, one can extend the valuation to the field extension)?
| https://mathoverflow.net/users/13948 | Chevalley's valuation extension theorem and the axiom of choice | I think [this](https://mathoverflow.net/questions/30581/extension-of-valuation) more or less answers it?
| 1 | https://mathoverflow.net/users/25726 | 59747 | 37,093 |
https://mathoverflow.net/questions/59741 | 18 | The title says it all ... Obviously, any such triple must be of the form
$(4a+1,4a+2,4a+3)$ where $a$ is an integer. Has this problem
already been studied before ? The result would follow from Dickson's
conjecture on prime patterns, which implies that there are
infinitely many integers $b$ such that $4(9b)+1,2(9b)+1$ ... | https://mathoverflow.net/users/2389 | are there infinitely many triples of consecutive square-free integers? | To expand on the answer in my comment, the proportion of integers $a$ for which $4a+1,4a+2,4a+3$ are all squarefree is $\prod\_{p\not=2}(1-3/p^2)$, with the product taken over all odd primes $p$. As this product converges to a positive limit, there are infinitely many such $a$. A quick heuristic is to look modulo $p^2$... | 31 | https://mathoverflow.net/users/1004 | 59753 | 37,098 |
https://mathoverflow.net/questions/59745 | 13 | It is known that for metric graphs the concepts of *Gromov's hyperbolicity* and *strictly positive* Cheeger constant are related. Let us first recall the definition of the Cheeger constant. Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For a collection of vertices $A \subseteq V(G)$, let $\partial A$ d... | https://mathoverflow.net/users/13825 | Gromov's Hyperbolicity and Positive Cheeger Constant in Planar Graphs | If I add the assumption that the given graph has bounded degree, and the same holds for the dual graph, then the answer to your question is no. A positive Cheeger constant implies a linear isoperimetric inequality which in turn implies Gromov hyperbolicity.
However, if you allow the dual graph to have unbounded degr... | 9 | https://mathoverflow.net/users/1650 | 59754 | 37,099 |
https://mathoverflow.net/questions/59757 | 2 | It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See [Parry's Topics in ergodic theory p14](http://farm6.static.flickr.com/5291/5564946645_58634817f9_b.jpg)
Given a probability space... | https://mathoverflow.net/users/13838 | Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant? | For a counterexample to Jon's implicit rewording of your question, consider a purely atomic probability space in which the atoms all have different measure.
For a positive result, consider any purely non atomic probability space. It is measure isomorphic to $[0,1]$ with Lebesgue measure. I guess for non separable pur... | 6 | https://mathoverflow.net/users/2554 | 59761 | 37,101 |
https://mathoverflow.net/questions/59758 | 3 | Any map $A \to B$ of abelian varieties of the same dimension over a global field $K$ induces a map $\mathcal{A} \to \mathcal{B}$ on the corresponding Neron models over $X$ (where $X=Spec{\mathcal{O}\_K}$ for any number field $K$ or $X$ is a complete smooth curve over a perfect field with function field $K$) which furth... | https://mathoverflow.net/users/13628 | Maps on the identity components of Neron models | If I'm not mistaken, the map $Res^0$ is a bijection. Indeed, restriction induces maps
$$Hom\_X(\mathcal A,\mathcal B) \to Hom\_X(\mathcal A^0,\mathcal B^0)\to Hom\_K(A,B).$$
The composite is a bijection, by the universal mapping property of Neron models,
and both maps are injective, since $A$ is Zariski dense in $\mat... | 5 | https://mathoverflow.net/users/2874 | 59767 | 37,106 |
https://mathoverflow.net/questions/59770 | 19 | A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all simple using the concept of Riemann surfaces. But unfortunately, I just can't find it back. Does someone know if such a r... | https://mathoverflow.net/users/13625 | Are the nontrivial zeros of the Riemann zeta simple? | This is widely open. Moreover, I think we will prove the Riemann Hypothesis much earlier than the simplicity of the zeros (if true). The latter is somehow much more accidental, the only reasonable argument I know in favor of it is "why would two zeros ever coincide"? Note, however, that some automorphic $L$-functions d... | 35 | https://mathoverflow.net/users/11919 | 59777 | 37,112 |
https://mathoverflow.net/questions/59796 | 8 | Let $X$ be a contractible compact [edit: locally connected] topological space
(Hausdorff and second countable). Let $f\colon X\to X$
be a continuous map. Then (I suppose) $f$ has a fixed
point. Personally, I cannot think of a better generalization
of Brouwer's fixed point theorem, but is it true?
| https://mathoverflow.net/users/9833 | The generalization of Brouwer's fixed point theorem? | No. I believe the first counterexample is from:
Kinoshita, S. On Some Contractible Continua without Fixed Point Property. Fund. Math. 40 (1953), 96-98
which I unfortunately can't find online. Kinoshita's example is described on page 127 in this excellent [article by Bing](http://www.jstor.org/stable/2317258), howev... | 24 | https://mathoverflow.net/users/6950 | 59797 | 37,123 |
https://mathoverflow.net/questions/59717 | 11 | Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all sentences $\sigma,\tau\in Sent$, $\sigma$ has less consistency strength than $\tau$ (or $\sigma\leq\_{cons}\tau$) iff $... | https://mathoverflow.net/users/13938 | (Non?)-linearity of the consistency strength ordering in ZF | Marios, this is indeed a fascinating topic.
The consistency strength hierarchy is *not* linearly ordered. One can produce counterexamples by variants of Gödel sentences or of [Rosser sentences](http://en.wikipedia.org/wiki/Rosser%2527s_trick). It is actually an interesting exercise to produce explicit examples of a ... | 21 | https://mathoverflow.net/users/6085 | 59800 | 37,125 |
https://mathoverflow.net/questions/59784 | 3 | Suppose that Z\_1, ... , Z\_n are binomial distributions with E[Z\_i]=z\_i.
If (Z\_i) are pairwise independent, then, It's well known that the Chebyshev inequality can bound the tail distributions.
If (Z\_i) are i.i.d, then, one can use Chernoff bound to bound the tail distributions.
As we know, Chernoff bound gi... | https://mathoverflow.net/users/6326 | bound the tail distribution | Some tools from the theory of the classical moments problem are useful here. You can see how they are used and get some bounds on your question in my joint paper with Itai Benjamini and Ron Peled [here](http://www.math.ubc.ca/~origurel/papers/BGGP07abs.pdf) (this is an extended abstract, hopefully the paper itself will... | 4 | https://mathoverflow.net/users/1061 | 59801 | 37,126 |
https://mathoverflow.net/questions/59755 | 2 | Let $\mathit{H}$ be a (real or complex) Hilbert space and $U:\mathit{H}\rightarrow\mathit{H}$ be a unitary operator. What conditions can be placed on $U$ to guarantee a sequence $v\_n$ such that $|v\_n|=1$ and ($Uv\_n$,$v\_n$)$\rightarrow$0 as $n\rightarrow\infty$?
| https://mathoverflow.net/users/10476 | Almost Orthogonal Vectors given a Unitary Operator | I'll suppose this is a complex Hilbert space and you're using the convention that the inner product is linear in the first argument and conjugate-linear in the second. The set of all possible $\langle Uv, v\rangle$ for unit vectors $v$ is the numerical range of $U$. Since unitary operators are normal, the closure of th... | 6 | https://mathoverflow.net/users/13650 | 59805 | 37,129 |
https://mathoverflow.net/questions/59762 | 2 | Hello is there anyone that would know where I can find an example of a noetherian N-1 ring that is not a Nagata ring. (See the Wikipedia article "[Nagata ring](http://en.wikipedia.org/wiki/Nagata_ring)" for the definitions of N-1 ring and Nagata ring.)
| https://mathoverflow.net/users/13953 | An example of a noetherian N-1 ring that is not N-2 and/or a Nagata ring | There is a discrete valuation ring $R$ (hence trivially N-1) of characteristic $p>0$ whose completion $\widehat{R}$ contains an element $x\not\in R$ such that $x^p\in R$. Such a ring cannot be N-2.
| 4 | https://mathoverflow.net/users/7666 | 59809 | 37,131 |
https://mathoverflow.net/questions/59823 | 10 | Q1: Do you known examples, where the different is not a principal ideal?
Q2: Is there a good interpretation for the reason, why this happens?
See e.g. Neukirch, Proposition 2.4, page 197.
The reason why I ask: the definition of the canonical additive character $\psi:x \mapsto \mathrm{e}^{2 \pi i (\mathrm{Tr}\_{F ... | https://mathoverflow.net/users/10400 | When is the different in a number field a principal ideal? | A partial answer:
Regarding Q1: An example for this is the number field generated by third root of $175$ .
See e.g. a comment by KConrad on this question
[Which number fields are monogenic? and related questions](https://mathoverflow.net/questions/21267/which-number-fields-are-monogenic-and-related-questions)
or ... | 12 | https://mathoverflow.net/users/nan | 59831 | 37,142 |
https://mathoverflow.net/questions/59843 | 5 | Let $X$ be a hyperelliptic Riemann surface, and let $J$ be the hyperelliptic involution. Then consider the quotient surface $X/ < J > ,$ my question is whether $X/ < J > $ is a Riemann surface?
On the standard Riemann surface textbook, the answer is yes, $X/< J >$ is the Riemann sphere $S^{2}$. More generally, let $... | https://mathoverflow.net/users/13054 | Quotient Surface of A Hyperelliptic Involution | Let $X$ be the compact Riemann surface and $H$ a finite subgroup of $G=Aut(X)$. Then you can think of two different constructions.
1. One is taking the quotient in the category of complex manifolds $X/H$ (which corresponds to the [GIT quotient](http://en.wikipedia.org/wiki/Geometric_invariant_theory) $X//H$ in the al... | 8 | https://mathoverflow.net/users/4721 | 59850 | 37,155 |
https://mathoverflow.net/questions/52871 | 6 | Let $M$ be a connected smooth manifold, and assume that it is parallelisable; that is, its tangent bundle is trivial. Does $M$ admit an H space structure? That is, does there exist a smooth map $\mu:M\times M\to M$ and an identity element $e$ satisfying $\mu(e,x)=\mu(x,e)=x$?
The motivation for asking is the followin... | https://mathoverflow.net/users/12412 | Does a Trivial Tangent Bundle Induce a Multiplication? | Ryan Budney's comment pretty much killed the question, but anyway...
Let $X\_{m,n} = S^{2m}\times S^{2n+1}$, with $m\le n$ (strictly) positive integers.
**Lemma**: $X=X\_{m,n}$ is parallelisable.
*Proof*: this follows from playing around with vector bundles, the key facts being that $TX = \pi^\*(TS^{2m}) \oplus \... | 12 | https://mathoverflow.net/users/13119 | 59851 | 37,156 |
https://mathoverflow.net/questions/59830 | 7 | So the title says it all,
>
> >
> Q: Given a large odd integer $N>>0$, what can we prove about the smallest prime
> $p>N$ such that $gcd(p-1,N)=1$?
>
>
>
Note that such a prime exists: Given an integer $a$ coprime to $N$ we know that there are infinitely many primes $p$
with $p\equiv a\pmod{N}$. In particula... | https://mathoverflow.net/users/11765 | Given an odd integer N find the smalletst prime p > N such that (p-1,N)=1 | Your first question seems to follow fairly easily from the Bombieri-Vinogradov theorem; actually it seems only to need Renyi's original result.
By Moebius inversion, it should suffice to find an asymptotic formula of the form $\sum\_{d|N} \mu(d) \psi(x;d,1) = cx + O(x/\log{x})$ for some $c > 0$ and
$N < x \leq 3N/2... | 9 | https://mathoverflow.net/users/2627 | 59854 | 37,159 |
https://mathoverflow.net/questions/59724 | 5 | I would like to know which results have been obtained concerning Selberg's orthonormality conjecture. For example, has it been proven that for every pair of distinct primitive functions of the Selberg class $(F,G)$, $\displaystyle{\sum\_{p\leq x}\frac{a\_p(F)\overline{a\_p(G)}}{p}=o(\log\log x)}$?
Thank you in advance.... | https://mathoverflow.net/users/13625 | The current state of Selberg's orthonormality conjecture | Orthogonality has been proven for certain pairs of automorphic L-functions. The proof procedes like a proof of the Prime Number Theorem, replacing the logarithmic derivative of $\zeta(s)$ with that of $L(s,\pi\otimes\tilde\pi')$, where $\pi$ and $\pi'$ are automorphic representations.
The only proofs in print are fo... | 1 | https://mathoverflow.net/users/6753 | 59868 | 37,165 |
https://mathoverflow.net/questions/59862 | 12 | In Johnstone's book Topos Theory, he mentions an unresolved problem as to whether the 2-category $\mathbf{\text{Topos}}$ of Grothendieck toposes and geometric morphisms admits pseudo-colimits. $\mathbf{\text{Topos}}$ does admit lax colimits (including, famously, examples of Artin-Wraith gluing), and he remarks that ins... | https://mathoverflow.net/users/2926 | Does the 2-category of toposes admit pseudo-colimits? | The answer is yes; it seems to be due to Ieke Moerdijk in "The classifying topos of a continuous groupoid, I" (1988). There is also an exposition in section B3.4 of Johnstone's more recent book *Sketches of an Elephant*.
The idea is: a pseudo-limit in Cat of toposes and lex left adjoints "obviously" satisfies all the... | 13 | https://mathoverflow.net/users/49 | 59873 | 37,168 |
https://mathoverflow.net/questions/59869 | 5 | We are given a function $f: \mathbb R^d \to \mathbb R$. For simplicity we can assume that $f$ is smooth and compactly supported. Is the Sobolev norm of order $\frac{1}{2}$ strong enough to prove an inequality like this
$$|f(a)-f(b)| \leq \| f \|\_{H^{\frac 1 2}}$$
or a variant, where the left side depends only on the d... | https://mathoverflow.net/users/13970 | Can the Sobolev norm of order 1/2 detect "jumps"? | You need $s>\frac{d}{2}$ for this to hold. You will find this in any text on Sobolev spaces.
| 3 | https://mathoverflow.net/users/12120 | 59876 | 37,170 |
https://mathoverflow.net/questions/59861 | 16 | I'm currently reading the paper ["Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase"](http://adsabs.harvard.edu/abs/1983PhRvL..51.2167S) by Barry Simon.
Imagine a vector bundle with a connection $\nabla$. For simplicity, we assume that this is a $U(1)$ vector bundle. Parallel transport gives rise to the holo... | https://mathoverflow.net/users/8494 | Why is the integral of the second chern class an integer? | Greg,
I realize that I may as well write an answer rather than a series of comments.
Although Jessica has given a good answer, I'll try to say this as concretely as possible,
since I now think I understand the question more clearly. The question was actually
about the integrality of
$$\frac{1}{4\pi^2}\int\_M F\wedg... | 10 | https://mathoverflow.net/users/4144 | 59881 | 37,171 |
https://mathoverflow.net/questions/59880 | 4 | I have noticed that there are two common choices for the coefficients in defining the Chern-Simons lagrangian:
$
S(A) = \frac{k}{8\pi^2}\int\_M Tr(AdA + 2/3 A^3)
$
and
$
S(A) = \frac{k}{4\pi}\int\_M Tr(AdA + 2/3 A^3).
$
What is going on here, why the two different choices? In both cases, the parameter $k$ is always ... | https://mathoverflow.net/users/13132 | normalization of Chern-Simons lagrangian | In the first case, a gauge transformation changes S(A) by an integer. In the second case, by $2\pi$ times an integer. The second case is a useful normalization for physicists, who care about the behavior of $exp(iS(A))$. The first case is probably a more sensible convention for doing differential geometry, but I don't ... | 6 | https://mathoverflow.net/users/35508 | 59885 | 37,174 |
https://mathoverflow.net/questions/59739 | 13 | Given a Gaussian process on some topological space $T$, with a continuous covariance kernel
$C(\cdot,\cdot)\colon T\times T\to R$, we can associate a Hilbert space, which is the reproducing kernel Hilbert space of real-valued functions on $T$, with $C$ as kernel function. This contruction is given in, for instance, R J... | https://mathoverflow.net/users/6494 | Gaussian processes, sample paths and associated Hilbert space. | The question of continuity of a Gaussian process is a rich one with a lot of theory. Let $T$ be a compact index set, and suppose that $X\_t$ is a mean-zero Gaussian process with covariance function $c(t,s)$. The continuity properties of the process $X\_t$ are entirely determined by the covariance function.
One very ... | 7 | https://mathoverflow.net/users/238 | 59888 | 37,176 |
https://mathoverflow.net/questions/59859 | 7 | I'm trying to compute the coefficients of the following polynomial, where $\omega$ is a primitive $p$-th root of unity, for $p$ prime:
$$a(x) = \prod\_{i=0}^{p-1} f(\omega^ix).$$
It turns out that the $i$-th coefficient is always an integer, and non-zero only when $i$ is a multiple of $p$. So it seems to me like th... | https://mathoverflow.net/users/13965 | Computing a polynomial product over roots of unity | Notice that
$$
y^p-x^p=\prod\_{i=0}^{p-1} (y-\omega^i x)\ .
$$
Therefore
$$
\prod\_{i=0}^{p-1} f(x\omega^i)= Res\_y (y^p-x^p, f(y))\ .
$$
Where $Res\_y$ is the resultant of the polynomials in $y$. The above is just a particular case of the so called Poisson product formula. You can then compute the resultant using for ... | 15 | https://mathoverflow.net/users/7410 | 59890 | 37,178 |
https://mathoverflow.net/questions/59904 | 1 | Let G be a graph with vertices $1,2,...,n$ and $F(x)=\displaystyle{\sum\_{ij\in\rm{E(G)}}x\_ix\_j}$.
Let S be the subset of $\mathbb{R}^n$ given by $x\_i\ge 0$, $\sum x\_i=1$. We're interested in $\displaystyle{\max\_{x\in S}}$ $F(x)$.
Why is any local maximum of F in the interior of S also a global maximum?
--... | https://mathoverflow.net/users/10304 | Any local maximum of F in the interior of a simplex is also a global maximum? | It's because the Hessian quadratic form of $F$ restricted to the orthogonal complement of the vector of all ones is exactly the Laplacian of the graph $G$ (a good basis for the orthogonal complement is the set of vectors having $1$ in the first coordinates, and $-1$ in the $i>1$-st coordinate). The Laplacian matrix is ... | 5 | https://mathoverflow.net/users/11142 | 59907 | 37,186 |
https://mathoverflow.net/questions/59910 | 1 | I have the following question:
Let $K$ be an algebraic number field and $[K:Q]=n$. Let $O\_K$ be a full ring of integers of $K$.
Assume that $O \subset O\_K$ is a subring such that rank of $O$ over $Z$ is $n$ ($=\operatorname{rk}\_{Z}O\_K$).
Let $pO\_K=P\_1^{e\_1} ... P\_m^{e\_m}$ is a decomposition into prime ideals... | https://mathoverflow.net/users/4856 | Number of ideals in primary decomposition | You don't say which ideal in $O$ you are trying to decompose. I will guess you mean to decompose the ideal $pO$. The number of primary ideals in a *minimal* primary ideal decomposition of $pO$ in $O$ *need not* be the same as the number of prime ideal factors of $pO\_K$ in $O\_K$.
Example: Pick your favorite prime nu... | 3 | https://mathoverflow.net/users/3272 | 59913 | 37,187 |
https://mathoverflow.net/questions/59279 | 11 | I'm currently studying some problems about the Whitehead manifold $W$ (the open 3-manifold which is contractible but not homeomorphic to $\mathbb{R}^3$). Does there exists some survey paper on its properties ?
I'm particularly interested in geometric results in the spirit of "Taming 3-manifolds using scalar curvature... | https://mathoverflow.net/users/8887 | Geometry of Whitehead manifolds. | [McMillan proved](http://www.ams.org/mathscinet-getitem?mr=131280) that any contractible 3-manifold is obtained as a union of handlebodies, each of which is homotopically trivial in the next (this generalizes the method of construction of Whitehead).
Later he proved that there are [uncountably many topologically disti... | 15 | https://mathoverflow.net/users/1345 | 59917 | 37,189 |
https://mathoverflow.net/questions/59919 | 0 | The setting is here.
X: 6000x8000 non-sparse matrix
B: 8000x1 sparse vector with only tens of non-zeros
d: positive number
M: is sparsified X'X, i.e. thresholding the elements smaller than d in magnitude to be 0.
Only hundreds of elements are left. So (X' \* X - M) have many small elements and is not sparse.... | https://mathoverflow.net/users/13978 | How to accelerate/avoid multiplication for large matrices in Matlab? | Don't compute the entries that would be multiplied with the zero entries of $B$. That is, take the submatrix $X\_{nz}$ of $X$ consisting of those columns corresponding with the nonzero entries of $B$, and take $B\_{nz}$ to be the concatenation of all nonzero entries of $B$. Then compute the sparsification of $X^T \cdot... | 1 | https://mathoverflow.net/users/13000 | 59923 | 37,191 |
https://mathoverflow.net/questions/59921 | 5 | If we take a sequence of compact hyperbolic Riemann surface with k geodesic boundary components such that the lengths of the geodesic boundary components go to zero, then in the "limit", we should get a surface with k punctures/cusps. Is there a concept of distance ( in Teichmuller theory/ Riemannian geometry ) which w... | https://mathoverflow.net/users/6953 | Is there a concept of Combined Teichmuller space for surfaces with both geodesic boundary and punctures/cusps | If you have a compact hyperbolic surface with geodesic boundary $\Sigma$, then you may double the surface along its boundary to get a closed hyperbolic surface $D\Sigma=\Sigma\cup\_{\partial\Sigma}\Sigma$, which has an involution $\tau:D\Sigma\to D\Sigma$ which exchanges the two sides and fixes $\partial \Sigma$. One m... | 9 | https://mathoverflow.net/users/1345 | 59925 | 37,192 |
https://mathoverflow.net/questions/59915 | 5 | In Shimura's Intro to Arithmetic Theory of Automorphic Forms, he defines a cusp of a Fuchsian group $\Gamma$ as a point $s \in \mathbb{R} \cup \{ \infty \}$ that is fixed by a parabolic element of $\Gamma$.
I know the definition of the width of a cusp in the case $\Gamma = SL\_2(\mathbb{Z})$ or a congruence subgroup ... | https://mathoverflow.net/users/7313 | Cusp width for an arbitraty Fuchsian group | For subgroups $ \Gamma \subset SL\_2(\mathbb Z)$, every cusp is a covering space of the single cusp for the quotient of the upper half plane by $SL\_2(\mathbb Z)$ (that is, the (2,3,infinity orbifold), and the index of the covering is a natural concept that equals the width. There is no such structure in general: in fa... | 11 | https://mathoverflow.net/users/9062 | 59929 | 37,196 |
https://mathoverflow.net/questions/59931 | 2 | For $lim\_{n \rightarrow \infty} \frac{ f(n) \log n } { g(n) } = 0$
we can construct languages in $DTime(g(n))$ but not in $DTime(f(n))$.
We know how to prove $Space(n) \neq NP$. Since $x \Rightarrow x 1^{|x|^2}$ is closed under NP but not Space due to the Space hierarchy theorem.
Question: do we know of any langau... | https://mathoverflow.net/users/3609 | Language in Space(n) but not in NP | Neither part of your question is known. To see this, note that $PSPACE$ contains $SPACE(n)$ (and $NP$), so exhibiting a language in $SPACE(n)$ not in $NP$ would separate $PSPACE$ and $NP$ - this is not known.
$SPACE(n)$ contains $L$ (logspace), and it is open whether $NP=L$ (note that $NP$ contains $L$). Thus, giving... | 10 | https://mathoverflow.net/users/4416 | 59942 | 37,203 |
https://mathoverflow.net/questions/59943 | 4 | This comes from Hörmander's "An Introduction to Complex Analysis in Several Variables".
We defined the $A(\Omega)$-hull (analytic functions in an open set $\Omega$). $\hat{K}$ of a compact set $K\subset\Omega$ by $\hat{K}=\{z;z\in\Omega, |f(z)|\leq\sup\_K |f| \operatorname{for every } f\in A(\Omega) \}$.
The book s... | https://mathoverflow.net/users/13982 | Holomorphically Convex Hull a Subset of the convex hull of | The exponential function grows in module as the exponential of the real part. Therefore, the set of all $z$ such that $|exp(az)|\leq \sup\_K |exp(a\times\cdot)|$ is a half space containing $K$, and meeting $K$. You get all such half spaces, if you vary $a$ in $\mathbb{C}$ or even on the unit circle. Their intersection ... | 5 | https://mathoverflow.net/users/13700 | 59946 | 37,204 |
https://mathoverflow.net/questions/59918 | 10 | Let $\|\cdot\|\_F$ and $\|\cdot\|\_2$ be the Frobenius norm and the spectral norm, respectively.
I'm reading Ji-Guang Sun's paper '[Perturbation Bounds for the Cholesky and QR Factorizations](http://dx.doi.org/10.1007%2FBF01931293)' from BIT 31 in 1991.
While deriving the perturbation bound on Cholesky factorization ... | https://mathoverflow.net/users/11361 | Kind of submultiplicativity of the Frobenius norm: $\|AB\|_F \leq \|A\|_2\|B\|_F$? | This inequality is true. But let me first make a comment. When you say that any matrix norm is submultiplicative ($\|XY\|\le\|X\|\cdot\|Y\|$), you understate that a matrix norm over $M\_n(\mathbb C)$ is subordinated to a norm of $\mathbb C^n$:
$$\|A\|:=\sup\_{x\ne0}\frac{\|Ax\|}{\|x\|}.$$
But the Frobenius norm **is no... | 29 | https://mathoverflow.net/users/8799 | 59950 | 37,207 |
https://mathoverflow.net/questions/59951 | 3 | In the version of central limit theorem for strictly stationary but weakly dependent (for instance $\alpha$-mixing with fast decaying mixing coefficient) random variables $X\_1, X\_2, \cdots$, the [theorem in this Wikipedia page](http://en.wikipedia.org/wiki/Central_limit_theorem#CLT_under_weak_dependence) states (see ... | https://mathoverflow.net/users/9754 | The $\sigma > 0$ condition in the Central Limit Theorem | How is the general case different than this example?
If $\sigma=0$ then the variance of $S\_n/\sqrt{n}$ goes to 0 so $S\_n/\sqrt{n} \to 0$ in distribution.
| 8 | https://mathoverflow.net/users/1061 | 59955 | 37,210 |
https://mathoverflow.net/questions/59900 | 2 | Let $K\hookrightarrow K'$ be a regular extension of fields, and $K[x\_{1},\cdots,x\_{n}]\hookrightarrow K'[x\_{1},\cdots,x\_{n}]$ the corresponding ring extension. Does every prime ideal of the first ring expand to a prime ideal in the second?
| https://mathoverflow.net/users/12940 | Behaviour of Primes under Regular Coefficient Extensions | The answer is Yes.
A field extension $K \to K'$ is regular if and only if for every $K$-algebra $A$ which is an integral domain the $K'$-algebra $A' := A \times\_K K'$ is an integral domain (Bourbaki, Algebra, Chap. V, §17, No.3). Applying this to $A = K[x\_1,\dots,x\_n]/{\mathfrak p}$ for a prime ideal ${\mathfrak p... | 2 | https://mathoverflow.net/users/13302 | 59960 | 37,212 |
https://mathoverflow.net/questions/59972 | 14 | Let $\mathbb{A^+}$ be the set of non-negative [algebraic numbers](http://en.wikipedia.org/wiki/Algebraic_number). Consider the set of "polynomials" : $$\mathbb{P} = \lbrace a\_0 + a\_1x^{r\_1} + a\_2x^{r\_2} + a\_3x^{r\_3} +\cdots + a\_nx^{r\_n}| a\_0, a\_i, r\_i \in \mathbb{A}, r\_i > 0, i= 1,2,\cdots,n\rbrace$$ We ca... | https://mathoverflow.net/users/5627 | Transcendental numbers: yet another classification | $\mathbb P$ is countable. Moreover, any $f\in\mathbb P$ is analytic, hence it has only countably many zeros. Thus $\mathbb A\_E$ is countable, and in particular, extra-transcendental reals exist, and $\mathbb R^+\smallsetminus\mathbb A\_E$ has the power of continuum.
For a concrete example, the [Lindemann–Weierstrass... | 20 | https://mathoverflow.net/users/12705 | 59976 | 37,223 |
https://mathoverflow.net/questions/59938 | 30 | Does anyone have good examples of a space $X$ and a map $f: X \to X$ so that $f\_\*: H\_\*(X) \to H\_\*(X)$ is the identity but (e.g.) $f\_\*: H\_\*(X; \mathbb{F}\_2) \to H\_\*(X; \mathbb{F}\_2)$ is not the identity?
**Edit:** As mentioned in the comments, $f\_\*$ is an isomorphism on the mod-2 homology, but I don't ... | https://mathoverflow.net/users/5010 | Examples for non-naturality of universal coefficients theorem | To expand on my comment, let $M = \mathbf{R}P^2$ be the Moore space with (reduced) homology concentrated in dimension $1$. Let $f:M \to \Sigma M$ be the map
$$ M \to S^2 \to \Sigma M $$
given by collapsing the $1$-skeleton of $M$ and then including the bottom cell into $\Sigma M$. This map induces $0$ on $\tilde H\_\as... | 14 | https://mathoverflow.net/users/6023 | 59986 | 37,230 |
https://mathoverflow.net/questions/59978 | 26 | Does anyone know if there is a classification of the subgroups of the real numbers taken under addition? If not can anyone point me in the directiong of any papers/materials which discuss properties of or interesting facts about these subgroups?
| https://mathoverflow.net/users/13997 | Additive Subgroups of the Reals. | If you would like to classify the subgroups in the sense of Lebesugue measure, you may find the following facts helpful.
(1) Any measurable proper subgroup of the real line is of measure $0$.
(2) Any non-measurable subgroup $G$ of the real line charges fully everywhere, i.e., for any interval $I$, $m^{\ast}(G \cap ... | 22 | https://mathoverflow.net/users/13776 | 59988 | 37,231 |
https://mathoverflow.net/questions/59991 | 5 | Take all the $n\times n$ matrices of 0's and 1's and define an equivalence relation as follows: Two matrices are equal if there is a way to pass from the one to another by alternating the columns and the rows (acting by $S\_n$ on the columns and on the rows).
>
> Is there a good way to determine whether two such ma... | https://mathoverflow.net/users/14726 | Invariants on matrices | This is the [graph isomorphism problem](http://en.wikipedia.org/wiki/Graph_isomorphism_problem).
More precisely, if you have to permute the rows and the columns by the same permutation, then this is graph isomorphism (use $1$ to code "edge present" and $0$ to code "edge absent".) If you are allowed to use different ... | 18 | https://mathoverflow.net/users/297 | 59995 | 37,236 |
https://mathoverflow.net/questions/59996 | 12 | It's fairly classical that for $D>1$ and $(D,l)=1$ one has
$$\sum\_{\stackrel{p\leq x}{p\equiv l\; (mod \; D)}}\frac{\log p}{p} = \frac{\log x}{\phi(D)} + \textrm{O}(1)$$
where if I understand correctly the dependence on $D$ in the $\textrm{O(1)}$ is captured by something like
$$\frac{1}{\phi(D)}\sum\_{\textrm{non-prin... | https://mathoverflow.net/users/6194 | Sum of log p/p for p equivalent to l mod D | You're right that your proposed method for estimating the new sum, while correct, gives a worse error bound than we can obtain otherwise. Rather than quoting the classical result itself, I suggest going back to the proof of that classical result and modifying it to address the new sum.
Somewhat more precisely: the Le... | 16 | https://mathoverflow.net/users/5091 | 60006 | 37,243 |
https://mathoverflow.net/questions/60000 | 3 |
>
> **Possible Duplicate:**
>
> [Deriving Inverse of Hilbert Matrix](https://mathoverflow.net/questions/47561/deriving-inverse-of-hilbert-matrix)
>
>
>
The inverse of the [Hilbert Matrix](http://en.wikipedia.org/wiki/Hilbert_matrix) is made up entirely of integer entries, but I can't seem to
find an elementa... | https://mathoverflow.net/users/13958 | Elementary proof that the Hilbert Matrix is invertible with integer entries | You can look at M.-D. Choi's paper
<http://www.jstor.org/stable/pdfplus/2975779.pdf>
(American Math Monthly, 1983, "Tricks or Treats with the Hilbert Matrix") for this, and much more.
| 9 | https://mathoverflow.net/users/11142 | 60009 | 37,246 |
https://mathoverflow.net/questions/59944 | 2 | Sorry, I misuse the concept of quasi-isometry, I mean almost isometry(also called a Hausdorff approximation).
As we known, isometric Riemann manifolds have the same spectrum of Laplace-Beltrami. And it defined a class of isospectral manifolds which is a highly identical signature of manifold. However, in application,... | https://mathoverflow.net/users/6526 | Relationship between spectrum geometry and almost-isometry | The above comments are still mostly valid with $\epsilon$-isometries:
if $(M,g)$ and $(N,g)$ are Riemannian manifolds with diameters less than $D$, then
$f:M\to N$, $x\mapsto n\_0$ for some $n\_0\in N$ has
$$
|d\_N(f(x),f(x')) - d\_M(x,x')| = d\_M(x,x') \leq D
$$
and for all $y \in N$, because
$$
d\_N(y,n\_0) \... | 2 | https://mathoverflow.net/users/1540 | 60013 | 37,249 |
https://mathoverflow.net/questions/58169 | 12 | I am in a reading group studying Seidel's book (*Fukaya Categories and Picard-Lefschetz Theory*). All of the participants have backgrounds in symplectic topology/pseudoholomorphic curve methods. We are stuck in trying to understand the chapter presenting the algebraic background for Fukaya Categories.
Seidel makes th... | https://mathoverflow.net/users/477 | Why do A_\infty functors form an A_\infty category? | I can explain the pictures I usually draw to think of $A\_\infty$ functors,
but I don't know if they're standard. Anyway, I'll describe what is
just a rubric for ingesting the long formulas, nothing more.
Let's consider first the Yoneda embedding $Y$, which re-thinks an object $L$
in an $A\_\infty$-category $A$ as an... | 7 | https://mathoverflow.net/users/1186 | 60014 | 37,250 |
https://mathoverflow.net/questions/59941 | 1 | Many books on differential geometry develop the geometry in the setting of principal bundles or moving frames. But when it comes the time to do riemannian geometry they leave all that nice machinery and just talk about the Riemann tensor, sectional curvature, Jacobi's equation and the first and second variations of eng... | https://mathoverflow.net/users/13981 | How do I see sectional curvature in the principal bundle (or in Cartan's) approach to riemannian geometry? | Moving frames and differential forms are primarily useful for exact formal pointwise computations involving local differential invariants of a geometric structure (such as a Riemannian metric) and proving theorems that follow from such computations. An example might be the uniqueness of Riemannian metrics with constant... | 5 | https://mathoverflow.net/users/613 | 60016 | 37,252 |
https://mathoverflow.net/questions/58565 | 1 | Let M be a pseudo-Riemannian maniflod and H be the holonomy group of M at the point m. It is possible that M\_m has an H-invariant subspace if M\_m is indecomposable. Does it admit a decomposition M\_m=M\_1+M\_2 which M\_i is H-invariant?
| https://mathoverflow.net/users/13673 | Decomposition which is invariant under the action of holonomy group | Tom Krantz proved that if the holonomy group $H$ of an indecomposable pseudo-Riemannian manifold preserves a non-trivial decomposition $T\_xM=V\_1\oplus V\_2$,
then there exists also an $H$-invariant decomposition $T\_xM=U\_1\oplus U\_2$ into the direct sum of two totally isotropic subspaces, in particular, the manifol... | 0 | https://mathoverflow.net/users/14009 | 60017 | 37,253 |
https://mathoverflow.net/questions/59009 | 2 | I want to prove that there does not exist some complexity class that contains all recursive languages.
Any complexity class C is defined by a complexity measure $\Phi$ (according to Blum axioms) and a total recursive function f:N $\rightarrow$ N. So there does not exists C that contains all languages L for which the... | https://mathoverflow.net/users/13788 | No complexity class contains all recursive languages | Let $\langle M\_i\rangle$ be an enumeration of all Turing Machines (TM) and $\langle f\_i\rangle$ the corresponding ($f\_i=f\_{M\_i}$) enumeration of the functions in $RE$. Suppose there is an $f\in REC$ such that all the recursive languages are in the complexity class $\mathcal{C}(f)=${$f\_i\in RE:\forall x\ \Phi(i,x)... | 3 | https://mathoverflow.net/users/13938 | 60043 | 37,269 |
https://mathoverflow.net/questions/60044 | -1 | In an informal sense, groups are related to symmetry. I was wondering if there are groups that describe some sort of asymmetry. Does anyone know of such groups or is asymmetry and groups a contradiction in terms?
Thanks a lot
| https://mathoverflow.net/users/14022 | groups and asymmetry | Maybe it is really a contradiction: once you have a group acting on some set preserving some structure you have a group homomorphism from your group into the automorphism group of that structure. This is really a very general phenomenon.
However, to give you some example which goes perhaps more into your direction: i... | 1 | https://mathoverflow.net/users/12482 | 60045 | 37,270 |
https://mathoverflow.net/questions/60050 | 4 | Injective Banach spaces, with morphisms as contractive linear maps, have been classically studied (and are $C(K)$ spaces with $K$ Stonian). But what about projectives?
So $P$ will be projective if given Banach spaces $E$ and $F$ and a quotient map (aka metric surjection) $\psi:E\rightarrow F$, given any contractive $... | https://mathoverflow.net/users/406 | Projective Banach spaces | You essentially answered your first question yourself: the ground field is a (contractive) retract of any nonzero Banach space by Hahn-Banach. If there were a non-zero projective Banach space in your sense then the ground field would be projective as well.
On the other hand: It is a theorem due to Köthe and Pełczyńsk... | 3 | https://mathoverflow.net/users/11081 | 60052 | 37,275 |
https://mathoverflow.net/questions/60035 | 20 | Topological K-theory is usually defined by setting $K(X)$ to be the groupification of the monoid $Vect\_\mathbb{C}(X)$ of complex vector bundles over $X$ (with addition given by Whitney sum). However, we can alternatively declare that $[B]\sim [A]+[C]$ whenever $0\rightarrow A \rightarrow B\rightarrow C\rightarrow 0$ i... | https://mathoverflow.net/users/303 | holomorphic K-theory | Grothendieck proved that there is an analytification functor $X \mapsto X^{an}$ from schemes locally of finite type over $\mathbb C$ to the category of (non-reduced!) analytic spaces, which is fully faithful when restricted to *proper* schemes. This induces isomorphisms from $K-$ groups in the algebraic sense on $X$ to... | 28 | https://mathoverflow.net/users/450 | 60053 | 37,276 |
https://mathoverflow.net/questions/60034 | 2 | I was thinking about the statement "if f is continuous on the interval I, there is not necessarily an interval J in I on which f is monotone." and this led me to the question "does there exist a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ that has uncountably infinite turning points?" when I say turning p... | https://mathoverflow.net/users/13927 | Is there a continuous function on $f:\mathbb{R} \rightarrow \mathbb{R}$ with uncountably infinite turning points? | Of course you want to rule out the constant function, so you probably mean that there is a unique highest and lowest point in the neighborhood. Assuming this, with your new definition of turning point, you can choose your neighborhoods to be intervals with rational endpoints. This will force the number of turning point... | 4 | https://mathoverflow.net/users/6269 | 60061 | 37,279 |
https://mathoverflow.net/questions/60039 | 13 | Let $G$ be a reductive group over a number field $k$, with center $Z$. Let $P$ be a parabolic subgroup. Let $H$ be a reductive subgroup of $G$. To what extent can we understand the double coset space $P\_k\backslash G\_k/H\_k$? I'll give some examples, then my motivation for the question, and then I'll refine the quest... | https://mathoverflow.net/users/6753 | Double coset spaces of reductive groups and integral representations of L-functions | This is a very nice question!
All of your examples are spherical quotients. Please take a look at
<http://andromeda.rutgers.edu/~sakellar/rs.pdf> and Yiannis' other papers
for connections between spherical quotients and integral representations for L functions.
| 8 | https://mathoverflow.net/users/10458 | 60077 | 37,284 |
https://mathoverflow.net/questions/60074 | 10 | Not much is known about vector bundles on $\mathbb{P}^2$ but I wonder if the following is a tractable question:
If $E,E'$ are non-isomorphic vector bundles on $\mathbb{P}^2$, then is there always a smooth curve $C \subset \mathbb{P}^2$ such that $E|\_C$ and $E'|\_C$ are still non isomorphic?
A related and perhaps e... | https://mathoverflow.net/users/7 | Can curves differentiate vector bundles on P^2? | Any curve of large enough degree will do. Set $F:= E'\otimes E^{\vee}$; if $d$ is a very large integer, then $\mathrm H^1(F(-d)) = 0$. Take any curve $C$ of degree $d$, and suppose that $E\mid\_C$ and $E'\mid\_C$ are isomorphic; this isomorphism is given by a section of $F\mid\_C$. Since $\mathrm H^1(F(-d)) = 0$, this ... | 21 | https://mathoverflow.net/users/4790 | 60081 | 37,287 |
https://mathoverflow.net/questions/60075 | 55 | It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the [Erdős–Rényi random graph](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model) $G(n,p)$ is asymptotically almost surely connected. The way I know how t... | https://mathoverflow.net/users/4558 | Connectivity of the Erdős–Rényi random graph | A nice question. Here's a strategy that occurs to me, though it could fail miserably.
The basic problem seems to be what you said about variance: the appearances of different spanning trees are far from independent, since it is possible to make local modifications to a spanning tree and get another one. (For example,... | 14 | https://mathoverflow.net/users/1459 | 60091 | 37,291 |
https://mathoverflow.net/questions/60090 | 6 | This is probably something well known (either in the affirmative or in the negative) but I couldn't get this information easily:
Braid group:Symmetric group::?:Signed symmetric group
By "signed symmetric group" I mean the wreath product of the cyclic group of order two by the symmetric group with its usual action a... | https://mathoverflow.net/users/3040 | Braid group analogue for signed symmetric group? | There are braid groups attached to every [Coxeter group](http://en.wikipedia.org/wiki/Coxeter_group) which are obtained by forgetting that the generators in the standard presentation square to the identity but keeping the other relations. I believe the signed symmetric group is the Coxeter group of type $B\_n$.
| 13 | https://mathoverflow.net/users/290 | 60092 | 37,292 |
https://mathoverflow.net/questions/60093 | 1 | We have the theorem that if a partially ordered group is [directed](http://en.wikipedia.org/wiki/Directed_set) and [integrally closed](http://en.wikipedia.org/wiki/Integrally_closed) then it must already be commutative.
Now we have a partial order on real functions, which is: $f$ is finally greater than $g$, in symbo... | https://mathoverflow.net/users/10629 | non-comparable growth in a composition group of functions | What is the proof that the order is archimedean on your group? $f=x^2$, $g=x^2+x$, so $\alpha := f^{-1}\circ g \sim x+1/2+O(x^{-1})$ and more generally $\alpha^n \sim x+n/2+O(x^{-1})$. Thus $\alpha^n \lt\_\infty f$ for all $n$ but $\alpha \gt\_\infty x$.
| 2 | https://mathoverflow.net/users/454 | 60103 | 37,297 |
https://mathoverflow.net/questions/60101 | 13 | Recently I was thinking about some questions concerning $\mathbb{Z}[x]$ and realized that they might be a bit easier if I knew the relative densities of reducible polynomials.
Let $P\_d$ denote the set of all elements of $\mathbb{Z}[x]$ of degree $\leq d$. My basic question is:
>
>
> >
> > **Question:** What fr... | https://mathoverflow.net/users/12301 | Density of Irreducible Polynomials in $\mathbb{Z}[x]$ | I restore the following in clarified form as an over-sized comment; in a temporary (at least I hope it was only temporary) state of confusion I posted it as an answer, which it is not (I was not careful regarding the different notions of ir/reducibility, sorry about that).
The density of integral polynomials of fixed... | 9 | https://mathoverflow.net/users/nan | 60111 | 37,300 |
https://mathoverflow.net/questions/60108 | 142 | I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive arrows is zero seems like a fairly general notion, but I have not come across it in fields like biology, economics, etc. A... | https://mathoverflow.net/users/10930 | Occurrences of (co)homology in other disciplines and/or nature | Robert Ghrist is all about applied topology: Sensor Network, Signal Processing, and Fluid Dynamics. (homepage: <http://www.math.upenn.edu/~ghrist/index.html> ). For instance, we want to use the least number of sensors to cover a certain area, such that when we remove one sensor, a part of that area is undetectable. We ... | 54 | https://mathoverflow.net/users/12310 | 60116 | 37,304 |
https://mathoverflow.net/questions/60113 | 12 | Where can I find the proof of the following fact: If $M$ is a contractible manifold of dimension $n\ge 5$, then the direct product of $M$ and $\mathbb{R}^{n+1}$ is homeomorphic to $R^{2n+1}$ ?
| https://mathoverflow.net/users/nan | contractible manifolds | This was proved in the PL-setting in:
McMillan, D. R.; Zeeman, E. C.
*On contractible open manifolds.*
Proc. Cambridge Philos. Soc. 58 1962 221–224.
From MathReviews:
"An open manifold is defined to mean a non-compact space that is triangulable by a countable complex which is a combinatorial manifold without bou... | 15 | https://mathoverflow.net/users/8176 | 60117 | 37,305 |
https://mathoverflow.net/questions/60107 | 5 | I am reading Deligne: Hodge III, and am puzzled by a certain statement in section 10. If anyone could give a reference or a hint for how to prove this, I would be grateful. Maybe it is obvious and I just don't see why.
We consider an extension $G$ of an abelian variety by a torus. Then Deligne claims that the kernel ... | https://mathoverflow.net/users/349 | Describing the kernel of the exponential map as a homology group | The exponential map realizes Lie(G) as the universal covering space of G, and the kernel is group of covering transformations. Thus the kernel is $\pi\_1(G)$, which, being commutative in this case, equals $H\_1(G,Z)$. (I assume we are over the complex numbers.)
| 16 | https://mathoverflow.net/users/14013 | 60121 | 37,309 |
https://mathoverflow.net/questions/60088 | 2 | Hello everyone.
I am trying to establish a fractional integration lemma of this form.
For $\alpha\geq 0$, and
$1\leq p,q<\infty$ and $0\leq \frac{1}{q}-\frac{1}{p}=\frac{\alpha}{d}$
or $1\leq p,q\leq\infty$ and $0\leq \frac{1}{q}-\frac{1}{p}<\frac{\alpha}{d}$,
for $f$ a function of $t\in\mathbb{R}$ and $x\in\mathb... | https://mathoverflow.net/users/14029 | Fractional integration lemma | It is not clear to me why in your estimate you put a function f depending on t. The norms are only in x, the operators act only on the x variable, so t is just a parameter and if your estimate is true it must be true for a function f independent of t. Also, I think the sign in front of $3/2$ should be a plus.
Anyway,... | 1 | https://mathoverflow.net/users/7294 | 60125 | 37,312 |
https://mathoverflow.net/questions/60126 | 0 | Let $f: M \to N$ be a smooth maps between smooth manifolds. Then $f$ is a submersion (by definition) if its differential is also surjective. Now suppose $f$ is surjective. Is it possible that the surjective map $f$ fails to be a submersion on a set in $N$ of measure non-zero? If so, what is such a map?
Suppose the ma... | https://mathoverflow.net/users/14039 | Do surjections exist which are not submersions on a set of measure non-zero. | [Sard's Theorem](http://en.wikipedia.org/wiki/Sard%27s_theorem)
| 4 | https://mathoverflow.net/users/2788 | 60131 | 37,315 |
https://mathoverflow.net/questions/60156 | 5 | [Edited to include a dense orbit]
>
> Let $X=Spec(A)$ be a normal affine scheme over an algebraically closed field $k$, with an action of a linearly reductive group $G$. Suppose $x\in X$ is a $G$-invariant $k$-point and that $X$ contains a dense open $G$-orbit. Note that this implies that every $G$-orbit contains $... | https://mathoverflow.net/users/1 | If Spec(A) has a G-fixed point and a dense G-orbit, is Spec(A) a cone? | If you by "cone" mean exactly that $A$ should be isomorpic to
$\mathrm{gr}\_{\mathfrak m}A$ it seems that the following is counterexample: Let
$G=\mathbb G\_m$, $A=k[x,y,z]/(x^2+y^3+z^5)$ with $tx=t^{15}x$, $ty=t^{10}y$ and
$tz=t^{6}z$ (exponents chosen more or less at random). Then the tangent cone at
the origin (the ... | 6 | https://mathoverflow.net/users/4008 | 60159 | 37,331 |
https://mathoverflow.net/questions/60160 | 15 | This is not my area of research, but I am curious. Let $G=\left< X|R \right>$ be a finitely presented group, where $X$ and $R$ are finite. There are many questions which are undecidable for all such $G$, for example whether $G$ is trivial or whether a particular word is trivial in $G$. Is there any non-trivial question... | https://mathoverflow.net/users/5034 | Decidability in groups | The problem whether $G$ is perfect, that is $G=[G,G]$ is decidable because you need to abelianize all relations (replace the operation by "+" in every relation) and solve a system of linear equations over $\mathbb Z$. For example, if the defining relations are $xy^{-1}xxy^5x^{-8}=1, x^{-3}y^{-2}xyx^5 = 1 $, then the Ab... | 25 | https://mathoverflow.net/users/nan | 60165 | 37,333 |
https://mathoverflow.net/questions/60062 | 10 | We denote by $\otimes\_{\epsilon}$ the injective Banach tensor product.
Which is the asymptotic volume of the unit ball of the Banach space $\ell\_1^n\otimes\_{\epsilon}\ell\_1^n$?
| https://mathoverflow.net/users/5210 | volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$? | Here is an argument that is certainly overkill and introduces a logarithmic factor which is probably unnecessary.
Let $K$ be the unit ball in $\ell\_1^n \otimes\_\epsilon \ell\_1^n$ and $K^\circ$ the polar body (the unit ball in $\ell\_{\infty}^n \otimes\_\pi \ell\_{\infty}^n$). It is convenient to introduce the volu... | 9 | https://mathoverflow.net/users/908 | 60175 | 37,335 |
https://mathoverflow.net/questions/60179 | 7 | An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so these cones sometimes can be used to distinguish non-isomorphic varieties.
Another interesting class of divisors are the... | https://mathoverflow.net/users/10930 | Cones, monoids, and the space of (very) ample divisors | Regarding your second question, take $A$ and $X$, where $A$ is a general principally polarized Abelian surface and $X$ is a very general surface of degree $\geq 4$ in $\mathbb{P}^3$.
Then the Néron-Severi of both these varieties is cyclic of rank $1$, generated by the class $\Theta$ of a Theta divisor in the former ... | 7 | https://mathoverflow.net/users/7460 | 60183 | 37,341 |
https://mathoverflow.net/questions/60185 | 1 | Hello!
Let $v\_i\in R^d$ $(i=1,...,n,n>d)$ be unit-length vectors ($v\_i^Tv\_i=1$). Then $v\_iv\_i^T$ is an *orthogonal projection matrix*, which has many elegant properties. Now consider a linear combination of these orthogonal matrices
$$A=\sum\_{i=1}^n c\_i v\_i v\_i^T$$
where $c\_i$ are positive scalars and $\sum... | https://mathoverflow.net/users/12734 | Linear combination of orthogonal projection matrices | If you make the change Mikael suggests, you are basically looking at tight frames. $c\_i^{1/2} v\_i$ should be the image under an orthogonal projection of an orthonormal basis from some $n$ dimensional superspace.
| 5 | https://mathoverflow.net/users/2554 | 60190 | 37,343 |
https://mathoverflow.net/questions/60195 | 4 | The answer to the original question is no, see JSE!
Are the $\Gamma(N)$ the only normal congruence subgroups of $\mathrm{PSL}\_2(\mathbb{Z})$ with no finite subgroups (elliptic elements)? What about the normal subgroups of $\Gamma(4)$, which does not contain torsion elements. Here, $\gamma \in \Gamma(N)$, if $\gamma... | https://mathoverflow.net/users/10400 | Are the $\Gamma(N)$ the only normal congruence subgroups of $\mathrm{SL}_2(\mathbb{Z})$? | Almost but not quite. A congruence subgroup has to contain some Gamma(N). So if it is normal its image in SL\_2(Z) / Gamma(N) is a normal subgroup of SL\_2(Z/NZ). That group is almost simple but not quite. A proper normal subgroup need not be trivial; it could be +-1 for instance. Or if N is very small you have a few m... | 12 | https://mathoverflow.net/users/431 | 60196 | 37,346 |
https://mathoverflow.net/questions/60189 | 5 | Although the answer to my question is probably implicit in the answers to the question asked here: [Density of numbers having large prime divisors (formalizing heuristic probability argument)](https://mathoverflow.net/questions/14664/density-of-numbers-having-large-prime-divisors-formalizing-heuristic-probability), I c... | https://mathoverflow.net/users/6698 | Density of numbers not divisible by a large prime power | Your numbers have positive lower density. To see this let $z$ be a positive integer to be fixed later, and denote
$$ c:=\prod\_{p \leq z}(1-1/p). $$
Consider all square-free integers $x < n \leq 2x$ which are composed of primes $z < p \leq \sqrt{x}$. Note that these numbers satisfy the requirements. Their number, by ... | 4 | https://mathoverflow.net/users/11919 | 60198 | 37,347 |
https://mathoverflow.net/questions/60201 | 72 | What are good ways to think about Lagrangian submanifolds?
Why should one care about them?
More generally: same questions about (co)isotropic ones.
Answers from a classical mechanics point of view would be especially welcome.
| https://mathoverflow.net/users/2837 | What is a Lagrangian submanifold intuitively? | Lagrangian submanifolds arise naturally in Hamiltonian Mechanics, because of the classical Arnold-Liouville theorem. Let me state it here:
**Theorem (Arnold-Liouville).** Let $(M, \omega, H)$ be an integrable system of dimension $2n$ with integrals of motion $f\_1=H$, $f\_2, \ldots, f\_n$. Let $c \in \mathbb{R}^n$ be... | 40 | https://mathoverflow.net/users/7460 | 60206 | 37,350 |
https://mathoverflow.net/questions/59738 | 29 | The title quote is from p.221 of the 2010 book,
*The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions*
by Shing-Tung Yau and Steve Nadis. "Nash's theorem" here refers to the Nash embedding theorem
(discussed in an earlier MO question: "[Nash embedding theorem for 2D manifolds](ht... | https://mathoverflow.net/users/6094 | "The complex version of Nash's theorem is not true" | The failure is actually more profound than you might guess at first glance:
There are conformal metrics on the Poincare disk that cannot (even locally) be isometrically induced by embedding in $\mathbb{C}^n$ by any holomorphic mapping. For example, there is no complex curve in $\mathbb{C}^n$ for which the induced me... | 63 | https://mathoverflow.net/users/13972 | 60207 | 37,351 |
https://mathoverflow.net/questions/60203 | 4 | For a given sequence of real numbers $(x\_n)\_{n=1}^\infty \subset [0,1]$, let $A(a,b,N)$ to be the number of terms $x\_n$ of the sequence up to index $N$ such that $a \leq x\_n \leq b$. A sequence of real numbers $(x\_n)\_{n=1}^\infty$ is said to be uniformly distributed modulo 1 if $\displaystyle \frac{A(a,b,N)}{N}$ ... | https://mathoverflow.net/users/10898 | Is a sequence of the following type uniformly distributed modulo 1? | $H\_n$ is a asymptotic to $\log n+\gamma+O(1/n)$. This means that the values of $x\_n$ for $e^k \leq n\leq e^{k+1/2}$ all fall into the same interval of length about $1/2$. The sequence is not equidistributed (the proportion of $x\_n$ in that interval for $n\leq e^k$ and for $n\leq e^{k+1/2}$ differ much).
| 8 | https://mathoverflow.net/users/806 | 60213 | 37,355 |
https://mathoverflow.net/questions/60229 | 1 | My question concerns using Maximum Likelihood to estimate unknown parameters. I will sincerely appreciate if anyone can help me find out if my approach is flawed.
Assume we have a random vector $V$, and we can observe $M$ samples of it denoted by $V\_1,V\_2,\ldots,V\_M$. Define a scalar random variable $X = f(V\mid\t... | https://mathoverflow.net/users/9950 | Question on Maximum Likelihood Estimation | The correct usage is "a sample of size $M$", rather than "$M$ samples".
The function $f$ and the probability distribution of $V$ would completely determine the distribution of $X = f(V\mid\theta)$, and therefore would completely determine $\mu$ and $\sigma$. Hence form some function $h$ we have $h(\theta) = (\mu,\sig... | 1 | https://mathoverflow.net/users/6316 | 60236 | 37,364 |
https://mathoverflow.net/questions/60214 | 10 | Let $X\_0$ be a proper variety over a finite field $k.$ For each prime number $\ell\ne p,$ we have the $\ell$-adic intersection cohomology groups $IH^i(X).$ Due to Gabber, the alternating sum of these
$$
\sum\_{i=0}^{2\dim X\_0}(-1)^i[IH^i(X)],
$$
regarded as a virtual representation of the Weil group $W(\overline{k}/... | https://mathoverflow.net/users/370 | "geometric" interpretation of the alternating sum of intersection cohomology groups | It's a weighted sum over the $k$-rational points, where each point is weighted by the trace of Frobenius on its local intersection cohomology. This follows instantly from the Grothendieck trace formula. I'm not sure there's really a better description of it beyond that in general. Of course, what you can say about the ... | 1 | https://mathoverflow.net/users/66 | 60242 | 37,367 |
https://mathoverflow.net/questions/60228 | 15 | Can anyone suggest a survey (or surveys) that provides an update to *Tilings and patterns* by Grunbaum and Shepard? If there's a more recent book, that would be fantastic, but I don't see one.
I am most interested in the combinatorics of Wang tilings and other square tilings, with the motivation of applying those tec... | https://mathoverflow.net/users/9197 | Tiling survey that updates "Tilings and patterns"? | How is your German?
MR2219468 (2006m:05054) Ardila, Federico; Stanley, Richard P. Pflasterungen. (German) [Tilings] Math. Semesterber. 53 (2006), no. 1, 17–43.
MR2133310 (2006e:52036) Zong, Chuanming What is known about unit cubes. Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 2, 181–211 (electronic).
MR2087242 (... | 10 | https://mathoverflow.net/users/3684 | 60258 | 37,375 |
https://mathoverflow.net/questions/60257 | 5 | Hey all,
I read the meta, and I realize this question might be pretty elementary for this site, but I'm having trouble computing this, and I know it won't take too much insight for someone to give me an approximation.
Say I have a 5x5x5 tic-tac-toe board (noughts and crosses), where each of the 125 spaces on the cu... | https://mathoverflow.net/users/14068 | Approximate search space on a 5x5x5 cube with 3 different possible classes? | The set of boards which have any symmetry is microscopic compared with the total number of boards, so the number of boards up to symmetry is roughly $3^{125}/48 \approx 9.1\times 10^{57}$.
The probability that a random board has a particular $5$ in a row is $2/(3^5)$, and if I count correctly there are $94$ places to... | 3 | https://mathoverflow.net/users/2954 | 60267 | 37,382 |
https://mathoverflow.net/questions/60264 | 6 | Let $A$ be the smallest set of the following functions on the positive reals:
* The identity function is in $A$,
* for a function $f\in A$ also the inverse $f^{-1}$ is in $A$,
* for two functions $f,g\in A$ also the sum $f+g$, the product $f\cdot g$ and the composition $f\circ g$ is in $A$.
In each generation the f... | https://mathoverflow.net/users/10629 | linear order in a group of functions | Yes. Every function $f\in A$ is definable by a first-order formula in the structure $(\mathbb R,+,\cdot,\le)$. It follows that the set $F$ of its fixpoints is also definable. Since the structure is [o-minimal](http://en.wikipedia.org/wiki/O-minimal_structure), $F$ is a finite union of intervals (possibly degenerate). (... | 9 | https://mathoverflow.net/users/12705 | 60276 | 37,386 |
https://mathoverflow.net/questions/60030 | 19 | Let $z\_1,z\_2,\ldots,z\_n$ be i.i.d. random variables in the unit circle. Consider the polynomial
$$
p(z)=\prod\_{i=1}^{n}{(t-z\_i)}=t^n+a\_{1}t^{n-1}+\cdots+a\_{n-2}t^2+a\_{n-1}t+a\_n
$$
where the $a\_i$ are the symmetric functions
$$
a\_{1}=(-1)\sum\_{i=1}^{n}{z\_{i}}\hspace{0.3cm},\quad a\_{2}=(-1)^2\sum\_{1\leq i<... | https://mathoverflow.net/users/13825 | Estimates for Symmetric Functions | OK, here is my argument (sorry for the delay).
First of all, $Z$ is essentially the maximum of the absolute value of the polynomial $P(z)=\prod\_j(1-z\_jz)$ on the unit circumference (up to a factor of $n$, but it is not noticeable on the scale we are talking about).
Second, the maximum of the absolute value of a (... | 11 | https://mathoverflow.net/users/1131 | 60279 | 37,387 |
https://mathoverflow.net/questions/60272 | 1 | Let $F$ be a number field and let $o$ be its ring of integers. Let $o\_p$ resp. $F\_p$ be the completion at a prime ideal $p$ in $o$. Let $B$ be the group of upper triangular matrices in $GL\_2$. Let $\pi $ be a character of $B(F\_p)$.
How can we describe the irreducible which occur in the restriction of the induced ... | https://mathoverflow.net/users/10400 | Spectral decomposition of parabolic induced for GL2(Zp) | This is done in :
Casselman, William The restriction of a representation of ${\rm GL}\_2 (k)$
to ${\rm GL}\_{2}({\mathfrak o})$. Math. Ann. 206 (1973), 311–318.
Also see :
Silberger, A.: $PGL\_2$ over the p-adics. Lecture Notes in Mathematics 166, Berlin-
Heidelberg-New York: Springer 1970
Very roughly speakin... | 4 | https://mathoverflow.net/users/4767 | 60280 | 37,388 |
https://mathoverflow.net/questions/60197 | 5 | Let $G$ be a group (you preferred type: finite, compact, ...):
Mackey has a machinery to classify all irreducible representations of a locally compact group $G$ in terms surjective group homomorphism:
$$ \sigma : G \rightarrow N$$
by some irreducible representations of subgroups of $N$ and some projective representat... | https://mathoverflow.net/users/10400 | Classifying representation through extensions | Re: Q1, until *real* answers appear.
There is a survey article: "Projective representations and the Mackey obstruction - a survey", Contemporary Mathematics, v. 449 (2008), pp. 345-378.
Besides this article, there are a lot of interesting tidbits in [the above-mentioned volume](http://www.ams.org/bookstore-getitem/... | 2 | https://mathoverflow.net/users/6269 | 60283 | 37,390 |
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