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https://mathoverflow.net/questions/124558 | 1 | Let $\Gamma\subset {SL}\_2(\mathbb Z)$ be a congruence subgroup, and let $X(\Gamma)$ be the associated compact modular curve. The inclusion $\Gamma \subset SL\_2(\mathbb Z)$ induces a canonical projection $p:X(\Gamma)\to X(1)$ which is a non-constant holomorphic map of compact Riemann surfaces, where $X(1)=X(SL\_2(\mat... | https://mathoverflow.net/users/32209 | Algebraicity of the canonical projection $X(\Gamma)\to X(1)$ and of $X(\Gamma)$ | Let $X(\Gamma)\to \mathbf{P}^1\_{\mathbf C}$ be the composition of the natural map $X(\Gamma)\to X(1)$ associated to the inclusion $\Gamma\subset$ SL$\_2(\mathbf Z)$, and the $j$-invariant $j:X(1)\to \mathbf{P}^1\_{\mathbf C}$. This map is, as you said, ramified over at most three points. Since these three points are a... | 1 | https://mathoverflow.net/users/4333 | 124569 | 69,751 |
https://mathoverflow.net/questions/124565 | 1 | I am a grad student working on some independent research trying to derive some exact formulas for a particular class of power series. During my study I came across the following integral which would lead to an exact solution of my power series if I can find an exact anti-derivative.
$\int^x\_0 \frac{u^{r-1}}{1-u}\, ... | https://mathoverflow.net/users/32214 | For what values of the parameter does this function have an elementary anti-derivative? | Hi,
Your integral is the incomplete beta function, and has elementary expressions when $r$ is equal to an integer or half-integer. For example, you found the value for $r=1/2$, and when $r=3$ it is
$\frac{1}{2} \left(-2 x-x^2-2 \log(1-x)\right) $.
I haven't found a good reference for this yet; a quick check in Ma... | 1 | https://mathoverflow.net/users/8955 | 124574 | 69,753 |
https://mathoverflow.net/questions/124494 | 9 | In ZFC set theory (or better in NBG set theory, where the language is more flexible with proper classes), we have that every unbounded class of ordinal numbers is a proper subclass of the class On of all ordinals and that every such proper class has a unique bijection (by the enumeration function) with the proper class... | https://mathoverflow.net/users/30395 | Bijective-equivalent collections of proper classes in set theory | I will give a partial answer for models of $NBG$ in which local choice ($AC$) holds, namely:
>
> **Theorem**. There is a model $M$ of $ZFC$ whose natural expansion $(M,\cal{A})$ to a model of $NBG$ (where $\cal{A}$ is the collection of all parametrically definable classes of $M$) has at least 3 bijection-inequivale... | 4 | https://mathoverflow.net/users/9269 | 124584 | 69,756 |
https://mathoverflow.net/questions/122283 | 5 | We have a digraph D=(V,A) and its arc set A is partitioned into classes. We can flip the classes, which means changing the direction of all the arcs in the class.
Is there any result on when can we make the digraph acyclic with this operation?
Even such special case would be interesting for me such as:
* the unde... | https://mathoverflow.net/users/24634 | When can we make a digraph acyclic by fliping groups of arcs? | A teacher of mine (Zoltán Király) proved it to be NP-complete by reducing the 3-SAT problem to this problem: for each clause there should be an indirected circle with 4 edges. The classes and the starting orientation are:
* one edge from each circle as a referential edge directed clockwise.
* the other 3 edges of eac... | 3 | https://mathoverflow.net/users/24634 | 124600 | 69,764 |
https://mathoverflow.net/questions/124577 | 2 | It is a well known fact that the irreducible representations of $SL\_2(\mathbb{F}\_p)$ over $\overline{\mathbb{F\_p}}$ are given by the symmetric powers $Symm^k(V)$, where $V = \overline{\mathbb{F}\_p}^2$, for $k$ ranging in $0,\ldots,p-1$. In the case of $GL\_2(\mathbb{F}\_p)$ irreducible representations are exhausted... | https://mathoverflow.net/users/18478 | Char $p$ representations of $SL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$ | The short answer to your basic question is no; but a lot is known (and written down in various places including a 1978 paper in J. Algebra by D.J. Glover based on his A.N.U. thesis). The long answer is that the bookkeeping involved even for this small case gets quite involved, a little more so for the finite general li... | 7 | https://mathoverflow.net/users/4231 | 124607 | 69,766 |
https://mathoverflow.net/questions/124588 | 18 | My main research has been in hyperbolic geometry and geometric group theory. I always thought that the only real "application" of my work was that the universe is a 3-manifold.
But recently I found out that hyperbolic 3-space arises in a natural way from relativity: according to the work of Einstein and others, there... | https://mathoverflow.net/users/27933 | How does hyperbolicity of space time affect our lives? | I would suggest John Barrett's essay on [The Hyperbolic Theory of Special Relativity](http://arxiv.org/abs/1102.0462) as a comprehensive answer.
>
> The principle of relativity corresponds to the hypothesis that the kinematic space is a space of constant negative curvature. The value of the radius of curvature is t... | 17 | https://mathoverflow.net/users/11260 | 124609 | 69,768 |
https://mathoverflow.net/questions/124602 | 2 | Given $\sigma$ a shift map, $m$ - a Markov measure, $C\_a$, $C\_b$ - cylinder sets.
Suppose $P \in C\_b$. The problem is to show the following
\begin{equation}
m(C\_a \cap \sigma^{-1}(P)) = \frac{m(C\_a \cap \sigma^{-1}(C\_b))}{m(C\_b)} m(P).
\end{equation}
This seems to be obvious, but I don't see how to prove this in... | https://mathoverflow.net/users/32223 | The property of a Markov measure | There is a more conceptual explanation. This is a direct corollary of the Markov property formulated in "invariant form": **the past and and the future are conditionally independent with respect to the present.**
Your identity can be rewritten as
$$
\frac{m(C\_a \cap \sigma^{-1}(P))}{m(P)} = \frac{m(C\_a \cap \sigma... | 1 | https://mathoverflow.net/users/8588 | 124619 | 69,771 |
https://mathoverflow.net/questions/124613 | 3 | Suppose $(M,g)$ and $(N,g')$ are smooth Riemannian manifolds and $J^r(M,N)$ is the
smooth manifold of $r$-jets $j^r\_xf$ of smooth maps $f:M\to N$.
Is there an 'induced' Riemannian metric $g''$ on $J^r(M,N)$?
Of course the term 'induced' is conceptual vague here...
That's because, I don't want to restrict the q... | https://mathoverflow.net/users/21302 | Induced Riemannian metric on Jet-Manifold | There are, of course, several different functorially induced metrics on $J^r(M,N)$ when $M$ and $N$ are endowed with given Riemannian metrics.
For example, $J^0(M,N)=M\times N$ and one can just take the product metric.
Meanwhile $J^1(M,N)$ can be regarded as a vector bundle over $M\times N$ with fiber $T^\ast\_xM\... | 9 | https://mathoverflow.net/users/13972 | 124620 | 69,772 |
https://mathoverflow.net/questions/124627 | 2 | The last part of the paper *Located Sets and Reverse Mathematics* [Journal of Symbolic Logic 65 (1999), 1451–1480] by Giusto and Simpson involves a proof as follows:
Given $A$ an effectively immune set, i.e. there exists a recursive function $p$ such that $A$ is infinite and $W\_e\subseteq A$ implies $|W\_e|< p(e)$, ... | https://mathoverflow.net/users/23835 | Indices of r.e. sets | Edit: restoring my original answer, because I now believe it's correct.
It's not true, in general.
First, there's a total computable function $h$ such that if $\phi\_e(e)\downarrow$, then $\phi\_{h(e)}(h(e))\downarrow$ and outputs the modulus of $\phi\_e(e)$. Next, suppose $A$ could compute such a $g$. Then $A$ can... | 4 | https://mathoverflow.net/users/32178 | 124636 | 69,779 |
https://mathoverflow.net/questions/124623 | 1 | Let $S$ be a smooth, projective surface over $\mathbb{C}$ and let $\xi$ be a $0$-dimensional subscheme. I look at the sheaf $\underline{Ext}^2(O\_\xi,O\_S)$. Is it true that, if $\xi$ is curvilinear, then $\underline{Ext}^2(O\_\xi,O\_S)=O\_\xi$? What happens instead if $\xi$ is not curvilinear?
| https://mathoverflow.net/users/33841 | Curvilinear subschemes and Ext functors. | Yes, it is true. Indeed, if $\xi$ is curvilinear then it is lci, so locally there is a resolution
$$
0 \to O\_S \xrightarrow{ (g,-f) } O\_S \oplus O\_S \xrightarrow{ (f,g) } O\_S \to O\_\xi \to 0.
$$
Applying local $Hom$ to $O\_S$ one obtains an exact sequence
$$
0 \to O\_S \xrightarrow{ (f,g) } O\_S \oplus O\_S \xrigh... | 2 | https://mathoverflow.net/users/4428 | 124638 | 69,780 |
https://mathoverflow.net/questions/124633 | 13 | One important property of the [Robinson-Schensted correspondence](http://en.wikipedia.org/wiki/Robinson-Schensted_correspondence) (RS) is that the longest increasing subsequence of the permutation $\sigma$ is $\lambda\_1$, the first entry of the shape $\lambda(\sigma)$ of the tableaux associated to $\sigma$. [Greene's ... | https://mathoverflow.net/users/7717 | Generalization's of Greene's Theorem for the Robinson-Schensted correspondence | For RSK the answer is "well known". You can find the statements neatly arranged in an article by Christian Krattenthaler <http://arxiv.org/abs/math/0510676>.
I think the right framework for this question is Sergey Fomin's theory of dual graded graphs.
However, I don't think there are many other insertion algorithms w... | 12 | https://mathoverflow.net/users/3032 | 124651 | 69,788 |
https://mathoverflow.net/questions/124677 | 1 | Which are the finite groups $(G,\cdot)$ with the following property: for every $f \in Aut(G)$, there are $g,h\in Aut(G)$ such that $f(x)=g(x)\cdot h(x), \forall x\in G?$
I already have verified that:
(i) such a group is abelian;
(ii) all finite abelian groups of odd order have this property;
(iii) a finite abelia... | https://mathoverflow.net/users/17565 | A question on automorphisms of finite abelian groups | Here is an outline for odd order abelian $p$-groups:
The main point is that every non-zero element in a field with at least three elements is a sum of two non-zero elements. Using this, you can show that irreducible matrix in $GL\_n(\mathbf Z/p\mathbf Z)$ can be written as a sum of two non-singular matrices, for such... | 3 | https://mathoverflow.net/users/9672 | 124685 | 69,803 |
https://mathoverflow.net/questions/124697 | 1 | Let $G$ be a finite group. A chain of subgroups of $G$ of length $d$ is a sequence of subgroups of the form
$$ \{e\}=G\_0 \subsetneq G\_1 \subsetneq \ldots \subsetneq G\_{d-1} \subsetneq G\_d=G. $$
The length $l(G)$ of $G$ is, by definition, the length of the longest chain of subgroups of $G$.
Let $\Lambda(G)$ be the... | https://mathoverflow.net/users/31670 | length of a finite group versus number of conjugacy classes of subgroups | Let $G=(\mathbb{Z}/p\mathbb{Z})^n$, for $p$ a prime and $n>1$.
Then $l(G)=n$ but $\Lambda(G)$ can be made as large as you like by choosing $p$ large.
| 6 | https://mathoverflow.net/users/22989 | 124699 | 69,810 |
https://mathoverflow.net/questions/124702 | 9 | Is there a group $G$ with the property that $G$ is a smooth manifold, the multiplication map of $G$ is smooth, but the inversion map of $G$ is not smooth?
| https://mathoverflow.net/users/31858 | Group G hasn't all conditions of Lie group | [Robert L. Bryant "An Introduction to Lie Groups and Symplectic Geometry"](https://www.math.duke.edu/~bryant/ParkCityLectures.pdf) requires in the definition of a Lie group only that the multiplication map be smooth, and then proves that the inversion map must be smooth also. (Proposition 1, page 14.)
| 19 | https://mathoverflow.net/users/10503 | 124704 | 69,812 |
https://mathoverflow.net/questions/124637 | 14 | I recently heard a talk about these topics and found them very interesting.
The talk was centered on the formal structure and didn't really focus on examples.
So my question is: what is your favorite application of topological chiral homology? (or its other variants and specialisations)
| https://mathoverflow.net/users/25442 | Applications of topological chiral homology and factorization algebras (aka higher Hochschild cohomology) | It might be worth pointing out that topological chiral homology is a specialization to the topological (or locally constant) setting of a construction that originated (AFAIK) in conformal field theory, namely the functor of conformal blocks for a vertex algebra and its derived formulation
as chiral homology, due to Bei... | 14 | https://mathoverflow.net/users/582 | 124712 | 69,815 |
https://mathoverflow.net/questions/124708 | 13 | For a prime $p$, consider the graph on the vertex set ${\mathbb F}\_p$, in
which every vertex $z$ is adjacent to $z\pm 1$ and also to $z^{-1}$ (unless
$z=0$). I was told that this graph is known to be an expander, but the person
who told me this couldn't recall where exactly this graph has been studied.
Does anybody kn... | https://mathoverflow.net/users/9924 | An expander (?) graph | Here is Theorem 4.4.2 in Lubotzky's lovely book "Discrete groups, expanding graphs and invariant measures". On the projective line over $\mathbb{F}\_p$, connect $z$ to $z\pm 1$ and to $-\frac{1}{z}$; this is a family of 3-regular expander graphs. The proof is indeed based on Selberg's 3/16-theorem.
| 18 | https://mathoverflow.net/users/14497 | 124717 | 69,819 |
https://mathoverflow.net/questions/124720 | 1 | Is ZFC+Con(ZFC) powerful enough to show there isn't any standard model of ZFC? What you think about it?
| https://mathoverflow.net/users/29570 | Standard model of ZFC | Many set theorists define that a model $\langle M,E\rangle$ of set
theory is *standard* to mean that the set membership relation $E$
of the model is the actual set membership relation $\in$,
restricted to objects in $M$.
The existence of such a standard model of ZFC is equivalent to the
existence of a well-founded mo... | 19 | https://mathoverflow.net/users/1946 | 124723 | 69,821 |
https://mathoverflow.net/questions/124706 | 4 | Dear all,
I have the following question:
Let $M$ be a subgroup of $GL\_n(p)$ where $p$ is a prime and let $M'$ denotes the derived subgroup of $M$. Can we conclude that $|M:M'|\leq p^n-1$?
Any counterexample or reference is very much appreciated. Thank you in advance.
| https://mathoverflow.net/users/32259 | Subgroups of the general linear groups | REVISION: In fact, it is a consequence of the (now proved) so-called $k(GV)$-problem that the answer is indeed affirmative when $M$ has order prime to $p,$ and something rather stronger holds. There is a book about the $k(GV)$-problem by Peter Schmid. The $k(GV)$-problem is a special case of a problem of R. Brauer. The... | 8 | https://mathoverflow.net/users/14450 | 124728 | 69,822 |
https://mathoverflow.net/questions/124725 | 0 | Let $G$ be a Lie group, $K\subseteq G$ be a compact group and $N\subseteq$ be a nilpotent group s.t. $N\cap K= \{e\}$. Let $H=N\rtimes K$ be the semidirect product of $N$ and $K$ and let $\Gamma$ be a discrete subgroup of $H$. Is it true that $\Gamma$ has a nilpotent subgroup of finite index. Also, can we guaranty that... | https://mathoverflow.net/users/31071 | Nilpotent subgroups of uniform finite index | Start with this paper: A. I. Mal′cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949), 9–32 (look also [here](http://www.math.utoronto.ca/~vtk/NILPnew1.pdf)). This reduces your problem to the case when $\Gamma$ is a lattice in $N\rtimes K$. Next, observe that projection of $\Gamma$ to ... | 4 | https://mathoverflow.net/users/21684 | 124734 | 69,826 |
https://mathoverflow.net/questions/48790 | 3 | The category of simplicial presheaves on a small Grothendieck site $\mathcal{C}$ can be given a model structure by defining weak equivalences and cofibrations sectionwise. It's called the (global) injective model structure and has a mapping space functor $Hom$, given by $Hom(X, Y)\_n = hom(X\times \Delta^n, Y)$.
Usin... | https://mathoverflow.net/users/11437 | Local injective model structure for simplicial presheaves | Many statements in the question are incorrect, and so I want to put an answer here to prevent future visitors to this thread from being confused. First, it is **NOT TRUE that localizing at the local weak equivalences is the same as localizing at S** $= \lbrace X \to L^2X\rbrace$ where $L^2$ is sheafification. From the ... | 3 | https://mathoverflow.net/users/11540 | 124738 | 69,827 |
https://mathoverflow.net/questions/124726 | 5 | Let $D$ and $E$ be toposes and let $f\_{\ast}\colon D\to E$ be the direct image part of a geometric morphism $(f^{\ast},f\_{\ast})$ between them. Considered as categories, we have (covariant) power-object endofunctors on each:
$$P\_D\colon D\to D \hspace{.5in} P\_E\colon E\to E$$
where, for a morphism $\phi$ in $D$ we ... | https://mathoverflow.net/users/2811 | When does the direct image functor nicely push past the power/exists functor? | Since every power object is an internal Heyting algebra, and $f\_\*$ preserves the structure of internal Heyting algebras, there are trivial examples of such natural transformations corresponding to the constants $\top$ and $\bot$. Of course, this is uninteresting.
Let me write $P^{\mathcal{D}}$ and $P^{\mathcal{E}}... | 5 | https://mathoverflow.net/users/11640 | 124741 | 69,830 |
https://mathoverflow.net/questions/124744 | 2 | Suppose there are $r$ linearly independent vectors $v\_1,\dots,v\_r\in \mathbb{R}^n$, all of them have integer-valued entries and $\|v\_i\|\_\infty\leq m$ for some integer $m$.
My goal is to find an integral basis $w\_1,\dots,w\_{n-r}$ in the orthogonal complement to $\text{span}(v\_1,\dots,v\_r)$, such that $\max\_i... | https://mathoverflow.net/users/14432 | integral basis of orthogonal complement | The situation in which we seek a single vector in the orthogonal complement with small entries is addressed by [Siegel's lemma](http://en.wikipedia.org/wiki/Siegel%27s_lemma). Regarding the basis problem, there is a general and very sharp result of [Bombieri and Vaaler](http://link.springer.com/article/10.1007%252FBF01... | 4 | https://mathoverflow.net/users/630 | 124745 | 69,832 |
https://mathoverflow.net/questions/124655 | 3 | I'm aware of Bianchi's (local) classification of homogenous 3-manifolds into the Bianchi types I through IX, and I can follow the algebra for classifying the Lie algebras. However, I still can't visualize the different spaces, except for the simpler ones. For example, I can see that type I is just locally Euclidean $E^... | https://mathoverflow.net/users/26762 | Visualizing Bianchi type/homogenous spaces | For a different viewpoint from the excellent treatments by Scott and Thurston of 3-dimensional geometries, if you are trying to get a feel for the homogeneous Riemannian $3$-manifolds (which, as noted, were first classified by Bianchi), you might want to try looking at them from the point of view of their most basic in... | 8 | https://mathoverflow.net/users/13972 | 124751 | 69,834 |
https://mathoverflow.net/questions/124665 | 3 | Let $\mathbf{p}=(p\_1,\dots,p\_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X\_1,\dots,X\_m)$ be a vector of independently-distributed binomial random variables such that $X\_i\sim \text{Binom}(n,p\_i)$. Further, let $h(\mathbf{X})$ be a convex function on $[0,n]^m$.
**Question:** Is the real-valued function $g(... | https://mathoverflow.net/users/31735 | Is the Binomial Expectation of a Multivariate Convex Function Convex in the Vector p? | The answer is no, a counterexample is for $n=1$, $m=2$: let $h$ be zero on (0,0), (0,1), (1,0), and let $h(1,1)=1$. It is rather clear that you can get it for a convex $h$.
Then, $g(p\_1,p\_2)=p\_1 p\_2$, which is not a convex function on the unit square.
| 5 | https://mathoverflow.net/users/31371 | 124759 | 69,837 |
https://mathoverflow.net/questions/124747 | 1 | Hello
I am trying to solve a problem involving a cross-country ski trail map. I wish to travel every trail on the map, at least once, but no more than twice (so I can out-and-back on a destination, for example).
I don't believe this is a TSP problem, as TSP seems to be point oriented (ensuring all points on a map a... | https://mathoverflow.net/users/32268 | TSP, but for all routes not all points | Look up [Chinese postman problem](http://en.wikipedia.org/wiki/Chinese_postman_problem). This can be done if and only if your graph is connected.
Assuming you want to start and end at the same point, you look at the vertices of odd degree and find a minimum-cost perfect matching (where the cost is distance in the gr... | 4 | https://mathoverflow.net/users/13650 | 124762 | 69,838 |
https://mathoverflow.net/questions/122683 | 7 | Let $X = \operatorname{Spec}(R)$ be a conical Poisson variety over $\mathbb{C}$. This means
that $R$ is a non-negatively graded Poisson algebra over $\mathbb{C}$
with $R\_0 = \mathbb{C}$ and that the Poisson
bracket is homogeneous of negative degree. Assume that $X$ admits a conical symplectic resolution.
Examples to ... | https://mathoverflow.net/users/10273 | Fundamental groups of symplectic leaves | I do not know the answer. I believe that it is so. See <http://arxiv.org/abs/1301.1008> for algebraic fundamental groups.
If you consider quiver varieties of affine types, they are moduli spaces of instantons on ALE spaces. If you replace the base space by $\mathbb R^4$ and assume rank is $2$, then the finiteness fol... | 4 | https://mathoverflow.net/users/3837 | 124766 | 69,840 |
https://mathoverflow.net/questions/124765 | 3 | Consider a probability space $(X,F,\mu)$, and the quotient $G$ of the sigma-algebra $F$ by its null sets. Endow $G$ with the metric $d(A,B) = \mu(A \triangle B)$. Is $(G,d)$ a compact metric space?
When $F$ is generated by a (countable) partition, the answer is yes. Is there a general argument to prove that $(G,d)$ i... | https://mathoverflow.net/users/4661 | Compactness of sigma-algebra for the $L^1$ metrics | Take the classical Bernoulli scheme with the base $(1/2,1/2)$, and let $A\_n$ be the set where the $n$-th coordinate is 1. Then $d(A\_n,A\_m)=1/2$ whenever $n\neq m$. Since all purely non-atomic Lebesgue spaces are isomorphic, the same is applicable to any purely-non-atomic Borel measure on a Polish space.
| 7 | https://mathoverflow.net/users/8588 | 124783 | 69,847 |
https://mathoverflow.net/questions/124754 | 51 | This is more of a question about terminology than about math.
The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it is. Many sources use the term in different ways.
Any classical holomorphic modular form for $\mathrm{SL}\_2(\mathbb{Z... | https://mathoverflow.net/users/1355 | What is the difference between an automorphic form and a modular form? | Very briefly: until work of Hans Maass c. 1949, "modular" or "automorphic" both referred to holomorphic functions invariant-up-to-cocycle (that is, invariant holomorphic sections of a bundle) on a quotient $\Gamma\backslash X$. For $X$ the upper half-plane, these were *ellipic* modular forms, visible since the 19th cen... | 55 | https://mathoverflow.net/users/15629 | 124785 | 69,849 |
https://mathoverflow.net/questions/124760 | 3 | I am recently reading the paper "Natural triangulations associated to a surface" by B.H. Bowditch and D.B.A. Epstein (<http://www.sciencedirect.com/science/article/pii/0040938388900080#>), where they described a natural triangulation for moduli space of a surface. In the first part they produces a ideal triangulation (... | https://mathoverflow.net/users/9485 | Triangulation of moduli space. | Since $N$ is compact, $S - N$ is not compact. In particular $S - N$ is homeomorphic to a punctured surface: here a *punctured surface* is a compact surface without boundary minus a finite set of points.
It may help to copy out their definitions word-by-word. Then find an example for each definition.
| 5 | https://mathoverflow.net/users/1650 | 124786 | 69,850 |
https://mathoverflow.net/questions/124771 | 4 | Given a Polish space $X$, I note $C\_b(X)$ the set of the continuous bounded functions with the norm of the uniform convergence, and $(C\_b(X))^\star$ its topological dual with the $\*-$weak convergence $\sigma((C\_b(X))^\star, C\_b(X))$.
To a Borel probability $P$ on $X$ we can associate a $l(P) \in (C\_b(X))^\star... | https://mathoverflow.net/users/32274 | Convergence of probability measure and the *-weak convergence ? | I believe that the answer is affirmative. Weak$^{\ast}$ topology on probability measures can be metrized (e.g. via [Lévy–Prokhorov metric](http://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric)). When we deal with a Polish space $X$, then this metric is complete. Using this, I would like to conclude that the ... | 1 | https://mathoverflow.net/users/24953 | 124788 | 69,852 |
https://mathoverflow.net/questions/124779 | 4 | I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously diagonalised? In the paper, one operator, $A$, is self adjoint and positive definite and the other, $a$ is bounded and pos... | https://mathoverflow.net/users/32277 | When are two operators simultaneously diagonalisable? | Even one positive definite operator on an infinite-dimensional Hilbert space need not have any eigenvectors at all: it might have continuous spectrum. The more general statement is the Spectral Theorem. Commuting bounded normal linear operators generate a closed commutative
$\*$-subalgebra $A$ of ${\mathcal B}(H)$, and... | 6 | https://mathoverflow.net/users/13650 | 124799 | 69,857 |
https://mathoverflow.net/questions/124789 | 3 | Let $G$ be a finite group with normal subgroup $N$. Let $\chi$ be an irreducible complex character of $G$. Then Clifford's Theorem says that $\mathrm{res}^{G}\_{N}\chi = e(\eta\_1 + \cdots + \eta\_r)$ where $e$ is a positive integer and each $\eta\_i$ is a complex irreducible character of $N$ such that the $\eta\_i$'s ... | https://mathoverflow.net/users/7443 | Character fields and Clifford's theorem | I don't think there's any clear cut relationship. In particular, the inequality $e \ge [\mathbb Q(\eta) : K]$ needn't hold. For example, take $G=S\_5$, $N=A\_5$ and let $\chi$ be the unique irreducible $S\_5$-character of degree $6$. Then we have $\mathrm{res}^{G}\_{N} \chi = \eta + \bar{\eta}$, where $\eta$ and $\bar{... | 6 | https://mathoverflow.net/users/430 | 124800 | 69,858 |
https://mathoverflow.net/questions/124753 | 16 | My question begins with a caveat: I sometimes spend time with topologists, but do not consider myself to be one. In particular, my apologies for any errors in what I say below — corrections are encouraged.
My impression is that manifold topologists like to consider three main categories of (finite-dimensional, paraco... | https://mathoverflow.net/users/78 | Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds? | First, it follows from the work of Kirby and Siebenmann that in dimensions $\le 6$ PL and DIFF categories are equivalent. In particular, if you are working in dimension 4 (where gauge-theoretic invariants are mostly used) then the answer to your your question is negative. Starting in dimension 7, there are smooth manif... | 10 | https://mathoverflow.net/users/21684 | 124801 | 69,859 |
https://mathoverflow.net/questions/124306 | 2 | Let me first give the intuition for my question: Suppose that you want to use a ruler to mark $n$ points in a line on a page, with 1 cm distance between neighbor points. There are two ways:
1- Mark the $i+1$th point 1 cm next to the $i$th point.
2- Mark the $i+1$th point $i$ cm next to the $1$st point.
If there i... | https://mathoverflow.net/users/24930 | Reducing the error of Algorithms by assigning variables formulas instead of values | You mean replace *double* and *int* by expressions as in [Symbolic computation](http://en.wikipedia.org/wiki/Symbolic_computation)?
It's feasible as long as your expressions can be put in a standard (canonical) form, like polynomials with coefficients ordered by decreasing degree or linearised trigonometric sums. It... | 1 | https://mathoverflow.net/users/26510 | 124807 | 69,861 |
https://mathoverflow.net/questions/124810 | 1 | The title has it all. I'm looking for a proof/disproof of the fact that an algebraically closed field, say $\mathbb K$, has characteristic zero iff the following property (R) holds: For all $n,k \in \mathbb N^+$, every invertible $n$-by-$n$ matrix with entries in $\mathbb K$ has at least one $k$-th root. The question i... | https://mathoverflow.net/users/16537 | All and the only algebraically closed fields s.t. any regular n-by-n matrix has a k-th root for every k | wccanard's comment gives one direction: If the field has characteristic $p$, then there is no matrix $A$ with $A^p=\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}$.
For the other direction, let the field have characteristic $0$. We want to show that $A^k=B$ has a solution $A$ for any $B$. Without loss of generality, $B$ is ... | 5 | https://mathoverflow.net/users/18739 | 124814 | 69,865 |
https://mathoverflow.net/questions/124816 | 2 | Is the following true?
Let $\Sigma$ be a compact orientable hypersurface without boundary in $R^n$. Then $R^n\setminus\Sigma$ has at least two connected components.
| https://mathoverflow.net/users/32284 | Alexander duality theorem | Yes: whenever $U \subset M$ is an open subspace with complement $Z$, then there is a long exact sequence
$$ \ldots\to H^\bullet\_c(U) \to H^\bullet\_c(M) \to H^\bullet\_c(Z) \to H^{\bullet+1}\_c(U)\to \ldots $$
which in this case gives
$$ H^{n-1}\_c(\mathbf R^n) = 0 \to H^{n-1}\_c(\Sigma) \to H^n\_c(\mathbf R^n \setmi... | 5 | https://mathoverflow.net/users/1310 | 124817 | 69,867 |
https://mathoverflow.net/questions/124805 | 0 | My question start with the following observations:
1. If you have a finite number of topological spaces $X\_1, \dots , X\_n$ you can define a space that is the disjoint union of its $\sqcup\_{i=1}^n X\_n=Y$.
2. If you have a cardinal number $I$ and a family of topogical spaces $(X\_i)\_{i\in I}$ we can define the dis... | https://mathoverflow.net/users/31524 | Existence of a Sub-Category of the Category of Topological Spaces | Spaces of the sort you are looking for are called *universal*.
**Definition:** A space $X$ is *universal* for a class of spaces $\mathcal{C}$ if it belongs to $\mathcal{C}$ and every space in $\mathcal{C}$ embeds into $X$.
Here are some examples:
1. Let $\Sigma = \lbrace{\bot, \top\rbrace}$ be the Sierpinski spac... | 9 | https://mathoverflow.net/users/1176 | 124819 | 69,869 |
https://mathoverflow.net/questions/124821 | 7 | Berkovich mentioned the following result of Mann in his book on p-groups:
*The number of nonlinear irreducible characters of given degree in a p-group is divided by p-1.*
Do you know any reference for this statement? Also, I would like to ask you if a similar assertion holds for conjugacy classes.
| https://mathoverflow.net/users/13641 | Characters of p-groups | Here is a more elementary argument than Geoff's. The number of linear characters of $P$ divides the order of $P$ and is hence a power $p^r$ of $p$. All the other characters
have degrees $d\_i$ of the form $p^{a\_i}$. Hence
$$ p^n = p^r +\sum\_i p^{2a\_i}. $$
I claim that if $p^n$ is written as a sum of powers of $p$,... | 11 | https://mathoverflow.net/users/2807 | 124829 | 69,874 |
https://mathoverflow.net/questions/124842 | 0 | What is commonly meant by Hurwitz's construction of simple covers?
| https://mathoverflow.net/users/4096 | Hurwitz's construction of simple covers | I think most people understand under this term the following: It is a branched cover $X\to\mathbb P^1(\mathbb C)$ of a compact connected Riemann surface $X$ to the Riemann sphere $\mathbb P^1(\mathbb C)$ such that the monodromy generators belonging to the branch points are transpositions. Or equivalently: If $n$ is the... | 3 | https://mathoverflow.net/users/18739 | 124843 | 69,879 |
https://mathoverflow.net/questions/124844 | 1 | I was curious as to whether given any problem, lets say:
$x^2-1 = 0$
There exists a function that given this set of symbols as input return the exact set of symbols contained in the answer. In this case $x=-1,x=1$
Surely, given that the process of getting from the problem's set of symbols to the solution's is no... | https://mathoverflow.net/users/32294 | Does there exist a function (however complex) which given an input in the form of any problem which can be solved in a rigorous and non-random way can return the solution to that problem. | Your question seems to concern the issue of the computability of solutions of computable functions, and the larger context for such a question is the subject known as [computable analysis](http://en.wikipedia.org/wiki/Computable_analysis).
Carl Mummert has [a very nice blog post concerning the following theorems](ht... | 5 | https://mathoverflow.net/users/1946 | 124850 | 69,881 |
https://mathoverflow.net/questions/123533 | 1 | Recently, I met a problem related to p-adic matrices in my research, the key of the problem can be summarized in the following way:
1: whether there exist spectrum theorem for p-adic matrix.\
2: whether invertible p-adic matrix is dense in $$C\_{p}^{n\times n}$$
3:whether diagonalizable p-adic matrix is dense in $$... | https://mathoverflow.net/users/11966 | Spectrum theorem for p-adic matrix analysis | All three results are true.
Beginning with the third, it suffices to show that the set of $n\times n$-matrices in $\mathbb{C}\_p$ whose characteristic polynomials have distinct roots is dense since these will certainly be diagonalizable.
Now the characteristic polynomial of a matrix may be described as a polynomia... | 2 | https://mathoverflow.net/users/345 | 124857 | 69,883 |
https://mathoverflow.net/questions/124811 | 0 | Suppose X is a normal topological space. Suppose some metric space for example.
If {$A\_n$}$\_{n=1}^{\infty}$ is a collection of pairwise disjoint closed subsets of X, can we find a continuous function on X such that it takes the constant value $n$ on $A\_n$..?
| https://mathoverflow.net/users/31772 | Can we generalize the result of Urysohn's lemma to countable collection of pairwise disjoint closed subsets of a normal space..? | As evidenced in the comments and David´s answer, the problem is that even if each $A\_n$ is closed, there can be too much "interaction" between the $A\_n$'s. A more extreme example would be to take $A\_n=\{ q\_n \}$ where $\{q\_n\}\_{n=1}^\infty$ is some enumeration of the rational numbers.
However if $\{A\_n \}\_{n=... | 0 | https://mathoverflow.net/users/17836 | 124859 | 69,885 |
https://mathoverflow.net/questions/124841 | 0 | Good morning,
I'm just curious about the following. With a compact Kahler manifold, we can associate an Albanese torus. This helps us a lot study the manifold.
**My question:** *Are there other holomorphic objects associated with a compact complex manifold?* I'm interested in the objects whose shape is well unders... | https://mathoverflow.net/users/11376 | Holomorphic objects associated with a compact complex manifold? | Besides the intermediate Jacobians for Kaehler manifolds, any compact complex manifold has a rational morphism to an algebraic variety, called the algebraic reduction, so that the morphism is an isomorphism of the fields of rational functions: K. Ueno, Classification theory of algebraic varieties and compact complex spa... | 1 | https://mathoverflow.net/users/13268 | 124871 | 69,891 |
https://mathoverflow.net/questions/124840 | 21 | Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to $g$.
Question: Does there exist an embedding into some high dimensional $\mathbb R^N$ with the following properties:
*... | https://mathoverflow.net/users/26935 | Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood? | It does not hold for hyperbolic plane.
It follows since the volume growth of the hyperbolic plane is exponential, while volume growth of $\mathbb{R}^N$ is polynomial.
**Postcript.** Let us say that a Riemannian manifold $M$ has polynomial volume growth if there is a polynomial $p$ such that volume of any $r$-ball in ... | 31 | https://mathoverflow.net/users/1441 | 124878 | 69,893 |
https://mathoverflow.net/questions/124848 | 0 | Suppose $M$ is a complex manifold and $\Omega$ a (edit: bounded) pseudoconvex domain in $M$. Let $u:M\setminus\Omega\to\mathbb{R}$ be a pluriharmonic function. Is it true that $u$ has a pluriharmonic extension to $M$? edit: $dim\_{\mathbb{C}}(M)\geq 2$.
| https://mathoverflow.net/users/32296 | Extension of pluriharmonic functions | Your condition $\dim M>2$ does not save the situation: you can have many counterexamples
with $M=M'\times C^n$ where $\dim M'=1$ and your functions are independent of the second
variable. And in dimension $1$ you certainly have plenty of pluriharmonic (=harmonic) functions which do not extend anywhere.
| 5 | https://mathoverflow.net/users/25510 | 124879 | 69,894 |
https://mathoverflow.net/questions/124839 | 2 | The standard model of Mercator projection shows a cylinder wrapped around a spherical earth eg [Wiki](http://en.wikipedia.org/wiki/Mercator_projection).
[Many sites](http://docs.openlayers.org/library/spherical_mercator.html) describe the resulting square map like this:
"...spherical Mercator maps use an extent of the... | https://mathoverflow.net/users/32293 | Deriving the Mercator projection algorithm | In a [webpage](http://www.math.ubc.ca/~israel/m103/mercator/mercator.html) on Mercator's projection, [Robert Israel](https://mathoverflow.net/users/13650/robert-israel) offers the following suggestion:
>
> If you want a physical model of
> Mercator's projection, let the globe
> be a spherical balloon that is blow... | 7 | https://mathoverflow.net/users/15837 | 124893 | 69,901 |
https://mathoverflow.net/questions/124899 | 14 | This is a problem I have had for a while. For a triangle, the side opposite the largest angle has the largest length (and similarly for smallest angle). For a tetrahedron, the question is whether the face opposite the largest solid angle (trihedral angle - the area of the spherical triangle with angles equal to the dih... | https://mathoverflow.net/users/32307 | Solid angles of a tetrahedron | I think this statement does not hold. The following tetrahedron should give a counterexample:
$A=(-0.5, 0, 0)$, $B=(1,0,0)$, $C=(0,\varepsilon^2, \varepsilon)$, $D=(0,\varepsilon^2, -\varepsilon)$, $1>>\varepsilon>0$.
Here $BCD$ has largest area, but the spherical angles at $C$ and $D$ should be significantly large... | 18 | https://mathoverflow.net/users/943 | 124902 | 69,906 |
https://mathoverflow.net/questions/124892 | 9 | In [Foncteurs analytiques et espèces de structures](http://link.springer.com/chapter/10.1007%2FBFb0072514), Joyal defines *virtual species*, as a (quotient) of formal differences of functors $F,G:\mathbb{B}\rightarrow \mathsf{Set}$, and then proceeds to show that these form a (commutative) ring. The quotienting operati... | https://mathoverflow.net/users/3993 | On the category of virtual species | I had tried to do something like this around 1994, and then again a little later as I will explain in a moment. The first time around, I had tried formalizing this in the context of species valued in a category of "virtual sets" (finite sets, to avoid an Eilenberg swindle), guided by Joyal's pretty observation that the... | 10 | https://mathoverflow.net/users/2926 | 124908 | 69,911 |
https://mathoverflow.net/questions/124918 | 1 | I found in Awodey's "Theory Category", second edition p34, a set of poset axioms to define Boolean algebras, whereas I don't see how they can be sufficient.
Here are the axioms:
>
> A Boolean algebra is a poset B
> equipped with distinguished elements
> 0, 1, binary operations $a \vee b$ of
> "join" and $a \we... | https://mathoverflow.net/users/29853 | Poset axioms of Boolean algebra | As remarked by Joel, this is indeed very direct starting from $a\vee b \le a \vee b$ and using the third axiom. Many thanks to have unstucked me.
| 1 | https://mathoverflow.net/users/29853 | 124922 | 69,916 |
https://mathoverflow.net/questions/124838 | 9 | In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d\_N differentials"), starting at the HOMFLY homology of a knot, and converging to the sl\_N Khovanov-Rozansky homology of the knot.
My question is: are there expected to be "intermediate" sp... | https://mathoverflow.net/users/8041 | Khovanov-Rozansky homology and spectral sequences | Yes. I think the existence of these sequences is "general knowledge" amongst knot homologists, though I'll be honest, I'm not sure where they are written down at the moment. Had you asked me 10 minutes ago, I would have said they were in the paper of Rasmussen you cite, but they seem to not actually be written down (th... | 7 | https://mathoverflow.net/users/66 | 124925 | 69,917 |
https://mathoverflow.net/questions/124920 | 1 | Let $X$ be a Polish space and $(P\_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C\_b(X)$ which is continuous and bounded $$\lim \int\_X f(x) P\_n(dx)= \int\_X f(x) P(dx)$$ I further assume that there is a $A$ such that for any $n$ ... | https://mathoverflow.net/users/32274 | Weak convergence in measure for negligible sets. | The answer is obviously "no" (take for $P\_n$ the sequence of $\delta$-measures at the points $1/n$ on the real line). However, it is "yes" if the set $A$ is open.
PS I think giving the right answer should be a reasonable prerequisite for declaring a question to be too easy for "research level".
| 1 | https://mathoverflow.net/users/8588 | 124928 | 69,918 |
https://mathoverflow.net/questions/124930 | 3 | **Edit:** Zhen Lin incisively observes in a comment below that the category of compact Hausdorff spaces is monadic over the category of sets, hence is cocomplete. That answers the first part of question 1 as it was stated previously. I have changed that question accordingly.
**Final edit:** Chris Schommer-Pries answe... | https://mathoverflow.net/users/21095 | Finitely cocomplete categories of compact Hausdorff spaces | As Zhen Lin points out in the comments to your question, the category of compact Hausdorff spaces has all small colimits. However the inclusion functor from CHTop to Top does not preserve these colimits in general. It seems from your other questions that you are interested in a category, either CHTop or something simil... | 6 | https://mathoverflow.net/users/184 | 124934 | 69,921 |
https://mathoverflow.net/questions/72782 | 10 | In the seminal paper [$K\_2$ and algebraic cycles](http://www.jstor.org/pss/1970902), Bloch make the following conjecture :
>
> Suppose $A$ is a local Noetherian integral domain with quotient field $F$
>
>
> * $K\_2(A)$ → $K\_2(F)$ is injective
> * Assume in addition $A$ is normal, $K\_2(A)$ = $∩\_pK\_2(A\_p)$ wh... | https://mathoverflow.net/users/8932 | Current status of a conjecture of Bloch | The second statement is false (even if we modify it by replacing $K\_2(A)$ by its image in $K\_2(F)$). A counterexample is $A=k[x,y,z]\_(x,y,z)/(z^2-xy)$. See J. Reine Angew. Math. 381 (1987), 37–50.
| 12 | https://mathoverflow.net/users/32229 | 124938 | 69,922 |
https://mathoverflow.net/questions/124950 | 1 | *Conjecture:* Let be $f$ a modular form of weight $k$ and $j$ a strictly positive integer, then the set $f,f',...,f^{(j)}$ is $\mathbb C$-linearly independent in $A$.
Is that conjecture true or false? Do you know a counterexample or a proof?
Notation: Let $\Pi=\{x+iy\in \mathbb C|y>0\}$ be the upper half-plane, $Ho... | https://mathoverflow.net/users/32330 | $f,f',...,f^{(j)}$ is $\mathbb C$-linearly independent if $f$ is a modular form | The only functions $f$ for which $f,f
',\ldots, f^{(j)}$ are linearly dependent are linear combinations of exponential functions, as one learns in a basic course on ordinary differential equations.
But maybe you meant to ask instead about whether $f$ and its derivatives are *algebraically independent*. Then it turns... | 3 | https://mathoverflow.net/users/1310 | 124951 | 69,927 |
https://mathoverflow.net/questions/124943 | 6 | I am reading the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by R. Bryant (here the link: <http://arxiv.org/abs/math/9912246>) and there are certain things that are unclear to me.
1. Bryant defines on page 11 in his paper the set $V\_{n}(\mathcal{I},\pi) =$ {$E\in V\_{n}(\mathcal{I... | https://mathoverflow.net/users/32327 | Questions on R. Bryant's paper "Calibrated embeddings in the special Lagrangian and coassociative cases" | I'm afraid that that article does not do a lot of details in the introductory Section 0, just because more complete explanations were already available in earlier articles of mine. Here are some brief answers. Right now, I don't have the time to write out the explanations in greater detail.
1. This is a consequence o... | 16 | https://mathoverflow.net/users/13972 | 124952 | 69,928 |
https://mathoverflow.net/questions/124940 | 6 | Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int\_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the Banach space $C$ of the continuous funtions $$F : t\in [0,1] \to F(t)\in \mathbb{R}^d$$ with the norm of the uniform con... | https://mathoverflow.net/users/32274 | Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions? | Let $\phi:C\to[0,\infty]$ be defined for $F\in C$ as the norm of $F$ in $W^{1,1}$ if $F$ is absolutely continuous, and $+\infty$ otherwise. Then $\phi$ is lower semi-continuous for the topology of uniform convergence and $W^{1,1}=\{\phi<\infty\}$ is Borel measurable.
| 3 | https://mathoverflow.net/users/8966 | 124954 | 69,930 |
https://mathoverflow.net/questions/124921 | 26 | Here are three vague theorems rolled up in one.
>
> Let $X$ and $Y$ be sufficiently nice topological spaces and $f:X \to Y$ a sufficiently nice surjection. If for each $y \in Y$, the fiber $f^{-1}(y) \subset X$ is (topologically, homotopically, homologously) equivalent to a point, then $f$ induces a (topological, h... | https://mathoverflow.net/users/18263 | Is there a general theory of fiber theorems? | **Edit:** I have added some definitions and details to my answer.
In the most general form I can find, your third question is a consequence of two results regarding cell-like maps and fine homotopy equivalences. It is closely related to and makes use of Chapman's celebrated theorem on topological invariance of Whiteh... | 11 | https://mathoverflow.net/users/21095 | 124955 | 69,931 |
https://mathoverflow.net/questions/107621 | 5 | In his paper '$t$-analogue of $q$-characters of finite dimensional representations of quantum affine algebras' - <http://arxiv.org/abs/math/0009231> - H. Nakajima states a conjectural definition of the $t$-analogue of the $q$-character of a standard module $M\_{P}$ (with weight $P$) of a quantum affine algebra at level... | https://mathoverflow.net/users/9970 | Status of a conjectural definition of H. Nakajima | See <http://arxiv.org/abs/1004.2321>.
| 6 | https://mathoverflow.net/users/3837 | 124956 | 69,932 |
https://mathoverflow.net/questions/120840 | 7 | $\DeclareMathOperator{\Spec}{Spec}$
My question is concerned with vanishing cycles of a locally constant sheaf for a smooth morphism in the case $l = p$. In the case $l \neq p$ this is a statement in SGA7-II. See below for the precise question. First let me start with some
Background
----------
Let us assume that... | https://mathoverflow.net/users/31051 | Vanishing cycles of a locally constant sheaf for a smooth morphism in the $l = p$-case | $\DeclareMathOperator{\ord}{ord}$
$\DeclareMathOperator{\Spec}{Spec}$
This negative answer to my question is a slight expansion of an email sent to me by Brian Conrad. Since I merely added some details for my own benefit this post is community wiki.
As Damian Rössler pointed out in his comment the essential ingredi... | 6 | https://mathoverflow.net/users/31051 | 124960 | 69,934 |
https://mathoverflow.net/questions/114253 | 5 | Consider $2n$ coordinates $x\_1,\ldots,x\_n,y\_1,\ldots,y\_n$ and the quadratic form $q = \sum\_{i=1}^n x\_i y\_i$. Now call $O(q,A)$ (orthogonal group of $q$) the group of $(2n)\times(2n)$ matrices, with coefficients in a commutative ring $A$, which preserve $q$. (This is an algebraic group over $\mathop{\mathrm{Spec}... | https://mathoverflow.net/users/17064 | Explicit equation of Dickson invariant / quasideterminant / special orthogonal group over the integers | Hi Gro-Tsen ! The following is the outcome of a discussion with Olivier Taïbi.
Let V be the free module of rank $2n$ over $\mathbb{Z}$ with basis $f\_1,\dots,f\_n,g\_1,\dots,g\_n$ and equipped with the split quadratic form $q=\sum\_{i=1}^nx\_iy\_i$. Let $C(V,q)$ be the Clifford algebra of $(V,q)$ and $C^+(V,q)$ be it... | 3 | https://mathoverflow.net/users/2868 | 124963 | 69,935 |
https://mathoverflow.net/questions/124936 | 3 | I am looking for examples of CM fields whose Galois group is not abelian. By a CM field K I mean an imaginary quadratic extension of a totally real field $K\_0$. If the extension is not Galois I take the Galois closure $L/K$.
Obviously the degree of such a field is even.
When $[K:Q]=2,$ K is an imaginary quadratic ... | https://mathoverflow.net/users/32322 | Galois groups of CM fields | Let $K\_0$ be totally real of degree $2,3$ and $K/K\_0$ a totally imaginary quadratic extension. Let $L\_0$ and $L$ be the Galois closures of $K\_0$ and $K$ over ${\mathbb Q}$. Then $Gal(L)$ maps onto $Gal (L\_0)$ with kernel an abelian $2$ group (a vector space over ${\mathbb F}\_2$) .
So the issue is: given a tota... | 2 | https://mathoverflow.net/users/23291 | 124964 | 69,936 |
https://mathoverflow.net/questions/124965 | 22 | All free groups of finite or infinite countable rank are subgroups of the free non-abelian group $F\_2$, which is linear. However, a free group of infinite uncountable rank will not be a subgroup of $F\_2$. Is it linear, too ?
This might easily follow from model theory, but I could not find a proof in the literature so... | https://mathoverflow.net/users/32332 | Are all free groups linear, i.e., admit a faithful representation to GL(n,K) for some field K ? | Free group of rank $c$ embeds in $Sl(2, F(t))$ where $F$ is a field of cardinality $c$.
Edit: Here is the detailed argument which, as Yves noted in his comment, proves a stronger result.
Theorem. Let $L$ be a field which is not an algebraic extension of a finite field and let $c$ be the cardinality of $L$. Then t... | 25 | https://mathoverflow.net/users/21684 | 124972 | 69,938 |
https://mathoverflow.net/questions/124969 | 1 | I'm feeling quite a bit embarrassed to ask such a basic thing, but I can't seem to figure it out. Let $R$ be a commutative ring and $A$ be a commutative $R$-algebra. Is the fork
$$
A \xrightarrow{i} A \otimes\_R A \rightrightarrows A \otimes\_R A
$$
an equalizer, where $i(a) = a \otimes 1$ and the two parallel arrows a... | https://mathoverflow.net/users/4183 | An equalizer in commutative algebras | Yes, it is. Let a tensor $t$ be in the equalizer of $f$ and $g$. Then, $f\left(t\right)=g\left(t\right)$. If we write $t$ in the form $\sum\limits\_{j\in I} a\_j\otimes b\_j$ (with $I$ being a finite set, and $a\_j$ and $b\_j$ being elements of $A$), then this rewrites as $\sum\limits\_{j\in I} a\_jb\_j\otimes 1 = \sum... | 2 | https://mathoverflow.net/users/2530 | 124974 | 69,940 |
https://mathoverflow.net/questions/124933 | 8 | Question first asked on math.stackexchange here: <https://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse>
**On the convexity of element-wise norm 1 of the inverse**
Let us define $\|A\|\_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as $$\|A\|... | https://mathoverflow.net/users/32321 | On the convexity of element-wise norm 1 of the inverse | The answer is *Yes* when $n=2$,but **No** when $n\ge3$. Here is the analysis.
The differential $L\_A$ of $A\mapsto A^{-1}$ is $L\_A=-A^{-1}BA^{-1}$. Likewise, the Hessian is
$$H\_A[B]=2A^{-1}BA^{-1}BA^{-1}=\frac2{(\det A)^3}\hat A B\hat AB\hat A,$$
where $\hat A$ is the adjugate matrix (mind that $A$ being symmetric,... | 7 | https://mathoverflow.net/users/8799 | 124978 | 69,942 |
https://mathoverflow.net/questions/124999 | 5 | Short question:
Let $M$ and $N$ be smooth manifold, with appropriate smooth function algebras
$C^\infty(M,\mathbb{R})$ and $C^\infty(N,\mathbb{R})$.
Can we express the smooth function algebra of the cartesian product manifold
in terms of $C^\infty(M,\mathbb{R})$ and $C^\infty(N,\mathbb{R})$?
I know it is neithe... | https://mathoverflow.net/users/21302 | Smooth function algebra on cartesian product and beyond | If $M$ and $N$ are compact, then $C^\infty(M\times N)=C^\infty(M)\bar\otimes\_{i}C^\infty(N)$, the completed injective tensor product which coincides with the completed projective tensor product, since the locally convex spaces involved are nuclear.
Edit: This also holds for for non-compact $M,N$; see [Treves: Topolo... | 4 | https://mathoverflow.net/users/26935 | 125001 | 69,951 |
https://mathoverflow.net/questions/124877 | 2 | We know that for finding the solutions of PDE equations, one of methods is "reduction of PDE", . For nonlinear equation
$v\_t=(v^{-4/3}v\_x)\_x+\lambda v$ how can we compute the generators of Lie algebra symmetries of this equation?
| https://mathoverflow.net/users/nan | symmetry of generationg function of PDE | There are several methods, but let me describe (what is perhaps the simplest) one: On $\mathbb{R}^4$ with coordinates $(t,x,u,p)$, consider the pair of $2$-forms
\begin{aligned}
\Upsilon\_0 &= (du-p\ dx)\wedge dt,\\\\
\Upsilon\_1 &= du\wedge dx + d(u^{4/3}p)\wedge dt + \lambda v\ dx\wedge dt.
\end{aligned}
It is easy t... | 7 | https://mathoverflow.net/users/13972 | 125014 | 69,956 |
https://mathoverflow.net/questions/124961 | 8 | I have had this question on my mind for two decades. We know, after Heath-Brown, that one out (say) of 3, 5, 7 is a primitive root mod p for infinitely many primes p. We just don't know which one. (We presume each is, or some GRH goes wrong per Hooley.)
Now if I invoke proof theory, it must be in terms of the proof o... | https://mathoverflow.net/users/6153 | Proof theory and primitive roots | It's the Pigeonhole Principle in the final step which is not constructively valid, not Heath-Brown's main result which, after inspection, doesn't show any of the tell tale signs of non-constructiveness.
Let $P\_t$ be the set of all primes for which $t$ is a primitive root. Heath-Brown shows that if $q$, $r$, $s$ are ... | 12 | https://mathoverflow.net/users/2000 | 125029 | 69,962 |
https://mathoverflow.net/questions/125027 | 5 | This question is closely related to my [previous question about modules over truncated sphere spectra](https://mathoverflow.net/questions/124352), in particular, it has the same motivation.
Recall that every space (or ∞-groupoid) can be represented as the homotopy colimit of some simplicial diagram of 0-truncated spa... | https://mathoverflow.net/users/402 | Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces? | If you don't group complete, then free $E\_{\infty}$-spaces are $1$-truncated.
Consequently, for $k > 0$, the answer is "all $E\_{\infty}$-spaces". When $k=0$, you'll
get those which are homotopy equivalent to simplicial commutative monoids.
| 12 | https://mathoverflow.net/users/7721 | 125030 | 69,963 |
https://mathoverflow.net/questions/124992 | 10 | Let $X$ be a based space. Then the Moore loop space $MX$ is defined to be the topological monoid whose points are based loops $[0,a] \to X$ where $a \ge 0$ is allowed to vary. Composition is gotten by concatenating loops.
Since $MX$ is a topological monoid, we can form the *bar construction* $BMX$.
This is geometric ... | https://mathoverflow.net/users/8032 | On the naturality of the bar construction | I wrote down two quick and simple solutions in Lemmas 14.3 and 15.4 on pages 84 and 90 of
"Classifying spaces and fibrations ([15] on my web page). The first is the evident zigzag
of natural weak equivalences
$$ X \leftarrow B(PX,MX,\ast) \rightarrow B(\ast,MX,\ast) = BMX, $$
where $PX$ is the Moore path space.
The l... | 10 | https://mathoverflow.net/users/14447 | 125032 | 69,965 |
https://mathoverflow.net/questions/125013 | 3 | Hi,
Up until now I've kind of taken this issue for granted, but I'm trying to think about it now and nothing seems to be making the situation much clearer.
The problem is (in short), what the correct notion of the based loop space (or even based path space)should be for 1-connected spaces $X$ with a free $G$-action... | https://mathoverflow.net/users/27770 | Equivariant based loop space for free G-spaces | "Basepoints" of "based" $G$-spaces are required to be $G$-fixed. Conceptually, a "basepoint" in an object $X$ of any category with a terminal object wants to be a map from the terminal object into $X$, and the terminal object in $G$-spaces is the one-point $G$-space. As you have observed, free $G$-spaces can't have suc... | 4 | https://mathoverflow.net/users/14447 | 125036 | 69,967 |
https://mathoverflow.net/questions/124847 | 15 | Is there some irreducible $F \in \mathbb{Z}[x]$ such that $\mathbb{Z}[x]/(F)$ has no principal maximal ideal? Equivalently, is it possible that the $1$-dimensional integral domain $\mathbb{Z}[x]/(F)$ has no prime element?
The maximal ideals have the form $(p,f)$, where $p \in \mathbb{Z}$ is a prime and $f \in \mathbb... | https://mathoverflow.net/users/2841 | Principal maximal ideals in Z[x]/(F) | Let $A$ be an order in the ring of integers $O\_K$ of a number field $K$. We claim that there are infinitely many principal maximal ideals $P$ of $A$. By using localization at rational primes, we have a bijection between the sets of maximal ideals of $A$ and $O\_K$ with residue characteristic relatively prime to $N = [... | 7 | https://mathoverflow.net/users/30180 | 125041 | 69,968 |
https://mathoverflow.net/questions/125010 | 2 | We know that blowing up a point on a surface produces a $(-1)$ curve. Is there any such standard techniques to produce $(-2)$ curves in a smooth surface?
| https://mathoverflow.net/users/32151 | Producing $(-2)$ curves on a smooth surface | In case you want a curve of arbitrary genus with arbitrary negative self-intersection, you can do this: Let $C$ be a smooth projective genus $g$ curve on a smooth surface. Suppose $C^2=n$ and take $m$ (pairwise) different points on $C$ and blow them up. The strict transform of $C$ is isomorphic to $C$, so it has genus ... | 3 | https://mathoverflow.net/users/10076 | 125042 | 69,969 |
https://mathoverflow.net/questions/125043 | 6 | I'm trying to better understand the connection between the concepts of ramification of a field extension, and ramification of a quaternion algebra. I'm also trying to build a better understanding of the motivation behind this term. Bear with me as I summarize my understanding of the stuff...
If $K$ is a field with va... | https://mathoverflow.net/users/14835 | Ramified quaternion algebras | If $F/{\mathbf Q}$ is a quadratic field then all but finitely many places $v$ of ${\mathbf Q}$ are unramified in $F$, and we could interpret what that means in a couple of ways: prime ideal factorization (for nonarchimedean $v$), extensions of absolute values (any $v$), or base extension by ${\mathbf Q}\_v$ (any $v$). ... | 9 | https://mathoverflow.net/users/3272 | 125068 | 69,984 |
https://mathoverflow.net/questions/125076 | 8 | Let $L\_3$ be the third-order language of set theory with identity on the first sort. Variables $x$ are first-order, $y$ are second-order, and $z$ are third-order. In the style of Lévy and Bernays, we can extend the third-order theory $\mathrm{ZFC3}$ by reflection axioms of the form $\phi \rightarrow \exists x [\mathrm... | https://mathoverflow.net/users/8547 | Applications of higher-order reflection principles | I view these kinds of hypotheses as on a continuum stretching from
weak compactness up through all the levels of the indescribability
hierarchy, second order, third order and so on. This hierarchy of
indescribability continues transfinitely via the strongly
unfoldable cardinals, which can be thought of simply as the
tr... | 8 | https://mathoverflow.net/users/1946 | 125081 | 69,989 |
https://mathoverflow.net/questions/125079 | 2 | In the context of the Bingham probability distribution the ${ }\_1F\_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. Thus, it is of interest to evaluate this function effectively. For the more general class of hypergeometric functions ... | https://mathoverflow.net/users/19959 | Computing hypergeometric function of matrix argument | It would be useful, if you'd point out the precise "open questions" regarding these evaluations that are of interest to you.
But since you mentioned Plamen Koev's work, I am sure you have tried out his matlab code [for computing Hypergeometric functions of matrix argument](http://math.mit.edu/~plamen/software/mhgref.... | 2 | https://mathoverflow.net/users/8430 | 125089 | 69,996 |
https://mathoverflow.net/questions/125095 | 25 | When doing my language exams for my doctorate, I requested to translate articles relevant to my interests that had not previously been translated, and to be able to do them at home, with the agreement that they would be much more difficult than what you might do on the spot in an exam setting. This was a very enlighten... | https://mathoverflow.net/users/14835 | Is there a database somewhere for sharing translations of mathematical works? | Put it on arxiv.org, in the math.HO category ("HO" stands for "History and Overview"). I've seen (and read) contributions there that are no more than translations of Euler from Latin to English. You'll want to write it up like a paper, with an explanatory paragraph providing context before the actual translation. This ... | 21 | https://mathoverflow.net/users/7759 | 125112 | 70,005 |
https://mathoverflow.net/questions/125108 | 5 | Is anything known about the consistency strength of the following statement?
* $\kappa$ is a Mahlo cardinal and there is a stationary set of $a \in \mathcal{P}\_\kappa(\kappa^+)$ such that $a \cap \kappa$ is an inaccessible cardinal and the order type of $a$ is $(a \cap \kappa)^+$?
This statement follows from $\kap... | https://mathoverflow.net/users/1682 | Stationary many subsets of $\kappa^+$ whose order type is a cardinal and whose intersection with $\kappa$ is an inaccessible cardinal | Well, here is a very slight weakening of your $\kappa^+$-supercompactness upper bound, to the assumption merely that $\kappa$ is [*nearly* $\kappa^+$-supercompact](http://cantorsattic.info/Nearly_supercompact). This hypothesis is strictly weaker than $\kappa^+$-supercompactness, but still, under this assumption, the se... | 1 | https://mathoverflow.net/users/1946 | 125114 | 70,006 |
https://mathoverflow.net/questions/125120 | 5 | If $K$ is a field, then as is well known every finite separable extension $L$ of $K$ is of the form $L=K(\alpha)$ for some $\alpha \in L$.
A similar statement can be made about an extension of discrete valuation rings with separable residue field extension.
These statements very much resemble the statement "every p... | https://mathoverflow.net/users/6779 | Is the primitive element theorem a cohomological statement? | The vanishing of the cohomology group $H^1(Spec(R),GL\_n)$ doesn't actually say that all projectives of rank $n$ are free; it says only that all projectives of rank $n$ are isomorphic. Combining this with the observation that at least one such projective is free, we get that they're all free.
But in the case of field... | 12 | https://mathoverflow.net/users/10503 | 125121 | 70,007 |
https://mathoverflow.net/questions/125007 | 8 | Given a quadratic form $F$ in $n$ variables, there is an associated theta function $\theta\_F(z) = \sum\_{m \in \mathbb{Z}} q^{F(m)}$, which is a modular form of weight $n/2$. Letting $F(m) = m^2$ gives a theta function $\theta\_F(z) = 1 + 2 \sum\_{n = 1}^\infty q^{n^2} \equiv 1 \pmod{2}$ with weight $1/2$.
Does the... | https://mathoverflow.net/users/32344 | Does there exist a half-integer weight theta function which is is equivalent to 1 modulo 4? | This is nothing like a complete answer (EDIT-now it seems to be--see below), but it may suggest a fruitful attack, based on the comment (see below for a version incorporated into this answer) that I made earlier. Your question is related to my [question 124243](https://mathoverflow.net/questions/124243/are-these-empiri... | 6 | https://mathoverflow.net/users/6214 | 125128 | 70,009 |
https://mathoverflow.net/questions/125116 | 3 | Is there a rotation representation that can also represent "turns", instead of collapsing coincident rotations into the same representation?
In 2D, a simple angle satisfies this, as it can have additional multiples of $2\pi$. For example, rotating by a turn and a half would be $3\pi$.
Is there something similar for... | https://mathoverflow.net/users/9005 | 3D Rotation Representation for Multiple Turns | Since you haven't described what you plan to do with this representation, I'm not sure what method would work well.
One of the problems with representing "turns" in more than two dimensions is that you don't have much in the way of discrete invariants. This is because the fundamental group of $SO(n)$ only has two ele... | 5 | https://mathoverflow.net/users/121 | 125132 | 70,013 |
https://mathoverflow.net/questions/125125 | 9 | Let $\mathcal{P}$ be a (small) exact category. Without delving into any homotopy theory, we can provide characterisations of $K\_0(\mathcal{P})$ and $K\_1(\mathcal{P})$ as plain categorical constructions:
$K\_0(\mathcal{P})$ is the free abelian group on the objects of $\mathcal{P}$ under the relations $[P] = [P'] + [... | https://mathoverflow.net/users/19313 | Categorical description of the second K-group | Algebraic generators and relations for Quillen's K-group $K\_n(P)$ are given in this paper: "[Algebraic K-theory via binary complexes](http://dx.doi.org/10.1090/S0894-0347-2012-00743-7)". Those you mention don't give the Quillen K-group $K\_1(P)$ in general, but just for Quillen-exact categories where every short exact... | 9 | https://mathoverflow.net/users/15247 | 125136 | 70,015 |
https://mathoverflow.net/questions/125139 | 1 | I want to calculate the dual space of $\mathcal{C}\_0[a,b]$, that is the space of continuous functions on $[a,b]$ vanishing at $a$. I know that the dual space of $\mathcal{C}[a,b]$ (continuous functions) is the space of differences of Lebesgue-Stieltjes measure associated to increasing functions, left-continuous, and v... | https://mathoverflow.net/users/32388 | Dual space of $\mathcal{C}_0[a,b]$ | Each (bounded linear) functional on $C[a,b]$ is also a functional on $C\_0[a,b]$. Each function $f \in C[a,b]$ can be written as $f(x) = f\_0(x) + f(a) \cdot 1$, where $f\_0 \in C\_0[a,b]$. On the other hand, each functional on $C\_0[a,b]$ can be extended to a functional on $C[a,b]$ (Hahn-Banach). Therefore the answer ... | 3 | https://mathoverflow.net/users/nan | 125140 | 70,017 |
https://mathoverflow.net/questions/125080 | 3 | There is a large literature on harmonic analysis on locally compact group, that
I am just beginning to discover. However I have not seen so far anything that emphasizes the *central* functions on $G$. A good reference on this subject could thus be very useful,
as I'd like to understand generalizations to as large as po... | https://mathoverflow.net/users/9317 | Spectral synthesis for central functions on locally compact groups | *This a long comment, which indicates the difficulties and gives a decomposition of measures in terms of orbital integrals instead of irreducible reps.*
As you have noticed yourself, there do not exists many continuous functions, which are invariant under conjugation. This was my comment with the closure of conjugacy... | 2 | https://mathoverflow.net/users/10400 | 125143 | 70,018 |
https://mathoverflow.net/questions/125134 | 0 | Let $V$ be a finite dimensional vector space over $\mathbb{R}$ and denote the dual of $V$ by $V^\*$. Then for $E$ a subspace of $V$ and $\epsilon\in \Lambda^2E^\*$ clearly the space $$L=L(E,\epsilon):=\{X+\xi \in E\oplus V^\*: \left.\xi\right|\_{E}=\iota\_X\epsilon\}$$ is isotropic with respect to the inner product on ... | https://mathoverflow.net/users/24132 | Maximal isotropic subspaces of $V\oplus V^*$ | First question: The space $\lbrace X+i\_X\epsilon: X\in E\rbrace$ is the graph of $\epsilon$ viewed as a mapping $E\to E^\*$, so its dimension is the dimension of $E$. The freedom in the choice of $\xi$ is the dimension of $V/E$.
Second question: No, since $E = E\oplus 0$ is not a subset of $L$.
| 2 | https://mathoverflow.net/users/26935 | 125150 | 70,021 |
https://mathoverflow.net/questions/115669 | 0 | Hi
I just started working on degeneration and contractions, I would like to know:
why no Lie algebra degenerate to a rigid algebra?(rigid algebra:an algebra whose orbit is zariski open)
Why the closure of a rigid Lie algebra forms the irreducible component of variety of Lie algebras?
Thank you
| https://mathoverflow.net/users/29742 | why no Lie algebra degenerate to a rigid algebra? Why the closure of a rigid algebra forms the irreducible component of variety of Lie algebras? | For the second question suppose that $L$ is a geometrically rigid Lie algebra,
i.e., the orbit $O(L)$ is open. The algebra $L$, and hence $O(L)$ is contained
in some irreducible component $\mathcal{C}$. Since a nonemty open subset in an
irreducible space is dense, the closure of $O(L)$ equals $\mathcal{C}$. The firs... | 1 | https://mathoverflow.net/users/32332 | 125161 | 70,025 |
https://mathoverflow.net/questions/125142 | 2 | Let $L:\mathscr A \times \mathscr B \longrightarrow \mathscr C$ and $R\_1:\mathscr B^{op} \times \mathscr C \longrightarrow \mathscr A$ be two functors such that there is a bijection
$$ \mathscr C(L(A,B),C) \cong \mathscr A( A, R\_1(B,C)) $$ natural in $A,B,C$.
Is there any sufficient conditions to ensure existence... | https://mathoverflow.net/users/32299 | Is there an analog of adjoint functor theorem for adjunctions of two variables? | Firstly, note that it is enough to construct an isomorphism $$\mathcal{C}(L(A,B),C) \simeq \mathcal{B}(B, R\_2(A,C))$$ The third natural isomorphism then follows automatically. Secondly, standard application of Yoneda's lemma shows that if for each $A\in \mathcal{A}$ you have a right adjoint $R\_2(A,\cdot)$ to $L(A,\cd... | 5 | https://mathoverflow.net/users/10605 | 125169 | 70,027 |
https://mathoverflow.net/questions/125178 | 0 | The solutions to the Dirichlet problem of elliptic PDE with smooth enough coefficients below to H\_0^1 and also belong to C\_infinity on the interior. Does that mean the classical limit of the function on the boundary is 0?
| https://mathoverflow.net/users/30525 | H_0^1 and C_infinity on the interior, does that imply classical limit is 0 on the boundary? | Yes. By Theorem 9.17 in Brezis' book, let $\Omega$ be just a bit regular$^\*$ and let $u\in W^{1,p}(\Omega)\cap C(\bar{\Omega})$. Then $u\in W^{1,p}\_0(\Omega)$ if and only if $u(z)=0$ for all $z\in \partial \Omega$.
$^\*$ $C^1$ will do, but even less regularity might be really necessary, I guess.
| 1 | https://mathoverflow.net/users/26039 | 125186 | 70,035 |
https://mathoverflow.net/questions/125180 | 9 | Recall that for a subgroup $\Gamma \subset SL\_2(\mathbb{Z})$ a modular form $f$ of weight $k$ is a holomorphic function from the upper-half plane into the complex numbers such that for any
$\begin{pmatrix}
a & b \\
c& d
\end{pmatrix}\in \Gamma$
and any $z$ in the upper half-plane, we have
$f(\frac{az+b}{cz+d}) ... | https://mathoverflow.net/users/2039 | Modular interpretation of nebentypus | Yes, there is such an interpretation, which is standard (cf. Delinge-Rapoport or Katz-Mazur).
You begin by interpreting the modular forms of level $\Gamma\_1(N)$ and weight $k \in \mathbb N$ as section of $\omega^k$ (plus condition at infinity) on the moduli space
$Y\_1(N)$ (over $\mathbb C$) of all elliptic curves $E$... | 12 | https://mathoverflow.net/users/9317 | 125191 | 70,038 |
https://mathoverflow.net/questions/125187 | 2 | Can anyone help to find some information about these structures?
| https://mathoverflow.net/users/32403 | Algorithmically finite-dimensional (noncommutative) algebras. | All what I could found is a talk of Bakhadyr Khoussainov
<http://www.math.nsc.ru/conference/malmeet/11/plenary/2011MM_Khoussainov.ppt>
in which he considered "algorithmically finite universal algebras". Maybe this will useful for you.
| 1 | https://mathoverflow.net/users/18814 | 125199 | 70,043 |
https://mathoverflow.net/questions/125182 | 1 | Lets say you have a circular table that seats $n$ people and $b\lt n -1$ identitcal boys. If you were to divide the boys into $k$ teams of size $\geq 1$, how many ways are there to seat the boys so that the teams sit together and have at least 1 empty seat separating them? Consider the seats to be unique (e.g. if we ha... | https://mathoverflow.net/users/31805 | Counting seating arrangements at a circular table | Replace each occupied seat with a $1$ and each empty seat with a $0$.
Choose a seat $s\_0$. Copy $s\_0$ and break the cycle into a sequence of length $n+1$ so that $s\_0$ is occupied iff the last location $s\_n$ is occupied. This sequence is either a string of $k+1$ streaks of $1$s with $k$ streaks of $0$s between, ... | 4 | https://mathoverflow.net/users/2954 | 125205 | 70,045 |
https://mathoverflow.net/questions/125104 | 22 | I hope this this is not seen as too much as jumping on the band-wagon, but here goes.
Deligne's proof of the last of the [Weil conjectures](http://en.wikipedia.org/wiki/Weil_conjectures) is well-known and just part of a huge body of work that has lead to prizes, medals etc (wink wink). The other conjectures were prov... | https://mathoverflow.net/users/4177 | How many proofs of the Weil conjectures are there? | I guess that "just the Riemann hypothesis" means the statement about the eigenvalues of Frobenius acting on the $H^i$ of a smooth proper variety, and "all of then" means the full theory of weights : definition of mixed sheaves, how the 4 operations affect weights etc (from which you get the hard Lefschetz theorem, the ... | 46 | https://mathoverflow.net/users/31960 | 125207 | 70,046 |
https://mathoverflow.net/questions/125202 | 3 | Let $M$ be an $n$-dimensional Riemannian manifold with sectional curvature lower bound 1. Fix a point say $O\in M$, let $S(r)$ denote the distance sphere centered at $O$ with radius $r$. The classical Hessian comparison theorem says that the principle curvatures of $S(r)$ is less than that of standard sphere ${S}^n(1)$... | https://mathoverflow.net/users/1190 | Diameter estimate of distance sphere of positive curved manifold | I guess you want to ask is it true that
$$\mathop{\rm IntrinsicDiameter}[S(r)]\le\mathop{\rm IntrinsicDiameter}[\tilde S(r)],$$
where $\tilde S(r)$ denotes the sphere of radius $r$ in the standard sphere.
* This is true if $r\ge \tfrac\pi2$; it follows since $S(r)$ has bigger curvature than $\tilde S(r)$ in the sens... | 3 | https://mathoverflow.net/users/1441 | 125208 | 70,047 |
https://mathoverflow.net/questions/125214 | 10 | Consider positive integers $c$, $k$, and $s$. Does there exist some $N = N(c,k,s)$ such that the following holds?
>
> Take any $c$-coloring of the $k$-tuples of integers in $[1,N]$. Then there is an arithmetic progression of length $s$ such that all $k$-tuples of it have the same color. More precisely, there exist... | https://mathoverflow.net/users/32409 | Combining van der Waerden's theorem with Ramsey's theorem | The way I understand the question is that $k$-tuples of integers are considered so that each coordinate lies in $[1,N]$, so in other words one considers $[1,N]^k$ and colors all the elements of this set, so each poin $(a1,...,ak)$ gets some color.
There is a version of Szemerédi's Theorem for $\mathbb{Z}^k$, due to ... | 7 | https://mathoverflow.net/users/nan | 125218 | 70,051 |
https://mathoverflow.net/questions/125164 | 1 | Let $X$ be a topological space, $\mathcal{C}$ be a locally small category with "good properties" (such as having small inverse limit, small filtrant inductive limit...etc.) and $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$.
The + functor turns $\mathcal{F}$ to another presheaf $F^{+}$ such that: For any op... | https://mathoverflow.net/users/5482 | + functor (used to construct sheafification)'s property | You might be interested in the papers
>
> P. Freyd, M. Kelly, *Categories of continuous functors I*, J. Pure Appl. Algebra 2 (1972), 1-18.
>
>
> J.W. Gray, *Sheaves with values in a category*, Topology 3 (1965), 1-18.
>
>
>
| 2 | https://mathoverflow.net/users/2841 | 125221 | 70,053 |
https://mathoverflow.net/questions/125225 | 4 | For an ideal $I \subset R$ with relative K-groups $K\_i(R,I)$ we have an exact sequence
$K\_2(R) \to K\_2(R,I) \to K\_2(R/I) \to K\_1(R) \to K\_1(R,I) \to K\_1(R/I)$
$\to K\_0(R) \to K\_0(R,I) \to K\_0(R/I)$
This is reminiscent of the relative homotopy exact sequence and suggests - perhaps not alone - that we are... | https://mathoverflow.net/users/19313 | Intuition as to why the K-theory of a ring should be the homotopy theory of an H-space | For based spaces $X$ and $Y$ there is a map $\pi\_i(X)\times \pi\_j(Y)\to \pi\_{i+j}(X\wedge Y)$ given by
$$
(a:S^i\to X,b:S^j\to Y)\mapsto (a\wedge b:S^i\wedge S^j\to X\wedge Y).$$
This function of $a$ (resp. $b$) is a homomorphism if $i$ (resp. $j$) is positive.
Thus a space-level multiplication of the form $X\wedg... | 7 | https://mathoverflow.net/users/6666 | 125228 | 70,055 |
https://mathoverflow.net/questions/125233 | 0 | Consider a connected graph whose cycles have length at least 4, the maximum valence is 4, there are 2 vertices of valence 2 and 4 of valence 3. I would like to show graphs of this type are planar . I am not sure whether the result is true or not. I am essentially trying to show that any connected subgraph whose vertice... | https://mathoverflow.net/users/32416 | Determining whether a certain graph is planar | With the information given, you can not guarantee planarity. One counter example would be a graph consisting of the complete bipartite graph $K\_{4,4}$ (not planar) and any bipartite graph with two degree-2-vertices, four degree-three-vertices, and the remaining vertices degree 4, such that the degree-2-vertices are op... | 4 | https://mathoverflow.net/users/12487 | 125235 | 70,057 |
https://mathoverflow.net/questions/125200 | 15 | I'm trying to develop an intuition for Cech cohomology geometrically, but am currently failing. A lot of people seem to say that the groups $H^n$ measure obstructions to gluing local sections to get global ones. However I don't see how this works, and I think it's because I don't understand coboundaries.
I see that $... | https://mathoverflow.net/users/22337 | Coboundaries and Gluing in Cech Cohomology - Intuition? | Let $U$ denote the space we're working on and $\{U\_i\}$ an open cover of $U$ (obviously $U$ may be an open set of an ambient space, but that plays no importance here). Let's assume that there is a sheaf $\mathscr F$ on $U$ and for each $i$ a section $s\_i\in\mathscr F(U\_i)$.
Gluing the $s\_i$ means to find a sectio... | 14 | https://mathoverflow.net/users/10076 | 125236 | 70,058 |
https://mathoverflow.net/questions/125238 | 1 | (A qual problem) Let $\pi:S^{n}\rightarrow M$ be a covering map, $M$ being an orientable manifold. Show that $H\_{deR}^{k}(M)=0$ for $1\leq k < n $.
We can show $H\_{deR}^{1}(M)=0$ by the following argument. For a closed 1-form $\omega$ on $M$, $\pi^{\*}\omega $ is closed on $S^{n}$ thus exact, so $\pi^{\*}\omega$ i... | https://mathoverflow.net/users/24310 | deRham cohomology of a manifold with covering space $S^{n}$ | A finite covering map induces an injection on de Rham cohomology. Try searching for "integration along the fibers"; yours is an easy case, as you integrate on a finite set of points, which means you just sum.
| 4 | https://mathoverflow.net/users/4790 | 125240 | 70,059 |
https://mathoverflow.net/questions/125245 | 9 | Let $T$ be a measure-preserving transformation on a probability space $(\Omega,\mathcal B,\mu)$. Assume that for any pair of measurable sets $A,B\in\mathcal B$ with $\mu(A), \mu(B)>0$, one can find $N$ such that $T^{-n}(A)\cap B\neq\emptyset$ for all $n\geq N$. Is $T$ mixing with respect to $\mu$? This is likely to be ... | https://mathoverflow.net/users/37371 | Silly question about mixing | There is a condition known to be intermediate between the one you mention and mixing. A transformation is **lightly mixing** if $\liminf\_{n\to\infty} \mu(T^{-n}(A)\cap B) > 0$ for all $A$ and $B$ of positive measure. For a transformation which is lightly mixing but not mixing, see for example Friedman and King's paper... | 6 | https://mathoverflow.net/users/5963 | 125268 | 70,068 |
https://mathoverflow.net/questions/125231 | 3 | **Question is edited** Perhaps this formulation is clearer.
It is well known that if a power of a primitive (i.e. not a proper power) word $u$ contains two different occurrences of a word $v$, $|v|>|u|$, then the occurrences are shifts of each other by a multiple of $|u|$ (formally: $u^n\equiv pvq\equiv p'vq', |p|-|... | https://mathoverflow.net/users/nan | A property of periodic words | As you observe, we may assume that $p'$ is empty, i.e., that $v$ starts with a power of $u$. If $p$ is not a multiple of $u$, then writing $u$ as a circular word, it means you can find two district places in the circle where you can read $u$. So $u=xy=yx$ for some $x,y$ (namely if $|p|$ is $r$ mod $|u|$ take $x$ to be ... | 5 | https://mathoverflow.net/users/15934 | 125269 | 70,069 |
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