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https://mathoverflow.net/questions/38004 | 3 | Is there a classification theorem for "finitely generated abelian groups with a distinguished element"? If it helps, you can restrict this to the cases where the order of each element divides the order of the distinguished element.
My idea would be to try to use this for a classification theorem for rings with a fini... | https://mathoverflow.net/users/nan | slight extension to classification of finitely generated abelian groups | You say that we may assume that the order of every element divides the order of the distinguished element. That's the easy case. If the distinguished element generates a finite cyclic group $C$ of order $n$ and every element has order dividing $n$, then in fact $C$ is a direct summand. ("For the ring $\mathbb Z/n$ the ... | 3 | https://mathoverflow.net/users/6666 | 38012 | 24,427 |
https://mathoverflow.net/questions/38007 | 7 | I've been working through the heat-equation proof of the Atiyah-Singer index theorem. My question is what is the motivation for the definition of the index of an operator? I know there is the isomorphism between the homotopy classes of maps from a compact topological space to the space of Fredholm operators and the fir... | https://mathoverflow.net/users/1132 | Index of an Operator | Usually one really wants to know the dimension of the kernel of an operator rather than its index. The problem is that the dimension of the kernel is not a continuous function of the operator, so is very hard to compute in terms of topological data. The key point about the index is that it is a continuous function of t... | 16 | https://mathoverflow.net/users/51 | 38017 | 24,431 |
https://mathoverflow.net/questions/37849 | 5 | Is the (algebraic) span a finite set of vectors in a Hausdorff topological vector space over a complete field always closed?
I suspect yes, but I can't come up with a proof, and it seems like locally convex might be needed to get this.
| https://mathoverflow.net/users/nan | Closedness of finite-dimensional subspaces | This holds indeed for complete fields: see Theorem 2, Section I.2.3, of Bourbaki's "Espaces Vectoriels Topologiques".
Here is the argument.
Let $K$ be a (not necessarily commutative) field equipped with a complete nontrivial [absolute value](http://en.wikipedia.org/wiki/Absolute_value#Fields) $x\mapsto|x|$, let $n$... | 7 | https://mathoverflow.net/users/461 | 38030 | 24,439 |
https://mathoverflow.net/questions/37946 | 3 | Let $X\_t\in\mathbb{R}$ be an Ito diffusion process given by $$ dX\_t=a(b-X\_t)dt+\sigma dW\_t$$, then the characteristic operator of $X\_t$ is given by $$L=a(b-x)\frac{\partial}{\partial x}+\frac{\sigma^2}{2}\frac{\partial^2}{\partial x^2}$$
(more details about the characteristic operator can be found here <http://e... | https://mathoverflow.net/users/9059 | Characteristic operator | The wikipedia page cited in the question provides most of the answer: to get your operator compute
\begin{equation}
\lim\_{\delta \rightarrow 0} \frac{ {\mathbb E}[f(X\_\delta)] -f(x)}{\delta}
\end{equation}
The difference between your problem and the case covered in the wikipedia article is that $f$ in the above displ... | 2 | https://mathoverflow.net/users/3370 | 38041 | 24,446 |
https://mathoverflow.net/questions/38043 | 1 | Given the definition of subsets and equality of sets:
* A $\subset$ B, if x $\epsilon$ A $\rightarrow$ x $\epsilon$ B for every set x.
* A = B, if A $\subset$ B and B $\subset$ A
Why is it impossible to decide whether two circular sets I = {I} and J = {J} are equal.
I mean, the way is see it is that I is not an e... | https://mathoverflow.net/users/9081 | Equality of two circular sets | In ZF minus the axiom of foundation there is no way of proving
that all "circular" sets are equal. You could take a model of set theory
allowing ur-elements and replace some or all of these by circular sets.
If you accept the popular alternative to Foundation, Peter Aczel's
[Anti-Foundation axiom](http://en.wikipedia... | 5 | https://mathoverflow.net/users/4213 | 38045 | 24,448 |
https://mathoverflow.net/questions/37812 | 4 | I have been studying of late about formation of naked singularities in certain collapse scenarios in Einstein's theory. It seems to me that the canonical paper to read about how such a formation is established is the 1984 paper by Christodoulou in Communications in Mathematical Physics. ( <http://www.ams.org/mathscinet... | https://mathoverflow.net/users/2678 | Christodoulou's paper on naked singularities in inhomogeneous dust collapse | Your question is very broad, and so I'll just give some very broad answers too.
**Collapse scenarios** Are you absolutely sure you want to restrict yourself to dust collapse? In the case of a spherically symmetric scalar field, there is also <http://www.ams.org/mathscinet-getitem?mr=1307898> (and [this paper](http:/... | 8 | https://mathoverflow.net/users/3948 | 38055 | 24,453 |
https://mathoverflow.net/questions/38049 | 37 | I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [*and probably his own website, but I have a hard time seeing exactly what he's claiming*] Louis de Branges has claimed, numerous times, to have proved the Riemann Hypothesis; but clearly few people believe him. ... | https://mathoverflow.net/users/6651 | What, exactly, has Louis de Branges proved about the Riemann Hypothesis? | The paper by Conrey and Li "A note on some positivity conditions related to zeta and L-functions"
<https://arxiv.org/abs/math/9812166>
discusses some of the problems with de Branges's argument. They describe a (correct) theorem about entire functions due to de Branges, which has a corollary that certain positivity cond... | 49 | https://mathoverflow.net/users/51 | 38057 | 24,454 |
https://mathoverflow.net/questions/38037 | 3 | This context of this question is Rutten's [Universal Coalgebra](http://en.wikipedia.org/wiki/F-coalgebra), used for modelling systems. I'm interested in finding a description of a functor between different types of coalgebras corresponding to finding a certain subcoalgebra.
An $F+1$-coalgebra $\langle S,\alpha:S\to F... | https://mathoverflow.net/users/2620 | Maximal subcoalgebras of an $F+1$-coalgebra corresponding to an $F$-coalgebra | I think the construction you're looking for can be seen as a right adjoint, and hence the details of the construction can be seen as coming from general transfinite constructions of adjoints.
$\newcommand{\inl}{\mathrm{inl}} \newcommand{\Coalg}{\mathbf{Coalg}}$
There's a functor $\inl^\* : F$-$\Coalg \longrightarrow ... | 2 | https://mathoverflow.net/users/2273 | 38060 | 24,455 |
https://mathoverflow.net/questions/38047 | 13 | The ordinary notions of limit and colimit are universal solutions to a problem, specifically, finding terminal/initial objects in slice/coslice categories. In the context of homotopy right Kan extensions (it's not hard to show that the theory of homotopy limits reduces to this case (the same holds for left Kan extensio... | https://mathoverflow.net/users/1353 | What is the "universal problem" that motivates the definition of homotopy limits/colimits (and more generally "derived" functors)? | There is a notion of homotopical Kan extension defined for "homotopical categories" (cats with a class of weak equivalences satisfying the 2-out-of-6 property). I cannot go into much detail without making a bazillion definitions, but the reference is Dwyer, Kan, Hirschorn, and Smith's "Homotopy Limit Functors on Model... | 7 | https://mathoverflow.net/users/2468 | 38062 | 24,457 |
https://mathoverflow.net/questions/38059 | 9 | Is there a way of talking about continuity and smoothness for set valued functions? More precisely, consider $M$ and $N$ topological/smooth manifolds, and let $f$ a function that associates to each point $p\in M$ a subset $f(p) \subset N$ (I haven't made any assumptions on what target sets are allowed, but feel free to... | https://mathoverflow.net/users/3948 | Notion of smoothness for set-valued functions | My idea is that if we want to compare $f(p)$ and $f(q)$ for nearby points $p$ and $q$, then we need to be able to put $f(p)$ and $f(q)$ into the same space. To do this, I'm going to assume that $M$ is a finite-dimensional Riemannian manifold, so that we can make use of a connection $\nabla$ on $M$.
For all $p$, let $... | 4 | https://mathoverflow.net/users/238 | 38072 | 24,462 |
https://mathoverflow.net/questions/38064 | 7 | This question is somewhat related to [Differential inclusions for distributions](https://mathoverflow.net/questions/37524/differential-inclusions-for-distributions) but I am asking for something rather more specific, so I hope it is alright to leave this as a separate, new question.
Let $M$ be a smooth manifold, the... | https://mathoverflow.net/users/3948 | Hypersurfaces orthogonal to a cone | There are a global obstrutions in some cases when local solutions exist, even in the case that the cone is an open subset of the cotangent bundle. For example: let $M^3$ be the tangent line bundle to any Riemannian surface $N^2$. The Levi-Civita connection defines a 2-plane field, where curves tangent to these planes r... | 11 | https://mathoverflow.net/users/9062 | 38078 | 24,466 |
https://mathoverflow.net/questions/37127 | 2 | I've found (as have others), that for some analytic functions, a Padé approximant of it has an infinite convergence radius, whereas its associated Taylor series has a finite convergence radius. $f(x)=\sqrt{1+x^2}$ appears to be one such function. My questions are:
1) Is there any function where the Taylor series has ... | https://mathoverflow.net/users/8864 | Can Convergence Radii of Padé Approximants Always Be Made Infinite? | Don't be lured into thinking Pade approximates are 'nice'. Here is why:
**Notation:**
Let $[L/M]\_f(z)$ be the Pade approximation $\frac{p\_n(z)}{q\_m(z)}$ to $f$ where $\text{deg}(p\_n)\leq n$ and $\text{deg}(q\_m)\leq m$, $q\_m(0)=1$.
**Parital Answer:**
1) The partial theta function $h\_q(z)$ with $q=e^{iz}... | 4 | https://mathoverflow.net/users/2011 | 38087 | 24,469 |
https://mathoverflow.net/questions/38085 | 12 | Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and cocontinuous (in particular, exact). Since $R\text{-Mod}$ is both complete and cocomplete, $\operatorname{Forget}$ has both a l... | https://mathoverflow.net/users/78 | Why not _co_free modules? | This construction is used frequently (at least, I use it frequently in my work).
For example, it appears in the usual proof that module categories have enough injectives.
(In this case one studies $Cofree(\mathbb Q/\mathbb Z)$, as you anticipated.)
If we generalize slightly, and replace $\mathbb Z$ by the group ring... | 8 | https://mathoverflow.net/users/2874 | 38091 | 24,471 |
https://mathoverflow.net/questions/38089 | 19 | The following is some version of Tannaka-Krein theory, and is reasonably well-known:
>
> Let $G$ be a group (in Set is all I care about for now), and $G\text{-Rep}$ the category of all $G$-modules (over some field $\mathbb K$, say). It is a fairly structured category (complete, cocomplete, abelian, $\mathbb K$-enri... | https://mathoverflow.net/users/78 | What's so special about the forgetful functor from G-rep to Vect? | If $G$ is an affine algebraic group (for example a finite group), then the category of $k$-linear cocontinuous symmetric monoidal functors from $\mathsf{Rep}(G)$ to $\mathsf{Vect}\_k$ is equivalent to the category of $G$-torsors over $k$. In particular, not every such functor needs to be isomorphic to the identity. For... | 35 | https://mathoverflow.net/users/7721 | 38092 | 24,472 |
https://mathoverflow.net/questions/38086 | 20 | Consider $n$ circles with variable radii $r\_1,\ldots, r\_n$ that pack inside a fixed circle of unit radius. In other words, all $n$ variable-radius circles are contained in the unit radius circle and their interiors have empty intersections. The tangency graph of a packing comprises $n+1$ vertices, one for each circle... | https://mathoverflow.net/users/40739 | A circle packing conjecture | You have an elaborate set of ideas, and I haven't thought through all of what you outlined, but here's a suggestion:
Oded Schramm generalized the circle packing theory to include arbitrary convex shapes, and showed they work in much the same way. (The famous case of packing squares is one instance included in this gene... | 19 | https://mathoverflow.net/users/9062 | 38096 | 24,475 |
https://mathoverflow.net/questions/38066 | 16 | André Weil's likening his research to the quest to decipher the Rosetta Stone (see this [letter](http://www.ams.org/notices/200503/fea-weil.pdf) to his sister) continues to inspire contemporary mathematicians, such as Edward Frenkel in [Gauge Theory and Langlands Duality](http://arxiv.org/abs/0906.2747).
Remember tha... | https://mathoverflow.net/users/447 | Which languages could appear on Weil's Rosetta Stone? | All the 12 or more approaches to geometry over the field with one element are tentatives to create such intermediate languages. But you seemed to ask more about a pre-existing area of it's own which may serve as a bridge - in this direction there are
$\bullet$ Alexandru Buium's theory of [Arithmetic Differential Equa... | 12 | https://mathoverflow.net/users/733 | 38097 | 24,476 |
https://mathoverflow.net/questions/38082 | 6 | Is there an explicit bound on the number of tetrahedra needed to triangulate a hyperbolic 3-manifold of volume $V$? Or more generally a hyperbolic $n$-manifold of volume $V$?
| https://mathoverflow.net/users/4325 | Is there an explicit bound on the number of tetrahedra needed to triangulate a hyperbolic 3-manifold of volume V? | A couple of things are true:
1. If you have any Riemannian manifold of bounded infinitesimal geometry (curvature pinched above and below), its thick part, where the injectivity radius $> \epsilon$, can be triangulated with a number of simplices bounded by a constant times volume, where the constant depends on the curva... | 19 | https://mathoverflow.net/users/9062 | 38099 | 24,477 |
https://mathoverflow.net/questions/38080 | 9 | Fill in the blank, please :)
>
> Let $\mathcal C$ be a complete and cocomplete abelian category. A **generator** in $\mathcal C$ is an object $X \in \mathcal C$ such that every object $Y \in \mathcal C$ is a *colimit* of a (small) diagram made entirely of $X$s; in this way, $X$ knows everything there is to know abo... | https://mathoverflow.net/users/78 | What are examples of cogenerators in R-mod? | (This is closely related to Hailong's comment above.)
You can say (albeit rather abstractly) what *any* cogenerator must look like. The following can be found in T.Y. Lam's *Lectures on Modules and Rings*, Theorem 19.10. Let $\{V\_i\}$ be a complete set of simple right $R$-modules, with injective hulls $E(V\_i)$. The... | 7 | https://mathoverflow.net/users/778 | 38102 | 24,479 |
https://mathoverflow.net/questions/38084 | 14 | Say a monoid $M$ *has infinite products* if, for any (possibly infinite) sequence $(m\_1,m\_2,\ldots)$ of elements of $M$, there exists an element $m\_1m\_2\cdots\in M$, satisfying some good properties. First, if the sequence is finite, it should coincide with the usual product on $M$. Second, concatenation of sequence... | https://mathoverflow.net/users/2811 | Monoids with infinite products | If I understand the question, the short answer is "yes, you can freely and functorially adjoin infinite products to monoids". The basic idea is that algebraic theories can accommodate arbitrary arities (bounded above by some cardinal), and one can discuss relative free-forgetful adjunctions between categories of algebr... | 4 | https://mathoverflow.net/users/2926 | 38111 | 24,483 |
https://mathoverflow.net/questions/38118 | 2 | Hi all:
I'm wondering if there is a simple formula for this.
Simple Example:
```
x=cos(pt);
y=cos(qt);
where p,q are integers.
```
Question: How many intersection points are there?
0) only need to consider (p,q) are relatively prime.
1) Firstly, I thought it was just basic counting:
Let N(p,q) be the nu... | https://mathoverflow.net/users/8744 | Formula for number of intersection points for Lissajous curve? | EDIT: with both functions switched to cosine I get $$ \frac{(p-1)(q-1)}{2} $$ for both odd-odd and for even-odd.
ORIGINAL, both functins sine: For $p$ even and $q$ odd I get $$ 2 p q - p - q $$ which is the same as you have when both are odd.
Examples, I had to count over a few times,
(p,q,count) :
(2,1,1); (2,3,... | 1 | https://mathoverflow.net/users/3324 | 38124 | 24,491 |
https://mathoverflow.net/questions/38126 | 5 | Let $p>2$ be a prime, $C\_p$ be the additive group of integers mod $p$. Then the multiplicative group $\{1,...,p-1\}$ of units in the field $Z/pZ$ is cyclic of order $p-1$, it acts on $C\_p$ by left multiplication. Let $G\_p$ be the corresponding semi-direct product of order $p(p-1)$. Question: does this group admit a ... | https://mathoverflow.net/users/nan | a balanced presentation of a cyclic-by-cyclic group? | G5 has a presentation on generators a,b and relations ba=aab, abbabb=1.
| 7 | https://mathoverflow.net/users/3710 | 38135 | 24,498 |
https://mathoverflow.net/questions/38137 | 5 | I am forced to know the etale fundamental group of the grassmannian over the rational field. I searched it but couldn't find any hint. I am wondering whether there are some positive results or recipe to compute it.
Thank you!
| https://mathoverflow.net/users/9096 | Etale $\pi_1$ of Grassmannian | The answer is that any Grassmannian is geometrically simply connected, so the etale fundamental group over $\mathbb{Q}$ is simply [**edit**: !!] the absolute Galois group $\operatorname{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$ of $\mathbb{Q}$.
In more detail: let $X$ be a geometrically integral variety defined over $\... | 15 | https://mathoverflow.net/users/1149 | 38142 | 24,501 |
https://mathoverflow.net/questions/38098 | 12 | Often when people write about the geometrization conjecture they assume (for simplicity) that the manifold is orientable. I never seriously thought of non-orientable 3-manifolds, but while reading Morgan-Tian's [paper](https://arxiv.org/abs/0809.4040) I realized that they prove the geometrization for every compact 3-ma... | https://mathoverflow.net/users/1573 | Geometrization for 3-manifolds that contain two-sided projective planes | Most 3-manifold topologists tend to hypothesize away 2-sided projective planes. If a 3-manifold contains a 2-sided projective plane, then it must be non-orientable, and the preimage of the projective plane in the orientable double cover must be essential. Cutting along a maximal collection of disjoint essential embedde... | 12 | https://mathoverflow.net/users/1345 | 38146 | 24,502 |
https://mathoverflow.net/questions/38145 | 4 | A self-affine tile is a compact set $T$ in $\mathbb R^n$ of positive Lebesgue measure for which there is an $n\times n$ expanding matrix $A$ (i.e. all its eigenvalues have modulus greater than 1) such that the affinely inflated copy $A(T)$ of $T$ can be perfectly tiled with essentially disjoint translates of $T$.
Th... | https://mathoverflow.net/users/6766 | A question about self-affine tiles | The classification is given in section 5 of ["Integral Self-Affine Tiles in $\mathbb R^n$ I. Standard and Nonstandard Digit Sets"](http://jlms.oxfordjournals.org/content/54/1/161.abstract) by Lagarias and Wang (Theorem 5.2 and corollary 5.2a). Their result builds on the previous paper by A. M. Odlyzko, "Non-negative di... | 2 | https://mathoverflow.net/users/2384 | 38153 | 24,507 |
https://mathoverflow.net/questions/38155 | 0 | Can every solution of a homogeneous linear system be approximated by a solution in rational numbers?
In mathematical terms: Let $$Ax=0$$ be a homogeneous linear system in $n$ determinates for an $m\times n$-matrix $A$ (possibly $m>n$) with integer entries (say all entries $1,0,-1$ for simplicity).
Given a solution $... | https://mathoverflow.net/users/39082 | Rational solutions of homogeneous equations | Since A has integer entries, putting it in reduced row-echelon form shows that the solution-space is spanned by vectors with rational coordinates. Rational multiples of the spanning vectors are then dense in the solution-space, so vectors with rational coordinates are also dense in the solution-space. Therefore every s... | 4 | https://mathoverflow.net/users/nan | 38156 | 24,509 |
https://mathoverflow.net/questions/38160 | 15 | By Tennenbaum's theorem, PA itself does not have any computable nonstandard models. The integer polynomials which are 0 or have a positive leading coefficient form a computable nonstandard model of Robinson arithmetic, which also happens to make the order relation total. Since Presburger arithmetic is decidable, we can... | https://mathoverflow.net/users/nan | Computable nonstandard models for weak systems of arithemtic | One of the usual ways of proving Tennenbaum's theorem also
applies to many of the theories on your list, and so they
can have no computable nonstandard models.
The proof I have in mind is the following, which I also
explained in [this MO
answer](https://mathoverflow.net/questions/12426/is-there-a-computable-model-of-... | 11 | https://mathoverflow.net/users/1946 | 38162 | 24,511 |
https://mathoverflow.net/questions/38151 | 2 | How many different rectangles (in terms of area) can fit in a 20-unit-wide square? The rectangles can be squares, and their dimensions are integers.
| https://mathoverflow.net/users/9107 | How many different rectangles (in terms of area) can fit in a 20-unit-wide square? | If you're looking for the number of different areas realizable by fitting rectangles in a 20x20 square *with (integer-length) edges parallel to the coordinate axes*, the answer is the number of elements in {$ \{ x \times y | x,y \in \{ 1..20 \} \} $}. In Haskell, `length . List.nub . sort $ [x*y| x<-[1..20] , y<-[1..20... | 0 | https://mathoverflow.net/users/5790 | 38172 | 24,519 |
https://mathoverflow.net/questions/38161 | 62 | Obviously there exists a list of the finite simple groups, but why should it be a nice list, one that you can write down?
[Solomon's AMS article](https://www.ams.org/notices/199502/solomon.pdf) goes some way toward a historical / technical explanation of how work on the proof proceeded. But, though I would like somed... | https://mathoverflow.net/users/2362 | Heuristic argument that finite simple groups _ought_ to be "classifiable"? | It is unlikely that there is any easy reason why a classification is possible, unless someone comes up with a completely new way to classify groups. One problem, as least with the current methods of classification via centralizers of involutions, is that every simple group has to be tested to see if it leads to new sim... | 67 | https://mathoverflow.net/users/51 | 38174 | 24,521 |
https://mathoverflow.net/questions/37651 | 73 | I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of x,y, assuming that x and y are sufficiently close. By "explicit", I mean things like a closed form description in terms ... | https://mathoverflow.net/users/766 | Riemannian surfaces with an explicit distance function? | I'll briefly spell out what others have pointed to concerning geodesics on surfaces of revolution (or more generally, surfaces with a 1-parameter group of symmetries), because it's nice and not as widely understood as it should be.
Geodesics on surfaces of revolution conserve angular momentum about the central axis, ... | 53 | https://mathoverflow.net/users/9062 | 38183 | 24,525 |
https://mathoverflow.net/questions/38191 | -2 | Consider a markov chain matrix P of size n x n (n states).
P is known to be:
1- Not irreducible (i.e. there exist at least a pair of states i, j such that we cannot go from i to j)
2- Not all states are recurrent.
3- Aperiodic (the return to some states can occur at irregular times).
4- there are at least two... | https://mathoverflow.net/users/8021 | Convergence of a markov matrix | Suppose the answer is Yes. Then suppose we add two more states $i\ne j$ with $P\_{i,j}=1$ and $P\_{j,i}=1$, and no other state can go to states $i$ or $j$. Then for the new matrix the assumptions are still satisfied, but now the answer is No. Therefore the answer must be No.
| 0 | https://mathoverflow.net/users/4600 | 38194 | 24,530 |
https://mathoverflow.net/questions/38139 | 2 | I am having hard time to convert following set of differential equation to state space equation. I am a biologist and my math skills fall short as I don't know where to start. Any suggestion or feedback is highly appreciated. Thanks in advance.
**Updates**:
Based on comments I have updated the question (which is not ... | https://mathoverflow.net/users/9097 | Converting an ODE system to State space formulation | Ok, let me give this a shot:
* Because your system is nonlinear, I'm assuming you want the nonlinear state-space form. You can easily get the linear form by doing a Taylor series expansion on it around some equilibrium point.
* The fact that most of the states are not measurable is not a big problem. You can estimate... | 2 | https://mathoverflow.net/users/7851 | 38197 | 24,531 |
https://mathoverflow.net/questions/38188 | 6 | Is there an easy example of a (closed) hyperbolic 3-manifold that fibers over the circle but contains some totally geodesic surface?
(Of course such manifolds exist if the 'Virtually Fibered Conjecture' were correct, since a geodesic surface lifts to the fibered cover. But is there something more eplicit?)
| https://mathoverflow.net/users/39082 | Totally geodesic surfaces in fibered 3-manifolds | There are many specific known examples. Here is one construction:
Start with the 3-torus $T^3$, parametrize in the standard way as $R^3/Z^3$. It fibers over the circle in many ways.
Let $a$, $b$ and $c$ be three disjoint circles, coming form lines parallel to the x, y and z axes.
For most fibrations, these three circ... | 24 | https://mathoverflow.net/users/9062 | 38206 | 24,537 |
https://mathoverflow.net/questions/38193 | 34 | For simplicity, let me pick a particular instance of Gödel's Second Incompleteness
Theorem:
ZFC (Zermelo-Fraenkel Set Theory plus the Axiom of Choice, the usual foundation of mathematics) does not prove Con(ZFC), where Con(ZFC) is a formula that expresses that
ZFC is consistent.
(Here ZFC can be replaced by any oth... | https://mathoverflow.net/users/7743 | Interpretation of the Second Incompleteness Theorem | For the philosophical point encapsulated in (\*) in the question, it seems that corollaries of the second incompleteness theorem are more relevant than the theorem itself. If we had doubts about the consistency of ZFC, then a proof of Con(ZFC) carried out in ZFC would indeed be of little use. But a proof of Con(ZFC) ca... | 32 | https://mathoverflow.net/users/6794 | 38210 | 24,539 |
https://mathoverflow.net/questions/38219 | 17 | As I have been studying algebraic topology, something that I found puzzling was the existence of finite homotopy groups. For instance, $\pi\_{4}(S^{2})\cong\pi\_{5}(S^{4})\cong\mathbb{Z}/2\mathbb{Z}$. I was wondering if there was any kind of intuitive reason for why this might be true, and if there are spaces $X$ such ... | https://mathoverflow.net/users/6856 | Intuition on finite homotopy groups | The simplest (to understand) case of finite $\pi\_1$ is the group $SO\_3$. This can be illustrated using an arm or a belt! $SO\_3$ is the group of rotations in space and a based loop in $SO\_3$ can be thought of as a description of the motion of an object in such a way that it ends up back where it started. By attachin... | 24 | https://mathoverflow.net/users/45 | 38220 | 24,543 |
https://mathoverflow.net/questions/38141 | 7 | Suppose $K$ is a field endowed with a non-archimedian absolute value. Assume $K$ has characteristic $p>0$ and that $[K:K^p] < \infty$. Let $L$ be the completion of $K$ with respect to this absolute value. Is it always true that $[L:L^p] = [K:K^p]$?
| https://mathoverflow.net/users/9099 | Degree of $[K:K^p]$ and Completion | In fact, BCnrd's comment says that for dvr's, only for non-excellent one's $[L:L^p]\ne [K:K^p]$ can happen. Actually, suppose $d=[K:K^p]$ is finite. Consider the canonical map $K\otimes\_{K^p} L^p \to L$. The source is a $L^p$-vector space of dimension $d$, its image is therefore closed ($L^p$ is complete) and contains... | 8 | https://mathoverflow.net/users/3485 | 38229 | 24,550 |
https://mathoverflow.net/questions/37739 | 8 | For a large enough $n$, and a parameter $ m $ I'm looking for a subset of the prime numbers in the range $[n,2n]$ with a unique structure. I am looking for a prime $p$ and a set of $m$ positive (not necessary different) integers: $\Delta\_1,\Delta\_2,\ldots,\Delta\_m$ such that the following set consists only of prime ... | https://mathoverflow.net/users/9017 | Boolean Cube of Primes | The argument that gives you cubes in dense sets shows roughly speaking (via repeated applications of Cauchy-Schwarz) that the number of k-dimensional cubes in a set of density delta is at least something around $\delta^{2^k}n^{k+1}$, which is the number you would get in a random set. (I am in fact giving the result for... | 2 | https://mathoverflow.net/users/1459 | 38233 | 24,553 |
https://mathoverflow.net/questions/38187 | 1 | Is there a complete "infinite mixture of gaussians representation" for densities? What I mean is, is there, for any reasonably big class of densities $\phi(x)$ I can come up with a function $c(\mu, \tau)$ such that:
$\displaystyle\phi(x) = \int d\mu d\tau\; c(\mu, \tau) \exp\left(-\frac{\tau (x-\mu)^2}{2}\right)$??
... | https://mathoverflow.net/users/757 | Completeness of an "infinite mixture of gaussians" representation | How about this: the set of finite mixtures of (non-degenerate) Gaussians is weakly dense in the space of probability measures on $\mathbb{R}$. Proof: the set of finite mixtures of constants is certainly weakly dense. But a constant can be approximated as a Gaussian with small variance.
| 1 | https://mathoverflow.net/users/4832 | 38242 | 24,559 |
https://mathoverflow.net/questions/36673 | 2 | Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in [this paper](http://arxiv.org/abs/math/9201282), but I can't figure out one thing : can we say **all open subsets** of this boundary has dimension 2 ? Are there any references ? Thanks.
| https://mathoverflow.net/users/8779 | Hausdorff dimension of subsets of the Mandelbot set. | Yes - and it is indeed in Shishikura's paper, as in your comment.
The "open subsets" of the boundary with respect to which topology? That of the ambient space, or the relative topology of the set itself? Since the latter topology is extremely wild, it is much clearer to use the ambient topology, as Shishikura does.
... | 2 | https://mathoverflow.net/users/3993 | 38247 | 24,562 |
https://mathoverflow.net/questions/38262 | 5 | (This is related to my question at [Computable nonstandard models for weak systems of arithemtic](https://mathoverflow.net/questions/38160/computable-nonstandard-models-for-weak-systems-of-arithemtic) )
Is there a nontrivial computable discrete ordered ring with Euclidean division that is not isomorphic to Z?
If so... | https://mathoverflow.net/users/nan | Computable rings similar to Z | Berarducci and Otero in "A Recursive Nonstandard Model of Normal Open Induction" (Journal of Symbolic Logic v61, 1996)
give a discretely ordered ring $R$ with recursive operations having the following properties:
1. $R$ is integrally closed in its quotient field. (So elements with "rational" square roots are perfect ... | 6 | https://mathoverflow.net/users/5229 | 38265 | 24,576 |
https://mathoverflow.net/questions/38264 | 3 | I'm seeking a function which belongs to $W^{1,p}(\Omega)$ for $p < n$ which is *not differentiable a.e*. There is a standard theorem which shows that if $p > n$ then in fact any function in $W^{1,p}$ is differentiable a.e. I would like an example where a weak derivative exists, lies in $W^{1,p}$ for $p < n$ but fails t... | https://mathoverflow.net/users/8755 | A function in $W^{1,p}(\Omega)$ for $1 < p < n$ which is not differentiable a.e | Pick a number $\alpha$ with $0<\alpha< n/p-1$ and a smooth, nonnegative function $g(r)$ defined for $r>0$ with $g(r)=r^{-\alpha}$ when $r$ is small, $g(r)=0$ when $r$ is large. Then $x\mapsto g(|x|)$ belongs to $W^{1,p}(\mathbb{R}^n)$. Write $$f(x)=\sum\_{i=1}^\infty 2^{-i}g(|x-q\_i|)$$ where $(q\_i)$ is a dense sequen... | 3 | https://mathoverflow.net/users/802 | 38273 | 24,582 |
https://mathoverflow.net/questions/38283 | 5 | I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter $w$, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue of the corresponding matrix.
For small values of $w$, the corresponding matrix is small and I can use the so-called pow... | https://mathoverflow.net/users/9136 | Computing the largest eigenvalue of a very large sparse matrix | You could use the Arnoldi Iteration algorithm. This algorithm only requires the matrix A for matrix-vector multiplication. I'm expecting that you will be able to black-box the function v→Av. What you generate is an upper Hessenberg matrix H whose eigenvalues whose can be computed cheaply (by a direct method or Rayleigh... | 4 | https://mathoverflow.net/users/2011 | 38288 | 24,592 |
https://mathoverflow.net/questions/38238 | 43 | Recently I learned that there is a useful analogue of mathematical induction over $\mathbb{R}$ (more precisely, over intervals of the form $[a,\infty)$ or $[a,b]$). It turns out that this is an old idea: it goes back to Khinchin and Perron, but has for some reason never quite caught on and thus keeps getting rediscover... | https://mathoverflow.net/users/1149 | A principle of mathematical induction for partially ordered sets with infima? | Something very close to François' conditions achieves the
desired if-and-only-if version of the theorem for partial
orders, providing an induction-like characterization of the complete partial orders, just as Pete's theorem characterizes the complete total orders.
Suppose that $(P,\lt)$ is a partial order. We say th... | 15 | https://mathoverflow.net/users/1946 | 38292 | 24,595 |
https://mathoverflow.net/questions/38026 | 15 | Let $G = S\_n$ (the permutation group on $n$ elements).
Let $A\subset G$ such that $A$ generates $G$.
Is there an $n$-cycle $g$ in $G$ that can be expressed as
$g = a\_1 a\_2 ... a\_k$
where $a\_i\in A \cup A^{-1}$
and $k\leq c\_1 n^{c\_2}$, where $c\_1$ and $c\_2$ are constants?
What about $2$-cycles, or elem... | https://mathoverflow.net/users/398 | Generating n-cycles | That all elements in a symmetric group with a specified arbitrary generating set can be reached in a polynomial amount of steps is a known open problem, and has been investigated for some time now. This long-standing conjecture has been proven for most choices of generating sets. Heuristically one expects this to be tr... | 9 | https://mathoverflow.net/users/2384 | 38296 | 24,597 |
https://mathoverflow.net/questions/38304 | 1 | Let $F$ be a field, $n$ be a positive integer. Denote by $h\_{F}(n)$ the maximal dimension of a subspace $X\subset F^n$ such that $(x,y)=0$ for any two (not necessary distinct) vectors $x,y\in F^n$, where $(x,y)=x\_1y\_1+\dots+x\_ny\_n$ for $x=(x\_1,\dots,x\_n)$, $y=(y\_1,\dots,y\_n)$. For example, $h\_{\mathbb{R}}(n)=... | https://mathoverflow.net/users/4312 | maximal number of mutually orthogonal vectors | This is the question of finding maximal [isotropic subspaces](http://en.wikipedia.org/wiki/Isotropic_quadratic_form) of
an inner-product space. The results for finite fields of odd characteristic
are well-known and can be found in Serre's *Course in Arithmetic*.
Let's consider the quadratic form $Q=x\_1^2+\cdots+x\_n... | 2 | https://mathoverflow.net/users/4213 | 38306 | 24,603 |
https://mathoverflow.net/questions/38280 | 1 | Let $f$ and $g$ be two discrete signals. I want to find a monotone function h such that
$h=argmin\_{h}\sum\_{n\in[0,N]}{(f(n)-h(g(n)))^2}$
I don't really care about finding the global optimum, I just want a good fit. What would be a good representation of f to achieve that? Thanks!
| https://mathoverflow.net/users/180 | Finding an optimal monotone function? | Here is another try:
Assume w.l.o.g. that values of $g(n)$ are in increasing order, i.e. $g(0) \le g(1) \le \cdots\le g(n)$. Moreover, assume the first $n\_1$ values in that sequence are equal, then the next $n\_2$ values are equal, etc. and that there are $m$ distinct values
i.e.
$$g(0)=g(1)=\cdots=g(n\_1-1) < g(n... | 1 | https://mathoverflow.net/users/1184 | 38313 | 24,606 |
https://mathoverflow.net/questions/38305 | 4 | Hi all,
I would like to know if there is a way to compute some measure of similarity between two ordinary graphs with weighted edges. Graphs do not share vertices and can differ in number of vertices and edges.
Any hints, suggestions and thoughts are highly appreciated.
Best,
Jozef
| https://mathoverflow.net/users/9139 | Similarity of weighted graphs | If you view the weights as edge lengths then you can view each graph as a metric space, and then use the [Gromov-Hausdorff](http://en.wikipedia.org/wiki/Gromov%E2%80%93Hausdorff_convergence) distance between the two metric spaces. This may not be at all suitable for your application but it has been very useful in my ow... | 9 | https://mathoverflow.net/users/3401 | 38316 | 24,608 |
https://mathoverflow.net/questions/38307 | 23 | At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. Application: asymptotic properties of Schroedinger equations, scattering.
You have two convex (compact, smooth, everything) ... | https://mathoverflow.net/users/7294 | Trapped rays bouncing between two convex bodies | Yes, there is always a trapped ray. The simplest way to see it is to find the path between the two bodies that minimizes length. It is necessarily perpendicular to both surfaces.
EDIT: I see the question was edited to ask for more than this trivial answer, so the new answer: there is a unique trapped ray from any sta... | 34 | https://mathoverflow.net/users/9062 | 38320 | 24,612 |
https://mathoverflow.net/questions/38324 | 26 | In 1977, [Henry Pogorzelski](http://en.wikipedia.org/wiki/Henry_Pogorzelski) published what some believed was a claimed proof of Goldbach's Conjecture in [Crelle's Journal (292, 1977, 1-12)](http://www.deepdyve.com/lp/de-gruyter/goldbach-conjecture-frA6e4f0DS). His argument has not been accepted as a proof of Goldbach'... | https://mathoverflow.net/users/2594 | Did Pogorzelski claim to have a proof of Goldbach's Conjecture? | In the 1970's Pogorzelski published a sequence of four papers in Crelle concerning the Goldbach Conjecture (and various generalizations and abstractions):
>
> MR0347566 (50 #69) Pogorzelski, H. A. On the Goldbach conjecture and the consistency of general recursive arithmetic. Collection of articles dedicated to Hel... | 35 | https://mathoverflow.net/users/1149 | 38328 | 24,618 |
https://mathoverflow.net/questions/24131 | 19 | The group of $n\times n$ matrices with integer entries and determinant equal to 1, $SL(n,Z)$, is a finitely generated group (in fact, it is generated by 2 matrices). I am interested to know if the semigroup of the matrices in $SL(n,Z)$ where all the entries are nonnegative is also finitely generated. This is true at le... | https://mathoverflow.net/users/3960 | Is the semigroup of nonnegative integer matrices with determinant 1 finitely generated? | The question has been amply answered, but perhaps it's worth explaining the geometry of the situation. The positive monoid acts by projective transformations on the $n-1$-simplex (the view of the positve orthant as seen from the origin, and the partially order on the monoid is the order by inclusion of the image under ... | 16 | https://mathoverflow.net/users/9062 | 38332 | 24,622 |
https://mathoverflow.net/questions/38119 | 16 | I wrote a research paper "A mathematical model of the Mafia game" ([arXiv:1009.1031](http://arxiv.org/abs/1009.1031) [math.PR]). However, I do not know where to publish it. As an undergraduate studying majorly physics, I have little knowledge of mathematical journals. Moreover, its not easy for me to classify its subje... | https://mathoverflow.net/users/9093 | Where to publish a paper on the Mafia game? | I have only glanced at your paper, but one possibility is to submit it to [*The Mathematical Intelligencer*](http://www.springer.com/mathematics/journal/283?detailsPage=editorialBoard), in particular to Michael Kleber, who edits the "Mathematical Entertainments" column. This is a great place for serious mathematical an... | 18 | https://mathoverflow.net/users/3106 | 38337 | 24,623 |
https://mathoverflow.net/questions/38323 | 10 | I try to understand some of the topology of the space of pointed non-compact hyperbolic surfaces (with the pointed Gromov-Hausdorff topology). It is known that the fundamental
group of a non-compact surface is a free group, so I am interested in free Fuchsian groups
(discrete, free groups of direct isometries of the hy... | https://mathoverflow.net/users/4961 | fundamental domains for free fuchsian group. | Yes, this is true. Topologically, one may find a locally finite collection of properly embedded arcs in a connected surface whose complement is homeomorphic to $R^2$. Then make each of these arcs geodesic in the hyperbolic metric. The complement will be the fundamental domain of the type you want.
Addendum: I'll add... | 12 | https://mathoverflow.net/users/1345 | 38338 | 24,624 |
https://mathoverflow.net/questions/38315 | 3 | A pair $(X,O\_X)$ is a ringed space if $X$ is a topological space and $O\_X$ is a sheaf of rings. If every stalk $O\_{X,x}$ is a local ring, then we say that $(X,O\_X)$ is a locally ringed space.
In the case of $X$ being an abstract algebraic variety, not necessarily irreducible, and $O\_X$ its sheaf of regular funct... | https://mathoverflow.net/users/1887 | Ringed and locally ringed spaces | 1. No, all schemes are locally ringed spaces (cf. Robin Chapman's comment above). In fact, schemes are often defined by using locally ringed spaces.
2. It depends on what you mean by "this". If you have a pair of lines crossing, the local ring at the crossing has zero divisors, but it's still local.
3. Take a point, an... | 8 | https://mathoverflow.net/users/121 | 38339 | 24,625 |
https://mathoverflow.net/questions/38344 | 11 | Of course, no continuous real valued non-constant function can attain only rational or irrational values, but can there be a pair of nowhere-constant continuous functions f and g such that for all x, at least one of f(x) and g(x) is rational? Or maybe a countable collection of continuous functions, {f1, f2...} such tha... | https://mathoverflow.net/users/4903 | Can there be two continuous real-valued functions such that at least one has rational values for all x? | If you allow the functions to be constant on some intervals, then there are some easy examples, and Ricky has provided one.
But if you rule that out, then there can be no examples, even with countably many functions. To see this,
suppose that $f\_n$ is a list of countably many continuous functions which are never co... | 16 | https://mathoverflow.net/users/1946 | 38346 | 24,628 |
https://mathoverflow.net/questions/37976 | 5 | Given any subspace $A\subset X$ of a topological space with Lebesgue dimension $\le N$.
Let $\bar{A}$ denote the closure of $A$. Assume, that the pair $(\bar{A},A)$ satisfies the Z-set condition, i.e. there is a homotopy $H:\bar{A}\times [0;1]\rightarrow \bar{A}$, such that $H\_1=id$ , Image$(H\_t)\subset A$ for all... | https://mathoverflow.net/users/3969 | Lebesgue dimension of closures satisfying the Z-set condition | No for general topological spaces, yes for metrizable ones (and I believe the argument can be generalized to all normal spaces).
Bad example: $X=\{a,b,c\}$ with open sets $\emptyset$, $X$, $\{a\}$, $\{a,b\}$, $\{a,c\}$. Let $A=\{a\}$, then $\bar A=X$. The homotopy is given by $H\_1=id$, $H\_t\equiv a$ for $t<1$. The ... | 4 | https://mathoverflow.net/users/4354 | 38352 | 24,630 |
https://mathoverflow.net/questions/37993 | 21 | Let P be an arbitrary probability space.
I would like to find a compact topological group $G$ so that the Haar probability measure on $G$ admits a measurable map to the probability space $P$.
By a measurable map, I mean a function which lifts measurable sets to measurable sets of the same measure. That is, $f : Q \... | https://mathoverflow.net/users/9068 | Is every probability space a factor space of the Haar Measure on some group? | It is possible to find the following: A compact abelian group G with Haar measure $\mu\_G$, a subset $S\subseteq G$ of full outer Haar measure and a measurable function $f\colon S\to X$ with $\mu\_P(E)=\mu\_S(f^{-1}(E))$ for measurable $E\subseteq P$. In fact, as you mention, G can be taken to be a large enough product... | 9 | https://mathoverflow.net/users/1004 | 38362 | 24,636 |
https://mathoverflow.net/questions/38347 | 14 | Framed functions arose in the work of K. Igusa defining cohomology invariants for smooth manifold bundles (Igusa-Klein torsion). In the late 80's, he proved a strong connectivity result about the "space of framed functions" using Morse theory and conjectured that this space was, in fact, contractible. Jacob Lurie recen... | https://mathoverflow.net/users/7867 | The space of framed functions | It is possible to prove the contractibility directly (and thereby bypass the obstruction theory arguments sketched at the end of section 3 of my paper). The statement itself is an example of an h-principle: namely, one can show fairly easily that the framed function space of ${\mathbb R}^{n}$ is contractible, so the ge... | 15 | https://mathoverflow.net/users/7721 | 38368 | 24,641 |
https://mathoverflow.net/questions/38367 | 8 | Notation: Let$M$ be a smooth, closed manifold, $S$ any submanifold of $M$, $Diff(M)$ the group of diffeomorphisms of $M$ and $Imb(S, M)$ the group of smooth imbeddings of $S$ into $M$.
A classical result of R. Palais from the 1960 paper *Local triviality of the restriction map for embeddings* says that the map $Diff(... | https://mathoverflow.net/users/7867 | Restrictions of Diffeomorphisms | It's not clear what you mean by "various refinements and generalizations". Cerf has a huge paper published by IHES "Topologie de certains espaces de plongements" which goes into many related details. In a way it's more of a ground-up collection of basic information on the topology of function spaces.
Regarding your 2... | 9 | https://mathoverflow.net/users/1465 | 38372 | 24,645 |
https://mathoverflow.net/questions/38348 | 2 | Fix $N>0$. Let $b\_i=(b\_{i,1}, b\_{i,2}, b\_{i,3}, b\_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b\_{i,j}< N$. (The zero tuple is disallowed.)
Define $w\_i=(\prod\_{j=1}^4 (z-z\_j)^{b\_{i,j}})^{\frac1{N}}$.
Consider $w\_i$ as an element of the following vector space: the alge... | https://mathoverflow.net/users/5399 | Linear independence in the algebraic closure of $\mathbb{C}(z)$ | You can also use Galois theory or monodromy. Take a minimal linear dependence relation and apply the automorphimsm of the algebraic closure that fixes $(z-z\_j)^{1/N},j>1$ and fot $j=1$ multiplies the function by an N-th root of unity, thus getting a new relation and you can produce a shorter relation from those two.
... | 1 | https://mathoverflow.net/users/2290 | 38375 | 24,648 |
https://mathoverflow.net/questions/38376 | 4 | $\mathbf{n}$ is nilpotent Lie algebra with $N$ being the corresponding *algebraic* Lie group. Now one neat feature of this setting is that you can take the exponential map to be identity. In other words you can define a group structure on $\mathbf{n}$ using the Campbell-Hausdorff formula. I have the following questions... | https://mathoverflow.net/users/8811 | Nilpotent Lie algebras and unipotent Lie groups | I will suppose that $\mathfrak n$, etc is finite-dimensional.
If by "the corresponding Lie group" you mean "the corresponding connected simply-connected Lie group", then what you say is correct (in spite of Victor Prostak's comment). Let $\{\mathfrak n,[,]\}$ be a Lie algebra over a field $k$ of characteristic $0$. T... | 8 | https://mathoverflow.net/users/78 | 38388 | 24,653 |
https://mathoverflow.net/questions/38385 | 6 | Let R and S be commutative rings with a 1 different from zero. Let m and n be positive integers. Assume the ring of m-by-m matrices over R is isomorphic to the ring of n-by-n matrices over S. Does it follow that R is isomorphic to S? Does it follow that m = n? Does either of those follow from the other? I'm interested ... | https://mathoverflow.net/users/nan | Exotic isomorphism of matrix rings | Yes and yes. Let $T=M\_m(R)=M\_n(S)$.
The center of $T$ is isomorphic to both $R$ and $S$.
The $1\times m$ matrices over $R$ form an $(R,T)$-bimodule and the $n\times 1$ matrices over $S$ form a $(T,S)$-bimodule. Tensor these over $T$ to get an $(R,S)$-bimodule. As an $S$-module the direct sum of $m$ copies of thi... | 8 | https://mathoverflow.net/users/6666 | 38391 | 24,654 |
https://mathoverflow.net/questions/37721 | 4 | The topology of a closed surface can be constructed
by identifying edges of a [fundamental polygon](http://en.wikipedia.org/wiki/Fundamental_polygon) of an
even number $2n$ of edges.
Labeling the edges and using $\pm 1$ exponents to indicate
direction,
the construction can be specified by a string of $2n$ symbols:
$a b... | https://mathoverflow.net/users/6094 | Fundamental polygons with infinite pairwise identifications | Identification spaces with infinitely complicated identifications do occur naturally in various contexts. Here are a few more remarks:
1. If you define an involution on a countable set of intervals on the boundary of a surface, as in Victor Protsak's remark, and extend the equivalence relation to finite equivalence c... | 10 | https://mathoverflow.net/users/9062 | 38394 | 24,657 |
https://mathoverflow.net/questions/38380 | 4 | When solving the heat equation on say $\mathbb{R}$ (or $[0,2\pi]$, whichever is easier to talk about) we are posing Cauchy data on the surface $t=0$. My understanding is that $t=$constant are precisely the characteristic surfaces of the heat equation.
I realize this question may seem elementary to those who know the... | https://mathoverflow.net/users/8755 | Posing Cauchy data for the heat equation: $t=0$ a characteristic surface? | The concept of a non-characteristic surface for a PDE or a system of PDE's is useful primarily for only establishing the existence and uniqueness of real analytic or formal power series solutions to the initial value problem using the Cauchy-Kovalevsky theorem.
The generalization of this to the smooth category is the... | 8 | https://mathoverflow.net/users/613 | 38403 | 24,662 |
https://mathoverflow.net/questions/38414 | 26 | A friend of mine recently asked me if I knew any simple, conceptual argument (even one that is perhaps only heuristic) to show that if a triangulated manifold has a non-vanishing vector field, then Euler's formula (the alternating sum of the number of faces of given dimensions) vanishes. I didn't see how to get started... | https://mathoverflow.net/users/7311 | Euler Characteristic of a manifold with non-vanishing vector field, | Consider a straight simplex $\Delta^n$ in $\mathbb R^n$ and take a generic constant vector field $v$ (transversal to the faces of $\Delta^n$). Choose all faces of $\Delta^n$ such that the field moves the center of the face inside the simplex. Then the alternating sum of the numbers of these simplices (signed by the par... | 22 | https://mathoverflow.net/users/943 | 38417 | 24,670 |
https://mathoverflow.net/questions/38421 | 2 | Hello,
do you know any papers or books that use algebraic geometry in evolutionary game theory ?
| https://mathoverflow.net/users/9166 | Applications of Algebraic Geometry in Evolutionary Game Theory | Perhaps Datta's [papers](http://math.berkeley.edu/~datta/) and his thesis [Algebraic Methods in Game Theory](http://math.berkeley.edu/~datta/thesis.pdf) could be a good place to start.
| 2 | https://mathoverflow.net/users/2149 | 38423 | 24,673 |
https://mathoverflow.net/questions/38185 | 3 | Given a Lebesgue measurable set A with strictly positive measure, can we find an open interval (a,b) such that x belongs to A for almost every x in (a,b)?
Thanks in advance for any comments!
| https://mathoverflow.net/users/36814 | About measurable sets and intervals | The usual Cantor set constructed by removing 1/3 at each step is nowhere dense but has measure 0. However, there exist nowhere dense sets which have positive measure. The trick is to try to remove less, for instance you remove 1/4 from each side of [0,1] during the first step then 1/16 from each pieces etc...
The res... | 4 | https://mathoverflow.net/users/3859 | 38425 | 24,674 |
https://mathoverflow.net/questions/38308 | 3 | hi,
assume that I have a function $q$ which is a Fourier Multiplier of order zero, i.e.
$$
\left|\left( \frac{d}{dx}\right)^nq(x)\right|\lesssim \left(\frac{1}{1+|x|}\right)^n\quad \mbox{for all }n\geq 0.
$$
Can I conclude that $q$ is the Fourier Transform of a finite Radon Measure?
If not, what are the conditions ... | https://mathoverflow.net/users/6035 | fourier transform of radon measure | No, such function doesn't need to be a Fourier transform of a finite measure (and, thereby, doesn't need to be a multiplier in $L^1$ or $L^\infty$). This is well-known and the most classical counterexample is just a smoothed Heaviside function $q$ that is $0$ on $(-\infty,1]$, $1$ on $[1,+\infty)$ and whatever you want... | 5 | https://mathoverflow.net/users/1131 | 38428 | 24,676 |
https://mathoverflow.net/questions/35531 | 15 | For a category $\mathcal{C}$, let $\mathcal{C}-Set$ denote the category of functors $\mathcal{C}\to{\bf Set}$. Recall that given a functor $F\colon\mathcal{B}\to\mathcal{C}$, the ``composition with $F$" functor is denoted $F^\*\colon\mathcal{C}-Set\to\mathcal{B}-Set.$ It has a left and a right adjoint, $F\_!$ and $F\_\... | https://mathoverflow.net/users/2811 | First: upper-star, then: lower-star, finally: lower-shriek | First of all: yes, there's certainly a connection. See <http://ncatlab.org/nlab/show/polynomial+functor>. If the base category is $Set$, the composite
$$Set/W \stackrel{f^\ast}{\to} Set/X \stackrel{g\_\ast}{\to} Set/Y \stackrel{h\_!}{\to} Set/Z$$
first takes a $W$-indexed set $S\_w$ to an $X$-indexed set $T\_x = ... | 7 | https://mathoverflow.net/users/2926 | 38433 | 24,680 |
https://mathoverflow.net/questions/38434 | 1 | Suppose $A$ is a positive definite matrix and $B$ is a non-symmetric
matrix with all positive principal minors.
Is their product $AB$ a matrix with all positive principal minors?
I believe the answer is yes, and I have been trying to find a proof but got stuck along the way. The wiki page for minor gives a corolla... | https://mathoverflow.net/users/7154 | Principal Minors of Matrix Product | This isn't true even if $A$ and $B$ are both symmetric and positive definite. For example, let $$A=\begin{pmatrix} 1 & 2\\2 & 5\end{pmatrix}, \quad B=\begin{pmatrix} 1 & -2\\-2 & 5\end{pmatrix},\quad\text{then}\quad AB=\begin{pmatrix} -3 & 8\\-8 & 21\end{pmatrix}.$$
| 6 | https://mathoverflow.net/users/5740 | 38436 | 24,681 |
https://mathoverflow.net/questions/38437 | 31 | There is a question someone (I'm hazy as to who) told me years ago. I found it fascinating for a time, but then I forgot about it, and I'm out of touch with any subsequent developments. Can anyone better identify the problem or fill in the history, and say whether it's still unsolved? It's a challenging question if I'v... | https://mathoverflow.net/users/9062 | Is there a reset sequence? | I believe you are referring to the [Road coloring theorem](http://en.wikipedia.org/wiki/Road_coloring_problem). It was solved [in this preprint](http://arxiv.org/abs/0709.0099).
| 17 | https://mathoverflow.net/users/121 | 38440 | 24,682 |
https://mathoverflow.net/questions/38454 | 9 | In a bonus exercise last year, we were asked to compute the completion in general of such a stalk on a smooth manifold of dimension $n$ (it is isomorphic to the ring of formal power series over $\mathbb{R}$ in $n$ unknowns). It's clear that this is a bad case to work with, since smooth manifolds admit bump functions (w... | https://mathoverflow.net/users/1353 | What information does the completion of a stalk at its maximal ideal give us in the holomorphic, analytic, or algebraic cases? | First I think you are a little bit unfair when you say that the completion of
the ring of germs of $C^\infty$-functions "contains very little". Mapping a
function into that completion gives you the Taylor series of function which
contains a lot of data about the function even though you certainly miss a
significant amo... | 14 | https://mathoverflow.net/users/4008 | 38459 | 24,692 |
https://mathoverflow.net/questions/38450 | 8 | S is uncountable := |$\mathbb{N}$| < |S|
S is noncountable := |S| $\not\leq |\mathbb{N}|$
(X,$T$) is a nice space := (X,$T$) is a compact Hausdorff space without isolated points
Does [ ZF / ZF + Countable Choice ] prove that every nice space is [ uncountable / noncountable ] ?
If not, is it known to prove... | https://mathoverflow.net/users/nan | Compact Hausdorff spaces without isolated points in ZF | One of the usual ways of proving in ZFC that every compact
Hausdorff space $X$ without isolated points ([perfect
space](http://en.wikipedia.org/wiki/Perfect_space)) is
uncountable is by proving that there is a copy of the
Cantor space $2^\omega$ inside it, as follows. Pick two
points and separate them with neighborhood... | 11 | https://mathoverflow.net/users/1946 | 38467 | 24,697 |
https://mathoverflow.net/questions/38451 | 13 | Let $W:[0,1]\rightarrow\mathbb R$ be standard Brownian motion with $W(0)=0$.
Let $F\_n$ denote the collection of all the $2^n$ many piecewise linear continuous functions $f:[0,1]\rightarrow\mathbb R$ such that $f(0)=0$ and $f$ is linear with slope $\pm \sqrt{n}$ on the intervals $[\frac in,\frac{i+1}n]$ for $0\le i<n... | https://mathoverflow.net/users/4600 | Is the nearest walk to Brownian motion uniform? | My understanding of the question is this:
There is a function $V\_n$ that maps the $L^\infty([0,1])$ to the set $F\_n$ of $2^n$ piecewise linear functions as defined. $V\_n$ gives the Voronoi subdivision, taking each point of the big set to the nearest neighbor of the smaller set.
I believe you're asking whether $V... | 7 | https://mathoverflow.net/users/9062 | 38469 | 24,698 |
https://mathoverflow.net/questions/38435 | 6 | Alright, so [a similar question was recently asked](https://mathoverflow.net/questions/38026/generating-n-cycles) about the theoretical bound for generating certain permutations in polynomial time. I had been thinking about a related problem in algorithms (with applications to a specific problem in graph theory - namel... | https://mathoverflow.net/users/8345 | Testing permutations to see if they generate $S_n$ | The answer to (1) is yes, primitivity can be checked in O(n^3) time and practical computer implementations have been widely available for decades. See Butler's [Fundamental Algorithms for Permutation Groups](http://dx.doi.org/10.1007/3-540-54955-2) p.76 for this and various related algorithms (such as testing transitiv... | 5 | https://mathoverflow.net/users/3710 | 38474 | 24,702 |
https://mathoverflow.net/questions/27609 | 22 | A countable discrete group $\Gamma$ is said to be exact if it admits an amenable action on some compact space.
So clearly amenable groups are exact, but large familes of non-amenable groups are as well.
For many of the families that I know of (ex. linear groups, hyperbolic groups) that are exact, they also satisfy... | https://mathoverflow.net/users/5732 | An example of a non-amenable exact group without free subgroups. | I did not check the details, but most probably Gromov's random groups can be made torsion as well (and thus will not contain $F\_2$). Just impose the relations $u^{n\_u}$ on the steps with even numbers and do Gromov's construction of embedding the next graph of an expander on steps with odd numbers. Here $u$ runs over ... | 12 | https://mathoverflow.net/users/nan | 38476 | 24,703 |
https://mathoverflow.net/questions/38487 | 1 | I remember the following problem back from my undergraduate days:
>
> Suppose that $f\in C^1(\mathbb{R}^n)$ is a map such that for all *p*, we have $df(p)\in SO(n)$. Then, $df$ is a constant rotation, or in other words, $f$ is an affine rotation.
>
>
>
There is a clever proof of this fact using local inversion... | https://mathoverflow.net/users/8212 | Generalisation of a multivariable calc problem | There is also a generalization in a slightly different direction. The simplest version I know is that if $f:\mathbb R^n\to \mathbb R$ and $|\nabla f|=1$ everywhere (which is true for each coordinate mapping in your original problem), then $f$ is linear. The solution is to go along the gradient accent/descent curves and... | 6 | https://mathoverflow.net/users/1131 | 38493 | 24,712 |
https://mathoverflow.net/questions/38492 | 2 | I would like to show that for $s \in \mathbb{R}$ and a nonnegative integer $k$
$$
\triangle^k ((1+n)^s) \lesssim (1+|n|)^{s-k}
$$
where $\triangle$ is the discrete derivative, i.e. $\triangle^1 ((1+n)^s) = (2+n)^s - (1+n)^s$.
---
This is easy when $s \in \mathbb{Z}$, and in the continuous analogue because
$$
... | https://mathoverflow.net/users/1540 | Elementary proof of bounds on discrete derivative applied to $(1+n)^s$ | There's a generalization of the mean value theorem that can be
applied here, namely that if $f$ is smooth enough then for each $x$
there is $y$ such that
$$\Delta^kf(x)=f^{(k)}(y)$$
and $x < y < x+k$.
**Added** see
[Is it true that all the "irrational power" functions are almost polynomial ?](https://mathoverflow.net... | 3 | https://mathoverflow.net/users/4213 | 38494 | 24,713 |
https://mathoverflow.net/questions/38475 | 11 | Does any additive, covariant functor preserve direct sum?
| https://mathoverflow.net/users/9141 | Additive, covariant functor preserve direct sum? | I'm not sure what is being asked, but I'll try rephrasing what I suspect the question is: if $A$ and $B$ are enriched in abelian groups, and if $F: A \to B$ is a functor such that all the maps $\hom(a, a') \to \hom(F(a), F(a'))$ are homomorphisms, then is it true that $F$ takes finite sums (coproducts) in $A$ to finite... | 19 | https://mathoverflow.net/users/2926 | 38496 | 24,714 |
https://mathoverflow.net/questions/38217 | 4 | I'm creating a system that will allow people to rate images.
My idea is to use an Elo Rating system (<http://en.wikipedia.org/wiki/Elo_rating_system>) for each image and then use crowdsourcing to have people say if an individual image is better than another i.e
Is A better than B
This will be used to updated th... | https://mathoverflow.net/users/9120 | Elo Rating System Help with the Maths around number of matches | By the way, Elo ratings are named after Élő Árpád, so only the first letter should be capitalized.
1) An implicit assumption of the Elo rating system is that if you know the true advantage of A over B, and of B over C, then you know the true advantage of A over C. I don't think that should hold for the preferences f... | 10 | https://mathoverflow.net/users/2954 | 38500 | 24,715 |
https://mathoverflow.net/questions/38498 | 8 | A fundamental construction in a first course on manifolds is to build a smooth function $\psi\colon \mathbb{R} \to \mathbb{R}$ with the property that for some $0<\delta<\epsilon$ we have
* $\psi(x)=1$ for $|x|\leq\delta$;
* $\psi(x)=0$ for $|x|\geq\epsilon$.
Using this "bump function", one can do all sorts of "glui... | https://mathoverflow.net/users/5701 | Gluing two diffeomorphisms together | The group $GL(n,R)$ has an affine structure, coming from the coefficients of the group.
It has an easily described and fairly large convex neighborhoods of the identity, satisfying that for no vector $V \ne 0$ is $A(V)$ negative real multiple of $V$ (*i.e.*, no negative real eigenvalues).
The first derivative of a co... | 11 | https://mathoverflow.net/users/9062 | 38502 | 24,716 |
https://mathoverflow.net/questions/38497 | 5 | Hey everyone, I was reading about obstruction theory, here's a bit of a summary. Take a cellular space $X$ and a fibre bundle $p:E \to X$ with fiber $F$; consider the problem of extending a section $s$, defined on the $(n-1)$-skeleton over to the $n$-skeleton. We work cell by cell pulling back the bundle via the charac... | https://mathoverflow.net/users/9187 | Obstruction Cocycles | You should have a look at the paper given in the answer to [my earlier question on obstruction theory](https://mathoverflow.net/questions/31147/obstruction-theory-for-non-simple-spaces). It gives a very nice and direct proof that the obstruction cochain is a cocycle that also works in the case of non-simple spaces (a s... | 3 | https://mathoverflow.net/users/396 | 38505 | 24,717 |
https://mathoverflow.net/questions/38506 | -2 | In an algorithm book once the first example was how to compute a multiplication in a loop (only that, so I just remembered, and wanted to do it programmatically but with all operations)
Multiplication was simple, say 10 \* 4:
base = 0, x = 10, y = 4:
```
While y != 0:
base + x = 10, y--
base + x = 20, y--
base +... | https://mathoverflow.net/users/9191 | Mul + div using only add/sub ? | Division is repeated subtraction. Take 11 \div 3 as an example.
11 - 3 = 8.
Increment the quotient by 1.
Since 8 > 3, keep going.
8 - 3 = 5.
Increment the quotient by 1.
Since 5 > 3, keep going.
5 - 3 = 2.
Increment the quotient by 1.
Since 2 < 3, set the remainder to 2 and terminate.
I'll leave it to you to fi... | 0 | https://mathoverflow.net/users/8871 | 38509 | 24,719 |
https://mathoverflow.net/questions/38527 | 25 | Here is a rather pathetic question. In a [comment](http://gowers.wordpress.com/2010/08/21/icm2010-ngo-laudatio/#comment-9720) on Tim Gower's weblog, I tentatively stated that the fundamental lemma was necessary for the [work of Skinner and Urban](http://www.mathunion.org/ICM/ICM2006.2/Main/icm2006.2.0473.0500.ocr.pdf) ... | https://mathoverflow.net/users/1826 | The fundamental lemma and the conjecture of Birch and Swinnerton-Dyer | Dear Minhyong,
My understanding, based on recalling talks of Skinner and also briefly looking over the ICM paper that you linked to is that, yes, they do rely on the fundamental lemma, namely, the fundamental lemma for unitary groups as proved by Laumon and Ngo. Unfortunately, I'm not sufficiently educated in their w... | 20 | https://mathoverflow.net/users/2874 | 38530 | 24,730 |
https://mathoverflow.net/questions/38447 | 3 | What is an example of an action of a *linearly reductive* group variety acting on an affine variety with the property that there exists a closed orbit that is not separable?
To be more precisely, let's work over a fixed algebraically closed field $k$. Suppose that we are given an affine variety $X$ and a group variet... | https://mathoverflow.net/users/5337 | What is a closed orbit that is not separable? (ANSWERED) | BCnrd writes: "This happens for every G of positive dimension, since all one needs is a non-smooth subgroup scheme (such as kernel of Frobenius). Indeed, if H is a closed k-subgroup scheme of G then let X=G/H equipped with the natural left G-action (so X is smooth, since G is smooth, regardless of how "bad" H may be). ... | 2 | https://mathoverflow.net/users/66 | 38533 | 24,732 |
https://mathoverflow.net/questions/38544 | 3 | Let $U (\mathbf{R})$ be the standard unipotent subgroup of $SL(3, \mathbf{R})$. So $U(\mathbf{R})$ is the group of 3 by 3 upper triangular matrices with 1s on the diagonal. I am interested in the quotient space $U (\mathbf{R})/ U (\mathbf{Z})$. I think it is just the cube, $[0,1]^3$, but I am having difficulty writing ... | https://mathoverflow.net/users/8811 | Quotients of unipotent groups | The group $U(\mathbb Z)$ is nilpotent, but not isomorphic to $\mathbb Z^3$.
The quotient space is not the torus $T^3$ (= cube with opposite sides identified).
The fundamental domain which glues together to make the quotient can be described, but I'm not sure it's what you really want or need: there are easier ways to u... | 16 | https://mathoverflow.net/users/9062 | 38545 | 24,736 |
https://mathoverflow.net/questions/38539 | 2 | Let $G$ be a (discrete, say) group and $\mathbb K$ a field. The *regular representation* $G^{\mathbb K}$ is the vector space of all functions $G \to \mathbb K$. It is a (left, say) $G$-module: given $g\in G$ and $f: G \to \mathbb K$, the action is $g\cdot f: x \mapsto f(xg)$. Then $G^{\mathbb K}$ is a commutative algeb... | https://mathoverflow.net/users/78 | Are there other algebra structures on the regular representation of a group? | **Summary** Yes, there are many such examples, even for finite groups.
**Construction** Let $W<GL(V)$ be a complex reflection group, $A=\mathbb{C}[V]$ be the algebra of polynomial functions on $V$ and $A^W$ be the subalgebra of $W$-invariants. Then by the [Chevalley–Shephard–Todd theorem](http://en.wikipedia.org/wiki... | 6 | https://mathoverflow.net/users/5740 | 38548 | 24,738 |
https://mathoverflow.net/questions/38541 | 1 | I am desperately searching for a paper I once have read, but cannot find anymore. It was about a counting function f(n) - the only thing I remember is that f's expression contained n! and 2n - and the question in the title was something like "What does f(n) count?". The paper compared three or so different approaches, ... | https://mathoverflow.net/users/2672 | "What does f(n) count?" - in search for a paper | Could it be [*The answer is $2^n·n$! What's the question?*](http://dx.doi.org/10.2307/2589493) by Gary Gordon (Amer. Math. Monthly **106** (1999), no. 7, 636–645)?
| 10 | https://mathoverflow.net/users/989 | 38552 | 24,740 |
https://mathoverflow.net/questions/38549 | 10 | I am trying to understand what exactly Ngo's support theorem says about how the cohomology of compactified Jacobians varies over the versal deformation of a curve with plane singularities.
That is, let $C$ be a compact complex curve with singularities of embedding dimension 2. Let $\mathcal{C} \to \mathbf{V}$ be a v... | https://mathoverflow.net/users/4707 | Ngo's support theorem & the Jacobian over the versal deformation of a planar curve | Very partial answer - I don't think I can comment yet...
I found it helpful to rephrase the statement of the support theorem like this:
Let $R\pi\_\ast \mathbb Q = \bigoplus \_i IC\_{Z\_i}(L\_i)[d\_i]$ be the decomposition of the pushforward sheaf. If Z is one of the $Z\_i$ appearing in the sum above, then $Z$ is t... | 6 | https://mathoverflow.net/users/7762 | 38557 | 24,744 |
https://mathoverflow.net/questions/1988 | 19 | There is supposed to be a strong analogy between the arithmetic of number fields and the arithmetic of elliptic curves. One facet of this analogy is given by the class number formula for the leading term of the Dedekind Zeta function of K on the one hand and the conjectural formula for the leading term of the L functio... | https://mathoverflow.net/users/493 | Regulators of Number fields and Elliptic Curves | Let me first give you a heuristic "reason", why the regulator in the class number formula looks different from the regulator in the Birch and Swinnerton-Dyer conjecture. It is often more convenient (and more canonical) to combine the regulator and the torsion term in the Birch and Swinnerton-Dyer conjecture: one choose... | 16 | https://mathoverflow.net/users/35416 | 38558 | 24,745 |
https://mathoverflow.net/questions/38556 | 2 | Hi,
consider the following random walk on the lattice $\{0,\dots,n\}^2$. It starts at $(0,0)$ and then move either up or right, with probability respectively $p$ and $1-p$. Once it reaches the right border (respectively the up border), it goes up (respectively it goes right) to $(n,n)$. What properties do we know abo... | https://mathoverflow.net/users/9203 | Random walk on a two-dimensional uniform grid | Turn your square one fourth turn to the left and project it down.
To give some details: consider a simple random walk $Y$ on $\mathbb{Z}$ that is constrained to go left when yours goes up, and to go right when yours goes right. When your walk is on the boundary, then draw randomly independently the direction of $Y$. ... | 4 | https://mathoverflow.net/users/4961 | 38564 | 24,751 |
https://mathoverflow.net/questions/38577 | 1 | If a group G has a subgroup H of finite index which is torsion free, then does it satisfy $H\_\ast (G,Q) = H\_ \ast (H,Q)$? (probably it is very well known...)
| https://mathoverflow.net/users/9205 | a small question about group homology | No, it is not true. For example, let $\mathbf{Z}/2$ act on $\mathbf{Z}$ by inversion, and $G$ be the semidirect product. Then $\mathbf{Z}$ is a torsion-free finite index normal subgroup of $G$, but one easily computes that the rational cohomology of $G$ is trivial, by the Hochschild-Leray-Serre spectral sequence for th... | 6 | https://mathoverflow.net/users/318 | 38584 | 24,759 |
https://mathoverflow.net/questions/38586 | 7 | The $n$-th Mersenne number $M\_n$ is defined as
$$M\_n=2^n-1$$
A great deal of research focuses on Mersenne primes. What is known in the opposite direction about Mersenne numbers with only small factors (i.e. smooth numbers)? In particular, if we let $P\_n$ denote the largest prime factor of $M\_n$, are any results kno... | https://mathoverflow.net/users/8938 | Smoothness in Mersenne numbers? | I can give you a slightly better upper bound. Recall that $2^n - 1 = \prod\_{d | n} \Phi\_d(2)$ where $\Phi\_d$ is a cyclotomic polynomial. Now,
$$\Phi\_d(2) = \prod\_{(k, d) = 1} (2 - \zeta\_d^k) \le 3^{\varphi(d)}$$
so that in particular the largest prime factor of $2^n - 1$ is at most $3^{\varphi(n)}$. By taking... | 3 | https://mathoverflow.net/users/290 | 38591 | 24,764 |
https://mathoverflow.net/questions/38551 | 2 | Consider two countable families of objects, given as unions of finite subfamilies:
$F^k = \bigcup\_{n \in \mathbb{N}} F^k\_n$, *k* = 1,2.
Let there be a bijection $f: F^1 \rightarrow F^2$ such that $x \in F^1\_n \Leftrightarrow f(x) \in F^2\_n$ for all *n* $\in \mathbb{N}$.
This means: The two families have the s... | https://mathoverflow.net/users/2672 | Equivalence of families of objects with the same counting function | If you adopt Philippe Nadeau's proposal to define "essentially the same" in terms of isomorphism of species, then a standard example of "accidental" is the following. For any finite set $S$, there are just as many linear orderings of S as there are permutations of $S$ (i.e., bijections from $S$ to itself), namely $|S|!... | 7 | https://mathoverflow.net/users/6794 | 38592 | 24,765 |
https://mathoverflow.net/questions/38570 | 7 | Let $L$ be the constructible universe and $x,y \in L$ such that $x \subseteq y$. Is then $x \in D(y)$, i.e. $x$ a definable subset of $y$?
If this is not true, do we at least have the following: If $x,y \in L$, then $\rho(x) \leq \rho(y)$? Here $\rho$ denotes the $L$-rank.
And if this is also false, how do you prov... | https://mathoverflow.net/users/2841 | Inclusions of constructible sets are *not* definable | If you just want to prove that $L$ satisfies the power set axiom, you don't need condensation; Kunen's comment is correct. Given $x\in L$, to show that $L\cap P(x)\in L$, first consider an arbitrary $y\in P(x)\cap L$ and observe that, since $y\in L$, there is a first ordinal $\alpha(y)$ such that $y\in L\_{\alpha(y)}$.... | 7 | https://mathoverflow.net/users/6794 | 38594 | 24,766 |
https://mathoverflow.net/questions/38599 | 2 | Let X be a variety and $E$ an ample vector bundle on $X$. Let $G=G(r+1,E)$ be the Grassmann bundle over $X$ whose fiber over $x\in X$ is the Grassmannian of the $r+1$-dimensional subspaces of $E\_x$. Let $U$ denote the universal subbundle on $G$. Under which hypothesis is the dual of $U$ ample on $G$?
| https://mathoverflow.net/users/33841 | Ampleness of the universal subbundle | I think this is essentially never true, again by restricting to a fiber over $x\in X$. The problem is that (somewhat counterintuitively) the universal quotient bundle on $Gr(k,n)$ is not ample, and for the same reason, neither is the dual of the universal sub. (Except of course when $k=1$!) See Examples 6.1.5 and 6.1.6... | 5 | https://mathoverflow.net/users/5081 | 38602 | 24,768 |
https://mathoverflow.net/questions/38529 | 4 | What is a quotient of an affine scheme that is not a universal quotient? Let's recall some terminology.
Suppose that $k$ is an algebraically closed field and $G$ is a reductive group acting on an affine scheme $X$. Theorem 1.1 of [Geometric Invariant Theory](http://books.google.com/books?id=dFlv3zn_2-gC&printsec=fro... | https://mathoverflow.net/users/5337 | Uniform Quotient vs Universal Quotient | Here is an example, which is in some sense the simplest one. Suppose that $k$ has characteristic $p > 0$; set $X := \mathop{\rm Spec} k[x,y]$. Let $G$ be a cyclic group of order $p$ acting via $(x,y) \mapsto (x, x+y)$. The ring of invariants is $k[u,v] := k[x, y^p - x^{p-1}y]$. Consider the point $\mathop{\rm Spec} k =... | 3 | https://mathoverflow.net/users/4790 | 38604 | 24,770 |
https://mathoverflow.net/questions/38019 | 27 | It was asked in the Putnam exam of 1969, to list all sets which can be the range of polynomials in two variables with real coefficients. Surprisingly, the set $(0,\infty )$ can be the range of such polynomials. These don't attain their global infimum although they are bounded below. But is it also possible that such po... | https://mathoverflow.net/users/9075 | Zeros of Gradient of Positive Polynomials. | $(1+x+x^2y)^2+x^2$
| 52 | https://mathoverflow.net/users/1131 | 38634 | 24,776 |
https://mathoverflow.net/questions/34724 | 16 | ### Overview
For integers n ≥ 1, let T(n) = {0,1,...,n}n and B(n)= {0,1}n. Note that |T(n)|=(n+1)n and |B(n)| = 2n.
A certain set S(n) ⊂ T(n), defined below, contains B(n). The question is about the growth rate of |S(n)|. Does it grow exponentially, like |B(n)|, so that |S(n)| ~ cn for some c, or does it grow superex... | https://mathoverflow.net/users/8201 | Counting certain arrangements of n triangles. Does the count grow superexponentially? | Your sequence is bounded by $(125+\epsilon)^n$. Obviously, this isn't close to a good bound, but it answers the question.
We start by bounding a different question: Let $\Gamma\_n$ be the convex hull of $(0,0)$, $(0,n)$ and $(n,n)$. (So $\Gamma$ is rotated $180^{\circ}$ with respect to your $\Delta$.) Let $q\_n$ be t... | 6 | https://mathoverflow.net/users/297 | 38637 | 24,778 |
https://mathoverflow.net/questions/38581 | 5 | Let
\begin{equation}
R(x) = \sum\_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k})
\end{equation}
where $\mu$ is the Mobius function and
\begin{equation}
li(x) = \int\_0^x \frac{dt}{\log t}
\end{equation}
Is there a proof in the literature of
\begin{equation}
\pi(x)=R(x)-\sum\_{\rho}R(x^{\rho})
\end{equation}
where $\pi$ is... | https://mathoverflow.net/users/2011 | Proof in the literature of an equality involving the prime counting function | Stopple, A Primer of Analytic Number Theory, proves a theorem which looks something like the one under discussion. On page 248, he has $$\pi(x)=R(x)+\sum\_{\rho}R(x^{\rho})+\sum\_{n=1}^{\infty}{\mu(n)\over n}\int\_{x^{1/n}}^{\infty}{dt\over t(t^2-1)\log t}$$
You say that the literature treats your formula as a fact, ... | 2 | https://mathoverflow.net/users/3684 | 38646 | 24,785 |
https://mathoverflow.net/questions/38633 | 2 | This may not be precise enough for MO, but I'll give it a go.
Let $M$ be a symmetric closed monoidal model category with unit $u\in Ob(M)$. We define the vertices of an object $A$ to be points $x\in \operatorname{Hom}\_M(u,A)$. We define the fibre of $f:B\to A$ over a vertex of $x$ of $A$ to be the pullback of $f$ by... | https://mathoverflow.net/users/1353 | In what generality does the following statement hold: A fibration is acyclic if and only if all fibres are contractible fibrant objects. | Consider sSet×sSet with the induced structure from sSet. The monoidal unit is (1,1) (which is trivially fibrant, as is the unit in any cartesian monoidal model category), and so an object of the form (X,0) has no vertices at all (with your definition). Hence any map (X,0)→(Y,0) has (vacuously) all its fibers contractib... | 3 | https://mathoverflow.net/users/49 | 38653 | 24,788 |
https://mathoverflow.net/questions/38622 | 3 | Let $Dom$ be a uniform space, and $\hspace{.04 in}f$ be a continuous function from $Dom$ to itself satisfying:
1. For all non-empty open subsets $U$ and $V$ of $Dom$, there exists a natural
number $n$ and a member $x$ of $U$ such that $f^n(x)$ is a member of $V$.
2. The periodic points of $f$ are dense in $Dom$.
... | https://mathoverflow.net/users/nan | Chaos in uniform spaces | I believe the following works. Follow the proof in the reference supplied by Matthew Daws: choose two distinct periodic orbits and choose a compatible pseudometric $\rho$ such that all points in those orbits are at least $1$ apart under this pseudometric.
The proof establishes that the entourage $\lbrace(x,y):\rho(x,y)... | 2 | https://mathoverflow.net/users/5903 | 38667 | 24,795 |
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