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https://mathoverflow.net/questions/179225 | 5 | In Duke, Friedlander and Iwaniec's Erratum on "Bounds for automorphic L–functions. II"
They have the following estimates for derivatives of Bessel functions: For $k \geq 2$
\begin{align}
& z^{l}J^{(l)}\_{k-1}(z) \ll z(1+z)^{-3/2} \\ &
z^{l}Y^{(l)}\_{0}(z) \ll (1+ |\log z|)(1+z)^{-1/2}\\ &
z^{l}K^{(l)}\_{0}(z) \ll (1+ |... | https://mathoverflow.net/users/18950 | Estimates on derivatives of Bessel function | I think these estimates are wrong. They are true for $z\ll 1$, but for $z\gg 1$ the $z^l$ factors should be omitted. See, for example, Appendix C in Kowalski-Michel-VanderKam: Rankin-Selberg L-functions in the level aspect, Duke Math. J. 144 (2002), 123-191.
**Added.** I think this error does not cause any further pr... | 5 | https://mathoverflow.net/users/11919 | 179228 | 90,091 |
https://mathoverflow.net/questions/179177 | 4 |
>
> Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point?
>
>
>
The question is inspired by and related to [this one](https://mathoverflow.net/q/173391/24165).
Surely, globally non-topologizable infinite groups [do]... | https://mathoverflow.net/users/24165 | Are infinite groups "locally topologizable"? | Having slept on it, I realise that the answer is **No** (even if we omit the condition on the inverse). The first example of infinite countable non-topologizable group due to Olshanskii (1980) is actually locally non-topologizable.
Indeed, Olshanskii's group $G$ has exponent $p^2$ and the cyclic centre of order $p$.... | 5 | https://mathoverflow.net/users/24165 | 179233 | 90,092 |
https://mathoverflow.net/questions/179153 | 4 | In this [MO answer](https://mathoverflow.net/questions/83421/voevodskys-counterexample-to-the-existence-of-a-motivic-t-structure?rq=1) of M. Bondarko, he says:
>
> "the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..."
>
>
>
and in [Remarks on Grothe... | https://mathoverflow.net/users/56910 | Standard conjectures on positive characteristic | Over finite fields, the Tate conjecture by itself doesn't imply the standard conjectures, but together with the Hodge conjecture for CM abelian varieties it does (Milne 2002, 2009).
In a little more detail: Milne showed that the Hodge conjecture for CM abelian varieties implies the Tate conjecture for all abelian var... | 8 | https://mathoverflow.net/users/57398 | 179238 | 90,095 |
https://mathoverflow.net/questions/179205 | 10 | In "F-isocrystals on open varieties results and conjectures" Faltings says:
>
> "Finally, we extend the theory of weights and show as much as possible of the crystalline analogue of the Weil conjectures."
>
>
>
**My questions are:**
What is the current state of the crystalline analogue of the Weil conjectur... | https://mathoverflow.net/users/56910 | What is the current state of the crystalline analogue of the Weil conjectures? | As far as I can tell, [Kedlaya solved it 12 years ago.](http://arxiv.org/abs/math/0210149)
| 9 | https://mathoverflow.net/users/121 | 179241 | 90,097 |
https://mathoverflow.net/questions/179136 | 4 | If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference.
My question is regarding accessing data in non-trivial cohomological degree of a cosimplicial space. Ultimately, for me, this comes down to questions about loop... | https://mathoverflow.net/users/11546 | Higher Degree Data in a Cosimplicial Quasicategory and Delooping | So, as far as I can tell, this question has a lot of high-falootin' vocabulary in it, but is actually pretty basic. It just took me a while to think about the right way. I should mention that my understanding was greatly clarified by talking to [Adeel](https://mathoverflow.net/users/2503/adeel "Adeel") in the [Homotopy... | 1 | https://mathoverflow.net/users/11546 | 179249 | 90,099 |
https://mathoverflow.net/questions/179227 | 16 | Jean-Yves Girard writes at the end of his paper
"[Towards a Geometry of Interaction](http://jb55.com/linear/pdf/Towards%20a%20geometry%20of%20interaction.pdf)", page 105, that we have three intuitions about the nature of time:
1. time is logic modulo the order of rules,
2. time is the cut elimination process,
3. tim... | https://mathoverflow.net/users/57368 | Time in Girard's Geometry of Interaction | I did my PhD thesis in Girard's team in Marseille (my supervisor was Laurent Regnier, himself a student of Girard's) so I have quite a bit of experience with his "excentric" way of communicating and I can attempt an exegesis ( :-) ) of this particular sentence (besides, I am quite familiar with both the philosophical a... | 34 | https://mathoverflow.net/users/45027 | 179258 | 90,102 |
https://mathoverflow.net/questions/179250 | 4 | I am currently working on a problem that may be interpreted as recovering an unknown function from its Radon transform.
Unfortunately I don't have any background in Radon transform, but need to quickly get what I need for solving my problem.
The questions are:
* are there properties (e.g. degree of smoothness) ... | https://mathoverflow.net/users/31310 | Basic Questions about Radon Transforms | to follow up on my comment with a few more pointers:
Even if you will not be using the ready-to-use [MATLAB toolbox](http://www.mathworks.nl/help/images/the-inverse-radon-transformation.html) for Radon transform inversion, you will likely want to use MATLAB as a platform for the development and experimentation with ... | 3 | https://mathoverflow.net/users/11260 | 179262 | 90,105 |
https://mathoverflow.net/questions/179245 | 8 | In a survey paper of [Korkmaz](http://journals.tubitak.gov.tr/math/issues/mat-02-26-1/mat-26-1-8-0203-13.pdf) it is stated that $H\_2(\mathrm{Mod}\_3)$ is either $\Bbb Z$ or $\Bbb Z \oplus \Bbb Z\_2$, but I was not able to find out a precise computation of this group (resolving the ambiguity). Is this group known? Any ... | https://mathoverflow.net/users/23193 | Second homology of mapping class group of genus 3 | In his paper "Lagrangian mapping class groups from a group homological point of view" (available [here](http://arxiv.org/abs/0910.5262)), Sakasai proves that the desired homology group is $\mathbb{Z} \oplus \mathbb{Z}/2$.
| 11 | https://mathoverflow.net/users/317 | 179270 | 90,109 |
https://mathoverflow.net/questions/179244 | 2 | Can any one provide a hint to prove the following statement? :
Let $H$ be a complex reductive subgroup (not necessarily connected) contained in $SO(n,\mathbb{C)})$. Consider the map $H \rightarrow Aff^\*$, where $Aff^\*$ is the affine $(n-1)$-dimensional subspace not including the hyperplane ${x\_n =0 }$, defined by se... | https://mathoverflow.net/users/57428 | Characteristic polynomials of reductive subgroup over C | I'm not sure where your "statement" comes from, or why the specific type of embedding of $H$ is assumed here. But the closest relative of this situation I'm aware of goes back to work of Kostant (in characteristic 0) and later Steinberg (more generally), in which they consider a sort of adjoint quotient of a semisimple... | 0 | https://mathoverflow.net/users/4231 | 179288 | 90,114 |
https://mathoverflow.net/questions/179285 | 3 | Is there a finite nonabelian group satisfying all of the following identities?
$$
(x^py^p)^2 = (y^px^p)^2, \quad p = 2,3,5,7,11,\ldots (\text{primes})
$$
I thank you all in advance.
| https://mathoverflow.net/users/57297 | Existence of finite nonabelian groups satisfying certain identities | Assuming you want that to hold for all $x,y \in G,$ any (non-Abelian) extra-special $2$-group $G$ will have that property. For there are only two squares in $G$: the identity, and the unique element of order $2$ in $Z(G).$ However, note that $(x^{p}y^{p})^{2}$ and $(y^{p}x^{p})^{2}$ are conjugate in $G$, so since they ... | 14 | https://mathoverflow.net/users/14450 | 179289 | 90,115 |
https://mathoverflow.net/questions/178815 | 4 | If $(M\_t)\_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I\_0=1$, $I\_1(t)=M\_t$ and for $n \geq 2$ $$I\_{n}(t) = \int\_0^t I\_{n-1} (s) \mathrm{d} M\_s.$$ By noting that $n! I\_n(t) = H\_n(M\_t,\langle M,M \rangle\_t)$, where $H\_n(x,t)=t^{n/2} h\_n(x/\sqrt{t})$ and $h\_n$ is th... | https://mathoverflow.net/users/57022 | Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals? | The case $n=1$ is straighfoward. Now, applying Ito's formula and the definition of $\{I\_n\}$ to integrate by parts we have:
\begin{align\*}
I\_n = \int\_0^t I\_{n-1} (s) \ d M\_s = I\_{n-1}M - \int\_0^t I\_{n-2}M\ dM - \int\_0^t I\_{n-2}\ d\langle M,M \rangle\ = \
= I\_{n-1}M - \int\_0^t I\_{n-2}M\ dM - (\ I\_{n-2}... | 3 | https://mathoverflow.net/users/47322 | 179295 | 90,119 |
https://mathoverflow.net/questions/179297 | -1 | Is there a general solution for first-order partial differential equations of the form
$$m(x) \partial\_x f(x,y) = n(y) \partial\_y f(x,y)$$
for given $m(x),n(y)$ and reasonable boundary conditions (e.g. $f(x,0)=0$ etc.)?
| https://mathoverflow.net/users/3441 | Solution to simple first-order partial differential equations | Change the variables in $x$ and $y$, i.e., $\tilde x =\tilde x(x)$ and $\tilde y = \tilde y(y)$ to make $m(x) = n(y) = 1$. Then equation can be written as
$
\partial\_{\tilde x}f =\partial\_{\tilde y}f.
$
The equation above is equivalent to that $f= f(\tilde x+ \tilde y)$.
| 2 | https://mathoverflow.net/users/24798 | 179301 | 90,121 |
https://mathoverflow.net/questions/87484 | 8 | In Zsolt Tuza's [Unsolved combinatorial problems I](http://www.brics.dk/LS/01/1/BRICS-LS-01-1.pdf), Problem 46 is the following conjecture:
Let $G$ be a graph on $n$ vertices. Let $\alpha\_1$ be the maximum number of edges of $G$ such that every triangle in $G$ contains at most one of these edges. Let $\tau\_1$ be th... | https://mathoverflow.net/users/2530 | What is the correct statement of this Erdös-Gallai-Tuza problem generalizing Turan's triangle theorem? | I hope I am not violating any kind of MathOverflow etiquette by responding to a question 2 years after it has been asked, but your question is answered in the following preprint of mine that just hit the arXiv: <http://arxiv.org/abs/1408.5176>
The preprint shows that any vertex-minimal counterexample to the Erdős--Ga... | 9 | https://mathoverflow.net/users/6322 | 179306 | 90,122 |
https://mathoverflow.net/questions/179282 | 4 | Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$.
Moreover, let $M \colon \mathbb{R}^n \to \mathbb{R}^p$ be a linear map for $p \leq n$.
I'm interested in computing a $\mathcal{H}$-representation of $M \cdot P = \{Cx ... | https://mathoverflow.net/users/56325 | $\mathcal{H}$-polyhedron under a linear map | Yes, by the [Motzkin Double Description Method](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.8343) you compute the $V$-representation of $P,$ hence the $V$-representation of $M P,$ then, by the double description method (again) the $H$-representation of $M P.$
| 3 | https://mathoverflow.net/users/11142 | 179307 | 90,123 |
https://mathoverflow.net/questions/179298 | 5 | *This is a purely idle question, but one I'm increasingly interested the more thought I put into it:*
For $\mathcal{A}$ a universal algebra (that is, nonempty set together with some named functions), a *congruence* of $\mathcal{A}$ is an equivalence relation on $\mathcal{A}$ which respects the named functions: $$\ove... | https://mathoverflow.net/users/8133 | When are the congruence lattices nicer? | A congruence lattice of a universal algebra is a complete (therefore bounded) lattice, and is also algebraic (every element is a join of compact elements). As such, the collection C of congruence lattices of algebras of some class is a variety only when C is the collection of one element lattices.
The ' operation abo... | 3 | https://mathoverflow.net/users/35626 | 179314 | 90,128 |
https://mathoverflow.net/questions/179248 | 4 | Given a(n $\infty$-)category, there is a process called "stabilitazion" which spits out a stable $\infty$-category (as one can read about in either Higher Algebra or the nlab).
The famous example is Spectra, which is the stabilization of Top (and I guess the various enhancements of derived categories, out of categori... | https://mathoverflow.net/users/46690 | what is the stabilization of pointed sets? | The answer is in the comments: the stabilization is the trivial stable category.
| 4 | https://mathoverflow.net/users/46690 | 179318 | 90,131 |
https://mathoverflow.net/questions/179164 | 17 | The generalized homology theory of the Thom spectrum $MO=\varinjlim\Sigma^nMTO\_n$ is bordism theory:\begin{equation\*}\pi\_k(MO\wedge X)=\Omega^O\_k(X)\end{equation\*}These groups form the ring of (unoriented) $X$-cobordism classes of (unoriented) manifolds.
But what information do the Madsen-Tillmann spectra $MTO\_... | https://mathoverflow.net/users/51107 | Homology theory represented by Madsen-Tillmann spectra | This is an exercise in understanding the Pontrjagin--Thom correspondence. The group $\pi\_k(MTO(n) \wedge X\_+)$ is in bijection with tuples of
1. a $(n+k)$-manifold $M$,
2. an $n$-dimensional vector bundle $V \to M$,
3. a stable isomorphism $\varphi: V \oplus \epsilon^k \oplus \epsilon^\ell \cong TM \oplus \epsilon... | 23 | https://mathoverflow.net/users/318 | 179327 | 90,134 |
https://mathoverflow.net/questions/179325 | 6 | I've searched Google, but it seems that only research journal papers appear in search results, where some new, improved, or specialized extragradient method is discussed. I've also searched Wikipedia and Wolfram MathWorld.
I would like to perhaps know the straightforward definition of that, instead of deducing it fro... | https://mathoverflow.net/users/57464 | What is an extragradient method? | This is the key reference: G.M. Korpelevich, "The extragradient method for finding saddle points and other problems." *Ekonomika i Matematicheskie Metody* **12** (1976): 747-756.
I have not found this article online, but you can find a brief description [here](http://www.fixedpointtheoryandapplications.com/content/pd... | 7 | https://mathoverflow.net/users/11260 | 179328 | 90,135 |
https://mathoverflow.net/questions/179322 | 1 | I'm interested in classes C of $R^1$-valued random variables which possess the following properties:
1) the sum of two independent random variables from class C belongs to class C;
2) for any $\lambda \in R^1$, $\xi \in C$ we have $\lambda\xi \in C$;
3) any random variable from class $C$ has tails which are heav... | https://mathoverflow.net/users/47796 | A special class of random variables | There are many such classes. For example, take a single symmetric random variable satisfying 3 and 4, and let $C$ consist of all possible variables that you get from it by applying scalings and convolutions. Or let $C$ be the set of of all symmetric random variables for which there are $c\_1,c\_2>0$ and $k\_1,k\_2>3$ s... | 0 | https://mathoverflow.net/users/56624 | 179333 | 90,136 |
https://mathoverflow.net/questions/179330 | 3 | What are the complex rational homogenous spaces $G/P$ ($G$ a semi-simple complex Lie group, $P$ a parabolic subgroup) such that the set of real points $(G/P)(\mathbb R)$ is a (compact) riemannian symmetric space?
This is certainly well-known by the experts, but I'm not one of them...
Thanks for any help!
| https://mathoverflow.net/users/36575 | Rational homogenous spaces and symmetric spaces | Compact Riemannian symmetric spaces admitting a Lie group of
diffeomorphisms $G$ properly containing the isometry group are essentialy (up to covers) the symmetric $R$-spaces, which are of the form $G/P$. This is a celebrated theorem of Nagano [Nag, Theorem 3.1].
The list of these spaces is e.g. in the appendix of [I... | 2 | https://mathoverflow.net/users/6818 | 179336 | 90,138 |
https://mathoverflow.net/questions/179305 | 6 | In the context of decomposition matrices for Hecke algebras of finite Coxeter groups at a root of unity (such as the tables at the end of the book "Hecke algebras at a root of unity" by Geck-Jacon or in Geck-Pfeiffer), what is meant by the defect of a block? Is there a simple explanation of what the defect keeps track ... | https://mathoverflow.net/users/57454 | What does the defect of a block measure? | Originally ( in the context of group algebras of finite groups, which background is necessary to put later generalizations in context), the defect of a block was defined by Brauer as an arithmetical quantity. If $F$ is an algebraically closed field of prime characteristic $p,$ and $G$ is a finite group whose Sylow $p$-... | 16 | https://mathoverflow.net/users/14450 | 179337 | 90,139 |
https://mathoverflow.net/questions/179303 | 4 | Describe **all** the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map
$$
N = \begin{pmatrix}
0 & 1 & & \\
0 & 0 & & \\
& & 0 & 1 \\
& & 0 & 0 \\
\end{pmatrix}
$$
Equivalently, describe **all** fixed points of the unipotent map $u = \exp(N)$ acting on the grassmanian of pl... | https://mathoverflow.net/users/12170 | Invariant planes of a nilpotent matrix with two Jordan blocks of size two | I think it is easiest to just do it in Plucker coordinates. Write a $2$ plane in $4$ space as the row span of $\begin{pmatrix} w\_1 & x\_1 & y\_1 & z\_1 \\ w\_2 & x\_2 & y\_2 & z\_2 \end{pmatrix}$. Then the unipotent group action is:
$$\begin{pmatrix} w\_1 & x\_1 & y\_1 & z\_1 \\ w\_2 & x\_2 & y\_2 & z\_2 \end{pmatrix}... | 3 | https://mathoverflow.net/users/297 | 179342 | 90,140 |
https://mathoverflow.net/questions/179188 | 1 | For an irreducible finite depth finite index subfactor $(N \subset M)$, there is a structure of fusion category given by the even part of its principal graph. The simple objects $(X\_i)\_{i \in I}$ of depth $0$ or $2$ correspond to the projections $p\_i = I\_{H\_i}$ on the $2$-boxes space $\mathcal{P}\_2(N \subset M) =... | https://mathoverflow.net/users/34538 | What's the relation between fusion and coproduct? | In the formula, $tr(b)$ should be replaced by $tr(|b|)$, i.e., $\|b\|\_1$.
$$b\_\alpha\*b\_\beta=\sum\_\gamma\frac{\|b\_\alpha\|\_1\|b\_\beta\|\_1\bar{c}\_{\alpha\beta}^\gamma}{\sqrt{n}\|b\_\gamma\|\_1}b\_\gamma.$$
Unfortunately the proof needs the irreducibility. I do not know how to generalize it to weak Kac alg... | 5 | https://mathoverflow.net/users/57468 | 179351 | 90,145 |
https://mathoverflow.net/questions/179341 | 0 | It is known that [Polya's conjecture](http://en.wikipedia.org/wiki/Polya_conjecture) is false and the smallest counter-example is about $10^9$.
Assuming that we are searching for a counter-example not knowing that it exists. What useful information can I use to speed up my search ?
**Motivation :** If we try brute... | https://mathoverflow.net/users/46025 | Counterexample to Pólya's conjecture | [Lehman's paper](http://dx.doi.org/10.2307%2F2003890) is very clear about what he is doing. One can express $L(x)=\sum\_{n\leq x}\lambda(n)$ as a rather short sum whose terms are expressible easily from the values of $\mu(m)$, $\lambda(k)$, $L(y)$ for $m,k,y$ much smaller than $x$.
The starting identity is (4) in th... | 7 | https://mathoverflow.net/users/11919 | 179365 | 90,153 |
https://mathoverflow.net/questions/179189 | 0 | I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. Additionally, the distribution is defined by right invariant vector fields.
Due to the abundance of right invariance, I... | https://mathoverflow.net/users/41654 | equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution | OK. While I could write an exposition of the now-standard method of setting up and solving these sorts of right-invariant constrained variational problems on Lie groups, I think that it will be better for you to see a full exposition.
I recommend the book *Exterior Differential Systems and the Calculus of Variations... | 1 | https://mathoverflow.net/users/13972 | 179369 | 90,155 |
https://mathoverflow.net/questions/179320 | 2 | How can I prove that the following triangular kernel function defined in $[0, 1] \subset R^1$
$k(x, x') = (1 - 2|x-x'|)$
is a positive semidefinite function?
It turns out to be psd function when using a numerical simulation tool (Matlab) by checking psd of a kernel matrix..
I found out that $k(x, x') = (1 - |... | https://mathoverflow.net/users/57458 | How can I prove that the negative biased triangular kernel is positive semidefinite | Actually the Fourier coefficients of $1 - 2 |t|$ on $[-1,1]$ are
$$ \int\_{-1}^1 (1 - 2 |t|) \exp(-\pi i t n)\; dt = \cases{ 0 & for even $n$\cr 8/(n^2 \pi^2) & for odd $n$\cr}$$
so as a kernel on $[0,1]$ this is positive semidefinite, i.e. the operator
on $L^2[0,1]$ given by $$ Tf(x) = \int\_0^1 k(x,y) f(y)\; dy$$ is ... | 1 | https://mathoverflow.net/users/13650 | 179370 | 90,156 |
https://mathoverflow.net/questions/178806 | 6 | Is the following statement true, and if it is, does someone have a reference?
>
> Let $X$ be a compact (i.e., compact and Hausdorff) topological space. Then the Gleason space (=Iliadis absolute, =Stone dual of the Boolean algebra of regular open sets) of $X$ is (naturally homeomorphic to) the projective limit of $\... | https://mathoverflow.net/users/17064 | Is the absolute of a compact space the projective limit of the Stone-Čech compactifications of its open dense subsets? | Yes, your space, let's call it $GX$, is the projective cover $EX$ of $X$ in the category of compact spaces. Recall that the projective cover $p:EX\to X$ can be characterized by the properties:
* $EX$ is projective, that is every epimorphism $f:Y\to EX$ has a right inverse.
* $p$ is irreducible, that is for every clos... | 4 | https://mathoverflow.net/users/16678 | 179375 | 90,158 |
https://mathoverflow.net/questions/179377 | 15 | Consider the $k \times k$ block matrix:
$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \ddots &
\vdots \\ B & B & B & \cdots & A
\end{array}\right) = I\_k \otimes (A - B) + \mathbb{1}\_k \otimes B$$
where $A$ and $B$ are size $n \times n$ and $\mathbb... | https://mathoverflow.net/users/57474 | Determinant of a $k \times k$ block matrix | We can just manipulate $C$ in the usual way by row operations: Subtract the last "row" from all the other "rows" (this is really several traditional row operations done at once). This produces
$$
\begin{pmatrix} A- B &0& 0 & \ldots & 0 &B-A \\
0 & A-B &0 &\ldots & 0 & B-A\\
&& \ldots &&&\\
B & B & B & \ldots & B & A \e... | 18 | https://mathoverflow.net/users/48839 | 179379 | 90,160 |
https://mathoverflow.net/questions/179396 | 2 | In all that follows, we are working over $\mathbb{C}$. Let $B \subseteq P \subseteq {\rm GL}(n)$ be a parabolic subgroup. Can you say anything in general about the representations of $P$? I suspect the answer is no because I couldn't find anything about this in the standard books or using google.
If $P = B$ then ever... | https://mathoverflow.net/users/4002 | Representations of parabolic subgroups of the general linear group over the complex numbers | I think the idea is to write $P$ as a semidirect product of its unipotent radical $N$ and its (maximal reductive) Levi subgroup $M$. (If $P$ is a Borel, then $M$ is a maximal torus.) You can then restrict a complex representation of $P$ to $M$ and decompose it into irreducibles.
| 4 | https://mathoverflow.net/users/25358 | 179397 | 90,168 |
https://mathoverflow.net/questions/179406 | 9 | I am new to methods for simulating Markov chains in order to sample from the target, unknown distribution. After a couple days of reading, I found out that even though people have realized that non-reversible (or, irreversible) Markov chains usually converge faster to their invariant distributions. There are not many p... | https://mathoverflow.net/users/14390 | Markov chain Monte Carlo: why is non-reversible MC MC not as popular? | I will take a stab at this, even though I no longer have the full eloquence of a working mathematician. First of all reversible Markov chains are really self-adjoint operators in disguise, whose spectral properties are therefore well-understood at least in theory. But even in this nice setting, many seemingly simple qu... | 10 | https://mathoverflow.net/users/4923 | 179411 | 90,174 |
https://mathoverflow.net/questions/179418 | 2 | Let $C$ be a local complete intersection projective curve in $\mathbb{P}^3$. Assume that $C$ is integral. Let $\mathcal{L}$ be a line bundle on $C$ of negative degree. We know that if $C$ is smooth then there are no global sections of $\mathcal{L}$. Is this still true if $C$ is not smooth?
| https://mathoverflow.net/users/54369 | Negative degree line bundles over a singular projective curve have no sections? | Yes. Of course it depends how you define the degree of $\mathcal{L}$; I recommend Mumford's *Lectures on curves on an algebraic surface*, Lecture 11, for a very nice approach. It implies $\deg \mathcal{O}\_C(D)=\deg D :=\dim H^0(\mathcal{O}\_D)$ for an effective Cartier divisor $D$. In particular, if $\mathcal{L}$ has ... | 2 | https://mathoverflow.net/users/40297 | 179420 | 90,177 |
https://mathoverflow.net/questions/179425 | 1 | If the set system $(X,S)$ has the $(p,q)$-property does its dual system also have the property? (Possibly, for different $p$ and $q$.)
Explicitly, I am asking about the equivalence of the following two properties:
* Out of any $p$ sets in $S$ we can find $q$ with nonempty intersection.
* Out of any $p$ points in $... | https://mathoverflow.net/users/57467 | dual (p,q)-property | The properties are not equivalent. Let $S$ consist of (at least two) subsets of $X$, which have one point entirely in common, but which are otherwise disjoint (and which have other points than that common point). This has the $(p,q)$ property, since all sets in $S$ have nonempty intersection. But if $p,q>1$, then you c... | 3 | https://mathoverflow.net/users/1946 | 179426 | 90,180 |
https://mathoverflow.net/questions/179419 | 4 | Let $T$ be a tournament with $n$ vertices (i.e., between every pair of vertices there exists an edge in exactly one direction.) For any $k$, the vertices $A\_1,A\_2,...,A\_k$ form a **transitive path** if there exists an edge from $A\_i$ to $A\_j$ for all $i<j$. The **number of transitive paths** in $T$ is the sum of t... | https://mathoverflow.net/users/57070 | Minimum number of transitive paths in tournament | As pointed out in the comments, the established name for what you call a transitive path is "transitive subtournament". Call a transitive subtournament "maximal" if it isn't included in a larger transitive subtournament. [This paper](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2899028&fileI... | 7 | https://mathoverflow.net/users/9025 | 179432 | 90,184 |
https://mathoverflow.net/questions/179434 | 7 | It is known that the sum and the product of two Dedekind-finite cardinals are also Dedekind-finite cardinals. What about cardinal exponentiation ?
Question: Let A and B be two Dedekind-finite cardinals, let C be the cardinal A power B (i.e:let x be a set with cardinal A and y be a set with cardinal B and let C be the c... | https://mathoverflow.net/users/30395 | Exponentiation and Dedekind-finite cardinals | The answer is no, not necessarily, because if there are infinite Dedekind finite sets, then the class of Dedekind finite sets is not closed under power set, and hence not closed under $A\mapsto 2^A$.
To see this, simply note that if $A$ is any infinite set, then $P(A)$ has the singletons, the doubletons, the subsets... | 12 | https://mathoverflow.net/users/1946 | 179436 | 90,186 |
https://mathoverflow.net/questions/179407 | 4 | As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}\_q$, where $q=p^n$ and $p$ is a prime greater than $2$. The Heisenberg group is defined to be $$\left\{\left.\begin{pmatrix}1 & \mathbf{x}& t\\0 & I\_n &\mathbf{y} \\ 0 & 0 ... | https://mathoverflow.net/users/31342 | Irreducible representation of Heisenberg group with characteristic 2? | Heisenberg groups over a finite field $\mathbb{F\_q}$ with $q=2^m$ are abelian and its representations are all one-dimentional, i.e., characters. The classification of irreducible representations is given (among other references) in section $3$ of the article [The Representations of the Heisenberg Group over a Finite F... | 3 | https://mathoverflow.net/users/32332 | 179439 | 90,188 |
https://mathoverflow.net/questions/179445 | 7 | It is easy to give examples of continuous functions $f:[0,1]\to \mathbb R\_+\cup\{0\}$ non-zero but vanishing on a Cantor set (ex: [Can Cantor set be the zero set of a continuous function?](https://mathoverflow.net/questions/24034/can-cantor-set-be-the-zero-set-of-a-continuous-function)). It is clearly non-true for ana... | https://mathoverflow.net/users/39115 | Non-zero smooth functions vanishing on a Cantor set | A Cantor set $C\subset[0,1]$ is closed, and that is all we need.
Therefore $f(x)=d(x,C)$ (distance from $x$ to $C$) vanishes on and only on $C$.
It is also Lipschitz-continuous.
For your third question, you can take $g(x)=f(x)^2$.
This is $C^1$ apart from peaks in the middle points of the intervals of the complement ... | 19 | https://mathoverflow.net/users/55893 | 179448 | 90,189 |
https://mathoverflow.net/questions/167707 | 2 | I'm reading Milne's Introduction to Shimura varieties (<http://www.jmilne.org/math/xnotes/svi.pdf>) and there is something I don't get.
Let $G$ be a connected semisimple algebraic group $G$ over $\mathbb{Q}$ and let $D$ be an hermitian symmetric domain such that $G(\mathbb{R})^+$ (the $^+$ denote the identity compone... | https://mathoverflow.net/users/38416 | surjective homomorphism with compact kernel (Milne's note on Shimura varieties) | You will need to add a condition that $D$ has noncompact type.
Since we have a surjective homomorphism $G(\mathbb{R})^+ \to Hol(D)^+$ with compact kernel, it suffices to show that $Hol(D)^+$ has trivial centre (so this homomorphism factors through $H\_{nc}(\mathbb{R})^+$) and that $Hol(D)^+$ has no compact factors. T... | 1 | https://mathoverflow.net/users/1046 | 179450 | 90,190 |
https://mathoverflow.net/questions/179446 | 0 | Let $E,F$ and $G$ be three complex $C^{\infty}$-vector bundles of rank $r,s$ and $rs$.
(I am using the notation from Kobayashi - Differential geometry of complex vector bundles, VI §2)
Assume we have an isomorphism $f: E\otimes F \stackrel{\sim}{\longrightarrow} G$.
If $E$ and $G$ are equipped with Hermitian str... | https://mathoverflow.net/users/43247 | Can we always solve this equation in the space of Hermitian structures on a complex vector bundle? | If $E$ has rank $0$ or $1$, 'yes', otherwise, 'no'. Just do a dimension count. You'll find that you have (many) more unknowns than equations and, for general $h$ and $h''$, there will be no solution $x$.
The answer for your modified question is still 'no, in general when $r$ and $s$ are both greater than $1$.' This i... | 4 | https://mathoverflow.net/users/13972 | 179457 | 90,192 |
https://mathoverflow.net/questions/179454 | 1 | Hopcroft and Ullman's [definition of a Turing machine](http://en.wikipedia.org/wiki/Turing_machine#Formal_definition) seems to be standard. This definition defines a Turing machine to be a 7-tupel $T = \langle Q,\Gamma,b,\Sigma,\delta,q\_0,F \rangle$ obeying some requirements among which is:
>
> The input alphabet... | https://mathoverflow.net/users/2672 | Rationale behind an requirement on Turing machines | Your proposed treatment of having machines use binary input with the alphabet $\{0,1\}$, where $0$ counts as a blank symbol (so that the input is padded with infinitely many additional $0$s, is not Turing complete, since there will be sets of natural numbers that are decidable by ordinary Turing machines (including tho... | 7 | https://mathoverflow.net/users/1946 | 179465 | 90,196 |
https://mathoverflow.net/questions/179473 | 5 | Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align\*}
f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align\*} where $[r]$ is the largest integer that is less than or equal to $r$ for $r\in\mathbb{R}$.
Let $N=\left[\frac{1}{\alpha}\right]$ be the... | https://mathoverflow.net/users/57519 | Dynamics in the integers - Floor function | The answer to the first question is: yes. The function $f(x)$ jumps $0$ or $1$ at every integer (because $0\leq\alpha\leq 1$), and also $f(0)=0$, hence $A\_1(n)=f(n+1)$. Therefore
$$\frac{A\_1(n)}{n}=\frac{[(n+1)\alpha]}{n}=\frac{n\alpha+O(1)}{n}=\alpha+O\left(\frac{1}{n}\right)=\alpha+o(1). $$
Regarding the second q... | 7 | https://mathoverflow.net/users/11919 | 179475 | 90,201 |
https://mathoverflow.net/questions/178920 | 9 | Let $E\xrightarrow{p} \Sigma X$ be a principal G-bundle over a suspension. Write $\Sigma X= C\_+X\cup\_X C\_-X$. Then there are trivialisations of the restrictions $E|\_{C\_+X}\cong C\_+X\times G$, $E|\_{C\_+X}\cong C\_-X\times G$, and the transition function between them over their intersection X is defined by a map $... | https://mathoverflow.net/users/54788 | Clutching functions and Classifying maps | Here is one possible way of answering the question, using the simplicial model for $EG$. In this context, we can show very explicitly that the map $\Sigma X\to BG$ built (via adjointness) from the clutching function for $E$ classifies $E$ (and then if $X$ is a CW complex, any other classifying map for $E$ is homotopic ... | 6 | https://mathoverflow.net/users/4042 | 179478 | 90,202 |
https://mathoverflow.net/questions/179479 | 1 | Let $\{ B\_i \}\_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$.
If I want to estimate
$$
\frac{ \lambda \left( \cup B\_i \right) }{\lambda\left( C \right) }, \tag{1}
$$
where $\lambda$ is Lebesgue Measure, is it ok to sample uniformly in the cube then count the number of point which fall in any ... | https://mathoverflow.net/users/nan | Estimating the volume of a union of balls | A naive Monte Carlo will always work, probabilistically, by the Law of Large Numbers. The problem only arises if you want guaranteed correctness, 100% chance as opposed to say 99.999%.
| 1 | https://mathoverflow.net/users/4600 | 179488 | 90,205 |
https://mathoverflow.net/questions/179487 | 11 | My research involves geometric shapes in $R^2$, and I need a metric with several properties such as:
* Families of similar shapes, such as squares, are closed in this metric. Also more general families, such as the family of 2-fat objects, are closed in this metric.
* Converging sequences of interior-disjoint shapes,... | https://mathoverflow.net/users/34461 | A metric space of geometric shapes | The following paper gives an overview on Riemannian geometries on shape spaces and diffeomorphism group.
* Martin Bauer, Martins Bruveris, Peter W. Michor: Overview of the Geometries of Shape Spaces and Diffeomorphism Groups. Journal of Mathematical Imaging and Vision, 50, 1-2, 60-97, 2014. [(pdf)](http://www.mat.uni... | 4 | https://mathoverflow.net/users/26935 | 179492 | 90,209 |
https://mathoverflow.net/questions/179484 | 4 | Let $(M,g)$ be a Riemannian manifold. Assume that $X$ is a non vanishing vector field tangent to $M$.(Or assume that we have a one dimensional foliation of $M$). Under what geometric conditions we are sure that the codimension one distribution on $M$ orthogonal to $X$ (orthogonal to $F$) is integrable? Is there a globa... | https://mathoverflow.net/users/36688 | Complementary integrable vector fields | The obstruction against integrability of the orthogonal is called curvature
for the (Ehresmann) connection given by orthogonal projection onto the distribution generated by $X$. See section 17 of
* Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Societ... | 4 | https://mathoverflow.net/users/26935 | 179495 | 90,211 |
https://mathoverflow.net/questions/179468 | 3 | Let $S$ be a scheme. We consider the functor, called *affine hull,* from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to the inclusion functor between these two full subcategories of the category of $S$-schemes (cf. EGA I.9.1.21). This func... | https://mathoverflow.net/users/11025 | Affine hulls and base change | First of all, I think that the affine hull functor can be extended to the category of all $S$-schemes, that is, not only the quasicompact quasiseparated ones [**EDIT** this may be true but the argument given here is not correct as it stands, see below]. The assumption qcqs is useful in that it ensures that the sheaf $A... | 5 | https://mathoverflow.net/users/17988 | 179497 | 90,212 |
https://mathoverflow.net/questions/179504 | 2 | Briefly, prove that every odd number having only three distinct prime factors cannot be a perfect number.
I know there are results much stronger than the one above, but I am looking for an answer without computer cracking (which means the computation can be carried out by a person), thanks.
| https://mathoverflow.net/users/57532 | A Problem Concerning Odd Perfect Number | Here is a proof, perhaps not the simplest. Let $n$ be an odd perfect number with three distinct prime factors. As observed in the comments, $n$ is of the form $3^a5^bp^c$, where $p\in\{7,11,13\}$.
It is a simple fact (observed by Euler) that exactly one of the exponents $a$, $b$, $c$ is odd. Let $q^r\parallel n$ be t... | 7 | https://mathoverflow.net/users/11919 | 179505 | 90,216 |
https://mathoverflow.net/questions/179480 | 3 | In
`http://131.220.77.52/files/preprints/diophantine/bruedern/wpminicubefinal.pdf`, Bruedern and Wooley mention the following fact on the bottom of page 6:
Let
$$\Phi(\alpha) = \sum\_{h\le 6H}\sum\_{P<x\le 2P}e(\alpha h(3x^2 + 3xh + h^2))$$
They then assert that $$\int\_{0}^1\left|\Phi(\alpha)\right|^4d\alpha\ll H^3... | https://mathoverflow.net/users/40983 | Exponential Sum Bound | First, the paper is Bruedern and Wooley!
Next ... by Cauchy's inequality, one has
$$|\Phi(\alpha)|^2\le 6H\Psi(\alpha),$$
say, where
$$\Psi(\alpha)=\sum\_{h\le 6H}\left| \sum\_{P<x\le 2P}e(\alpha h(3x^2+3xh+h^2))\right|^2.$$
Thus the mean value in question is bounded above by $6H$ times
$$\int\_0^1\Psi(\alpha)|\Phi(\... | 8 | https://mathoverflow.net/users/37998 | 179508 | 90,218 |
https://mathoverflow.net/questions/179507 | 4 | Pictures in introductory texts to Morse theory are often drawn as to interpret a Morse function as a height function. Typically, an embedding of a torus into $\mathbb{R}^3$ is drawn, and the Morse function is then the height function by projecting onto one component (call the projection $\pi$).
This is a great pictur... | https://mathoverflow.net/users/13767 | Is a Morse function always the height function of some embedding? | The formal answer is yes. Moreover, the function does not have to be Morse, just any smooth function.
Indeed let $f\colon M\to \mathbb{R}$ be any smooth function. Let us fix an imbedding $i\colon M\to \mathbb{R}^{n-1}$; for large $n$ it always exists. Consider the imbedding $(i\times f)\colon M\to \mathbb{R}^{n-1}\t... | 9 | https://mathoverflow.net/users/16183 | 179510 | 90,220 |
https://mathoverflow.net/questions/179512 | 0 | I apologize for the problem too simple, but I'm not able at solving it.
Consider the Cauchy problem
$$
\left\{
\begin{array}{l}
\dot x=x(t)^2+t\\
x(0)=0
\end{array}
\right.
$$
Show that its solution is not defined in $[0,3]$.
| https://mathoverflow.net/users/45729 | Blow up solution for a Riccati's equation | Assume that x(t) is a solution: then, as its derivative is non negative, $\forall t$, $x(t) \geq 0$, so $x(1) \geq 0= \tan 0$. Note moreover that $\forall t \geq 1$, $x(t) \geq x'(t)^2 +1$. This forces $x(t)$, for $t \geq 1$ to be bigger than $\tan (t-1)$, which is the solution of the problem $$y(1)=0 \ \ \ y'(t)=y(t)^... | 6 | https://mathoverflow.net/users/46104 | 179514 | 90,221 |
https://mathoverflow.net/questions/179344 | 10 | In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ground field, which is a number. Such invariants are always multiplicative under disjoint union, this is a consequence of t... | https://mathoverflow.net/users/13767 | What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist? | If the dimension of $Z(S^{n-1})$ is greater than 1, then the TQFT is not even approximately multiplicative under connect sum.
If $Z(S^{n-1})$ is 1-dimensional, then a simple cut and paste argument shows that
$$ Z(M\_1 \sharp\, M\_2) = Z(M\_1 \sqcup M\_2) \,/\, Z(S^n) = Z(M\_1) \cdot Z(M\_2) \,/\, Z(S^n) . $$
So in th... | 6 | https://mathoverflow.net/users/284 | 179519 | 90,223 |
https://mathoverflow.net/questions/179516 | 3 | I am looking for a real modular function $F(q,\bar{q})$ such that in the limit of small $q,\bar{q}$ it behaves as:
$F(q,\bar{q})=(a\_0 + a\_1 (q + \bar q)+...)\log q \bar q+ (b\_0 + b\_1 (q + \bar q)+...)$
${\it i.e.}$ only non-negative powers of $q$ and $\bar{q}$ arise, with possibly a logarithmic divergence (but ... | https://mathoverflow.net/users/41940 | Existence of real modular function with specific behavior as $q\to 0$ | For any holomorphic cuspform $f$ vanishing to order exactly $1$ at $i\infty$, $F(x+iy)=\log|f(x+iy)|$ is of the form you want.
The extreme simple case of this construction is the (in)famous $f(z)=q\prod (1-q^n)^{24}$, the log of whose absolute value appears in the Laurent expansion of the Eisenstein series at $s=1$ (... | 2 | https://mathoverflow.net/users/15629 | 179526 | 90,228 |
https://mathoverflow.net/questions/179527 | 1 | The generators of a cubic Cayley graph always include at least one involution, since 3 is odd. There are then two possibilties for the other two generators: either (A) they are also (distinct) involutions, or (B) they are each other's inverses and of order >2. Note that
* non-isomorphic Cayley graphs can be isomorphi... | https://mathoverflow.net/users/4336 | Cubic Cayley (undirected) graphs | Everything that is not obviously forbidden can happen.
For example, a single group can have two Cayley graphs that are isomorphic (as graphs) but which are not of the same type. Such examples will definitely not be isomorphic as Cayley graphs (in your terminology).
I think if you take the dihedral group $D\_4=\lang... | 6 | https://mathoverflow.net/users/22377 | 179533 | 90,231 |
https://mathoverflow.net/questions/177267 | 9 | What is the earliest possible reference for definition and basic properties of Clifford algebras associated to quadratic modules over a ringed space? The ringed space does not need to be locally ringed, but I am willing to assume that $2$ is invertible in the sheaf of rings if that helps. Searching mathematical reviews... | https://mathoverflow.net/users/50846 | Clifford algebras for quadratic modules over ringed spaces | Maybe Asher Auel's thesis? But maybe it's still schemes, not sure! (but that thesis is still awesome)
<http://users.math.yale.edu/~auel/papers/docs/AUEL_thesisSS.pdf>
| 1 | https://mathoverflow.net/users/46690 | 179539 | 90,234 |
https://mathoverflow.net/questions/179542 | 4 | Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban
"The Iwasawa main conjecture for $\mathrm{GL}\_2$"?
I am interested in partucular in the case of elliptic curve (as in the end of section 3 of Skinner-Urban's work).
Thank you!
| https://mathoverflow.net/users/57348 | Reference for Skinner-Urban on the Iwasawa main conjecture for $\mathrm{GL}_2$ | Wan's introduction (in Beenakker's answer) is freely available <http://www.match.uni-heidelberg.de/publications.php> thanks to an agreement of Heidelberg math department with Springer.
| 7 | https://mathoverflow.net/users/56594 | 179552 | 90,236 |
https://mathoverflow.net/questions/179566 | 2 | Let $G$ be a (finite or infinite) simple graph. We let $\mathrm{Ind}(G)$ be the collection of independent sets. For any cardinal $\kappa$ and coloring map $\chi: G\to \kappa$ we have $\chi^{-1}(\{\beta\}) \in \mathrm{Ind}(G)$ for all $\beta \in \kappa$.
Let $\mathrm{Max}(\mathrm{Ind}(G))$ be the set of maximal indepe... | https://mathoverflow.net/users/8628 | Coloring graph such that the coloring classes are not maximal independent sets | Of course not. Consider any graph $G$ and any coloring map $\chi:G\to\kappa\_0$. Choose $\beta\in\kappa\_0$, extend the independent set $\chi^{-1}(\beta)$ to a maximal independent set $S$, and define a new coloring map $\chi':G\to\kappa\_0$ by setting $\chi'(v)=\beta$ if $v\in S$ and $\chi'(v)=\chi(v)$ otherwise.
If ... | 2 | https://mathoverflow.net/users/43266 | 179571 | 90,242 |
https://mathoverflow.net/questions/179562 | 0 | Let $A$ be an $m \times n$ matrix, and define:
\begin{align\*}
U &= {\rm diag} \{ \frac{1}{\beta\_j} \}, \beta\_j = \sum\_{k=1}^m |a\_{kj}|, j = 1 \dots n \\
V &= {\rm diag} \{ \frac{1}{\alpha\_i} \}, \alpha\_i = \sum\_{k=1}^n |a\_{ik}|, i = 1 \dots m.
\end{align\*}
i.e. $\beta\_j$ is the 1-norm of the $j$th column ... | https://mathoverflow.net/users/57562 | Proving that the eigenvalues of a certain matrix product are positive | That matrix is the product of two positive definite matrices, $M=UW$, with $W=A^TVA$. Hence it is similar to the symmetric matrix $U^{1/2}WU^{1/2}$, which is congruent to $W$ and thus positive definite. For a slightly more general result, see Horn, Johnson, *Matrix analysis*, 1st ed, Theorem 7.6.3.
| 7 | https://mathoverflow.net/users/1898 | 179575 | 90,243 |
https://mathoverflow.net/questions/179561 | 3 | Let $f:Y\rightarrow\mathbb{P}^1$ be an elliptic $K3$ surface, then the holomorphic 2-form $\Omega\_Y$ vanishes when restricted to an elliptic fiber $f^{-1}(b)$ with $b\in\mathbb{P}^1$. After a hyperkahler rotation, we get a new $K3$ surface $X$, and the map $f:X\rightarrow S^2$ becomes a special Lagrangian fibration wi... | https://mathoverflow.net/users/43423 | When will the mirror of a K3 surface be an elliptic K3? | A K3 surface admits an elliptic fibration if and only if there is an isotropic vector in the Neron-Severi lattice (the elliptic fiber corresponds to the isotropic vector after a sequence of reflections by $(-2)$-curves). So the mirror K3 surface is elliptic if and only if there is an isotropic vector orthogonal to the ... | 4 | https://mathoverflow.net/users/21014 | 179577 | 90,244 |
https://mathoverflow.net/questions/179578 | 8 | Let $G$ be a closed subgroup of $U(n,{\bf C})$, not necessarily connected. Regard ${\bf C}^n$ as a complex $G$-module $M$.
>
> **Q.** Suppose $M$ is irreducible as a $G$-module (equivalent, I think, to the condition that the linear span of $G$ is all of $M\_n({\bf C})$). Does there always exist a *finite* subgroup ... | https://mathoverflow.net/users/763 | Do compact groups acting irreducibly have finite subgroups which do the same? | No. Consider an irreducible representation of $\mathrm{SU}(2)$ of large dimension, say $n\ge n\_0$. Then it cannot be irreducible for a finite subgroup, because these are either abelian or dihedral (with only 1 or 2-dimensional irreducible reps) or in a short finite list, with finitely many possible dimensions of irred... | 8 | https://mathoverflow.net/users/14094 | 179584 | 90,245 |
https://mathoverflow.net/questions/179498 | 9 | Because addition and multiplication of two order types are non-commutative operations, we have that for every integer n, given n ordinals, there are at most n! distinct possible values for the sum (or the product) of these n ordinals when all permutations are considered.
In fact, A. Wakulic (Fund. Math., XXXVI, pp 255-... | https://mathoverflow.net/users/30395 | What is the maximal number of distinct values of the product of n permuted ordinals | Simply take the ordinals $\omega+1,...,\omega+n$ and one obtains $n!$ distinct products (This solution was taken from Chapter 8 Problem 39 and Chapter 9 Problem 66 in the book Problems and Theorems is Classical Set Theory).
This follows from the easily provable fact that for natural numbers $r\_{1},...,r\_{n}$ we hav... | 13 | https://mathoverflow.net/users/22277 | 179593 | 90,247 |
https://mathoverflow.net/questions/179595 | -1 | Let $M$ denote a Riemannian manifold, $\gamma$ a geodesic of $M$ defined on $\mathbb{R}$. Let $t\_{0} \in \mathbb{R}$ and $(\alpha,\beta) \in \mathbb{R}^{2}$. I define the *reparametrized* geodesic $\tilde{\gamma} \, : \, t \, \mapsto \, \gamma(\alpha t + \beta )$. Let $v \in T\_{\tilde{\gamma}(t\_{0})}M$. $\mathrm{P}\... | https://mathoverflow.net/users/37383 | Parallel transport along a reparametrized geodesic | This is correct. More is true: In your notation, for any continuous piecewise smooth curve
$c$ in $M$ and reparameterization $f$ one has:
$$
P\_{t\_0, f(t\_1), c} = P\_{t\_0, t\_1, c\circ f}\circ P\_{t\_0, f(t\_0), c}.
$$
See 24.1 of [here](http://www.mat.univie.ac.at/~michor/dgbook.pdf), e.g.
The proof is similar to y... | 1 | https://mathoverflow.net/users/26935 | 179597 | 90,249 |
https://mathoverflow.net/questions/179598 | 6 | Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in sup-norm).
(For simplicity we may just consider $M = \mathbb{R}^n$ with the usual Euclidean metric. Assuming bounded geometry i... | https://mathoverflow.net/users/13356 | Tensor product of certain Sobolev spaces on non-compact manifolds | The answer to the first question is no for a simple reason: I guess you do not want to complete the tensor product, but then then the left hand side is only dense in the right hand side.
For your second question, I think that the answer is yes, because $L^1(M)\hat\otimes L^1(N) \cong L^1(M\times N)$. But I have not ... | 4 | https://mathoverflow.net/users/26935 | 179606 | 90,252 |
https://mathoverflow.net/questions/179572 | 2 | Is there an agreed way of expressing undirected integration in formulas?
my idea of doing so would be to use the absolute value of the differential
$$\int\_a^b f(x)|dx| = \int\_b^a f(x)|dx|$$
but I would like to get some feedback before actually using it.
A typical situation, in which undirected integrals are... | https://mathoverflow.net/users/31310 | How to Express Undirected Integration | I quite like $f(x)|dx|$. Namely, $f(x)dx$ is a 1-form, whereas $f(x)|dx|$ is a density:
In differential geometry densities are used to integrate on non-orientable manifolds. They are sections of the line bundle with transition functions $|\det(d (u\circ v^{-1}))|$, from the chart $u$ to the chart $v$.
Also useful are h... | 3 | https://mathoverflow.net/users/26935 | 179609 | 90,254 |
https://mathoverflow.net/questions/177408 | 15 | Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any results giving a formula for the number of orbits of this action (or equivalently a formula for the number of ideal clas... | https://mathoverflow.net/users/46987 | Ideal classes fixed by the Galois group | I assume you want $K$ to be Galois over $\mathbb{Q}$. More generally, let $L/K$ be a Galois extension of number fields. The the class group $C\_K$ of $K$ maps to $C\_L^{G\_{L/K}}$, the part of $C\_L$ fixed by the Galois group of $L/K$, and you seem to be asking what the quotient $C\_L^{G\_{L/K}}/C\_K$ looks like.
Tak... | 10 | https://mathoverflow.net/users/11926 | 179614 | 90,256 |
https://mathoverflow.net/questions/179550 | 0 | Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl\_{\Delta}X\times\_Y X$.
a.Does there exist a collection of **smooth** morphisms of schemes $X\_i\rightarrow Y\_i$ such that
1.the schemes $W\_i:=Bl\_{\Delta\_i}X\_i\times\_{Y\_i} X\_i$ are *... | https://mathoverflow.net/users/38952 | irreducible etale cover of a blowup | Let $Y$ be the nodal cubic curve and let $X$ be its finite etale double cover. Then $X \times\_Y X$ is just a union of two copies of the finite etale double cover. Blowing up changes nothing.
You have a slight problem here - the finite etale double cover is union of two $\mathbb P^1$ which intersect at $0$ and $\inft... | 1 | https://mathoverflow.net/users/18060 | 179620 | 90,259 |
https://mathoverflow.net/questions/179617 | 10 | Let $E/\mathbb{Q}$ be an elliptic curve and suppose that the trace of Frobenius values are such that $a\_{p}(E) \equiv 0 \pmod{2}$ for all odd primes avoiding the conductor. Is it the case that $E$ contains a nontrivial rational two torsion point? Why?
| https://mathoverflow.net/users/57582 | Elliptic curves with trace of Frobenius values always congruent to 0 modulo 2 | Yes. Let $P(x) \in {\mathbb Q}[X]$ be the cubic whose roots are
the $x$-coordinates of the $2$-torsion points. The hypothesis says that
$P$ has a root mod $p$ for all but finitely many primes $p$.
If $P$ were irreducible then its Galois group would contain a $3$-cycle,
and then there would be infinitely many $p$ such ... | 12 | https://mathoverflow.net/users/14830 | 179621 | 90,260 |
https://mathoverflow.net/questions/179574 | 2 | Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions.
Are there positive integers $m<n \in \mathbb{N}$, such that for every $u,v\in U$ there are $u',v'\in U$ such that $|\frac{m}{n} - (u'+v')| < |\frac{m}{n} - (u+v)|$?
| https://mathoverflow.net/users/8628 | Approximating rational values in $]0,1[$ by a sum or difference of unit fractions | I suppose the intended inequality is
$$
\left| \frac{m}{n} - |u+v| \, \right| < \left| \frac{m}{n} - |u'+v'| \, \right|
$$
as **Wlodzimierz Holsztynski** suggests. But
there are no such $m,n$. Indeed for any real $r>0$,
rational or not, there exists a unique element of
$$
U + U := \{ u\_1 + u\_2 \mid u\_1,u\_2 \in U \}... | 3 | https://mathoverflow.net/users/14830 | 179623 | 90,261 |
https://mathoverflow.net/questions/179616 | 12 | Let $\frak T$ be a totally ordered set of topologies on $\Bbb R$.
Is $|\frak T|\le |\Bbb R|$?
| https://mathoverflow.net/users/47958 | Maximum length of a chain of topologies on $\Bbb R$ | The answer is no. As Ashutosh notes in the comments, a modification of my original answer shows in ZFC that there is a chain longer than $|\mathbb{R}|$.
**Theorem.** There is a linearly order chain of topologies on $\mathbb{R}$ of size larger than $\mathbb{R}$.
Proof. Let $\kappa$ be the smallest cardinal so that $... | 12 | https://mathoverflow.net/users/1946 | 179631 | 90,263 |
https://mathoverflow.net/questions/179611 | 23 | I have a history question for which I've had trouble finding a good answer.
The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et al. put the blame squarely on this axiom and its non-constructive character. It was noticed however that some amount o... | https://mathoverflow.net/users/11145 | construction of nonmeasurable sets | Paul Cohen posed the question of getting a model of "All Sets Lebesgue Measurable" in his early talks on his own results. (He did not mention the principle of Dependent Choices. Adding that to the problem was my idea.) I know of no work trying to prove the Vitali result constructively. Certainly Cohen's conjecture (whi... | 45 | https://mathoverflow.net/users/16371 | 179633 | 90,264 |
https://mathoverflow.net/questions/179634 | 9 | In "On the Groups J(X) - IV", Adams introduces the $e$-invariant, which turns out to be closely connected to the image of the $J$-homomorphism, but he introduces it in a more general setting. He has $d$- and $e$-invariants of a map $X\rightarrow Y$ with respect to some exact functor $k$ from Top into an abelian categor... | https://mathoverflow.net/users/47658 | Adams e-invariant | Yes, this has been done. Laures has defined an f-invariant using elliptic cohomology in [The topological q-expansion principle](http://www.ruhr-uni-bochum.de/topologie/lang.pdf). The explanation in Section 2 of his [more recent paper on Toda brackets](http://arxiv.org/pdf/1102.3783.pdf) might be a bit easier to read.
... | 11 | https://mathoverflow.net/users/2039 | 179639 | 90,267 |
https://mathoverflow.net/questions/179632 | 12 | The most usual definition of the completely positive map (c.p.) between two C\*-algebras (say, unital) is the following: $\sigma: A \to B$ should satisfy $\sigma(1)=1$ and for each $n \in \mathbb{N}$ the map $\sigma\_n: M\_n(A) \to M\_n(B)$ defined ``cordinatewise''should be positive.
However I met also the followin... | https://mathoverflow.net/users/24078 | Completely positive maps-equivalent definition | The converse follows from two observations:
1) If $a=[a\_{ij}]$ is a positive element in $M\_n(A)$, then $a=x^\*x$ for some $x=M\_n(A)$, and if we write this out in matrix entries we have
$$
a\_{ij} = \sum\_k x\_{ki}^\*x\_{kj}.
$$
2) The matrix $[a\_{ij}]$ is positive in $M\_n(A)$ if and only if it is positive in e... | 13 | https://mathoverflow.net/users/13360 | 179650 | 90,270 |
https://mathoverflow.net/questions/179547 | 2 | In truth, I do not need in the general case.
Let $R$ be a ring (associative, with unit element) and let $\mathrm{Comp}(R)$ the *category of complexes* over $\mathrm{Mod}\ R$.
If $\mathbb{N}$ is the natural numbers with the usual partial order and $\{X\_i^\bullet, \varphi\_{i,j}\colon X\_j^\bullet \to X\_i^\bullet\}... | https://mathoverflow.net/users/41035 | Is a inverse limit of indecomposable again indecomposable? | Suppose $M$ and $M'$ are two $R$-modules with a common submodule $N$ that has a descending chain of submodules
$$N=N\_0\supseteq N\_1\supseteq N\_2\supseteq\dots$$
with zero intersection.
Then if $X^\bullet\_i$ is the two term complex
$$\dots\to 0\to N\_i\to M\oplus M'\to 0\to\dots,$$
where the differential embeds $... | 6 | https://mathoverflow.net/users/22989 | 179652 | 90,271 |
https://mathoverflow.net/questions/84819 | 9 | Let $X$, $Y$, and $Z$ be topological spaces. Let $f:X \rightarrow Y$. Further assume that for every continuous function $g:Y \rightarrow Z$, $g \circ f$ is continuous.
Question: Under what conditions on the topology of $Y$ and/or $Z$ can we conclude that $f$ must then be continuous?
This is easily achieved if $Y$ ... | https://mathoverflow.net/users/3993 | Topological conditions forcing continuity | It's possible to find a sufficient condition without putting any restraint on the spaces $X$ or $Y$.
Let $Z$ be the Sierpinski space, that is $Z$ has $\{0,1\}$ as a base set, and $\tau = \{\emptyset, \{0\}, \{0,1\}\}$ as a topology.
If $f: X\to Y$ is not continuous, then there is $V\subseteq Y$ open such that $f^{-... | 11 | https://mathoverflow.net/users/8628 | 179658 | 90,275 |
https://mathoverflow.net/questions/179664 | 16 | As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, what I really want to know is applications of the section conjecture. For example, can we derive properties on the set of... | https://mathoverflow.net/users/44006 | Why is the section conjecture important? | You can start here ["Fermat's last theorem" and anabelian geometry??](https://mathoverflow.net/questions/1036/fermats-last-theorem-and-anabelian-geometry)
In particular, I mention there that: At some point Deligne thought he had a proof that the section conjecture implied Mordell, but the proof doesn't work. This is ... | 20 | https://mathoverflow.net/users/2290 | 179666 | 90,276 |
https://mathoverflow.net/questions/161037 | 8 | In the quantum harmonic oscillator, there exists a family of states called *coherent states* which form an overcomplete set of states. They are regarded as "the states most resembling classical states", in that their parameter satisfies the classical equation of motion of the harmonic oscillator. Multiple generalisatio... | https://mathoverflow.net/users/13767 | Does the notion of a "coherent state" exist in TQFTs? (ETQFTs?) | Yes they do. In the geometric quantization approach to the 3d Chern-Simons TQFT, the vector space assigned to a closed surface is a space of holomorphic sections of a certain line bundle. In this context, we have the notion of "coherent state".
If you're worried that your notion of coherent state (coming from the har... | 7 | https://mathoverflow.net/users/401 | 179667 | 90,277 |
https://mathoverflow.net/questions/179430 | 2 | Suppose I have a discrete time Markov chain $\boldsymbol{X}$ with state space $\mathbb{R}^+$. The chain is $\psi$-irreducible, aperiodic, atomless and has an invariant measure $\pi$.
If $\pi$ is finite, does that imply this measure is unique and that it is the stationary distribution of the Markov chain? I.e. does i... | https://mathoverflow.net/users/57020 | Stationary distribution of Markov chain | Yes. Let $T$ denote the transition kernel. Suppose $\pi$ and $\nu$ are distinct stationary distributions. Let $\tau'$ be defined by $\tau'(A) = \min(\pi(A),\nu(A))$: then $\tau'$ is a finite measure. Let $p = \tau'(\mathbb{R}^+)$; since $\pi \neq \nu$, we have $p < 1$, and since the chain is irreducible, $p > 0$. Let $... | 1 | https://mathoverflow.net/users/2660 | 179674 | 90,281 |
https://mathoverflow.net/questions/179661 | 11 | In [Cantor's Attic](http://cantorsattic.info/Erdos) it is stated that an $\omega$-Erdős cardinal is a stationary limit of ineffable cardinals, and Jech book is given as a reference, but I cannot find this result in that book. I have seen proofs in other sources using that Erdős cardinals are subtle, but since Jech does... | https://mathoverflow.net/users/41274 | Erdős cardinals and ineffable cardinals | To see that the set of ineffables is stationary, let $C\subseteq \eta\_\omega$ be any closed and unbounded set. It suffices to show that the good indiscernibles arising in the definition of $\omega$-Erdos for the structure $(V\_{\eta\_\omega}, \in, C)$ are in $C$ since by what you have already shown such indiscernibles... | 13 | https://mathoverflow.net/users/6942 | 179676 | 90,283 |
https://mathoverflow.net/questions/179688 | 13 | I have two related questions about smooth complete algebraic curves (over $\mathbb{C}$).
1. Does there exist a smooth complete algebraic curve $X$ that cannot be embedded as a complete intersection in $\mathbb{P}^n$ for any $n$ (nb: this is different from asking if every smooth curve in $\mathbb{P}^n$ is a complete i... | https://mathoverflow.net/users/57620 | Realizing algebraic curves as complete intersections | 1) The genus of a complete intersection of multidegree $(d\_1,\ldots ,d\_{n-1})$ in $\mathbb{P}^n$ is $g=1+\frac{1}{2} d\_1\cdot \ldots \cdot d\_{n-1}(\sum d\_i-n-1)$ (just compute the degree of the canonical bundle). This gives very particular values for $g$: $0, 1, 3, 4, 5, 6, 9, 10, 13, 15, 16,\dotsc $ . Any curve w... | 25 | https://mathoverflow.net/users/40297 | 179690 | 90,289 |
https://mathoverflow.net/questions/178447 | 9 | I posted it on MSE and didn't receive any comments or answers, so I thought I would post it here.
(See <https://math.stackexchange.com/questions/895549/keisler-order-saturated-ultrapowers>)
Keisler's paper "Ultraproducts which are not Saturated" states the following theorem as a corollary to a (much more) generali... | https://mathoverflow.net/users/51323 | Saturated Ultrapowers | W.W. Comfort, S. Negrepontis, *The Theory of Ultrafilters*, section 13, in particular Theorem 13.7 and Corollary 13.8, might be useful to you. It contains a textbook presentation of the relevant proofs (without games).
| 4 | https://mathoverflow.net/users/57583 | 179700 | 90,292 |
https://mathoverflow.net/questions/179709 | 7 | Consider the following linear system
$$\dot{x}=\sum\limits\_{i=1}^{m}{{{\alpha }\_{i}}}\left( t \right)\cdot {{A}\_{i}}\cdot x \quad (1)
$$
where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, $\sum\limits\_{i=1}^{m}{{{\alpha }\_{i}}}\left( t \right)=1$, ${{\alpha }\_{i}}\left( t \right)$are functions and $... | https://mathoverflow.net/users/57631 | What are the upper bound and stability conditions for the following simple linear system? | This is a more general case than the one of a [switched dynamical system](https://en.wikipedia.org/wiki/Joint_spectral_radius#Applications): it is obtained as the case in which $\alpha\_i(t)$ are constant along discrete time steps of length $h$, and at each of these time steps one of the $\alpha\_i(t)$ is one and all t... | 3 | https://mathoverflow.net/users/1898 | 179713 | 90,296 |
https://mathoverflow.net/questions/179581 | 6 | Let $\lambda$ and $\sigma$ be partitions of $n$: $\lambda\_1+\lambda\_2+\cdots+\lambda\_l=n$ and $\sigma\_1+\sigma\_2+\cdots+\sigma\_s=n$
Let $M\_{\lambda \sigma}$ be the number of ways to colour the parts of lengths $\lambda\_1,\lambda\_2,\cdots,\lambda\_l$ with $s$ colours such that
1) each part is coloured with ... | https://mathoverflow.net/users/41522 | Coloring summands of given n-partition with given weights of colors | Your number $M\_{\lambda\sigma}$ seems to be the same as $L\_{\lambda\mu}$,
defined in (6.9) page 103 in Macdonalds book on symmetric functions (second edition)
if I am not mistaken.
These numbers are coefficients in the transition matrix between power sum polynomials $p\_\lambda$ and monomial symmetrix polynomials, ... | 6 | https://mathoverflow.net/users/1056 | 179718 | 90,298 |
https://mathoverflow.net/questions/179739 | 6 | For given $n$, the following $n\times n$ real matrix $M=M^{T}$ is called *positive*, if
$x^{T}M x\geq 0$
holds for all *non-negative* real $x\_1,x\_2,\cdots,x\_n$,
where $x=(x\_1,x\_2,\cdots,x\_n)^T$.
Notice here, the definition of positive is not the same as usual defination of positive semidefinite.
Now the ... | https://mathoverflow.net/users/4987 | On the positivity of matrices | Such matrices are called [copositive](http://en.wikipedia.org/wiki/Copositive_matrix) in the literature.
Moreover, the statement you want to show is known to be false.
While I don't recall a counterexample right away,
an intuition for this is that it's NP-hard to check copositivity, but computationally easy to chec... | 16 | https://mathoverflow.net/users/11100 | 179744 | 90,306 |
https://mathoverflow.net/questions/179596 | 5 | (This question is inspired by Matthieu Romagny's answer to my [previous question](https://mathoverflow.net/questions/179468) about base change properties of affine hulls.)
Given a scheme $S$, it is well-known (cf. EGA I.9.1.21) that the inclusion of the category of affine $S$-schemes into the category of quasicompact... | https://mathoverflow.net/users/11025 | Existence of affine hulls | First, some general comments.
If $f:X\to S$ is an $S$-scheme, put $A(X):=f\_\*(\mathscr{O}\_X)$. If $X$ and $Y$ are $S$-schemes, we have a natural map $$\mathrm{Hom}\_S(X,Y)\to\mathrm{Hom}\_{\mathscr{O}\_S-\text{algebras}}(A(Y), A(X))$$ which is bijective if $Y$ is $S$-affine (EGA I, (9.1.5); this is true even if $X... | 7 | https://mathoverflow.net/users/7666 | 179761 | 90,309 |
https://mathoverflow.net/questions/179759 | 5 | Consider the elliptic PDE
$$(CR)\;\;\;\;\;\;\begin{cases} U\_{xx}=V\_{yy}\\U\_{yy}=-V\_{xx} \end{cases}$$
And its consequence $$(LAP)\;\;\;\;\;\;U\_{xxxx}+U\_{yyyy}=0$$.
Somehow, these equations are similar to the classical Cauchi Riemann and Laplace equation.
>
> 1.Assume that $U$ satisfies the "LAP" equation.... | https://mathoverflow.net/users/36688 | A question on certain elliptic PDE | 1.If $U$ satisfies LAP then there exists a $V$ such that $(U,V)$ satisfies CR. In fact, $V$ is unique up to the addition of a term of the form $a + bx + cy + d xy$, where $a$, $b$, $c$, and $d$ are constants. This is an elementary consequence of the Frobenius Theorem.
2.You need to specify what you mean by 'algebra'... | 10 | https://mathoverflow.net/users/13972 | 179767 | 90,311 |
https://mathoverflow.net/questions/179762 | 10 | Several theorems regarding differential equations are true not only in finite dimension but also in infinite dimension Banach spaces. This is in particular the case for the **Cauchy–Lipschitz theorem**.
On the other end the Peano existence theorem is false for Banach space with infinite dimensions. See [here](http://... | https://mathoverflow.net/users/41060 | Examples of continuous differential equations with no solution | Here is an example in $C([-1,1],R)$, which is a continuous analogue to the discrete example you pointed to:
$$
{du(t,x)\over dt} = \operatorname{sign}(u(t,x))\sqrt{|u(t,x)|} + x\;,\qquad u(0,x) = 0\;.
$$
For any $t > 0$, the solution (in $L^\infty$) develops a discontinuity at the origin, so that it doesn't belong to $... | 11 | https://mathoverflow.net/users/38566 | 179768 | 90,312 |
https://mathoverflow.net/questions/179760 | 2 | Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions $X = (x\_1, x\_2, ..., x\_N)$, we have
$X'' = A\*X$
How can this system become chaotic by introducing an extra term? For instance, would it become chaoti... | https://mathoverflow.net/users/57655 | Making a system of second-order ODEs chaotic | linearly coupled anharmonic oscillators exhibit chaotic dynamics, a simple example studied by [Steep, Louw, and Villet](http://www.publish.csiro.au/?act=view_file&file_id=PH870587.pdf) is
$$\ddot{x}\_1=-A x\_1-ax\_1^3-cx\_2$$
$$\ddot{x}\_2=-Ax\_2-ax\_2^3-cx\_1$$
| 2 | https://mathoverflow.net/users/11260 | 179771 | 90,315 |
https://mathoverflow.net/questions/179747 | 8 | It seems rather well known that given a Laplace-Beltrami operator $\mathcal{L}\_{M}$ on a manifold $M$ we can approximate its spectrum by that of a graph Laplacian $L\_{G}$ for some $G$ (where $G$ is usually a triangulation of $M$). See [here](https://mathoverflow.net/questions/66892/is-the-laplacian-on-a-manifold-the-... | https://mathoverflow.net/users/22051 | Can the graph Laplacian be well approximated by a Laplace-Beltrami operator? | Yes, embed the graph into a compact Riemannian surface with a Riemannian metric which is concentrated near the embedded praph and induces the graph distances on the graph. Away from a tubular neighborhood of the graph the metric should be uniformly small. Then the Laplace Beltrami operator of the the surface approximat... | 6 | https://mathoverflow.net/users/26935 | 179775 | 90,317 |
https://mathoverflow.net/questions/179745 | 2 | *Definition*: A group $G$ is **indecomposable** if: $G = G\_1 \times G\_2 \Rightarrow \exists i \ G\_i = 1$.
We can generalize the notion of indecomposable from groups to inclusion of groups as follows:
*Definition*: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A\_B ... | https://mathoverflow.net/users/34538 | Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable | If I have understood the definition of an indecomposable inclusion correctly, then any core-free subgroup $H$ of a direct product $G=G\_1 \times G\_2$ in which $H$ is indecomposable, and is not contained in any direct factor of $G$ isomorphic to $G\_1$ and $G\_2$ will be an example.
There are many such examples, such... | 2 | https://mathoverflow.net/users/35840 | 179789 | 90,323 |
https://mathoverflow.net/questions/179787 | 1 | If $A$ is a $n \times n$ complex matrix then the **Schur norm** of $A$ is given by $$ || A||\_S := \max\_{||B||=1} ||A\*B||,$$ where $||. ||$ is the operator norm and $\*$ is the Hadamard (entry-wise) product:
$$(A\*B)\_{ij}= A\_{ij} B\_{ij}.$$
In the case that $A$ is self-adjoint may we restrict the minimization o... | https://mathoverflow.net/users/56547 | Schur norm for self-adjoint operators | R. Bhatia has answered this question:
The answer is yes, and it's equation 3.7 of
<http://www.ams.org/journals/proc/1993-117-04/S0002-9939-1993-1116267-7/S0002-9939-1993-1116267-7.pdf>
| 1 | https://mathoverflow.net/users/56547 | 179791 | 90,324 |
https://mathoverflow.net/questions/179730 | 12 | **Question:** Is there any book of topology in the modern language of topos theory?
**Motivation:**
* In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the study of (the cohomology of) sheaves, and the sheaves "sited" on a given Grothendieck topology formed a topos - ... | https://mathoverflow.net/users/56910 | Reference request: Book of topology from "Topos" point of view | "Topology via Logic" is only half way there. It is firmly rooted in classical mathematics and makes no connections with toposes.
Mac Lane and Moerdijk is a good suggestion. As for reading it backwards: that is pretty much the aim of my "Locales and Toposes as Spaces" (Chapter 8 in "Handbook of Spatial Logics" (ed. Ai... | 20 | https://mathoverflow.net/users/57668 | 179797 | 90,326 |
https://mathoverflow.net/questions/179783 | 6 | This is a claim in Shelah's Proper and Improper Forcing, more specifically Claim 2.3(1) of Chapter X (p. 484). The proof of the claim is "Easy" but I cannot quite figure it out. There must be some trick that I am missing.
A forcing notion $P$ is $RUCar^{V}$-semiproper iff there is a cardinal $\kappa$, such that for a... | https://mathoverflow.net/users/29231 | $RUCar^{V}$-semiproperness implies properness | Given such an $N$ and $q$, you want to see that $q$ is $(N, P)$-generic. This is equivalent to $q$ forcing $N[G]\cap On = N\cap On$. (Corollary 2.13 page 106 of Proper and Improper Forcing)
Suppose $\dot\alpha$ is a $P$-name for an ordinal with $\dot\alpha\in N$. Then $\{\xi\in On: \text{ some $r\in P$ forces $\dot\a... | 8 | https://mathoverflow.net/users/18128 | 179799 | 90,327 |
https://mathoverflow.net/questions/179758 | 5 | I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form
$
\left(
\begin{array}{cc}
a+b & -a \\
a & b%
\end{array}%
\right),
$
where a and b are rational integers. By calculate I mean calculation ... | https://mathoverflow.net/users/20282 | Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q) | The matrix in the question is the representing matrix of multiplication with $b-a\zeta\_3$ on $\mathbb{Q}(\zeta\_3)$, for the basis $\{\zeta\_3,1\}$.
Therefore, the group of matrices in the question is $R\_{\mathbb{Q}(\zeta\_3)/\mathbb{Q}}(\mathbb{G}\_m/\mathbb{Q}(\zeta\_3))$, the Weil restriction of the multiplicative... | 3 | https://mathoverflow.net/users/50846 | 179801 | 90,328 |
https://mathoverflow.net/questions/179751 | 1 | Here is my main question: what is the difference between "straight" and "piecewise linear" and "continuous" embeddings of graphs/complexes in d-dimensional space?
Moreover I would like to know if any other type of embedding is defined that is in some sense similar to "straight" or "piecewise linear" or "continuous" e... | https://mathoverflow.net/users/27400 | Difference between straight and piecewise linear and continuous embeddings of graphs / complexes in d-dimensional space? | An embedding of an $n$-dimensional simplicial complex $K$ (a graph if $n=1$) into ${\mathbb R}^d$ is *linear* if the restriction to each simplex of $K$ is a linear map. In the stable range, $d>2n+1$, by general position any $K^n$ linearly embeds into ${\mathbb R}^d$.There are various Ramsey-type results, combining comb... | 7 | https://mathoverflow.net/users/25011 | 179804 | 90,330 |
https://mathoverflow.net/questions/179108 | 15 | Consider the sequence of [Apéry numbers](http://oeis.org/A005259)
$$
A\_n = \sum\_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum\_{j=0}^k \binom{k}{j}^3
= \sum\_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist [Alan Sokal](http://www.physics.nyu.edu/faculty/sokal/) conjectures that it is a [Stieltjes momen... | https://mathoverflow.net/users/454 | Is the sequence of Apéry numbers a Stieltjes moment sequence? | Here are some computations that may be useful. Here below, $\big[ \,f\,\big]\_a^b$ denotes $f(b)-f(a)$, and $n^{(k)}:=n(n-1)\dots(n-k+1)$.
**1. (An integration by parts).** *Let $w$ be a solution of the third order linear ODE on the interval $[a,b]$:
$$(x^4-34x^3+x^2)w''' + 3(2x^3-51x^2+x)w''+(7x^2-112x+1)w'+(x-5)w=0... | 3 | https://mathoverflow.net/users/6101 | 179805 | 90,331 |
https://mathoverflow.net/questions/179811 | 4 | Following Grothendieck, let us say that a category is **aspheric** if its nerve is weakly contractible and a functor $u : \mathcal{A} \to \mathcal{B}$ is **aspheric** if for every object $b$ in $\mathcal{B}$, the comma category $(u \downarrow b)$ is aspheric.
**Question.** Consider a pullback diagram in $\mathbf{Cat... | https://mathoverflow.net/users/11640 | Aspheric functors and Grothendieck fibrations | You may find this in [Maltsiniotis'Astérisque 301](https://www.imj-prg.fr/~georges.maltsiniotis/ps/prstnew.pdf): it is equivalent to prove the dual version (using Proposition 1.1.22), and then, Example 3.2.2 show that this is a particular case of Corollary 3.2.13 of loc. cit. This kind of ideas is developped systematic... | 7 | https://mathoverflow.net/users/1017 | 179814 | 90,336 |
https://mathoverflow.net/questions/179824 | 4 | Let $G\_{\bullet}$ be a simplical group. I denote by $D\_n \subset G\_n$ the subgroup generated by all the degeneracy maps $s\_i:G\_{n-1} \to G\_n$.
I also denote $$M\_n = \bigcap\_{i>0} \mathrm{Ker} d\_i \subset G\_n$$
and $$Z\_n = \bigcap\_{i\geq 0} \mathrm{Ker} d\_i \subset G\_n$$
When $G\_{\bullet}$ is a simplicial... | https://mathoverflow.net/users/43850 | In a Simplicial group the Degenerate elements don't intersect the kernel of the face maps. | Tomer: The claim you make about $M\_n\cap D\_n$ being trivial in the non abelian case is not true. In fact that condition is equivalent to the Moore complex of $G$ being a crossed complex in the sense of Brown and Higgins. This seems to be first proved in the thesis of Ashley (N. Ashley, Simplicial T-Complexes: a non a... | 6 | https://mathoverflow.net/users/3502 | 179827 | 90,341 |
https://mathoverflow.net/questions/179637 | 17 | Browkin and Brzezinski, in "Some remarks on the $abc$-conjecture", Math. Comp. **62** (1994), no. 206, 931–939, state the following generalization of the $abc$ conjecture to more than three variables:
>
> Given $n\ge3$, let $a\_1,\dots,a\_n \in \Bbb Z$ satisfy $\gcd(a\_1,a\_2,\dots,a\_n)=1$ and $a\_1+\dots+a\_n=0$,... | https://mathoverflow.net/users/5091 | Probing the generalization of the abc conjecture to more than 3 variables | In [A more general abc conjecture](http://arxiv.org/abs/math/9806171), p. 7 Paul Vojta conjectures exponent $1 + \epsilon$ *outside a proper Zariski-closed subset*
---
Your question appears in [The abc-conjecture and the n-conjecture,Coen Ramaekers](http://alexandria.tue.nl/extra1/afstversl/wsk-i/ramaekers2009.pd... | 7 | https://mathoverflow.net/users/12481 | 179833 | 90,343 |
https://mathoverflow.net/questions/179834 | 9 | We know that Ore's Conjecture is a theorem now. Does anyone know a counterexample to Ore's conjecture for perfect groups? I believe there should be one, but I dont have a counterexample. Thanks.
| https://mathoverflow.net/users/57278 | Ore's Conjecture for perfect groups | The counterexample referred to in the comment by NAME\_IN\_CAPS is the split extension of the so-called deleted permutation module for $A\_5$ over ${\mathbb F}\_2$. that is, the $4$-dimensional irreducible component of the $5$-dimensional permutation module. This group has $960$ elements, only $840$ of which are commut... | 18 | https://mathoverflow.net/users/35840 | 179841 | 90,347 |
https://mathoverflow.net/questions/163751 | 2 | Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H\_x$ of a point $x \in X$. It is the subgroup of $G$ that leaves $x$ invariant. We then have $G/H \cong X$ as $G$-spaces.
Also, given the representation category of a ... | https://mathoverflow.net/users/13767 | From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps? | I understand it for finite groups now, and I suspect that it's similar for semisimple Lie groups.
These statements are equivalent, and true for finite groups:
* The adjunction $\operatorname{Res} \dashv \operatorname{CoInd}$ is monadic (for compact $G/H$), its monad is $\mathrm{L}^2(G/H) \otimes -$.
* $H$-represent... | 0 | https://mathoverflow.net/users/13767 | 179854 | 90,352 |
https://mathoverflow.net/questions/179851 | 6 | This question is motivated by model theory, but it's really an analysis question (which means it may have an easy analysis answer that I just don't have the background for). Here's the main question, followed by some definitions and motivation:
>
> Is there a partial function $f:\mathbb R\to \mathbb R$, defined on ... | https://mathoverflow.net/users/15735 | Are there superexponential Pfaffian functions? | No, [1] proved more generally that the Pfaffian closure of any o-minimal polynomially bounded structure is exponentially bounded.
[1] Jean-Marie Lion, Chris Miller, and Patrick Speissegger, [*Differential equations over polynomially bounded o-minimal structures*](http://dx.doi.org/10.1090/S0002-9939-02-06509-7), Proc... | 7 | https://mathoverflow.net/users/12705 | 179860 | 90,354 |
https://mathoverflow.net/questions/179857 | 3 | The cycle containing $1$ of a uniform permutation has length which is uniformly distributed. I was wondering if the converse is true:
>
> Suppose $\sigma$ is a permutation on $\{1,\dots,n\}$ and let $u \in \{1,\dots,n\}$ be a uniformly chosen random variable, independent of $\sigma$. Let $V$ be the size of the cycl... | https://mathoverflow.net/users/18279 | Uniformly permutation and the length of a size biased cycle | Observe that the distribution of $V|\sigma$ depends only on the lengths of the cycles in $\sigma$. Therefore, the distribution of $V$ does not change if we replace the random variable $\sigma$ with $f(\sigma)$, where $f:S\_n\to S\_n$ is any function that maintains the lengths of the cycles.
Example: let $\sigma$ be t... | 3 | https://mathoverflow.net/users/48859 | 179866 | 90,357 |
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