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https://mathoverflow.net/questions/50712 | 24 | Given a Riemannian manifold $M$, let $\gamma: (a,b) \to M$ be a geodesic and $E$ a parallel vector field along $\gamma$. Define $\varphi: (a,b) \to M$ by $t \mapsto \exp\_{\gamma(t)}(E(t))$. Is there a "nice" expression for $\varphi'(t)$?
This question originates in an attempt to understand the proof of corollary 1.3... | https://mathoverflow.net/users/4345 | Derivative of Exponential Map | Let $x(u,t) = \exp\_{\gamma(t)}(u E(t))$. For fixed $t$, as $u$ ranges from 0 to 1, the curve $x(\cdot, t)$ is a geodesic segment from $\gamma(t)$ to $\exp\_{\gamma(t)}(E(t))$. Then $\phi'(t) = J(1)$ where $J$ is the Jacobi field along this geodesic segment, with the initial conditions $J(0) = \gamma'(t)$ and $(\nabla\... | 12 | https://mathoverflow.net/users/9301 | 50734 | 31,834 |
https://mathoverflow.net/questions/50739 | 3 | Consider $n$ points equally spaced around the unit circle, joined by all possible combinations of lines to make a complete graph. Let $g(n)$ be the number of triangles formed in the resulting diagram.
For example, $g(3) = 1$, $g(4) = 8$, $g(5) = 35$, $g(6) = 110$.
What is the general formula for $g(n)$?
You can s... | https://mathoverflow.net/users/11727 | The Mystic Rose | <http://oeis.org/A006600>
| 4 | https://mathoverflow.net/users/11853 | 50740 | 31,835 |
https://mathoverflow.net/questions/50581 | 3 | Hello,
assuming Selberg's orthonormality conjecture, let's consider bijective maps $f$ from Selberg's class $\mathcal{S}$ to itself such as:
1) $f$ maps a primitive function of $\mathcal{S}$ to a primitive function of $\mathcal{S}$,
2) $f$ maps a function of degree $d$ to a function of degree $d$,
3) for every ... | https://mathoverflow.net/users/11542 | Automorphisms of the Selberg class | Most elements in $\mathcal{S}$ probably don't have algebraic Dirichlet coefficients (even after scaling), e.g. this is conjectured for the $L$-functions of non-CM Maass forms. So I don't see that the Galois group in question acts on $\mathcal{S}$. I think it is fair to conjecture that the only automorphism of $\mathcal... | 7 | https://mathoverflow.net/users/11919 | 50748 | 31,839 |
https://mathoverflow.net/questions/50762 | 0 | Let $n>0$ be an even integer divisible by $4$
Let $R(t)=r\_0+r\_1t+ \cdots + r\_{n-1}t^{n-1}$
be a polynomial with nonzero integer coefficients in $\{-1,1\}$
such that
$R(\omega)$ is a nonzero integer for all complex $\omega$ $\notin$ $\{-1,1\}$
such that
$$
\omega^n=1
$$
Can we deduce that all these integers $... | https://mathoverflow.net/users/11016 | sign of values of an integer polynomial on roots of unity | $(1+x+x^2+x^3+x^4+x^5+ \ldots + x^{n-1})-2x^{n/2}$.
EDIT: It is customary after someone answers your question not to change the question. As it happens, the slightest possible change to my answer *still* provides an answer to the latest version of your question (at the time of this edit). The little icon produced fo... | 5 | https://mathoverflow.net/users/nan | 50765 | 31,850 |
https://mathoverflow.net/questions/44457 | 19 | Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to find a reference for *non-orientable* surfaces.
For orientable surfaces, the sphere with $g$ handles has homology $\ma... | https://mathoverflow.net/users/2233 | Simple curves on non-orientable surfaces. | The complete answer follows from the result of the paper referenced below (as the math review points out the result was also obtained slightly earlier by McCarthy and Pinkall).
@article {MR2161731,
AUTHOR = {Gadgil, Siddhartha and Pancholi, Dishant},
TITLE = {Homeomorphisms and the homology of non-orientable surfac... | 8 | https://mathoverflow.net/users/11142 | 50781 | 31,861 |
https://mathoverflow.net/questions/50790 | 3 | Suppose given a smooth morphism $f:X\to Y$ between varieties over $\mathbb{C}$ whose fibres are $\mathbb{P}^n$. Then I have an equality of Hodge polynomials
$H(X) = H(Y)H(\mathbb{P}^n)$, say because the hyperplane class generates the cohomology of $\mathbb{P}^n$ and hence $f\_\* \mathbb{Z}\_X$ cannot have monodromy.
... | https://mathoverflow.net/users/4707 | Is every projective space bundle locally trivial in the Zariski topology? | It is not necessarily trivial in the Zariski topology. Consider for instance the plane quadric $\{x^2+sy^2+tz^2\}\subseteq \mathbb P^2\times\mathrm{Spec}\mathbb C[s,s^{-1},t,t^{-1}]$ as a family of $\mathbb P^1$'s over $\mathrm{Spec}\mathbb C[s,s^{-1},t,t^{-1}]$. It is not even isomorphic to $\mathbb P^1$ over the gene... | 14 | https://mathoverflow.net/users/4008 | 50792 | 31,869 |
https://mathoverflow.net/questions/50794 | 6 | Oracle finding all integral points on genus 0 curves is a factoring oracle (e.g. $xy=n$ and $x^2-y^2=n$
I asked [Can the number of solutions $xy(x−y−1)=n$ for x,y,n∈Z be unbounded as n varies?](https://mathoverflow.net/questions/50479/can-the-number-of-solutions-xyx-y-1n-for-x-y-n-in-z-be-unbounded-as-n-var) and occu... | https://mathoverflow.net/users/11847 | Would an oracle for integral points on elliptic curves be a factoring oracle? | An integral point (actually, a rational point in the affine plane will do) on an elliptic curve $y^2 = x(x^2 + ax + b)$ comes (by the standard technique of simple 2-descent) from a rational point on some quartic
$$ N^2 = b\_1M^4 + aM^2e^2 + b\_2e^4, $$
where $b\_1b\_2 = b$. Thus if you want to factor an integer $N$, a... | 8 | https://mathoverflow.net/users/3503 | 50797 | 31,872 |
https://mathoverflow.net/questions/48009 | 4 | I would like to know if the following inequality is satisfied by all probability distributions (or at least some class of probability distributions) for all integer $n \geq 2$.
$\int\_0^{\infty} F(z)^{n-1}(1-\frac{F(z)}{n})\left[zF(z)^{n-2} - \int\_0^z F(t)^{n-2}dt\right]f(z)dz$
$\leq \int\_0^{\infty} F(z)^{n-1}\le... | https://mathoverflow.net/users/11247 | Inequality on probability distributions | The inequality holds for every $n\ge2$ (integer or not) and every probability distribution. Here is a proof. We begin with two easy facts.
**Fact 1:** For every $z\ge0$ and every $k\ge1$,
$$
zF(z)^k-\int\_0^zF(t)^k\mathrm{d}t=k\int\_0^ztF(t)^{k-1}f(t)\mathrm{d}t.
$$
**Fact 2:** For every $t$ and every $k\ge0$,
$$
(... | 8 | https://mathoverflow.net/users/4661 | 50802 | 31,875 |
https://mathoverflow.net/questions/50798 | 19 | What are the pairs $(P,Q)$ of subsets of $\mathbb N$ for which the map
\begin{eqnarray\*}
P\times Q & \rightarrow & {\mathbb N} \\\\
(p,q) & \mapsto & p+q
\end{eqnarray\*}
is a bijection ?
Obvious examples are $P=\mathbb N$ with $Q=\{0\}$, or $P=2\mathbb N$ with $Q=\{0,1\}$. Are there others ?
This question is rela... | https://mathoverflow.net/users/8799 | The sum of integers being a bijection | To comment on Qiaochu's answer, one can show that all such factorizations come from [mixed radix](http://en.wikipedia.org/wiki/Mixed_radix) representations (different bases, factorial base etc.). That is if $$\frac{1}{1-x}=P(x)Q(x)$$ then there must be a sequence $1=a\_0\le a\_1 \le a\_2\le\cdots$ so that $a\_i$ divide... | 25 | https://mathoverflow.net/users/2384 | 50803 | 31,876 |
https://mathoverflow.net/questions/50716 | 7 | A theorem of Reznick states that if $f>0$ is a real homogeneous polynomial in several polynomials that is positive away from the origin of ${\mathbb{R}}^n$, then for large $N$, the form $(\sum x\_i^2)^N f$ is a sum of squares.
Is there a characterization (or at least some work done) of when $(\sum x\_i^2)^N f$ is a s... | https://mathoverflow.net/users/nan | Is this Negativstellensatz with uniform denominators known? | Hi. I don't usually read this site, but a friend who does told me about your question. Let me give a couple of answers. The first is that my proof completely doesn't work if $f$
has non-trivial zeros. The second is that the answer depends on n. Hilbert himself proved in 1893 that if $n=3$ and $f(x,y,z)$ is psd with de... | 16 | https://mathoverflow.net/users/11935 | 50811 | 31,882 |
https://mathoverflow.net/questions/50800 | 2 | Given a set of points $S$ on the Euclidean plane, *Onion Peeling* determines the nested set $H$ of convex hulls on $S$. Define an analytical formula on $S$ which produces a point, not necessarily in $S$, that falls inside the innermost convex hull in $H$. The formula must not use $H$.
| https://mathoverflow.net/users/11934 | Finding points inside innermost convex hull | To proceed further along the lines of Gerry Myerson's comment and Joseph O'Rourke's illustration and remarks, there is no real analytic formula to produce a point inside the innermost core as a function of $n$ points, when $n \ge 4$:
If you have a triangle with 1 point inside, then the only possible solution is that ... | 9 | https://mathoverflow.net/users/9062 | 50816 | 31,886 |
https://mathoverflow.net/questions/50814 | 1 | Dear colleagues,
I want to know if there are some results on the bounds of modified Bessel functions $I\_\alpha(x)$ and $K\_\alpha(x)$? Especially, I need the exponential bounds for them, that is to say if it is possible to get a result like
$|I\_\alpha(x)|\leq C exp(b x), x\geq 0$
where $C$ is a constant or a po... | https://mathoverflow.net/users/11936 | About the exponential bounds for modified Bessel function | There are some bounds of that form in [this paper](http://www.bepress.com/cgi/viewcontent.cgi?article=1064&context=mdandersonbiostat). See also the first reference at the end of the paper.
| 1 | https://mathoverflow.net/users/136 | 50817 | 31,887 |
https://mathoverflow.net/questions/23011 | 6 | I'm guessing that the answer to this question is well-known, but I'm struggling to find anything to help me.
Let $X,Y$ be compact manifolds of dimension $n,m$ respectively. Let $f:X \to Y$ be a smooth map. Then one can consider the graph $\Delta\_f$ of $f$ as a cycle in $X \times Y$.
Firstly what is "known" about ... | https://mathoverflow.net/users/5101 | Graphs of maps between manifolds as cycles and intersection theory | A little late to the party, but here's what I think happens.
I will assume $X$ is closed and oriented, with fundamental class $[X]\in H\_n(X)$. I will also assume we're using field coefficients (or $X$ has torsion-free homology) so there is an isomorphism $H\_\*(X)\otimes H\_\*(X)\cong H\_\*(X\times X)$ given by cros... | 5 | https://mathoverflow.net/users/8103 | 50821 | 31,890 |
https://mathoverflow.net/questions/49737 | 2 | Let $\mathfrak{g}$ be a Kac-Moody Lie algebra (actually, I'd already be happy with an answer addressing the case where $\mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$).
**1st ?**: I'm wondering what results are known about the characters of $L(\lambda)$ (the irreducible module of highest weight $\lambda$) wh... | https://mathoverflow.net/users/8552 | Character formulas for non-integrable modules? | As one special case you may consider the Kac-Wakimoto admissible representations for affine Kac-Moody algebras. These representations are a bit more general than the integrable ones and satisfy modular invariance properties.
Kac-Wakimoto gave a character formula for them in <http://www.pnas.org/content/85/14/4956.sho... | 1 | https://mathoverflow.net/users/9899 | 50829 | 31,896 |
https://mathoverflow.net/questions/43433 | 24 | It is a classical theorem of Freyd that if a small category is complete (has all small limits—in fact, having small products suffices), then it is a preorder (has at most one morphism between any two objects). The proof of this theorem (which can be found [here](http://ncatlab.org/nlab/show/adjoint+functor+theorem) or ... | https://mathoverflow.net/users/49 | Small complete categories in a Grothendieck topos | Hi. I mentioned that I had thought about this on nForum a while back - sorry I didn't get back to you sooner. The following sketch of a proof is mainly due to Colin McLarty.
Two features which distinguish a Grothendieck topos from a more general topos are
1. That it has a geometric morphism to Sets, namely the glob... | 13 | https://mathoverflow.net/users/1106 | 50834 | 31,899 |
https://mathoverflow.net/questions/35925 | 5 | Riemann surfaces provide interesting examples of 1-types - interesting as they have roles in diverse areas. However, apart from 2-dimensional lens spaces, I can't readily bring to mind natural examples of spaces with non-trivial first two homotopy groups (non-trivial firt $k$-invariant optional, I suppose). Given a cro... | https://mathoverflow.net/users/4177 | Natural examples of finite dimensional spaces with interesting 2-type | The 2-type of a 4-manifold is an extremely interesting invariant. In fact, work of Hambleton and Kreck shows that in many cases it determines the homotopy type (if one adds the intersection form as an obvious additional invariant). As a consequence, such 2-types have a very rich structure.
It's quite tricky to figur... | 13 | https://mathoverflow.net/users/4625 | 50835 | 31,900 |
https://mathoverflow.net/questions/50580 | 17 | Let $X$ be a complex manifold with quotient singularities, and let $\tilde X$ be its resolution (that exists, for example, by Hironaka). Then I am pretty sure that $\pi\_1(X)\cong \pi\_1(\tilde X)$.
* Question. Is there a reference for such a statement? At least in dimension 3?
The reason why this should be true is... | https://mathoverflow.net/users/943 | Comparing fundamental groups of a complex orbifolds and their resolutions. | A reference is Theorem 7.8 of the article by Kollar: "Shafarevich maps and plurigenera of algebraic varieties", Invent. Math. 113. This proves the equality of fundamental groups for quotient singualrities in all dimensions and also for algebraic fundamental groups in the case of klt
singularities.
| 7 | https://mathoverflow.net/users/519 | 50853 | 31,914 |
https://mathoverflow.net/questions/50723 | 5 | Let $M^n$ and $N^n$ be two compact complex (or complex projective) manifolds. Let $f: M\to N$
be a holomorphic map of degree one. How to prove that for each $x\in N$ the set
$f^{-1}(x)$ is simply-connected?
**Added.** The above statement would follow from a different one:
There is an $\varepsilon$-neighbourhood $... | https://mathoverflow.net/users/943 | Topology of the preimage of a point for degree one holomorphic maps | [This is just a recap/editing of comments that seem to give an answer]
A suggestion about the second statement : follow the gradient of $\phi=d^2\_x∘f$, where $d\_x$ is the distance to $x\in N$ with respect to a real analytic metric on N, the gradient being taken with respect to another such metric on M.
For this ... | 6 | https://mathoverflow.net/users/6451 | 50862 | 31,919 |
https://mathoverflow.net/questions/50479 | 24 | Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in\mathbb Z$ be unbounded as $n$ varies?
$x,y$ are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of $n$ (assuming $n$ can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.... | https://mathoverflow.net/users/11847 | Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in\mathbb Z$ be unbounded as $n$ varies? | *Although this is not a mathematical answer I will put the results of my brute force search as an aswer as requested by jerr18. I didn't get anywhere with the thinking part.*
Code
----
You can find the (non-optimal) C-code I wrote under [my webpages](http://users.jyu.fi/~tamaraja/temp/solu.c). The biggest limitatio... | 10 | https://mathoverflow.net/users/11716 | 50863 | 31,920 |
https://mathoverflow.net/questions/50604 | 5 | I have been able to find, and understand reasonably well, expressions and derivations for the average wait times for (1) $s$ independent $M/M/1$ queues each with arrival rate $\lambda/s$ and service rate $\mu$, and (2) an $M/M/s$ queue with arrival rate $\lambda$ and service rate $\mu$. (See [this recent discussion](ht... | https://mathoverflow.net/users/10346 | Average wait time for multiple queues where arrivals enter shortest queue | There is also a paper by Vvedenskaya, Dobrushin, and Karpelevich "Queueing System with Selection of the Shortest of Two Queues: An Asymptotic Approach" (Problems of Information Transmission, 1996, 32:1, 15–27; a Russian original is freely available at <http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paper... | 1 | https://mathoverflow.net/users/11854 | 50866 | 31,921 |
https://mathoverflow.net/questions/50880 | 1 | Suppose $E[X]=E[Y]=0$, and $E[X^2]=E[Y^2]=1$. Can you show that $E[X^2Y^2] = 1 + 2\operatorname{cov}(X,Y)^2$? I am not even sure if this expression is correct, I found it in a geostatistics paper, which used this result to show something else. (Note that under these conditions $\operatorname{cov}(X,Y)$ is simply equal ... | https://mathoverflow.net/users/6696 | Calculating $E[X^2Y^2]$ given $E[X^2]$, $E[Y^2]$, $E[X]$, $E[Y]$, and that $X$, $Y$ are Gaussian. | The result holds if, additionnally to the conditions of the post, one assumes that the **vector** $(X,Y)$ is Gaussian. Then, $Y=aX+\sqrt{1-a^2}Z$ with $a=\mathrm{cov}(X,Y)$ and $Z$ standard Gaussian such that $X$ and $Z$ are independent. Using $E(X^4)=3$, $E(Z)=0$, $E(X^2)=E(Z^2)=1$ and the independence of $X$ and $Z$,... | 14 | https://mathoverflow.net/users/4661 | 50885 | 31,934 |
https://mathoverflow.net/questions/50892 | 3 | I have a copy of *Linear Algebra Done Right*, which I worked through years ago in college. I have been using that book to refresh my knowledge, but it does not have an applied or computational aspect to it at all. What would be a good follow-up or companion to *Linear Algebra Done Right* to introduce me to applied line... | https://mathoverflow.net/users/11953 | Applied linear algebra textbook? | "Numerical Linear Algebra" by Trefethen and Bau is IMO the single best book to start learning from. It is lucidly written, concise and relatively inexpensive. Perhaps its main drawback is an unconventional presentation starting from singular value decomposition (SVD) and presenting the other standard transformations as... | 9 | https://mathoverflow.net/users/1674 | 50894 | 31,940 |
https://mathoverflow.net/questions/50888 | 8 | Consider the set of formulas of a logic. If there was only one sort of "unary" deduction $\phi \Rightarrow \psi$ - like $(\forall x)\phi(x) \Rightarrow \phi(a)$ - we would immediately have a *category of formulas* (with deductibility $\Rightarrow$ as morphism). But alas, there are other ("higher") deduction rules, e.g.... | https://mathoverflow.net/users/2672 | Categories of logical formulae | Classical propositional logic is basically a boolean algebra, which may be viewed as a poset, which may be viewed as a category. We at the very least need to fix the primitive predicates; then the objects of the category are the well-formed formulae, and we have a morphism $P \to Q$ if and only if $\{ P \} \vdash Q$, w... | 6 | https://mathoverflow.net/users/11640 | 50898 | 31,943 |
https://mathoverflow.net/questions/50848 | 8 | Let me first recall a known fact. Suppose $X$ is a complex algebraic variety, $\mathscr{L}$ is a line bundle on $X$ and $L^\times$ is the total space of $\mathscr{L}$ with the zero section removed. Write $\pi:L^\times\longrightarrow X$ for the projection. Then $\mathscr{A}=\pi\_\*\mathscr{O}\_{L^\times}$ is naturally a... | https://mathoverflow.net/users/4384 | Gerbes and Z-graded symmetric monoidal categories | This question can be approached abstractly through the general Tannakian formalism,
as laid out e.g. [here](http://arxiv.org/abs/math/0412266), or very concretely by hand. You construct maps in both directions. To a $G\_m$ gerbe assign its tensor category $QC(Gerbe)$ of sheaves (I'll speak of quasicoherent sheaves out ... | 7 | https://mathoverflow.net/users/582 | 50899 | 31,944 |
https://mathoverflow.net/questions/50931 | 3 | If $X$ is an algebraic space of finite type over a finite field $k$, then I think it is true that the set of $k$ rational points of $X$ is finite.
This is of course true for $X$ is a scheme. I wish it is also true for the case when $X$ is an algebraic space.
I guess this is because any rational point of $X$ is "s... | https://mathoverflow.net/users/11964 | Rational points of an algebraic space over finite field | Take the complement of a non-empty open subscheme, and use noetherian induction.
| 6 | https://mathoverflow.net/users/4790 | 50937 | 31,960 |
https://mathoverflow.net/questions/50924 | 2 | [Rényi's Parking Constants](http://mathworld.wolfram.com/RenyisParkingConstants.html) comes up when one puts down unit length cars on a interval, such that the probability of covering any two interval is the same.
Are there any published results when the distribution is non-uniform?
| https://mathoverflow.net/users/6886 | Random parking problem on a probability distribution | Here is one: Jean-François Marckert, *Parking with density*, Random Structures and Algorithms 18 (4), 364-380 (2001). A preprint version is available on [this page](http://www.labri.fr/perso/marckert/papers.html).
| 6 | https://mathoverflow.net/users/4661 | 50949 | 31,968 |
https://mathoverflow.net/questions/50952 | 5 | Sierpinski showed that, on the assumption of CH (in fact, equivalently to it), each point in the plane can be coloured (say) black or white so that every section of the plane parallel to the $x$ axis is "almost" white $-$ in the sense that all but countably many points of it are white $-$ while every section parallel t... | https://mathoverflow.net/users/7458 | Can Sierpinski's anisotropic bicolouring of the plane, assuming the continuum hypothesis (CH), be extended to three dimensions? | The answer to the last sentence in the question is no, for a silly reason, namely that a plane through the origin and a sphere about the origin have an uncountable intersection, in which only countably many points could be white and countably many black. The obvious way to evade that silliness is to replace planes and ... | 7 | https://mathoverflow.net/users/6794 | 50955 | 31,971 |
https://mathoverflow.net/questions/50932 | 6 | Is there a function $f(a,b)$ which maps ordered pairs to lines in a plane in a continuous, bijective manner?
Here is the definition I am using for the limit with lines: a sequence of lines $L(1), L(2), \dots$ is said to approach another line $L$ if, for any point $p$ on $L$, the limit as $n\to\infty$ of the distance... | https://mathoverflow.net/users/9712 | Continuous bijective way of representing a line on a plane | There is no such bijection.
A line in the plane is almost the same as a plane through the origin in 3-space (by intersecting with the plane at height 1), except there's one plane through the origin that doesn't give you a line (the z=0 plane). So the space of lines in the plane is homeomorphic to $\mathbb{RP}^2$ minu... | 11 | https://mathoverflow.net/users/1 | 50974 | 31,983 |
https://mathoverflow.net/questions/50806 | 19 | I have often seen discussions of what actions to take in the context of rare events in terms of expected value. For example, if a lottery has a 1 in 100 million chance of winning, and delivers a positive expected profit, then one "should" buy that lottery ticket. Or, in a an asteroid has a 1 in 1 billion chance of hitt... | https://mathoverflow.net/users/9896 | Expected value as decision criterion in the context of rare events | The [Kelly criterion](http://en.wikipedia.org/wiki/Kelly_criterion) is *the* optimal betting strategy for a player with limited resources (and if one had an infinite amount of capital they probably would not be interested in buying lottery tickets anyway).
A Kelly player bets a fractional amount of their capital $X$... | 10 | https://mathoverflow.net/users/5371 | 50976 | 31,985 |
https://mathoverflow.net/questions/50962 | 10 | Let $K/\mathbf{Q}$ be an imaginary quadratic field, with $\sigma \in \mathrm{Gal}(K/\mathbf{Q})$ a generator. Suppose $\pi$ is a cuspidal automorphic representation of $GL\_2 / K$ with central character $\omega\_{\pi}$. If $\pi\_{\infty}$ has Langlands parameter $z\to \mathrm{diag}(z^{1-k},\overline{z}^{1-k})$ for $k \... | https://mathoverflow.net/users/1464 | Automorphic forms and Galois representations over imaginary quadratic fields: generalizing Taylor's theorem? | Various groups of people have thought about/are thinking about this. The natural source of the desired Galois reps. is a $U(2,2)$ Shimura variety. The problem is that the cohomology of this variety is not so easy to understand. The fundamental lemma certainly plays some role in controlling it, but I don't think that by... | 6 | https://mathoverflow.net/users/2874 | 50988 | 31,996 |
https://mathoverflow.net/questions/50997 | 3 | Let me start with two observations.
* In the classification of quadratic forms with rational coefficients, one has the following statement: a quadratic form in five indeterminates represents $0$ over $\mathbb Q$ if and only if it represents $0$ over $\mathbb R$.
* For every $n\ge2$, there exists a division ring $R\_... | https://mathoverflow.net/users/8799 | $n$-forms representing zero (versus division rings) | Take Terjanian's example of a form of degree 4 in 18 variables over $\mathbb{Q}$ without nontrivial zero over $\mathbb{Q}\_2$ (hence over $\mathbb{Q}$), as explained e.g. in Serre, Cours d'arithmétique, chap. 4. It is easy to see that this form represents zero over $\mathbb{R}$: in fact it is negative at the point (1,0... | 10 | https://mathoverflow.net/users/7666 | 51001 | 32,004 |
https://mathoverflow.net/questions/50143 | 8 | Is there any literature, especially when doing mathematics without the axiom of choice, that discusses using collections of cardinal numbers in place of individual cardinal numbers, when discussing cardinal numbers with certain properties?
That's a little vague, and I will presently give a few motivational examples t... | https://mathoverflow.net/users/8508 | Cardinal numbers vs collections of cardinal numbers | As mentioned, in constructive set theory we generally talk about large *sets* rather than large cardinals.
The rough idea is: a regular set is basically a transitive model of Replacement. An inaccessible set is basically a regular model of Exponentiation: if $a \in X$ and $b \in X$ then ${}^ab \in X$ (absent the law ... | 3 | https://mathoverflow.net/users/6787 | 51005 | 32,005 |
https://mathoverflow.net/questions/51008 | 12 | Begin with a polygon $P\_0$.
Place two points on every edge of the polygon such that they divide each side equally into three parts. Create a new polygon $P\_1$ by connecting all new points with lines.
If we begin with a square and iterate this process, what is the limit as the number of iterations approaches infin... | https://mathoverflow.net/users/11984 | Limit of a sequence of polygons. | **Corrected per Thorny's comment:**
The limit curve does not appear smooth, as is visible in this picture of the first 12 polygons obtained by the process, which become visually indistiguishable at the end:
[alt text http://dl.dropbox.com/u/5390048/chippedsquare.jpg](http://dl.dropbox.com/u/5390048/chippedsquare.jpg)... | 8 | https://mathoverflow.net/users/9062 | 51017 | 32,013 |
https://mathoverflow.net/questions/51003 | 0 | In recent days, I learned a linear algebra problem from one of my friends.
It can be stated as follows.
Given four matrices $A,B,C,D$, find three matrices $E,G,F$, not simultaneously zero, such that the following conditions (1), (2), (3) are satisfied:
$$
\begin{align\*}
(1) &\quad AE=EA, \cr
(2) &\quad BG=GB, \cr
(3... | https://mathoverflow.net/users/11966 | Whether the system of matrix equations is always solvable | Let $E=x I\_n$, $G=y I\_n$, then 1-2 are satisfied and the 3rd is a system of $n^2$ linear homogeneous equations with total number of variables equals to $n^2 + 2$, thus there are simultaneously non-zero solutions. of course, one can do better estimates on the dimension of the solution.
| 9 | https://mathoverflow.net/users/8699 | 51027 | 32,016 |
https://mathoverflow.net/questions/51018 | 7 | Here a solenoid is a dynamical system $(N,f)$ where $N$ is the solid torus $N=\mathbb{D}^2\times S^1$ with boundary $S^1\times S^1$, and $f:N\to N$ is a smooth embedding whose image is wrapped twice in $N$. For example Smale solenoid $f(z,w)=(\frac{1}{4}z+\frac{1}{2}w,w^2)$.
I am wondering if we can glue two solenoid... | https://mathoverflow.net/users/11028 | Glue two solenoids along their boundaries | Yes, there are diffeomorphisms of $S^3$ as you suggest. Here's one slightly more general construction: start from a regular neighborhood of any link in $S^3$ made from two unknotted circles. Two examples are shown below, the first associated with your example is the $(4,2)$-torus link, the second is the Whitehead link.... | 11 | https://mathoverflow.net/users/9062 | 51042 | 32,025 |
https://mathoverflow.net/questions/49836 | 2 | Let be $\Lambda$ a hosrseshoe for a $C^r$ ($r\geq2$) dipheomorphism $f:M\to M$ where $M$ is a two dimensional manifold. A well-known result is the following:
for each $x\in \Lambda$ let be $W^u(x)$ and $W^s(x)$ the unstable and stable manifolds respectively of $f$ in $x$. Then exists neighborhoods $\mathcal{V}\_1$ an... | https://mathoverflow.net/users/nan | Local Product Struture | You can find an alternative proof (to the one of Palis Takens) in section 6.4 of [this survey](http://arxiv.org/PS_cache/arxiv/pdf/0912/0912.2896v1.pdf).
| 0 | https://mathoverflow.net/users/5753 | 51045 | 32,027 |
https://mathoverflow.net/questions/51047 | 4 | By *symplectic manifold* I mean a pair $(M^{2n},\omega)$ consisting of a smooth, connected, even dimensional manifold and a non-degenerate $2$-form. I am interested in compact, boundarlyess examples where $\chi(M)=0$. If none such exist, can anyone provide a simple proof (understandable to a Topologist who knows a litt... | https://mathoverflow.net/users/8103 | Do there exist closed symplectic manifolds with Euler characteristic zero? | Yes. $T^2 \times T^2$ with the sum of the volume forms on each factor.
| 9 | https://mathoverflow.net/users/66 | 51048 | 32,028 |
https://mathoverflow.net/questions/51025 | 11 | Write $L^1$ for the Banach space $L^1([0, 1])$. Given $f \in L^1$, define $f\_1, f\_2 \in L^1$ by
$$
f\_1(x) = f(x/2),
\qquad
f\_2(x) = f((x + 1)/2).
$$
Let $I = \int\_0^1$. Then $I$ is the unique bounded linear functional on $L^1$ satisfying the equations
$$
I(\text{constant function at }1) = 1,
\qquad
I(f) = (I(f\_1)... | https://mathoverflow.net/users/586 | Who first found this characterization of Lebesgue integration? | I don't know if this is of help, but I have seen this idea for defining integration elsewhere, specifically on pages 10-11 of Reed and Simon's Functional Analysis. It would go something like this: let $S$ be the space of step functions obtained as linear combinations of characteristic functions of half-open intervals $... | 4 | https://mathoverflow.net/users/2926 | 51049 | 32,029 |
https://mathoverflow.net/questions/51046 | 7 | Given an algebraically closed field $k$, a smooth group scheme $G$ over $k$
and a principal $G$-bundle $X \rightarrow Y$, which is locally trivial in the fppf topology.
Is this bundle also locally trivial in the etale topology?
Or do we need some extra conditions for this to be true? Literature references about th... | https://mathoverflow.net/users/3233 | Does local triviality in the fppf topology imply local triviality in the etale topology? | The answer is yes. Since smoothness is preserved by flat descent, $X$ is smooth over $Y$. This implies that it has sections locally for the étale topology.
No ground field is needed: $Y$ could be any scheme and $G$ any smooth $Y$-group scheme.
[EDIT] to answer a comment by Keerthi Madapusi Sampath: if you don't ass... | 17 | https://mathoverflow.net/users/7666 | 51051 | 32,031 |
https://mathoverflow.net/questions/51019 | 4 | Recall that chromatic number of $\mathbb{R}^2$ is the least $n$ such that there exists a function $f$ from $\mathbb{R}^2$ into a set of colors ${C\_1,\ldots,C\_n}$ with $f(x)\neq f(y)$ for $||x-y||\_2=1$.
As far as I know, the problem which number this exactly is is still open. I was wondering whether this number is ... | https://mathoverflow.net/users/3118 | Is the chromatic number of the real plane invariant under the norm? | For the $L^\infty$ norm (or equivalently the $L^1$ norm rotating by 45 degrees) the chromatic number is easy to calculate: 4. Just colour translates of $[0,1)\times[0,1)$ by $\mathbb Z^2$ in a $2\times 2$ repeating pattern. It must be at least 4 because ${0,1}\times{0,1}$ are all distance 1 apart.
There's certainly n... | 11 | https://mathoverflow.net/users/11054 | 51052 | 32,032 |
https://mathoverflow.net/questions/51055 | 0 | I'm doing a bit of research for a tech presentation that touches on the subject of [mathematical duality](http://en.wikipedia.org/wiki/Duality_%28mathematics%29). (To be clear, my presentation is not on mathematics or duality, but mentions duality in passing.)
My question for the math folks is, **is it correct to say... | https://mathoverflow.net/users/3596 | Is division considered the mathematical dual of multiplication? | You misunderstand duality. A better example is: planes and lines through the origin (in 3-dimensional space). To each plane you can assign the line perpendicular to it, and vice versa. This is more just a pairing of two kinds of objects, it has interesting properties. For example, a collection of lines are contained in... | 5 | https://mathoverflow.net/users/11919 | 51059 | 32,035 |
https://mathoverflow.net/questions/51054 | 24 | The Bohr–Mollerup theorem characterizes the Gamma function $\Gamma(x)$ as the unique function $f(x)$ on the positive reals such that $f(1)=1$, $f(x+1)=xf(x)$, and $f$ is logarithmically convex, i.e. $\log(f(x))$ is a convex function.
What meaning or insight do we draw from log convexity? There's two obvious but less ... | https://mathoverflow.net/users/6756 | What does log convexity mean? | From the comments, log convexity leads one to conclude that Riemann Hypothesis implies Lindelof hypothesis. The implication of log convexity comes from Hadamard Three Circle Theorem
| 5 | https://mathoverflow.net/users/934 | 51061 | 32,037 |
https://mathoverflow.net/questions/51053 | 16 | I cannot seem to find any results at all on characteristic subgroup growth, even of free groups (and even of $F\_2$). By contrast, the growth function of all subgroups of finite index is well-understood, as is the growth function of all normal subgroups of finite index (it's the same as enumerating finite groups). The ... | https://mathoverflow.net/users/11142 | Counting characteristic subgroups | **4 Jan 2011: Edited to fix discussion of verbal subgroups**
I think $F\_2$ is expected to behave differently from higher free groups. For a finite simple group $G$, I think it's expected (known?) that all epimorphisms $F\_n \rightarrow G$ are equivalent up to $Aut(F\_n)$ when
$n > 2$, but there are many orbits for $... | 16 | https://mathoverflow.net/users/9062 | 51062 | 32,038 |
https://mathoverflow.net/questions/51066 | 10 | Students have asked me few times if I could recommend them a book with solved problems in algebraic topology. Unfortunately, the only one that springs to mind is Terry Lawson's *Topology: A Geometric Approach*, but I'm not quite satisfied with the exercises it contains.
On the other hand, there are lots of good exerc... | https://mathoverflow.net/users/6159 | Literature with solved problems in algebraic topology | Dear Jankir,
1) The [book](http://books.google.fr/books?id=8vu6sCdyAykC&printsec=frontcover&dq=matveev+Algebraic+topology&source=bl&ots=B2MLNt9qjp&sig=1tIFwjY77B5p7TK84ytQjsjeiBg&hl=fr&ei=OWoiTf2IFsql8QPKxfTBBQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CB0Q6AEwAA#v=onepage&q&f=false) by Sergey V. Matveev
*Lectu... | 17 | https://mathoverflow.net/users/450 | 51069 | 32,042 |
https://mathoverflow.net/questions/50992 | 25 | This question is motivated by the current interest of Mathematics and Physics community in Wall Crossing. My questions are :
1. What is wall crossing in Physics, what are the reasons for current interest in it.
2. What is wall crossing in terms of mathematics, what is the reason for interest, is it just physics or so... | https://mathoverflow.net/users/9534 | Wall Crossing in Physics and Mathematics | Very roughly speaking, "wall-crossing" refers to a situation where you construct a would-be "invariant" $\Omega(t)$, that would naively be independent of parameters $t$ but actually depends on them in a piecewise-constant way: so starting from any $t\_0$, $\Omega(t)$ is invariant under small enough deformations, but ju... | 43 | https://mathoverflow.net/users/580 | 51074 | 32,047 |
https://mathoverflow.net/questions/51065 | 4 | The Chern class yields an isomorphism $K\_0(X)\otimes \mathbb Q\cong \bigoplus\_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to $Hom\_{Chow\otimes \mathbb Q}(M(X),M(\mathbb P^n))$ for $n\ge \dim X$. Is there a conceptual explanation for the (compo... | https://mathoverflow.net/users/2191 | Explain the relation between $K_0$ and morphisms of Chow motives | If we consider just the question of constructing the map from $K\_0(X)$ to $Hom\_{Chow\otimes \mathbb Q}(M(X),M(\mathbb P^n)) $, and leave the question of proving that it is an isomorphism to later, I would say that this follows directly from the expression for the Chern classes of a vector bundle via the splitting pri... | 4 | https://mathoverflow.net/users/10849 | 51075 | 32,048 |
https://mathoverflow.net/questions/7090 | 12 | Let $C$ be a category. I'd like to say that a property $P$ of objects of $C$ (or rather isomorphism
classes of objects) is a "Yoneda property" or a "maps-in property" if there is a property $P'$ of
contravariant functors $h:C\to\mathrm{Set}$ such that the functor $\mathrm{Hom}(-,X)$ has $P'$ if and
only if $X$ has $P$.... | https://mathoverflow.net/users/1114 | What should the definition of "Yoneda property" be? | Hi Jim, not sure if this is the sort of thing you are after, but here is one possibility.
Let $K$ be a category and $F=(f\_i:A\_i\to B\_i)\_{i\in I}$ a family of morphisms in $K$. Say that an object $X$ is injective to $F$ if for each $i\in I$ and each $a:A\_i\to X$ there exists a morphism $b:B\_i\to X$ whose restri... | 4 | https://mathoverflow.net/users/10862 | 51076 | 32,049 |
https://mathoverflow.net/questions/51086 | 2 | What is the definition of relative differential forms of a family $\pi: X \to B$ of (nodal) curves, where $B$ is the base space.
| https://mathoverflow.net/users/11901 | What are the relative differential forms of a family of (nodal) curves? | I learnt the following from the paper *On the relative de Rham sequence* ([MathSciNet](http://www.ams.org/mathscinet-getitem?mr=681850) [JSTOR](http://www.jstor.org/stable/2043718)) by Nick Buchdahl. It is more general than the OP situation, but should easily specialise to it.
---
Let $f:X \to B$ be a smooth map ... | 5 | https://mathoverflow.net/users/394 | 51087 | 32,054 |
https://mathoverflow.net/questions/51079 | 2 | If $A$ is an $n\times n$ integer matrix, then trivially $S=A+A^t$ and $P = AA^t$
where $t$ is ``transpose", are both symmetric.
Assume that $A$ is also a "$\lbrace -1,1 \rbrace$" matrix, i.e., the square of each entry in $A$ is equal to $1$.
Is there some rational-coefficient symmetric polynomial $P(x,y)$ (dependin... | https://mathoverflow.net/users/11016 | Symmetric polynomials preserving $-1,1$ matrices | As already mentioned in the comments above, it's not a big deal to find such a polynomial for a particular $\pm1$ matrix. If the question is about a "universal" polynomial (that is, depending only on $n$), then I would expect the answer "no". For $n=2$, take the matrix
$$
A=\left(\begin{matrix} 1 & -1 \cr 1 & 1 \end{ma... | 6 | https://mathoverflow.net/users/4953 | 51102 | 32,061 |
https://mathoverflow.net/questions/51095 | 26 | All rings in this question are integral.
It is known that flat modules are torsion-free. Conversely, torsion-free modules over Prüfer domain (in particular, Dedekind domain) are flat, please see [here](http://en.wikipedia.org/wiki/Pr%C3%BCfer_domain). My questions are:
>
> * Is there a general condition under whi... | https://mathoverflow.net/users/8932 | Flat module and torsion-free module | Dear liu,
1) If $A$ is a domain in which every finitely generated ideal is principal, then a module over $A$ is flat iff it is torsion free (Bourbaki, Comm.Alg.,I,§2, 4, Prop.3). Of course a PID has this property, but the ring $\mathcal O(U)$ of holomorphic functions over a connected open subset $U\subset \mathbb C$ ... | 21 | https://mathoverflow.net/users/450 | 51107 | 32,065 |
https://mathoverflow.net/questions/51090 | 4 | Consider simple diffusion $dX\_t = \sigma dw\_t$ and a parameter $a>0$ and $X\_0=x$. Let us denote $Y\_t = X\_{at}$ - thus we made a change of time. Let us denote an original measure as $P$. How to find measure $Q$ such that the process $Y$ is obtained through the process $X$ not with the change of time but with a chan... | https://mathoverflow.net/users/11768 | Change of time or change of measure | for example, if $\sigma$ is a constant, then $Y$ satisfies $dY\_t = \sqrt{a} \sigma dW\_t$ so that the law $\mathbb{Q}\_Y$ and $\mathbb{Q}\_X$ of the processes $Y$ and $X$ on the Wiener space $C([0;T],\mathbb{R})$ are generaly singular: in other words $\frac{d \mathbb{Q}\_Y}{d \mathbb{Q}\_X} = 0$ if $|a| \neq 1$.
| 6 | https://mathoverflow.net/users/1590 | 51109 | 32,067 |
https://mathoverflow.net/questions/51114 | 3 | Let $A$ be a normal ring with quotient field $K$. Let $L/K$ be a finite separable extension.
Let $E\_1/K$ and $E\_2/K$ be extensions of $K$ contained in
$L$. Let $B\_1$ (resp. $B\_2$) be the normalization of $A$ in $E\_1$ (resp. $E\_2$) and let $C$ be
the normalization of $A$ in $L$. Then
$$K\subset E\_i\subset L\ \mb... | https://mathoverflow.net/users/8680 | Composition and intersection of residue fields | a) Consider the canonical map $B\_1\otimes\_A B\_2\to C$. It is finite ($C$ is finite over $A$) and is surjective on the generic fiber. The spectrum of the lefthand side is a disjoint union of normal schemes because of the étale hypothesis. This implies that the spectrum of $C$ is actually one of these connected compon... | 6 | https://mathoverflow.net/users/3485 | 51117 | 32,071 |
https://mathoverflow.net/questions/51096 | 0 | Let $S$ be a riemann surface.
If S has idea boundary curves,then the intrinsic metric on $S$ can be defined by the restriction to $S$ of poincare metric of the double of $S$.
Also this metric can be derived from the restriction to $S$ of the poincare metric of $S^N$,where $S^N$ is the Nielsen extension of $S$.I don't k... | https://mathoverflow.net/users/11901 | Nielsen extension of Riemann surface | The below is a cut and paste from a paper from Noemi Goldberg, PAMS, 1986. I blame Preview for the typesetting quality (= not knowing about mathjax). However, the original poster could also have googled "nielsen extension".
Let So be a Riemann surface of genus g with n punctures and m holes. Assume that 6<7- 6 + 2n +... | 1 | https://mathoverflow.net/users/11142 | 51126 | 32,076 |
https://mathoverflow.net/questions/51128 | 5 | Euler's proof that the fifth Fermat number is composite begins with the following argument. If $p$ divides $F\_n$, then the order of $2$ in $(\mathbb Z/p\mathbb Z)^\*$ is exactly $2^{n+1}$. Hence $p\equiv1$ mod $2^{n+1}$. If $n\ge2$, this implies that $2$ is a square mod $p$, $2=\omega^2$. Then the order of $\omega$ is... | https://mathoverflow.net/users/8799 | is $2$ a $2^n$-th power mod $p$ ? | There is no $N$ such that $p \equiv 1 \mod N$ implies that $2$ is a fourth power modulo $p$.
**Proof:** Suppose otherwise. Without loss of generality, suppose that $4$ divides $N$.
Let $K$ be the field $\mathbb{Q}(2^{1/4})$. If $p \equiv 1 \mod N$, then by hypothesis $x^4-2$ has a root in $\mathbb{F}\_p$. Moreover,... | 12 | https://mathoverflow.net/users/297 | 51140 | 32,083 |
https://mathoverflow.net/questions/51139 | 3 | Hi,
Let $(f\_n)\_{n \geq 1}$ be a sequence of increasing functions defined on an interval, say $[0,1]$.
Suppose that $\sum\_{n=1}^{\infty}f\_n(x)$ converges for all $x \in [0,1]$. Let $f:=\sum\_{n=1}^{\infty}f\_n$.
It is well known that an increasing function defined on an interval is differentiable almost ever... | https://mathoverflow.net/users/11990 | Differentiation of a series of increasing functions | Yes, see Theorem 4.1 on p. 177 of [this book](http://books.google.co.il/books?id=5ddbKSkaL8EC&pg=PR8&lpg=PR8&dq=%2522differentiating+series+of+monotone+functions%2522&source=bl&ots=_L6toIMGqu&sig=oTTRArrbAIjIKSFBRs6Egb819Ts&hl=iw&ei=IWYjTbK0BYGyhAfOnby3Dg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAYQ6AEwAA#v=onepage... | 5 | https://mathoverflow.net/users/10227 | 51143 | 32,085 |
https://mathoverflow.net/questions/51144 | 3 | It is well-known that a secound-countable topological space is separable. The proof goes like this: Let $(B\_n)$ be a (at most) countable base for the topology. We may assume that $B\_n$ is nonempty for all $n$, and we may choose some $x\_n \in B\_n$. Then $D = \{x\_n\}$ is easily seen to be dense.
**Question** This ... | https://mathoverflow.net/users/2841 | Is "second-countable implies separable" equivalent to the Axiom of countable Choice? | This is form 8L of the axiom of choice at <http://consequences.emich.edu/CONSEQ.HTM>, and is known to be equivalent to countable choice. The proof is fairly straightforward: if $B\_1, B\_2, ...$ is a countable collection of nonempty sets, consider the topological space $X$ consisting of the disjoint union of the $B\_i$... | 11 | https://mathoverflow.net/users/290 | 51145 | 32,086 |
https://mathoverflow.net/questions/51155 | 1 | The complex moduli space of a Calabi-Yau manifold is a complex manifold (BTT). Is it also Kahler ?
| https://mathoverflow.net/users/9534 | is complex moduli space of a Calabi - Yau Kahler | Yes, see
<http://www.scholarpedia.org/article/Calabi-Yau_manifold#Moduli_of_high_dimensional_Calabi-Yau_manifolds>
| 11 | https://mathoverflow.net/users/11142 | 51158 | 32,093 |
https://mathoverflow.net/questions/51169 | 21 | A friend of mine asked me what is the flux of the electric field (or any vector field like
$$
\vec r=(x,y,z)\mapsto \frac{\vec r}{|r|^3}
$$ where $|r|=(x^2+y^2+z^2)^{1/2}$) through a Mobius strip. It seems to me there are no way to compute it in the "standard" way because the strip is not orientable, but if I think ab... | https://mathoverflow.net/users/7952 | Flux through a Mobius strip | As you say, there is no standard definition of the flux through a nonorientable surface. I will try to convince you that you shouldn't want to define this.
There are two standard facts about flux. The first is that, if $V$ is a three dimensional volume, with boundary $\partial V$, and $F$ is a vector field, then $\i... | 21 | https://mathoverflow.net/users/297 | 51173 | 32,100 |
https://mathoverflow.net/questions/51176 | 1 | Suppose $f(t)$ is a continuous-valued, zero-mean stochastic signal with Gaussian autocorrelation (with variance $\sigma^2$). Suppose I then pass this signal through a step function, producing a new $\pm$1 signal $g(t)$ that has value +1 wherever $f(t) \ge 0$, and -1 wherever $f(t) < 0$. This signal is similar to a rand... | https://mathoverflow.net/users/12025 | Autocorrelation of a ±1-valued random process with certain statistics | Assume $(f(t))\_t$ is Gaussian and centered with variance $\sigma^2$ and autocorrelation $E[f(t)f(s)]=c(t,s)$ and let $g(t)=\mathrm{sgn}(f(t))$. Then $(g(t))\_t$ is $\pm1$ and centered with autocorrelation
$$
E[g(t)g(s)]=\frac{2}{\pi}\arcsin\left(\frac{c(t,s)}{\sigma^2}\right).$$
Now I answer the OP's more general q... | 4 | https://mathoverflow.net/users/4661 | 51185 | 32,105 |
https://mathoverflow.net/questions/51195 | 16 | The game of chomp is an example of a game with very simple rules, but no known winning strategy in general.
I copy the rules from [Ivars Peterson's page](https://web.archive.org/web/20110801000000*/http://www.maa.org/mathland/mathtrek_03_24_03.html):
>
> Chomp starts with a rectangular array of counters arranged... | https://mathoverflow.net/users/9248 | Winning strategy at chomp (a chocolate bar game)? | Here are two papers, from 2002 and 2007; not sure if they are new to you.
Jan Draisma and Sander van Rijnswou show in their paper,
"[How to chomp forests, and some other graphs](http://www.math.unibas.ch/%7Edraisma/recreational/graphchomp.pdf)" (2002),
that Chomp can be completely solved when the underlying graph $G$... | 9 | https://mathoverflow.net/users/6094 | 51199 | 32,113 |
https://mathoverflow.net/questions/51190 | 2 | Let $X$ be a homogeneous Markov process in a continuous time with value in the set $E$. Suppose that for some $T>0,x\in E, A\subset E$ we have
$$
P\_x[X\_t\in A] = 0
$$
for all $t\in [0,T]$ but
$$
P\_x[X\_{t'}\in A] >0
$$
for some $t'> T$. I wonder how to find an example of such a process $X$. The simple one
$$
dX\_t ... | https://mathoverflow.net/users/11768 | Counterexample Markov process | This question was already asked [here](https://mathoverflow.net/questions/50337/). Homogenous Markov chains won't work because the probability distributions of $X\_t$ and $X\_s$ are mutually absolutely continuous for every positive $t$ and $s$. For a homogenous diffusion with nonzero diffusion term, $P[X\_{2t}\in A]\ne... | 3 | https://mathoverflow.net/users/4661 | 51200 | 32,114 |
https://mathoverflow.net/questions/51187 | 22 | I'm not a set theorist, but I understand the 'pop' version of set-theoretic forcing: in analogy with algebra, we can take a model of a set theory, and an 'indeterminate' (which is some poset), and add it to the theory and then complete to a model with the desired properties. I understand the category theoretic version ... | https://mathoverflow.net/users/4177 | What is the generic poset used in forcing, really? | The other answers are excellent, but let me augment them by
offering an intuitive explanation of the kind you seem to
seek.
In most forcing arguments, the main idea is to construct a
partial order out of conditions that each consist of a tiny
part of the generic object that we would like to add; each
condition should... | 27 | https://mathoverflow.net/users/1946 | 51202 | 32,115 |
https://mathoverflow.net/questions/51201 | 6 | A version of the Borsuk--Ulam theorem states that a continuous antipodal map from the M-sphere into euclidean N-space has a zero
provided that M is at least N. Clearly the general case follows from the case when M = N. But is the case when M >> N any easier to prove than the equidimensional case?
| https://mathoverflow.net/users/12031 | Borsuk--Ulam question | I don't think so, since any antipodal (non-existent) map $S^n\to S^{n-1}$ would easily be "suspended" to an antipodal map $S^{n+1}\to S^n$. Iterating and composing would then yield antipodal maps $S^m\to S^{n-1}$ with arbitrarily large $m$. The $n=2$ case is somewhat easier though.
| 4 | https://mathoverflow.net/users/6451 | 51206 | 32,116 |
https://mathoverflow.net/questions/43665 | 16 | Is there a computer algebra system that is able to compute the Lie algebra cohomology in a given representation? What if the Lie algebra is finite dimensional?
In my case I would like to be able to compute the cohomology in the following situation:
let $\mathfrak{g}\subset \mathfrak{h}$ be an inclusion of finite dimens... | https://mathoverflow.net/users/4821 | Compute Lie algebra cohomology | I have looked at this question a few years ago (with some more recent sporadic gilmpses), so I am definitely not uptodate. Here it goes, anyway, what I have learned back then:
* GAP.
GAP is a wonderful tool, but I would not call its cohomology computation capabilities efficient. As
far I understand, it implements... | 17 | https://mathoverflow.net/users/1223 | 51207 | 32,117 |
https://mathoverflow.net/questions/51203 | 6 | Here I am referring to Kasparov's KK-theory, a bivariant functor on the category of separable C\* algebras. It is well known that $KK(A, \mathbb{C})$ is K-homology and $KK(\mathbb{C}, B)$ is K-theory, both of which can be calculated for a huge collection of C\* algebras (often by topological methods).
I am wondering... | https://mathoverflow.net/users/4362 | Can anyone calculate KK(A,B) when neither A or B are the complex numbers? | There is Rosenberg-Schochet universal coefficient theorem, which says $KK(A,B)\simeq Ext(K\_{\ast}(A),K\_{\ast+1}(B))\oplus Hom(K\_{\ast}(A),K\_{\ast}(B))$ (not canonically) when $A$ is $KK$-equivalent to a commutative algebra. It was proved in *The Künneth theorem and the universal coefficient theorem for Kasparov’s g... | 8 | https://mathoverflow.net/users/9942 | 51211 | 32,120 |
https://mathoverflow.net/questions/51170 | 5 | Let $X$ be an alphabet, $u,v,p,q,r,s$ be words in the alphabet $X$. I am looking for four elements in the free associative ring $R$ (i.e. four linear combinations of words in $X$) $x,y,z,t$ such that $$u-v=x(p-q)y+z(r-s)t.$$ Is this problem decidable?
The problem is motivated by the need to define an analog of Dehn ... | https://mathoverflow.net/users/nan | An equation in the free associative ring | If I understand your question, the answer to your easier problem is yes. (I'm assuming that by the free associative ring over $X$ you mean to take coefficients of words over the integers.)
First, you can order words in the alphabet, initially by degree, and then by ordering elements in $X$ and using a lexicographical... | 4 | https://mathoverflow.net/users/3199 | 51226 | 32,130 |
https://mathoverflow.net/questions/51225 | 1 | The following theorem is well-known:
Let $A$ be a ring. Let $C$ be a cocomplete $Ab$-category and $X$ an $A$-module-object in $C$, i.e. an object endowed with a ring homomorphism $\sigma : A \to \text{End}(X)$. Then this yields a cocontinuous $Ab$-functor $F : \text{Mod}(A) \to C$ with $F(A)=X$ and the action of $F$ ... | https://mathoverflow.net/users/2841 | Direct construction of cocontinuous functors on Mod(A) | This is too long for a comment only, therefore I write it into an answer (all categories are assumed to be additive). This will not answer your question but maybe it will provide some insight.
Let me first show you how to construct $F$ in the specific situation of $\text{Mod}(A)$ -- this is essentially the proof of t... | 2 | https://mathoverflow.net/users/11081 | 51237 | 32,139 |
https://mathoverflow.net/questions/51239 | 2 | This is sort of a vague (I apologize in advance) question, but I'm interested in the representation theory of the following group
$A \rtimes B$, where
$A = (S\_1)^{m\_1} \times (S\_2)^{m\_2} \times \ldots \times (S\_r)^{m\_r}$,
$B = S\_{m\_1} \times S\_{m\_2} \times ... \times S\_{m\_r}$,
and $B$ acts on $A$ by permu... | https://mathoverflow.net/users/3058 | Representations of semidirect products of symmetric groups | The representations of wreath products of symmetric groups are known: for example, see section 4.3 of "The representation theory of the symmetric group" by James and Kerber.
| 5 | https://mathoverflow.net/users/51 | 51241 | 32,140 |
https://mathoverflow.net/questions/51252 | 9 | In the Deligne-Rapoport paper entitled "Les schemas de modules de courbes elliptiques" the following is written (I translated in english):
Let $E$ be an elliptic curve with $\Gamma(N)$-level structure defined over
$\mathbb{C}((T))$. Let $E'$ be the minimal model of $E$ over $\mathbb{C}[[T]]$.
It may happen that $E'$... | https://mathoverflow.net/users/11765 | Neron models of elliptic curves with level N structure? | Roughly speaking, the N-torsion defined over the base injects into the Neron model (in characteristic 0), so the special fiber of the Neron model needs to have a subgroup isomorphic to Z/NZ x Z/NZ, since by assumption there is full level N structure. The special fiber (since there's bad reduction) has the form C^\* x F... | 10 | https://mathoverflow.net/users/11926 | 51255 | 32,148 |
https://mathoverflow.net/questions/51256 | 7 | Denote by $\pi(x,a,q)$ the number of primes $p\le x$ of the form $p=qk+a$
and $E(x,a,q)=\phi(q)^{-1}\mathrm{Li}(x)-\pi(x,a,q)$.
What is the strongest conjectured bound on $E(x,a,q)$ in terms of $x,q$?
| https://mathoverflow.net/users/9304 | Primes in arithmetic progressions | As Felipe notes, the main term should be li(x)/\phi(q). Replacing this in your definition of $E(x,a,q)$ above we have that, the best that is know even on the GRH is that $E(x,a,q) = O(x^{1/2}\log x)$. This estimate doesn't get better when $q$ is large (compared to $x$) and a lot more is believed to be true. For instanc... | 12 | https://mathoverflow.net/users/630 | 51261 | 32,151 |
https://mathoverflow.net/questions/51264 | 0 | Let $V = \Pi\_{1 \le i < j \le n} (a\_j - a\_i)$ be the [determinant of the Vandermonde matrix](https://planetmath.org/DeterminantOfTheVandermondeMatrix) where $1 = a\_1 < \cdots < a\_n = d$ (with $d >> n$). What is the smallest prime $p$ (or the lower bound) such that $p \nmid V$? Preferably $p < n$.
| https://mathoverflow.net/users/4415 | Smallest prime that does not divide the Vandermonde determinant | Not really clear about what is being asked. If the $a\_i$ are all divisible by the same p (choose one) then this p does divide V. Suppose the $a\_i$ are 1 ... n, then if $p < n$ then then with $a\_i=1$ there is an $a\_j$ with $a\_j-a\_i=p$. If $p \ge n$ then $p \nmid V$. If $p < n$ then in any n numbers there are two w... | 4 | https://mathoverflow.net/users/9310 | 51266 | 32,153 |
https://mathoverflow.net/questions/51163 | 2 | Some time ago, in connection with trying to understand a construction of Amitsur (Embeddings of matrix rings, [Pac JM 36 (1971)](http://projecteuclid.org/euclid.pjm/1102971264), I stumbled across a short paper which considered some variant of the following question:
>
> given (forgetful) functors $H: {\mathcal C} \... | https://mathoverflow.net/users/763 | Reference for factorization of left adjoints? | This should probably be a comment but I don't know how to include links in comments.
Results of this type are often called *adjoint triangle theorems*. There are many such: see John Power's paper [A unified approach to the lifting of adjoints](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1988__29_1/CTGDC_1988__29_1... | 6 | https://mathoverflow.net/users/10862 | 51271 | 32,157 |
https://mathoverflow.net/questions/51259 | 6 | Hi,
Assume you have a cosimplicial group $G$, so that for each $n \ge 0$ there is a group $G\_n$, and you have the usual cofaces and codegeneracies.
>
> Is there a known way to associate to this a collection of homology/homotopy groups in a sensible way?
>
>
>
"Sensible" means at least that it should provi... | https://mathoverflow.net/users/37021 | Are there homology groups for cosimplicial groups? | If we put some (to me) pretty reasonable-looking axioms, then the answer is no.
For a cosimplicial group $G$ let $h\_0G$ be the equalizer of the maps $d\_0,d\_1:G\_0\to G\_1$.
Suppose there is another functor $h\_1$ from cosimplicial groups to groups such that for a short exact sequence $$1\to G\to H\to K\to 1$$ o... | 9 | https://mathoverflow.net/users/6666 | 51272 | 32,158 |
https://mathoverflow.net/questions/51257 | 30 | [von Neumann-Bernays-Gödel set theory](http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory) (NBG) is a conservative extension of ZFC which contains "classes" (such as the class of all sets) as basic objects. "Conservative" means that anything provable in NBG *about sets* can also be p... | https://mathoverflow.net/users/49 | Can ZFC → NBG be iterated? | There is a cheap way of doing this, which may not be the optimal approach when a subtle task (such as the foundational question you have in mind) is the goal. But, then again, this may suffice.
Working in an appropriately strong theory, to simplify, the standard way to check that NBG is conservative over ZFC is to s... | 17 | https://mathoverflow.net/users/6085 | 51273 | 32,159 |
https://mathoverflow.net/questions/51282 | 4 | For a second order PDE (lets say the Laplace equation), is there a problem with specifying neumann boundary conditions, which instead of being specified in the direction normal to the boundary are instead specified in some other direction.
For example, could one specify the derivative in the direction of the boundary... | https://mathoverflow.net/users/12052 | Non-Normal derivative boundary conditions for a PDE | Suppose for definiteness we work with Laplace's equation $\triangledown^{2}u =0$ on the
unit disk in $R^{2}$. Assuming things are somewhat smooth, suppose one specified the
tangential, instead of the normal, derivative of $u$, i.e. specified $\partial{u} / \partial{\theta}$ on the unit circle. Picking any point $\thet... | 3 | https://mathoverflow.net/users/8472 | 51286 | 32,166 |
https://mathoverflow.net/questions/51283 | 0 | We wish to prove that
$$e^{i\pi}+1=0.$$
The standard approach is to use Euler's formula (immediate, for example, from the series definition of the exponential, sine and cosine) and then to use the *facts* that,
$$\sin(\pi)=0\text{, and }\cos(\pi)=-1$$
to draw the necessary conclusion.
*But*, starting with the serie... | https://mathoverflow.net/users/8202 | Rigourous proof of Euler's identity. | Denote by $f(x)$ the point on the unit circle with angle $x$. A simple geometric consideration shows that $f'(x)=if(x)$ which expresses the fact that any tangent line to the unit circle is perpendicular to the corresponding radius. Now $f^{(n)}(x)=i^nf(x)$ by induction, hence $f^{(n)}(0)=i^n$, and Taylor's formula with... | 10 | https://mathoverflow.net/users/11919 | 51287 | 32,167 |
https://mathoverflow.net/questions/51289 | 3 | This is a somewhat speculative question, so bear with that (or not, as is your preference).
Let $X$ be a smooth projective variety, and let $\omega\_X$ be its canonical sheaf. The Euler class of this line bundle $e(\omega\_X)\in H^2(X;\mathbb Z)$ (which for simplicity we'll assume is torsion-free) defines a distingu... | https://mathoverflow.net/users/66 | Can the class of the canonical bundle be recovered from the total space of the cotangent bundle if one forgets that it is a cotangent bundle? | The reduction mod $2$ of $e(X)$ is the second Stiefel-Whitney class of $X$ which by Wu's formula can be recovered from the homotopy type of $X$ (Steenrod operations and the Poincaré duality for the mod $2$ cohomology algebra of $X$). This can be read off from the mod $2$ cohomology of $T^\ast(X)$. Note that this recons... | 10 | https://mathoverflow.net/users/4008 | 51291 | 32,168 |
https://mathoverflow.net/questions/51290 | 1 | I just realize that even though I know what normal bundes are, I dont know how to compute them. The main objective is to
show that a ration curve C on a quintic threefold doesnt move. If C is a line, then its normal bundle in the ambient space is $\mathcal O^{\oplus 3}(1)$. If we know that C is rigid on the quintic thr... | https://mathoverflow.net/users/4275 | Calculating normal bundle | ``Rigid'' can mean either that $C$ does not move (i.e., defines an isolated, but maybe non-reduced, point on the Hilbert scheme of lines on the smooth quintic $Q$) or that it defines an isolated reduced point. Moreover, it is possible for a line $C$ to move on $Q$; e.g., if there is a hyperplane section of $Q$ that is ... | 2 | https://mathoverflow.net/users/8726 | 51296 | 32,171 |
https://mathoverflow.net/questions/51292 | 5 | In the paper [A Riemannian framework for tensor computing](https://doi.org/10.1007/s11263-005-3222-z "Pennec, X., Fillard, P. & Ayache, N. Int J Comput Vision 66, 41–66 (2006). zbMATH review at https://zbmath.org/?q=an:1287.53031"), by Pennec et al., on page 46 the authors state a "distance" function on the manifold of... | https://mathoverflow.net/users/12028 | Proving triangle inequality for an affine invariant distance on $\mathcal{Sym}_n^+$ | Let $X$, $Y$ and $Z$ be positive definite Hermitian matrices (you ask about the real symmetric case, the Hermitian one includes it). Let the eigenvalues of $(X^\*)^{-1/2} Y X^{-1/2}$ be $e^{\alpha\_i}$, those of $(Y^\*)^{-1/2} Z Y^{-1/2}$ be $e^{\beta\_i}$ and those of $(X^\*)^{-1/2} Z X^{-1/2}$ be $e^{\gamma\_i}$. Set... | 6 | https://mathoverflow.net/users/297 | 51303 | 32,176 |
https://mathoverflow.net/questions/51301 | 6 | Let $H$ be a smooth projective hypersurface in $\mathbb{P}^n(\mathbb{C})$ where $n\geq 3$. Then by the Lefschetz hyperplane theorem we have that $H^1(H,\mathbb{C})=
H^1(\mathbb{P}^n(\mathbb{C}),\mathbb{C})=0$. It thus follows that $\pi\_1(H)^{ab}$ is a finite
abelian group.
Q.1 Do we have an example of an $H$ such th... | https://mathoverflow.net/users/11765 | On the fundamental group of hypersurfaces | Non-singular projective hypersurfaces are simply connected. By the Lefschetz theorem $\pi\_k(X)\to\pi\_k(\mathbf{P}^n(\mathbf{C}))$ is an isomorphism for $k\leq n-2$ where $X$ is a nonsingular complex hypersurface: as shown e.g. in Griffiths-Harris (chapter 1, second proof of the Lefschetz hyperplane theorem) if $M$ is... | 18 | https://mathoverflow.net/users/2349 | 51305 | 32,178 |
https://mathoverflow.net/questions/51263 | 44 | If $P(x,y,...,z)$ is a polynomial with integer coefficients then every integer solution of $P=0$ corresponds to a homomorphism from $\mathbb{Z}[x,y,...,z]/(P)$ to $\mathbb{Z}$. So there are infinitely many solutions iff there are infinitely many homomorphisms. If $P$ is homogeneous, we consider solutions up to a scalar... | https://mathoverflow.net/users/nan | Infinitely many solutions of a diophantine equation | **Revised**
Interesting question.
Here's a thought:
You can think of a ring, such as $\mathbb Z$, in terms of its monoid of affine endomorphisms
$x \rightarrow a x + b$. The action of this monoid, together with a choice for 0 and 1, give the structure of the ring. However, the monoid is not finitely generated, si... | 18 | https://mathoverflow.net/users/9062 | 51316 | 32,186 |
https://mathoverflow.net/questions/51166 | 6 | Let $S:\varDelta^{op}\to (cat)$ be a functor where the category on the right is the category whose objects are categories with cofibrations and morphisms are exact functors(from Waldhausen's paper, Algebraic K-Theory of spaces). Waldhausen talks of the geometric realization of such a simplicial category. In the case of... | https://mathoverflow.net/users/11395 | Geometric Realization of a Simplicial Category | As Buehler states in the comments, Waldhausen is taking the nerve degreewise, and then taking the diagonal of the resulting bisimplicial set. This is a model for the homotopy colimit of the simplicial diagram of nerves.
Waldhausen mentions the question of smallness himself in a remark on p. 14 of *Algebraic K-theory ... | 6 | https://mathoverflow.net/users/3049 | 51320 | 32,189 |
https://mathoverflow.net/questions/51317 | 4 | Q1. So what is the "simplest example" of a compact complex manifold of dimenension $2$, say $X$, for which $H\_B^2(X,\mathbb{Z})$ has non-trivial torsion.
Q2. How do we think about these torsion elements? What is the geometrical content behind it?
| https://mathoverflow.net/users/11765 | Torsion in the Betti cohomology of complex surfaces | The torsion of $H\_B^2(X,\mathbb{Z})$ is that of $H\_1(X,\mathbb{Z})=\pi\_1(X)^{ab}$ (universal coefficient theorem for cohomology), so the simplest case should be a simply connected complex surface quotiented by a fixed point free holomorphic involution (or a prime order automorphism).
I would propose an [Enriques s... | 7 | https://mathoverflow.net/users/6451 | 51321 | 32,190 |
https://mathoverflow.net/questions/51103 | 2 | We begin with example. For the Poisson process with an intensity $\lambda\_1$ there is an equivalent change of measure which makes it intensity to $\lambda\_2$.
I would like to find the conditions when is it possible to do with a homogeneous Markov process in a continuous time? It seems to be true for the Markov chain... | https://mathoverflow.net/users/11768 | Change of measure Markov process | To address a question asked by the OP in a comment: yes, the LLN yields almost sure results (no expectations here) which show that there exist disjoint sets of complete trajectories such that each Poisson distribution "sees" only one of them (namely $Q\_\lambda$ "sees" only the set $D\_\lambda$ and $Q\_\lambda(D\_\mu)=... | 3 | https://mathoverflow.net/users/4661 | 51327 | 32,192 |
https://mathoverflow.net/questions/51329 | 2 | During my research this integral has shown up
$ \frac{1}{2T} \int\_{-T}^T \left( 1 - \frac{|\tau|}{T}\right)e^{-\alpha\tau^2}\cos(2\pi f\_0 \tau) d\tau$
I tried to solved by taking the real part of a the complex exponential but it didn't work. Any help?
Cheers,
Mikitov
| https://mathoverflow.net/users/11825 | Integral and limit | The limit is 0, the integral from $0$ to $T$ is:
$$\frac{i \sqrt{\pi } e^{-\frac{\pi ^2 f^2}{a}}
\left(\text{erfi}\left(\frac{\pi f-i a
t}{\sqrt{a}}\right)-\text{erfi}\left(\frac{\pi f+i a
t}{\sqrt{a}}\right)\right)}{4 a^{1/2}}
$$
$$
-\frac{\pi ^{3/2} f e^{-\frac{\pi ^2
f^2}{a}} \text{erfi}\left(\frac{\pi f-i a t}{... | 0 | https://mathoverflow.net/users/11142 | 51334 | 32,197 |
https://mathoverflow.net/questions/51180 | 10 | Every non-singular complete surface is projective. On the other hand, there are non-projective complete surfaces (see e.g. Excercise II.7.13 of Hartshorne) - and there are such examples where the surface is also normal (see e.g. [this](http://reh.math.uni-duesseldorf.de/~schroeer/publications_pdf/on_non_proj.pdf) ). A... | https://mathoverflow.net/users/1508 | Are there non-projective normal surfaces which are rational? | Nagata constructs a normal complete rational surface in the paper [Existence theorems for nonprojective complete algebraic varieties](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-2/issue-4A/Existence-theorems-for-nonprojective-complete-algebraic-varieties/10.1215/ijm/1255454111.full) (see S... | 7 | https://mathoverflow.net/users/3996 | 51341 | 32,200 |
https://mathoverflow.net/questions/51339 | 4 | The following problem optimization problem arose in a project I am working on with a student. I would like to minimize the quantity:
$$ M=\frac{1}{12} + \int\_0^\frac{1}{2} \left( \tfrac{1}{2}-x \right)^2 Q(x)^2 \ dx - 4 \left[ \int\_0^\frac{1}{2} \left( \tfrac{1}{2}-x \right) Q(x) \ dx\right]^2 $$
over all continuou... | https://mathoverflow.net/users/3659 | An optimization problem | I'll change the variable $y=\frac 12-x$ to make typing easier. Since, as Peter already observed, the condition $Q(0)=0$ is worthless and since $\frac 1{12}$ is just an additive constant, we are just to minimize $\int\_0^{1/2}y^2Q^2-4\left(\int\_0^{1/2}yQ\right)^2$ under the condition $\int\_0^{1/2}Q^2=1$. Since everyth... | 9 | https://mathoverflow.net/users/1131 | 51356 | 32,209 |
https://mathoverflow.net/questions/51353 | 4 | So let $G$ be a compact real Lie group. Let $\rho:G\rightarrow GL\_n(\mathbb{C})$ be an
irreducible representation of $G$ and let $\chi\_{\rho}$ be the character associated to
$\rho$. Let $\Lambda\_{\rho}$ be the highest weight of $\rho$ (this of course depends on a choice of a labelling of the roots of $G$) then it se... | https://mathoverflow.net/users/11765 | On the Weyl character formula | The character formula should be viewed here as a purely formal statement about weight multiplicities in the irreducible representation, so the analytic-looking exponential notation for compact Lie groups doesn't really add anything significant to the combinatorics. (The roots and weights actually live in the dual of th... | 16 | https://mathoverflow.net/users/4231 | 51358 | 32,211 |
https://mathoverflow.net/questions/51335 | 1 | I am looking for an example of a stationary, infinite point process on $\mathbb R^n$ with a few simple properties. I would not be surprised to discover that there is a well-studied, canonical process with these features, but I don't know the field very well and have had no success in my search thus far.
The most impo... | https://mathoverflow.net/users/5678 | A point process for modeling location of trees in an infinite forest? | All your requirements are satisfied by the Poisson-Disk process. It's the limit of a uniform sampling process with a minimum-distance rejection criterion. The easiest way to describe it is as the limit of the following process: uniformly sample points in the area of interest, rejecting any points that are less than $r$... | 3 | https://mathoverflow.net/users/7759 | 51360 | 32,213 |
https://mathoverflow.net/questions/19308 | 96 | I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as curvature in the former context while curvature is everywhere in the latter (indeed, it is hard to produce nontrivial re... | https://mathoverflow.net/users/4362 | Is there an analogue of curvature in algebraic geometry? | An algebraic analog of Chern-Weil theory (explicitly taking symmetric polynomials of curvature) is given by the Atiyah class.
Given a vector bundle $E$ on a smooth variety we can consider the short exact sequence
$$ 0\to End(E) \to A(E) \to T\_X\to 0$$
where $T\_X$ is the tangent sheaf and $A(E)$ is the "Atiyah algebr... | 48 | https://mathoverflow.net/users/582 | 51369 | 32,218 |
https://mathoverflow.net/questions/50700 | 11 | Let $X\_0$ be a smooth projective variety over $\mathbb{C}$ and let $\Theta\_{X\_0}$ be the locally free sheaf of $O\_{X\_0}$-module corresponding to tangent space of $X\_0$.
Goal: To find a sufficient condition on $X\_0$ so that it admits a model over $\overline{\mathbb{Q}}$ (the field of algebraic numbers).
By s... | https://mathoverflow.net/users/11765 | Deformation theory over the field of algebraic numbers | I think your question has a positive answer: The Kodaira-Spencer maps at each point $p \in B$ fit together to give a map of sheaves $\Theta\_B \to R^1 f\_\* \Theta\_{X/B}$ and your condition implies that this is map is zero. One may then base change to $C$ and apply Kuranishi's theorem (On the locally complete families... | 4 | https://mathoverflow.net/users/519 | 51382 | 32,223 |
https://mathoverflow.net/questions/51378 | 7 | [Polignac's conjecture](http://en.wikipedia.org/wiki/Polignac%27s_conjecture) (PC) is that there exists infinitely many pairs of consecutive prime numbers that are a distance $d$ apart for some natural number $d$. The twin prime conjecture is the particular instance of this conjecture for $d = 2$. The fact that this co... | https://mathoverflow.net/users/11318 | Implication of Polignac's conjecture on prime distribution in models of PA | For simplicity, take $d=2$. Then the the existence of a model of PA with at least one non-standard pair of twin primes is equivalent to the assertion
For all (fixed, standard) primes $p$ the sentence $$\phi\_p:\forall x>p,\,\,\, x \textrm{ is not prime or } x+2 \textrm{ is not prime }$$
is not provable in PA.
Is i... | 4 | https://mathoverflow.net/users/5229 | 51409 | 32,241 |
https://mathoverflow.net/questions/51400 | 0 | Hi,
i'm reading the proof of the fact that $C^{\infty}(M,N)$ is dense in the sobolev space $W^{1,m}(M,N)$, where $M,N$ are compact riemannian manifolds of dimension respectively $m,n$.
I recall quickly the definition of $W^{1,m}(M,N)$. Embedding $N$ isometrically in a $\mathbb{R}^J$ we define
$$W^{1,m}(M,N)= \left\{ f... | https://mathoverflow.net/users/4971 | question on the proof of density of $C^{\infty}(M,N)$ in the sobolev space $W^{1,m}(M,N)$ | I don't have the time to work out a real answer to this question but estimates like this can usually be proved using the following:
$ F(y) - F\_{\epsilon}(x) = \int \phi\_\epsilon(x,z)(F(y) - F(z)) dz $
where $\phi\_\epsilon$ is the mollifier kernel function and then substituting in
$ F(y) - F(z) = \int\_0^1 \gam... | 1 | https://mathoverflow.net/users/613 | 51410 | 32,242 |
https://mathoverflow.net/questions/51404 | 1 | 1. It is well known that the elementary theory $Q\_{fin}$ of finite quasiorders is undecidable. To be more precise, it is undecidable whether a first-order sentence built using a binary relational symbol $R$ is valid in all finite structures where the interpretation of $R$ is reflexive and transitive. This was proved i... | https://mathoverflow.net/users/12082 | Elementary Theory of Finite Linear QuasiOrders | A finite linear quasi-order is just a partition of $1,...,n$ into intervals? That elementary theory should be decidable. In general, the elementary theory of one equivalence relation on a finite set is decidable.
**Update:** A partition of $\{1,...,n\}$ into intervals is equivalent to the pair $(n,A)$ where $A$ is ... | 3 | https://mathoverflow.net/users/nan | 51412 | 32,244 |
https://mathoverflow.net/questions/51415 | 12 | Let X be an infinite set.
Is it possible to show the existence of a countably infinite subset of X without using the Axiom of Choice?
| https://mathoverflow.net/users/7303 | Is it possible to show that an infinite set has a countable (infinite) subset, without using the Axiom of Choice? | Short answer: No.
By countably infinite subset you mean, I guess, that there is a 1-1 map from the natural numbers into the set.
If ZF is consistent, then it is consistent to have an amorphous set, i.e., an infinite set whose subsets are all finite or have a finite complement. If you have an embedding of the natur... | 20 | https://mathoverflow.net/users/7743 | 51416 | 32,247 |
https://mathoverflow.net/questions/51422 | 5 | Let $p: E\to B$ be a covering map of $C^\infty$ manifolds, where $E$ has a complex structure. There are many cases when we want to know whether $B$ has a complex structure (which is obviously unique) making $p$ an analytic map, for example in the construction of families of elliptic curves.
The difficulty is that giv... | https://mathoverflow.net/users/6950 | Pushing Complex Structure Forward | For 1): take a double covering $E\to B$, where $E$ and $B$ are compact oriented surface of genus 3 and 2 respectively, and give $E$ a structure of Riemann surface with trivial automorphism group.
About 2): well, in the example above you see that you can deform a complex structure with no compatible complex structure ... | 13 | https://mathoverflow.net/users/4790 | 51430 | 32,254 |
https://mathoverflow.net/questions/51445 | 1 | Let $x$ be a real number and $N$ a positive integer. Define
$E(N,\delta) = \{(p,q) \in \mathbb{Z}^2: |p - q x| \leq \frac{\delta}{N}, |p|, |q| \leq N \}$,
i.e., the set of solutions to rational approximation of $x$ with accuracy $\frac{\delta}{N}$.
I am interested in the behavior of the cardinality of $E(N,\delta... | https://mathoverflow.net/users/3736 | number of solutions of diophantine approximation | It's usually written $p-qx$. If $\delta\lt1/2$, then $|(p/q)-x|\le(1/2)q^{-2}$, which, if I remember right, implies, by a theorem of Hurwitz, that $p/q$ is a convergent to the continued fraction of $x$. It should not be hard to show that the number of convergents to $x$ with denominator not exceeding $N$ is little-oh o... | 1 | https://mathoverflow.net/users/3684 | 51450 | 32,266 |
https://mathoverflow.net/questions/51431 | 2 | Given the Poincare Disc $D$ and its ideal boundary $S^{1}$, I want to construct a homeomorphism between $S^{1}$ and the gromov boundary of $D$, $\partial D$, of equivalence classes of geodesics given some base point p.
I can construct a bijection between $S^{1}$ and $\partial D$ by identifying equivalence classes of ... | https://mathoverflow.net/users/12086 | Homeomorphism between the boundary of the Poincare disc S1 and its Gromov Boundary | Let me first reformulate the question once again. Given a basepoint $p$ on the hyperbolic plane and two boundary points $x,y$, there are two measures of $x$ and $y$ being close as seen from $p$: the angle $\alpha$ between the geodesic rays issued from $p$ in the direction of $x$ and $y$, respectively, and the Gromov pr... | 4 | https://mathoverflow.net/users/8588 | 51453 | 32,268 |
https://mathoverflow.net/questions/51435 | 5 | Does Corollary 3.4.4 in Raynaud's paper ``Schemas en Groupes de Type (p, ..., p)'' apply also to the case where G is quasi-finite? If not, what is the more general statement?
The corollary states:
``Soit G un K-schéma en groupes fini, commutatif, annulé par une puissance de p et qui se prolonge en un R-schéma en g... | https://mathoverflow.net/users/12087 | Generalization of Raynaud's (p, p, ... p) result | As James Borger writes in his comments above, the answer is surely *no*. The restrictions on the characters appearing in the Galois representation attached to $G$ are being forced by the fact that $G$ has a finite flat model. If you throw that assumption away, you lose any control on the ramification of the Galois acti... | 8 | https://mathoverflow.net/users/2874 | 51457 | 32,270 |
https://mathoverflow.net/questions/51458 | 1 | Recently, I've been looking into *motifs* in networks (directed graphs) -- small connected induced subgraphs that appear significantly more frequently than in a "similar random graph".
In practice, we need to enumerate the induced subgraphs (up to isomorphism) in thousands of large random networks, which can take a l... | https://mathoverflow.net/users/2264 | Is there a canonical labelling package optimised for small graphs? | Nauty really is optimized for small graphs, if you use the C library interface. McKay (and others, such as myself) frequently use this library to generate all graphs of a given small order, which requires canonical labeling of small graphs very quickly. This happens many many many times (see <http://oeis.org/A000088> f... | 3 | https://mathoverflow.net/users/4167 | 51460 | 32,271 |
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