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https://mathoverflow.net/questions/178700 | 10 | I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to find a proof [or counterexample!].)
We have a family $(P\_x^t)\_{x \in \mathbb{R},t \geq 0}$ of Borel probability measu... | https://mathoverflow.net/users/15570 | Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes | I have the answers to my two questions. (I've actually had them for a while; apologies for the delay in posting.) I will give them in reverse order:
The answer to Q2 is *yes*; the structure of the proof is exactly as I outlined in the update. Details can be found in section 5 (in particular, Corollary 100) of my note... | 3 | https://mathoverflow.net/users/15570 | 199539 | 96,439 |
https://mathoverflow.net/questions/199462 | 8 | Given $n$. Two players in turn write different real numbers $x\_1,x\_2,x\_3,\dots$
The player after whose turn there is a monotone subsequence of length $n$ loses.
I guess that the question 'who wins' may be hopeless (nice if not so), but possibly there exists a better estimate of the number of turns than Erdős–Szeker... | https://mathoverflow.net/users/4312 | Yet another Erdős–Szekeres game | As noted in the comments (but with not quite the right reference) the game is a first player win for $n \geq 4$. The question here is about the misere form, so this is a combination of Proposition 7, Theorem 10 and a bit of Proposition 9 in the paper [Monotonic Sequence Games](http://arxiv.org/pdf/math/0602630.pdf) by ... | 8 | https://mathoverflow.net/users/1907 | 199541 | 96,440 |
https://mathoverflow.net/questions/199518 | 5 | What are the automorphisms of real Clifford algebras $Cl\_{n,0}$? Of course, I'm interested in the case where they are not central simple.
| https://mathoverflow.net/users/34575 | Automorphisms of Clifford Algebras | $C\_{n,0}$ is either a full matrix algebra over $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$, or the direct sum of two such algebras that are isomorphic. The exact description depends on the residue class of $n$ (mod $8$) and can be found in textbooks or on wikipedia <http://en.wikipedia.org/wiki/Classification_of_Cliff... | 10 | https://mathoverflow.net/users/68305 | 199545 | 96,441 |
https://mathoverflow.net/questions/199549 | 4 | By using the Löwenheim–Skolem theorem & Mostowski collapse, in every model $V$ of $ZF+Con(ZF)$ there is a countable transitive set $M$ such that $(M,\in\_M) \models ZF$. Is the following "converse" true?
>
> In every model $V$ of $ZF$ and every transitive set $M \in V$ such that $(M,\in\_M) \models ZF$, there exist... | https://mathoverflow.net/users/35734 | Reverse Skolem's paradox | The answer to your question is no. For example, suppose $\kappa$ is inaccessible and $M = V\_{\kappa}$. Then $M$ is a model of $\sf ZF$ but $M$ cannot be countable in $N$ as it is not countable in $V$.
Of course there will always be a set generic extension of $V$ containing such an $N$ (just generically add a bijecti... | 9 | https://mathoverflow.net/users/8106 | 199552 | 96,443 |
https://mathoverflow.net/questions/199047 | 5 | I am looking for a reference for the following result (if it is true, which I would expect):
Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. Let $\Gamma\_0\subset\partial\Omega$ be sufficiently regular.
Let $V\_1:=\left\{\phi\in C^\infty\left(\overline{\Omega}\right):\phi\geq 0 ~\text{on}~ \Gamma\_0\... | https://mathoverflow.net/users/68804 | Density of smooth functions in Sobolev space, respecting nonnegative traces | More of a comment than an answer, but...
I don't know a reference, but perhaps you could prove it as follows. For $u \in W^{1,p}(\Omega)$ with nonnegative trace, write $u = u^+ - u^-$ which are both in $W^{1,p}(\Omega)$. Since $u^+$ is nonnegative everywhere, you should be able to approximate it by nonnegative smooth... | 4 | https://mathoverflow.net/users/4832 | 199577 | 96,455 |
https://mathoverflow.net/questions/199573 | 3 | Is there a *comprehensive* reference book on **inequalities** in the
spirit of the one written by G.H. Hardy, J.E. Littlewood, and G. Pólya**(\*)**, but more ***up-to-date*** (i.e., published in *more recent years* and with both well-established results and *novel developments*)?
---
**(\*)** **Clarification**
... | https://mathoverflow.net/users/nan | More recently published comprehensive reference on inequalities in the spirit of Hardy-Littlewood-Pólya | I think that the following book is a very nice 'grandson' of the one written by Hardy, Littlewood, and Pólya:
>
> D.J.H. GARLING, *Inequalities. A Journey into Linear Analysis*,
> Cambridge University Press, 2007.
>
>
>
| 6 | https://mathoverflow.net/users/69052 | 199578 | 96,456 |
https://mathoverflow.net/questions/199581 | 2 | Let $A$ be an abelian variety over a number field $K$. Let $\mathcal{A}$ be the Neron model of $A$ over $O\_K$. Let $\Omega\_{\mathcal{A}/O\_K}$ be the sheaf of invariant differential forms on $\mathcal{A}$.
In the formulation of the Tamagawa Number Conjecture for abelian varieties, for e.g., page 13 of the followin... | https://mathoverflow.net/users/69051 | integral basis for the Lie algebra of the Neron model of an abelian variety | The same problem already appears in the formulation of the Birch and Swinnerton-Dyer conjecture for say an elliptic curve over a number field. When there is no longer a global minimal Weierstrass model, then we do not have a invariant differential $\omega\_E$ that generates the $O\_K$-module of differentials on the Nér... | 2 | https://mathoverflow.net/users/5015 | 199587 | 96,458 |
https://mathoverflow.net/questions/199590 | 1 | I'm not so good on geometry, so I fear this is a relatively basic question.
For any $N \in \mathbb{N}$, let us identify the tangent bundle of $\mathbb{R}^N$ with $\mathbb{R}^{2N}$ in the obvious manner. Let $M$ be a smooth manifold. For any chart $\,\varphi:U \, \to \, \varphi(U) \! \subset \! \mathbb{R}^n\,$ on $M$,... | https://mathoverflow.net/users/15570 | Is there a name for the function on $TTM$ swapping the 2nd and 3rd coordinates? | It is called the **canonical flip**, (see for example Michor, *Topics in differential geometry*, 2008, § 8.13). It can be characterised as the unique map such that
$$
\frac{\partial}{\partial s} \frac{\partial}{\partial t} \sigma (t, s)
= \theta \frac{\partial}{\partial t} \frac{\partial}{\partial s} \sigma (t, s).
$... | 4 | https://mathoverflow.net/users/42047 | 199594 | 96,460 |
https://mathoverflow.net/questions/199599 | 4 | I've checked the factorization of $2^N - 1$ up through N = 120 for the largest prime factor, and it looks like the largest value of N where $2^N-1$ has a largest prime factor under 2500 is N = 60 (largest prime factor = 1321). As N gets larger, the largest prime factors get larger, even for the "abundant" numbers like ... | https://mathoverflow.net/users/1305 | $2^n$-1 consisting only of small factors | [Zsigmondy's theorem](http://en.wikipedia.org/wiki/Zsigmondy%27s_theorem) implies that when $n \ge 7$, $2^n - 1$ has a prime divisor not dividing $2^k - 1$ for any $k < n$. So pick $n\_0$ large enough that every prime number less than $2500$ divides $2^k - 1$ for some $k \le n\_0$, then take $n > n\_0$. This proves the... | 13 | https://mathoverflow.net/users/290 | 199601 | 96,463 |
https://mathoverflow.net/questions/199592 | 2 | Consider the the state of a system at time $n$, $X\_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X\_0$, so we have
$$X\_n = A\_n \cdots A\_1X\_0.$$
Suppose $\mathbb E\log \Vert A\_i \Vert <\infty$ and $A\_i$ takes the form
$$A\_i =
\left[\begin{array}{cc... | https://mathoverflow.net/users/48466 | Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS) | The theory is right! If $H\_i$ has negative Lyapunov exponents, then images of any vector under the transpose system will be exponentially close to a single vector (everything is being compressed in the initial components). There is then a co-dimension 1 subspace where the components of this vector cancel.
The diffic... | 5 | https://mathoverflow.net/users/11054 | 199605 | 96,466 |
https://mathoverflow.net/questions/199559 | 5 | Among some [permutation polynomials](http://en.wikipedia.org/wiki/Permutation_polynomial) I've been studying, $f(x)=(x+1)^n-x^n$ is one of the polynomials that I cannot grasp.
>
> **Question** : For a given **odd prime** $p$, how can we find **every positive integer** $n$ such that $f(x)=(x+1)^n-x^n$ is a permutat... | https://mathoverflow.net/users/34490 | Permutation polynomials mod $p$ of the form $(x+1)^n-x^n$ | The expected answer is correct, it is a theorem by Norman Johnson, see [here.](http://dx.doi.org/10.1007/BF01223263) Note that $(X+1)^n-X^n$ is a permutation polynomial if and only if $(X+a)^n-X^n$ is a permutation polynomial for each fixed nonzero $a$. This latter condition is a special case of planarity: A function $... | 10 | https://mathoverflow.net/users/18739 | 199607 | 96,467 |
https://mathoverflow.net/questions/199555 | 4 | Let $(X, \tau)$ be a topological space. We say that $(X,\tau)$ is $T\_{\text{inj}}$ if there is an injective map $f:X\to\tau$ such that $x\in f(x)$ for all $x\in X$.
It is not hard to see that $T\_1$ and all the stronger separation axioms imply $T\_{\text{inj}}$.
**Questions.** Does $T\_0$ imply $T\_{\text{inj}}$? ... | https://mathoverflow.net/users/8628 | Existence of injective neighborhood selection function as separation axiom | $\textbf{Finite $T\_{0}$-spaces are $T\_{inj}$}$
I claim that every finite $T\_{0}$-space is $T\_{inj}$. If $(X,\tau)$ is a finite $T\_{0}$-topology, then there exists a partial ordering $\leq$ on $X$ so that the open sets are precisely the upwards closed sets. Therefore, let $f:X\rightarrow\tau$ map the element $x$ ... | 8 | https://mathoverflow.net/users/22277 | 199609 | 96,468 |
https://mathoverflow.net/questions/199548 | 5 | Let $M$ be a von Neumann algebra, and let $\mathcal{P}$ be the set of nontrivial (not equal to $0$ or $e$) projections of $M$. Define $p,q \in \mathcal{P}$ to be equivalent if there exist projections $p\_1, \ldots, p\_n \in \mathcal{P}$ with $p\_1=p$, $p\_n = q$, $p\_i \perp p\_{i+1}$ and $p\_i + p\_{i+1} < e$ for $1 \... | https://mathoverflow.net/users/46472 | Question about projections in von Neumann algebras | This is true. Since $p, q \in \mathcal P(M)$ are non-trivial and not maximal, then under your definition they are equivalent to any non-trivial subprojection. Also, we have $M \not= \mathbb M\_2(\mathbb C)$, and so there exists a non-trivial projection $z \in \mathcal P(M)$ commuting with $p$ and $q$, such that $zp \no... | 8 | https://mathoverflow.net/users/6460 | 199614 | 96,472 |
https://mathoverflow.net/questions/199571 | 0 | I keep finding references to $G$-Set graphs and I cannot find a definition anywhere.
They are usually mentioned at the same time as the random graph generator "rudy," so I believe they refer to a method of generating graphs but I cannot find out any more than that.
I apologise in advance if I have missed something... | https://mathoverflow.net/users/69045 | What is a G-Set Graph? | I believe these are the 57 graphs labelled $G\_1$ to $G\_{57}$ in Table 7 of a preprint by Helmberg and Rendl (<http://opus4.kobv.de/opus4-zib/files/306/SC-97-37.pdf>). As you thought, they were generated by rudy, and the table lists the data needed to get them from that program.
Apparently, these graphs are accepte... | 1 | https://mathoverflow.net/users/68305 | 199620 | 96,473 |
https://mathoverflow.net/questions/199589 | 6 | Let $R$ be an irreducible root system with a basis $\Pi$.
We obtain the Dynkin diagram $D$ and the extended Dynkin diagram ${\widetilde{D}}$ of $R$ with respect to $\Pi$.
Let $Q^\vee\subset P^\vee$ denote the coroot lattice and the coweight lattice, respectively.
It is known that the finite abelian group $P^\vee/Q^\vee... | https://mathoverflow.net/users/4149 | The action of the center on the extended Dynkin diagram | The question is perhaps best answered not in the context of Lie theory but in the related setting of affine Weyl groups, where an irreducible root system in Bourbaki's sense leads to an extended Dynkin diagram and corresponding Coxeter group. Here the isomorphic finite abelian groups $P/Q$ and $P^\vee/Q^\vee$ may be in... | 4 | https://mathoverflow.net/users/4231 | 199626 | 96,476 |
https://mathoverflow.net/questions/199630 | 2 | Consider the quantity
$$h\_n:=\frac{S\_{n-2}}{V\_{n-2}} \int\_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$$
where $S\_{n-2}$, $V\_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb R^{n-1}$.
**Question:** How can we understand $h\_n$? In particular, is $h\_n$ increasing?
Does it tend to $1$ as $n$ g... | https://mathoverflow.net/users/32906 | understanding the average height of a unit hyper-semisphere | Your expression equals $$\frac{\sqrt{\pi } (n-1) \Gamma \left(\frac{n-1}{2}\right)}{4 \Gamma
\left(\frac{n}{2}+1\right)},$$ which goes to zero as $n$ goes to infinity.
| 2 | https://mathoverflow.net/users/11142 | 199632 | 96,477 |
https://mathoverflow.net/questions/199631 | 2 | Suppose that $V$ is a $\mathbb C$-vector space. I'm eventually interested in the infinite-dimensional case, but let's say for now that it's finite dimensional. Suppose that $\mathscr S$ is a collection of rank-1 projectors on $V$, not necessarily commuting or respecting any interesting inner product, such that $\sum\_{... | https://mathoverflow.net/users/2383 | Traces and projectors | Counterexample:
$V$ is $2$-dimensional with basis $\left(e,f\right)$. The projectors are projecting on $\mathbb C e$ and $\mathbb C f$, each with kernel $e+f$. The map $T$ projects on $\mathbb C \left(e+f\right)$.
| 2 | https://mathoverflow.net/users/2530 | 199633 | 96,478 |
https://mathoverflow.net/questions/199610 | 5 | I am trying to estimate the integral $\int \mathbb{e} ^{-d(x\_0,x)^2} \mathbb{d}x$ on a Riemann manifold $(M,g)$, for some arbitrary fixed $x\_0 \in M$ and $d$ the usual distance. The only thing that I can think of is to use some coarea theorem, leading to $\int \_0 ^\infty \mathbb{e} ^{-r^2} A(x\_0, r) \mathbb{d}r$, w... | https://mathoverflow.net/users/54780 | Area of metric spheres in Riemannian manifolds | The integral might be infinite.
Indeed, the function $A(r)$ can grow as fast as you want, but in this case the Ricci curvature in some of the radial direction have to go to $-\infty$ as $r\to\infty$. (Needless to say $A$ can grow faster than any polynomial.)
If Ricci curvature is bounded below then $A(r)$ has at most... | 9 | https://mathoverflow.net/users/1441 | 199634 | 96,479 |
https://mathoverflow.net/questions/199628 | 4 | Let $M$ be an invertible symmetric $2n \times 2n$ matrix with entries in the finite field $\mathbb{F}\_2$. Is $\mathrm{Ker}\ (M^2 - I\_{2n})$ necessarily even dimensional?
| https://mathoverflow.net/users/63877 | Fixed space of the square of a symmetric matrix over $\mathbb{F}_2$ | The matrix $M = \left(\begin{array}{clcr} 1&1&0&0\\1&1&1&0\\0&1&1&0\\0&0&0&1\end{array} \right)$ is an example where the dimension of the space in question is odd.
(Later edit: However if $M = M^{t}$ and also $M \in {\rm Sp}(2n,2),$ then we in fact have $M^{2} = I\_{2n}$ as Darij Grinberg and Noam Elkies implicitly n... | 8 | https://mathoverflow.net/users/14450 | 199658 | 96,486 |
https://mathoverflow.net/questions/199661 | 1 | Let $G=(V,E)$ be a graph. For $v\in V$ we set $N(v) = \{w\in V: \{v,w\}\in E\}$.
Let us say that a graph is *neighborly* if there is an injective function $f: V\to V$ such that $f(v)\in N(v)$ for all $v\in V$. (Is there standard terminology for this? Any pointers appreciated!)
If a graph has a Hamiltonian cycle, let'... | https://mathoverflow.net/users/8628 | Generalization of Hamiltonian cycle | I think this is true and trivial, and not really research-level.
Since $V$ is finite and $f$ is injective, it is a permutation.
Write it as a union of disjoint cycles.
Assuming the graph has no loop, $f$ has no fixed point and thus every cycle has length at least two.
The graph induced on each cycle is clearly HC... | 2 | https://mathoverflow.net/users/22377 | 199663 | 96,488 |
https://mathoverflow.net/questions/199671 | 3 | Although the classification of simple Lie Algebras and their representations is fully understood, I wonder whether there is **some book with exhaustive tables describing explicit irreducible representations in low dimensions**. For instance, it would be very helpful for me to know which are the first irreducible repres... | https://mathoverflow.net/users/62367 | First Explicit Irreducible Representations | I think "Group Theory for Unified Model Building" by R. Slansky qualifies. As the title suggests it is written with an application in physics (beyond my understanding) in mind, but the tables are very useful for purely mathematical purposes as well. (Disclaimer: it is quite some years ago that I last read it.)
| 7 | https://mathoverflow.net/users/41139 | 199673 | 96,491 |
https://mathoverflow.net/questions/199641 | 2 | If we write a manifold or CW-complex $X$ as a subset of $\mathbb{R}^n$, in expression of coordinates, for example, \begin{multline}
F(S^2,k+1)=\{(x\_1,x\_2,x\_3,\cdots, x\_{3k+1},x\_{3k+2},x\_{3k+3})\in\mathbb{R}^{3k+3}\\ \mid
x\_1^2+x\_2^2+x\_3^2=1,\cdots, x\_{3k+1}^2+x\_{3k+2}^2+x\_{3k+3}^2=1,\\
\text{ for }i\neq ... | https://mathoverflow.net/users/41075 | cohomology algebra of submanifold in euclidean space | It surely depends on what expressions you want to allow,
but what comes to my mind is that even the much more basic question whether
a given set is empty or not
* is NP-complete, so don't expect an effective algorithm,
* is computable by Tarski's theorem.
Maybe a logician could say more.
| 1 | https://mathoverflow.net/users/69091 | 199676 | 96,492 |
https://mathoverflow.net/questions/197834 | 7 | I'm following a solution of an SDE from here
<http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf>
Start with the SDE
$$
dX\_t = \delta dt + 2\sqrt{X\_t} dW\_t
$$
consider a deterministic time change
$$
\tau = \frac{\sigma^2}{2\nu(2-\delta)}\left(1-\exp\left(-\frac{2\nu t}{2-\delta}\right)\right)
$$
the process ... | https://mathoverflow.net/users/68230 | Change of time variable in Wiener process | The time change described in the question may be handled as follows. Recall that if $W(t)$ is a standard Brownian motion then
$$
W(\tau(b))-W(\tau(a))
$$ has the same distribution as
$$
\int\_a^b \sqrt{\tau'(t)} dW(t)
$$ where $a \le b$. Therefore the time-changed process $\tilde X\_t = X\_{\tau(t)}$ satisfies:
$$
d... | 8 | https://mathoverflow.net/users/64449 | 199682 | 96,494 |
https://mathoverflow.net/questions/199683 | 2 | Let $|\cdot|$ denote the usual norm in $\mathbb{Z}^d$. Given a finite subset $S \subset \mathbb{Z}^d$, let $\varphi(S) = \sum\_{z \in S}|z|^2$. Given $m \in \mathbb{N}$, what is the size of $\varphi^{-1}(m)$? In particular, a rough upper bound would suffice.
| https://mathoverflow.net/users/23661 | How many finite subsets in $\mathbb{Z}^d$ have a given sum of squares? | $|\varphi^{-1}(m)|$ is a coefficient at $x^m$ in the product
$$
\prod\_{s\in \mathbb{Z}^d} (1+x^{|s|^2}):=F(x)
$$
Hence for $0<x<1$ we have $$|\varphi^{-1}(m)|\leq x^{-m} F(x).$$
Minimising RHS in $x$ we get a reasonable upper bound. To be more specific, denote $x=e^{-t}$, $t>0$. Then
$$
\log |\varphi^{-1}(m)|\leq mt+\... | 7 | https://mathoverflow.net/users/4312 | 199685 | 96,496 |
https://mathoverflow.net/questions/199691 | 1 | We discuss on the field of complex numbers. Let $X$ be a smooth projective hypersurface of dimension $n \geq 4$ in $\mathbb{P}^{n+1}(\mathbb{C})$. Assume that for a general point $x \in X$, there exists a 4-dimensional linear subspace $L\_x$ in $\mathbb{P}^{n+1}$ passing through $x$ such that the set-theoretic intersec... | https://mathoverflow.net/users/69076 | Special linear sections of a hypersurface | The answer is "no" if the degree $d$ is a multiple of $3$, say $d=3e$. I sketched the argument in the comments, but now that comment has disappeared. For a point $x$ in $X$, there is a Grassmannian $G= \mathbb{G}(\mathbb{P}^3,\mathbb{P}^{n-1})$ parameterizing linear spaces $L\_x$ of dimension $4$ that contain $x$. If $... | 0 | https://mathoverflow.net/users/13265 | 199696 | 96,501 |
https://mathoverflow.net/questions/199560 | 3 | It is known that for a compact metric space $X$ without isolated points the set of nonatomic Borel probability measures on $X$ is dense in the set of all Borel probability measures on $X$ (endowed with the Prokhorov metric). In particular if $X$ is a product space $X=X\_1\times\cdots\times X\_n$ (each $X\_i$ a compact ... | https://mathoverflow.net/users/69039 | Nonatomic probability measures | I think that Dave's argument (as well as the reference to the Hilbert cube) make this question more complicated than it actually is.
Let's take for a starting point the claim already formulated by the topicstarter: for a compact metric space $X$ without isolated points the set of non-atomic Borel probability measure... | 3 | https://mathoverflow.net/users/8588 | 199698 | 96,502 |
https://mathoverflow.net/questions/199713 | 0 | By playing around with assoc. Legendre polynomials, I arrived at
$$((l+1)+m) (P\_l^m(x))^2+((l+1)-m)(P\_{l+1}^m(x))^2 = 2(l+1)x P\_l^m(x)P\_{l+1}^m(x).$$
Now, I want to show that we don't have equality for $x \in (-1,1).$
I undertook quite some computations in order to be sure that this is really the case, but I ... | https://mathoverflow.net/users/69110 | Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero | Let $y\_1< z\_1< y\_2< z\_2< \dots< y\_{l-m}\leq z\_{l-m}< y\_{l-m+1}$ be roots of polynomials $f=(1-x^2)^{-m/2} P\_l^m$ and $g=(1-x^2)^{-m/2} P\_{l+1}^m$ ($y$'s are roots of $g$, $z$'s are roots of $f$). They are real, belong to $(-1,1)$ and alternate as written because $f$, $g$ are orthogonal polynomial in weighted $... | 1 | https://mathoverflow.net/users/4312 | 199720 | 96,509 |
https://mathoverflow.net/questions/199740 | 1 | Assume the chain $\{X\_n\}\_{n\in\mathbb{N}}$ on the statespace $(S,\mathcal{F})$ (we may assume it is countable) is aperiodic, irreducible and positive recurrent. We denote with $\pi$ its (unique) stationary distribution. Is it true that for any $x\in S$ $$\underset{n\rightarrow\infty}{\text{lim}}~\underset{\text{A}\i... | https://mathoverflow.net/users/69123 | Variation of Markov Chain Convergence Theorem | You are asking about the (one half of) the total variation distance between the measure $\mathbf P\_\pi$ and the shifted by $n$ measure $\mathbf P\_x$. The latter measure is Markov with the initial distribution $\delta\_x P^n$ (here $P$ is the transition operator of the Markov chain). Now, for Markov measures determine... | 2 | https://mathoverflow.net/users/8588 | 199744 | 96,518 |
https://mathoverflow.net/questions/199739 | 2 | There are many results (usually connected to specification-like properties) about density of periodic measures in the space of all invariant ones. However some questions that seem to be easy (at first glance at least) always puzzle me.
Consider dynamical system generated by a continuous self-map $f$ of a connected co... | https://mathoverflow.net/users/67034 | Density of periodic points and density of periodic measures | For (1) and (3), as mentioned by Christian, consider the identity map $I:S^1\to S^1$.
For (2), consider the map $f:S^1\to S^1$ fixing $1\in S^1$, and $f^n x\to 1$ (as $n\to\pm\infty$) for all other points. Then $\mathcal{M}\_p=\mathcal{M}\_i=\{\delta\_1\}$, but there is no other periodic point beside $1\in S^1$.
| 2 | https://mathoverflow.net/users/11028 | 199754 | 96,523 |
https://mathoverflow.net/questions/199667 | 11 | Let $X$ and $Y$ be independent random variates with the same probability distribution, $P(x)$. Assuming that the product $Z=XY$ is a random variate with normal distribution, say $$f\_Z(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} x^2}$$ what is the form of $P(x)$? Does it exist in closed form?
**EDIT:** I just realized... | https://mathoverflow.net/users/18598 | Square root of normal distribution | First: No helvio, the density of the product of two iid random variables (r.v.'s) is not necessarily divergent at 0. E.g., take the product of two iid r.v.'s each having a Gamma distribution with shape parameter 2.
Second: Let $U:=\ln|Z|$ and $W:=\ln|X|$. Then the characteristic function (c.f.) $f$ of $U$ is given b... | 10 | https://mathoverflow.net/users/36721 | 199755 | 96,524 |
https://mathoverflow.net/questions/199721 | 3 | Consider a morphism of commutative rings $h\colon R\rightarrow S$. This yields the two functors $h\_\*\colon{\sf Mod}(S)\rightarrow{\sf Mod}(R)$ (scalar restriction) and $h^\*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$ (scalar extension), and $h^\*$ is left adjoint to $h\_\*$. The unit of this adjunction is for an $R$-m... | https://mathoverflow.net/users/11025 | Scalar restriction and scalar extension | Such morphisms are called pure, there is an extensive literature about them. Faithfully flat morphisms are pure, but there are more pure morphisms: see [this question](https://mathoverflow.net/questions/152877/pure-morphisms-which-are-not-faithfully-flat) and its answers. – abx 6 hours ago
| 2 | https://mathoverflow.net/users/40297 | 199758 | 96,525 |
https://mathoverflow.net/questions/164526 | 5 | Let $G$ be a compact Lie group and $\frak g$ be its Lie algebra. Then by Marsden-Weinstein reduction theory we know that if we take $M=T^\*G$ and $J \colon M\to \frak g^\*$ be its moment map then the reduced space $$S=J^{-1}(\mu)/G\_\mu$$ is exactly $G/G\_\mu$ where $\mu\in \frak g^\*$ and $G\_\mu$ is the isotropy subg... | https://mathoverflow.net/users/nan | A question about Marsden-Weinstein reduction theory | I found the answer of my question. This question is well known, but I didn't know this fact.
Consider the right action of the Lie subgroup $H$ to $G$ : $(g,h)\to gh$, $g\in G$, $h\in H$. If we identify $\mathfrak h\cong \mathfrak h^\*$ we get the moment map $\mu:T^\*G\to \mathfrak h$, $\mu(g.\zeta)=\text{pr}\_\mathf... | 2 | https://mathoverflow.net/users/nan | 199761 | 96,526 |
https://mathoverflow.net/questions/199767 | 5 | In some notes on derived stacks, in describing categories of fibrant objects, the author drops this parenthetical:
>
> (Grothendieck said in his famous letter to Quillen that the choice of
> $\mathscr F$ is like the choice of a basis of a module.)
>
>
>
Question 1: Where can I find the quote that this refers ... | https://mathoverflow.net/users/35714 | Choice of fibrations is like a choice of a basis of a module | I guess that 'the letter' is meant to be Grothendieck's [*Pursuing Stacks*](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps.html), which started as a letter to Quillen (as one can read in the document) and then evolved in a kind of book/diary addressed to the reader. I haven't found that precise quote (the djvu fil... | 9 | https://mathoverflow.net/users/12166 | 199773 | 96,529 |
https://mathoverflow.net/questions/199775 | 4 | I know that, in the presence of $CH$, Namba forcing does not add reals. But when $CH$ fails, is it consistent that it still does not add reals?
| https://mathoverflow.net/users/29231 | "Namba forcing adds reals" independent of $ZFC + \neg CH$? | If CH fails, then Nm adds reals (or equivalently, new $\omega$-sequences of reals).
Proof: Let $f:\omega\_2\to 2^\omega$ be 1-1. Let $\bar \alpha:=(\alpha\_n:n\in \omega)$ be the name for the Namba sequence in $\omega\_2$. Then $x:=(f(\alpha\_n):n\in \omega)$ is the name of a new sequence of reals. The sequence $x$ ... | 6 | https://mathoverflow.net/users/14915 | 199776 | 96,530 |
https://mathoverflow.net/questions/101146 | 3 | We know
* Laplace equation (elliptic equations)
$ Δ u = 0$
* Heat equation (parabolic equations)
$u\_t − Δu = 0$
* Wave equation (hyperbolic equations)
$u\_{tt} − Δu = 0$
we have
- Hyperbolic geometric flow (hyperbolic equations)
$$\frac{∂^2}{∂t^2}g\_{ij}(t)=-2R\_{ij}$$
----------------------------------------
... | https://mathoverflow.net/users/nan | Short time existence on Hyperbolic Ricci flow in non-compact case | For the first question take local harmonic coordinates the equation takes the form
$$ \partial^2\_{tt} g\_{ij} - g^{kl} \partial^2\_{kl} g\_{ij} = l.o.t. $$
and so using finite speed of propagation local existence holds in the non-compact case provided that the data is not too wild near infinity.
In fact, for a... | 0 | https://mathoverflow.net/users/3948 | 199783 | 96,533 |
https://mathoverflow.net/questions/199781 | 6 | As explained in:
[Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds](https://mathoverflow.net/questions/195592/classification-of-su2-principal-fibre-bundles-over-four-dimensional-manifold)
principal $SU(2)$ bundles $P\_{SU(2)}$ over a four-dimensional manifold $M$ are classified by ... | https://mathoverflow.net/users/66688 | Classification of $SU(2)$-bundles versus the classification of $SO(3)$-bundles | You are correct that locally, there is no difference between SU(2) and SO(3) bundles, but there are important global differences. In particular, SO(3) bundles may have non-trivial $w\_2$, which obstructs their lifting to an SU(2) bundle. The Dold-Whitney theorem (Classification of oriented sphere bundles over a 4-compl... | 9 | https://mathoverflow.net/users/3460 | 199786 | 96,536 |
https://mathoverflow.net/questions/199771 | 4 | Let G be a finite group and S be the set of Sylow p-subgroups of G for a
prime p dividing the order of G. Assume that |S|>1.
Let U and V be two disjoint non-empty subsets of S such that,
$$\bigcup\_{P\in U}P=\bigcup\_{P\in V}P=\bigcup\_{P\in S}P$$ and
$$\bigcap\_{P\in U}P=\bigcap\_{P\in V}P=\bigcap\_{P\in S}P$$ .... | https://mathoverflow.net/users/47344 | About the set of Sylow-$p$ subgroups of $G$ | Yes I have found an example where this occurs.
The group $G$ is an extension $3^3:2^2$, of an elementary abelian group of order $27$ by one of order $4$, where the generators of the group of order $4$ act as the diagonmal matrices $(-1,-1,1)$ and $(1,-1,-1)$ on the group $3^3$. Then $G$ embeds into $S\_9$ with genera... | 10 | https://mathoverflow.net/users/35840 | 199788 | 96,537 |
https://mathoverflow.net/questions/199772 | 2 | This is a follow-up question to [Existence of injective neighborhood selection function as separation axiom](https://mathoverflow.net/questions/199555/existence-of-injective-neighborhood-selection-function-as-separation-axiom).
Let $(X, \tau)$ be a topological space. If there is an injective map $f:X\to\tau$ such tha... | https://mathoverflow.net/users/8628 | Critical topological spaces | The answer is no. Consider the space $X=\{0,1,2\}$ with the topology
$\tau=\{\emptyset,\{0\},\{1,2\},X\}$. There are precisely two
injective neighborhood selectors, both of which are almost surjective in your sense. Namely, we must
map $0\mapsto\{0\}$ and then $1$ and $2$ get mapped to $\{1,2\}$
and $X$, in either way... | 4 | https://mathoverflow.net/users/1946 | 199801 | 96,542 |
https://mathoverflow.net/questions/199809 | 2 | I remember that the recursion
$r(0)=0, \ \
r(n+1)=\frac{1}{2 [r(n)]+1-r(n)}$
produces a sequence of rational values $ 0 \mapsto 1 \mapsto 1/2 \mapsto 2 \mapsto 1/3 \mapsto ... $ which exausts the positive fractions (and of course every fraction can only appear once).
Unfortunately I do not remember the reference ... | https://mathoverflow.net/users/7979 | Enumerating positive fractions (reference missing) | Have a look at chapter 19 ("Sets, Functions and the continuum hypothesis") in "Proofs from THE BOOK" (5th edition) by Aigner/ Ziegler.
The sequence originates in the paper "Recounting the Rationals" by Calkin/Wilf but in this paper you can't find the formula you mention.
| 3 | https://mathoverflow.net/users/50551 | 199812 | 96,546 |
https://mathoverflow.net/questions/199814 | 23 | The p-adic integers $\mathbb{Z}\_p$ can be thought of as a subgroup of the direct product group $P = \prod\_{n \geq 1} \mathbb{Z}/p^n\mathbb{Z}$. Are they a direct summand of this group? That is, is the inclusion $\mathbb{Z}\_p \hookrightarrow P$ split?
| https://mathoverflow.net/users/6481 | Are the p-adics a direct summand of the direct product of the groups $\mathbb{Z}/p^n\mathbb{Z}$? | Assuming the axiom of choice, yes. Choose a non-principal ultrafilter on $\mathbb N$. This gives a consistent way to, given a function from $\mathbb N$ to a finite set, choose an element of the finite set, that doesn't depend on any finite subset of $\mathbb N$.
You can take an element of $\prod\_{n\geq 1} \mathbb Z/... | 28 | https://mathoverflow.net/users/18060 | 199816 | 96,548 |
https://mathoverflow.net/questions/199805 | 10 | Finite dimensional Lie groups have the nice property that if $V$ is a small neighborhood of the identity, and $U \subset V$ another neighborhood, then $V$ is covered by $U^k$ (the set of all products of $k$ elements of $U$) for some $k$. This follows from the fact that V is relatively compact.
Does the same property... | https://mathoverflow.net/users/69151 | Neighborhoods of the identity in diffeomorphism groups | What helps a little is that for any (second countable connected) smooth manifold $M$, the connected component $Diff\_c(M)\_0$ of the group $Diff\_c(M)$ is perfect (Epstein) and simple (Thurston). On the other hand, $\exp: \mathfrak X\_c(M)\to Diff\_c(M)$ is far from being locally surjective. Grabowski has shown, that f... | 6 | https://mathoverflow.net/users/26935 | 199817 | 96,549 |
https://mathoverflow.net/questions/199583 | 9 | A **discrete** distribution $p$ over $\mathbb{N}$ is said to be *log-concave* if it satisfies the following conditions:
1. The support of $p$ is a contiguous interval, i.e. $\exists a \leq b$ s.t. $p\_i > 0$ iff $a\leq i \leq b$.
2. for all $i\in\mathbb{N}$, $p\_i^2 \geq p\_{i-1}p\_{i+1}$.
(in the literature, condi... | https://mathoverflow.net/users/37266 | Reference on (discrete) log-concave probability distributions | There is a 67 page review from last year, [Log-concavity and strong log-concavity: a review](http://arxiv.org/abs/1404.5886), A. Saumard, J.A. Wellner (2014):
>
> We review and formulate results concerning log-concavity and
> strong-log-concavity in both discrete and continuous settings. We show
> how preservatio... | 12 | https://mathoverflow.net/users/11260 | 199820 | 96,550 |
https://mathoverflow.net/questions/199825 | 0 | I am searching for the name of the following distribution on the set of positive integers (including zero).
Let $C\in \mathbb{Z}\_+$ and $n\in \mathbb{N}$ are fixed.
Vector $p = (p\_1,\ldots,p\_n)$ is such that
1) $p\_i\in \mathbb{Z}\_+$.
2) $\sum\_{i=1}^{n}p\_i = C$.
The distribition of vectors $p$ is such tha... | https://mathoverflow.net/users/39752 | Name of distribution | You can put the uniform distribution on any nonempty finite set whatsoever. In this case, you could call it the uniform distribution on [the set of nonnegative integer solutions to an equation](https://math.stackexchange.com/questions/919676/the-number-of-integer-solutions-of-equations).
| 1 | https://mathoverflow.net/users/4600 | 199826 | 96,551 |
https://mathoverflow.net/questions/199833 | 5 | Given a knotted arc $A \subset D^3$ (whose endpoints are, say, at $(\pm 1,0,0)$), the spun knot on this arc is $$\partial\left((D^3, A) \times D^2\right), = (\partial(D^3,A) \times D^2) \cup ((D^3,A) \times \partial D^2),$$ a smoothly embedded 2-sphere in $S^4$.
The Gluck twist on a smoothly embedded 2-sphere $S^2 \h... | https://mathoverflow.net/users/40804 | Why does the Gluck twist on a spun knot give the standard $S^4$? | This was shown by Gluck, in the cited paper; see section 22. The basic point is explained in section 17. (What we now call) the Gluck twist will produce an equivalent knot if the circle action on the 2-sphere extends over some 3-manifold that the knot bounds; this is Theorem 17.1 but I recommend that you try to prove i... | 8 | https://mathoverflow.net/users/3460 | 199840 | 96,557 |
https://mathoverflow.net/questions/199838 | 1 | Let $\gamma\_1,\gamma\_2: \mathbb{S}^1 \to \mathbb{R}^2$ be two smooth, closed, convex curves that their [(special)affine curvature](http://en.wikipedia.org/wiki/Affine_curvature), $\mu\_1,\mu\_2$ are equal, that is $\mu\_1(\theta)=\mu\_2(\theta)$, for any $\theta\in \mathbb{S}^1$. Are$\gamma\_1,\gamma\_2$ related to e... | https://mathoverflow.net/users/61460 | Uniqueness affine curvature | Note that closed curves made by two arcs of ellipses of the given areas, say $a$ and $b$ may be not related to each other by an affine transformation.
They have constant affine curvatures on two arcs, so for suitable parametrization the curvatures are the same.
The example is $C^{1,1}$-smooth, but it can be $C^\inft... | 0 | https://mathoverflow.net/users/1441 | 199844 | 96,559 |
https://mathoverflow.net/questions/199863 | 5 | 2-bridge knots (aka rational knots) $K(p,q)$ are described by a rational number $\frac{p}{q}$ or likewise its continued fraction expansion $\left[a\_1,a\_2,\ldots,a\_k\right]$.
Has somebody worked out a list to identify the 2-bridge knots in the Rolfsen's table or the Callahan-Hildebrand-Weeks census or [some other k... | https://mathoverflow.net/users/39082 | 2-bridge knots in the Rolfsen's table | Cha and Livingston's [KnotInfo](http://www.indiana.edu/~knotinfo) includes the ability to list the bridge index of knots with at most 11 crossings, and so covers all of Rolfsen's table. The 2-bridge knots are those with bridge index 2. Additionally, their table includes the other invariants that you mention (Alexander ... | 12 | https://mathoverflow.net/users/3121 | 199864 | 96,565 |
https://mathoverflow.net/questions/199870 | 1 | Let $K$ be a simplicial set and let $\Delta K$ be the *category of simplices*, i.e the category where the objects are simplicial maps
$$
\Delta[n]\to K
$$
and the maps $\phi\: : \: (\Delta[n]\to K)\to (\Delta[m]\to K)$ are simplicial maps $\phi\: : \: \Delta[n]\to \Delta[m]$ such that the triangle commute. Now let $L$ ... | https://mathoverflow.net/users/41970 | Canonical colimit and cartesian product of simplicial sets | You can do this for any category of presheaves. Let $\mathcal{C}$ be a small category. For a presheaf $X$ on $\mathcal{C}$, we write $\mathbf{El} (X)$ for the category of elements of $X$, i.e. the comma category $(h \downarrow X)$ where $h : \mathcal{C} \to [\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ is the Yoneda embeddi... | 1 | https://mathoverflow.net/users/11640 | 199873 | 96,568 |
https://mathoverflow.net/questions/193769 | 6 | I'll begin by asking a general question, and then specializing to the situation I really care about.
Let $G$ be a group and let $(B, N)$ be a $BN$-pair in $G$ (see, for instance, page 39 of Tits' "Buildings of Spherical Type and Finite BN-pairs). Then $(B,N)$ has associated to it a chamber complex $X$ with a number o... | https://mathoverflow.net/users/30726 | Buildings associated to generalized $BN$ pairs | You may either work with an extended building as
L. Spice suggests in his answer, or consider the following based on the fact that a generalized $BN$-pair contains a genuine $BN$-pair. With you notation, the group $N$ writes as a semidirect product $\Omega\ltimes N\_0$ for some invariant subgroup $N\_0$ of $N$. Then $... | 4 | https://mathoverflow.net/users/4767 | 199882 | 96,573 |
https://mathoverflow.net/questions/199859 | 6 | There are concise and elegant characterisations of the real line as a topological space and as an ordered space in the literature. I am interested in the harder case of characterising subsets of the reals in this manner. There are satisfactory answers to the topological version (e.g., de Groot, Mary Ellen Rudin) which ... | https://mathoverflow.net/users/61738 | Characterising subsets of the reals as ordered spaces | The suggestion in the comments that a linear order embeds into
$\mathbb{R}$ just in case it has a countable dense set is not
quite true. For example, let $2\times\mathbb{R}$ be the *doubled real line* ($\mathbb{R}$ copies of $2$), the order arising from the reals by replacing each real number with two copies, a
lower o... | 13 | https://mathoverflow.net/users/1946 | 199883 | 96,574 |
https://mathoverflow.net/questions/199874 | 49 | From the point of view of formal math, what would constitute an appropriate statement of the classification of finite simple groups? As I understand it, the classification enumerates 18 infinite families and 26 sporadic groups and asserts that a finite group is simple iff it is in one of these families. Now the 18 infi... | https://mathoverflow.net/users/34444 | How do you *state* the Classification of finite simple groups? | There are really two separate questions that you seem to be conflating here.
The first is how to state the CFSG in a way that could be mechanically formalized. The second is how to state the CFSG that adequately reflects how human mathematicians think about it.
For the former question, one straightforward possibili... | 37 | https://mathoverflow.net/users/3106 | 199885 | 96,576 |
https://mathoverflow.net/questions/199821 | 3 | Are there examples of algebraic singularities which may be smoothed analytically but not algebraically? It certainly seems possible, but if not, why? Are there conditions under which this becomes true, e.g. what if the singularity is isolated?
| https://mathoverflow.net/users/8003 | Analytically but not algebraically smoothable singularity | For isolated singularities, the answer is **no**.
In fact, it follows by a result of Elkik that any deformation of an isolated singularity is algebraizable. See
R. Elkik, *[Solutions d'équations à coefficients dans un anneau hensélien](https://eudml.org/doc/81927)*, Ann. Sci. Ecole Norm. Sup. **6** (1973), 553-603,... | 2 | https://mathoverflow.net/users/7460 | 199888 | 96,578 |
https://mathoverflow.net/questions/195964 | 2 | I'm looking for results (or some ideas) on the following kind of pseudo-differential evolution equation:
$$
\frac{\partial u(t,x)}{\partial t} = \int\_{-\infty}^{t} B(t-s,x)\, A(x,D\_{x})u(s,x)\,ds \; ;\quad u(0,x) = u\_{0}(x)
$$
where $A$ is a $\Psi DO$ with symbol in $S^{m}(\mathbb{R}^{n}\times \mathbb{R}^{n})$ ... | https://mathoverflow.net/users/62513 | Pseudo-differential evolution equation | To get some feeling for the problem, we make the following simplifying
assumptions:
1. $B$ is independent of $x$, say $B \in \mathcal S(\overline{\mathbb R}\_+)$.
2. $A$ admits a (holomorphic) functional calculus.
3. There is a (holomorphic) function $h\colon \sigma(A)\to\overline{\mathbb H}\_-$ such that $\bigl(i\ta... | 4 | https://mathoverflow.net/users/69194 | 199893 | 96,580 |
https://mathoverflow.net/questions/199896 | 2 | Let $G$ be the general linear group $\operatorname{GL}(n,\mathbb{C})$ and $P$ a parabolic subgroup with Lie algebra $\mathfrak{p}$. Consider the vector bundles
$$
\mathcal{P} = G\times\_P \mathfrak{p} \subset G/P \times \mathfrak{gl}
$$
and
$$
\mathcal{T}\_{G/P} = G\times\_P \mathfrak{g/p}.
$$
I would like to unde... | https://mathoverflow.net/users/56926 | What is the cohomology of the tangent bundle of a flag variety? | The answer might be in
Michel Demazure. Automorphismes et déformations des variétés de Borel. Invent. Math., 39(2):179–186, 1977
| 2 | https://mathoverflow.net/users/48866 | 199910 | 96,582 |
https://mathoverflow.net/questions/199908 | 3 | I was checking an example of canonical singularities from surface.
We consider the surface $X:(xz=y^2)\subset \mathbb A^3$. The only singular point is the origin. We write down one affine piece of the blow-up of $\mathbb A^3$ at the origin, which is the map $\sigma:\mathbb A^3\to \mathbb A^3$ given by
$$x=u,y=uv,z=u... | https://mathoverflow.net/users/62798 | pull-back of canonical divisor under blow-up of a singular point | Here is the computation:
let us consider the action:
$$
\begin{array}{ccc}
\mu\_{2}\times\mathbb{A}^{2} & \longrightarrow & \mathbb{A}^{2}\\
(\epsilon,x\_{0}, x\_{1}) & \longmapsto & (\epsilon x\_{0},\epsilon x\_{1})
\end{array}
$$
The ring of invariants is given by:
$$k[x\_0^2,x\_0x\_1,x\_1^2]\cong k[y\_0,y\_1,y\_2]... | 6 | https://mathoverflow.net/users/14514 | 199912 | 96,583 |
https://mathoverflow.net/questions/199904 | 2 | When writing a paper, I feel like to point out exact references to the following seemly easy facts concerning flat structures on a closed surface $\Sigma$ with negative Euler characteristic:
1. The universal cover $\widetilde{\Sigma}$ equipped with the pullback of any flat metric with conic singularity is quasi-isome... | https://mathoverflow.net/users/17294 | Reference request: flat surfaces | For the first fact, a good reference is Jim Cannon's survey in Bedford/Keane/Series (you use fact the universal cover is quasi-isometric to the fundamental group, which is proved at great length by Cannon).
| 2 | https://mathoverflow.net/users/11142 | 199913 | 96,584 |
https://mathoverflow.net/questions/198396 | 6 | Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has
$$(\*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)}
+\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi,
$$
where $f$ is the characteristic function of a probability distribution $\mu$, so that $f(t)=\int\_{-\infty}^\infty e^{itx}\mu(dx)$ for all real $t$... | https://mathoverflow.net/users/36721 | Bound on the sum of arguments | Let's prove that
$$
\arg \frac{1-zf(s-u)}{1-zf(s+u)}< \pi/2- \arg(1-z\bar{z}f(2u)).
$$
Then summing this up with an analogous inequality
$$
\arg \frac{1-zf(t+u)}{1-zf(t-u)}< \pi/2- \arg(1-z\bar{z}f(-2u))
$$
we get what we need.
Denote $zf(s-u)=A$, $zf(s+u)=\bar{B}$, $z\bar{z}f(2u)=C$. Then for functions $\varphi\_1... | 5 | https://mathoverflow.net/users/4312 | 199919 | 96,586 |
https://mathoverflow.net/questions/199766 | 1 | Consider a convex function $f$ defined on a $d$-dimensional hypercube $[0,1]^d$. Now for fixed $m \in \mathbb{N}$, consider the grids $\mathcal{G}\_m=\{(i\_1/m,\cdots,i\_d/m)\}$ where $i\_\alpha\in\{0,1,\cdots,m\}$ for all $\alpha$, and do linear interpolation of $f$ on these grids to get a piecewise affine function $\... | https://mathoverflow.net/users/67107 | number of affine pieces of linear interpolation of convex functions in high dimension | Let's think of choosing the values of $f$ at the grid points one grid point at a time, in such a way that the result is convex.
If you work systematically through the grid points in an order so that each time you add a lattice point it is outside the convex hull of the lattice points you have already added, then you... | 0 | https://mathoverflow.net/users/468 | 199937 | 96,598 |
https://mathoverflow.net/questions/199926 | 86 | Perhaps under the influence of a recent question
on *[perverse sheaves](https://mathoverflow.net/q/198656/6094)*,
in conjunction with the impending $\pi$-day (3/14/15 at 9:26:53),
I recalled a long-ago parody of abstruse mathematical language
that I can no longer remember in detail nor find by searching.
I am not see... | https://mathoverflow.net/users/6094 | Parodies of abstruse mathematical writing | Is this what you're looking for?
<http://thatsmathematics.com/mathgen/>
Mathgen is an random math paper generator, based on [SCIgen](http://pdos.csail.mit.edu/scigen/) which does the same for computer science papers. It will provide you with an unlimited supply of abstruse nonsense: definitions, theorems, proofs, r... | 91 | https://mathoverflow.net/users/613 | 199940 | 96,600 |
https://mathoverflow.net/questions/199917 | 5 | It is claimed in the Hamkins and Lewis founding article "Infinite time Turing machines" (proof of the gap existence theorem 3.4) that for $\omega$ steps of a computation of a machine performing a dovetailing on input $0$, $\omega$ steps are also performed in each of the simulated machines.
How does this simulation w... | https://mathoverflow.net/users/69205 | Hamkins infinite time Turing machines: dovetailing ordinal time | I'm glad to hear that you're reading [my paper](http://jdh.hamkins.org/ittms/). This particular method of dovetailing infinitely many computations into one, however, is relatively standard in computability theoretic constructions, and doesn't have anything essentially to do with infinitary computability.
Let me expl... | 6 | https://mathoverflow.net/users/1946 | 199941 | 96,601 |
https://mathoverflow.net/questions/172148 | 38 | The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows from the paper "Number fields unramified away from $2$" by John W. Jones in the Journal of Number Theory in 2012.
The exis... | https://mathoverflow.net/users/48142 | Degree 17 number fields ramified only at 2 | Thank you for calling this problem to my attention.
I computed $K$ en route to AWS (though this year's
[topics](http://swc.math.arizona.edu/index.html)
are a rather different flavor of number theory...).
After some simplification (**gp**'s $\rm polredabs$), it turns out that
the field $K$ is generated by a root of
$$
f... | 25 | https://mathoverflow.net/users/14830 | 199946 | 96,604 |
https://mathoverflow.net/questions/199916 | 1 | Let $\mathbb F\_q$ be the finite field with $q$ elements. Suppose $V$ is a linear space of dimension $n$ over $\mathbb F\_q$, and $r<n$. What is the maximal $k$ such that for arbitrary $k$ subspaces $W\_1,W\_2,\dots,W\_k$ of $V$ of dimension $r$, there always exists a subspace $U$ of $V$ of dimension $n-r$ which satisf... | https://mathoverflow.net/users/37096 | What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null | I think we have $k = q$ (for all $r$).
First, suppose $r = n - 1$. The vector space $V$ can be covered by $q + 1$ codimension-one hyperplanes $W\_1,W\_2, \ldots,W\_{q+1}$ (but no fewer). (If we think of the projective space $[V]$, the most efficient covering is by taking the hyperplane at infinity, plus a set of $q$... | 5 | https://mathoverflow.net/users/68305 | 199947 | 96,605 |
https://mathoverflow.net/questions/199953 | 5 | Does there exist an uncountable collection $\Lambda$ of infinite subsets of the set of natural numbers such that (i) any two distinct subsets in the collection have a finite intersection and (ii) the sum of the reciprocals is divergent for each $A \in \Lambda$?
| https://mathoverflow.net/users/69214 | Sets of natural numbers with finite intersections and divergent sums of reciprocals | This can be done by a fairly simple diagonalization argument. Call a subset of $\mathbb{N}$ large if its sum of reciprocals is infinite. It suffices to prove the following (for we can then construct an uncountable family by induction). Let $\Lambda$ be a countable collection of large sets such that for any finite subse... | 3 | https://mathoverflow.net/users/75 | 199957 | 96,609 |
https://mathoverflow.net/questions/199969 | 1 | As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.
| https://mathoverflow.net/users/69184 | Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? | Since I was unable to find where else in MO this was answered, I'll answer this here. You can find a proof in chapter I, section 5 (pp. 83-84) of Algebra IV: Infinite Groups, Linear Groups by Kostrikin and Shafarevich; here's a Google books [link](https://books.google.com/books?id=O9cATtl-fowC&pg=PA83&lpg=PA83&dq=matri... | 4 | https://mathoverflow.net/users/2926 | 199972 | 96,614 |
https://mathoverflow.net/questions/199960 | 1 | I'm working on a stochastic algorithm and considering it to apply in case of any curved space (manifolds). But in order to make the algorithm as efficient as possible I want to include in it some measure of global geometry. One of such a measure is second fundamental form for embedded (in $R^n$) manifolds. The first fu... | https://mathoverflow.net/users/21753 | Global geometry measures for Riemannian manifolds | If I understand correctly, you're interested in measuring the "extrinsic geometry" of the embedding. In some sense, the first and second fundamental forms are *exactly* what you need to know to understand this:
From a theoretical viewpoint, if you know the first and second fundamental forms, then you know *everythin... | 1 | https://mathoverflow.net/users/1540 | 200009 | 96,628 |
https://mathoverflow.net/questions/198656 | 14 | Suppose I am in the setting of the decomposition theorem, i.e., we have the decomposition of the direct image $f\_\*\mathbb Q\_\ell$, where $f:X\to Y$ is proper. Then the direct image decomposes into a direct sum of shifted $IC$ extensions of semisimple local systems.
I would like to know a kind of converse to [this... | https://mathoverflow.net/users/48554 | When does a perverse sheaf occur in the decomposition theorem? | In general it is a difficult problem. For example, the core of Ngô's proof of the fundamental lemma is his support theorem which implies that in the context of the Hitchin fibration, all simple constituents of the direct image have full support.
If the morphism is semismall, then one has a direct sum of IC's without ... | 16 | https://mathoverflow.net/users/14154 | 200010 | 96,629 |
https://mathoverflow.net/questions/199881 | 5 | Let $f:X \to Y$ be a finite surjective morphism of quasi-projective schemes over $\mathbb{C}$, $X$ is reduced and $Y$ is integral. Suppose that there exists an integer $n$ such that for every *closed* point $y \in Y$, the fiber $f^{-1}(y)$ is reduced and consists of $n$ distinct closed points. Is it true that $f$ is fl... | https://mathoverflow.net/users/46578 | Covering of schemes and flatness | Apply Hartshorne, Exercise II.5.8 to $f\_\*\mathcal{O}\_X$.
| 2 | https://mathoverflow.net/users/13265 | 200021 | 96,632 |
https://mathoverflow.net/questions/200019 | 9 | I'm trying to understand why on earth the first chern class of a line bundle in K-theory $c\_1(L) = 1-L$.
I understand that the first Chern class of the trivial bundle is zero, and that $H-1$ generates the reduced K-theory of $CP^1$ (where H is the canonical line bundle over $CP^1$), but there must be more reasoning... | https://mathoverflow.net/users/56462 | Why is the first chern class of a line bundle $c_1(L) = 1-L$ in complex K-theory? | This comes from the choice of the $K$-theory Thom class for complex vector bundles.
Firstly, recall that $K$-theory $K^0(X)$ can be described as the group of bounded chain complexes of vector bundles on $X$, modulo the relation of forcing short exact sequences of such chain complexes to split. This is related to the ... | 14 | https://mathoverflow.net/users/318 | 200023 | 96,633 |
https://mathoverflow.net/questions/185719 | 2 | Maybe too easy a question for most members of this site, but suppose whenever $F$ and $G$ belong to the Selberg class, then so does $F\otimes G$ where the considered tensor product of $F$ and $G$ is defined in [this paper](http://www.mast.queensu.ca/~murty/Murty-Zaharescu.pdf). Does this imply that the degree of $F\oti... | https://mathoverflow.net/users/13625 | Tensor product of two elements of the Selberg class | So far as I know, the "Rankin-Selberg" tensor product should be the same thing in the Selberg-class world as in the Langlands-conjecture world. Certainly all known cases are consistent, to my knowledge.
If we believe that, then the $p$th Euler factor for the Rankin-Selberg product would be $1/\det(1-p^{-s}A\_p\otime... | 3 | https://mathoverflow.net/users/15629 | 200025 | 96,634 |
https://mathoverflow.net/questions/200029 | 4 | Suppose $X$ is a smooth variety, $X\to \mathbf{P}^n$ is a closed immersion. For the fixed imbedding, we call $X$ is linearly normal if $\Gamma(\mathbf{P}^n,O(1))\to\Gamma(X,O\_X(1))$ is surjective; we call $X$ is projectively normal if $\Gamma(\mathbf{P}^n,O(k))\to\Gamma(X,O\_X(k))$ is surjective for all $k$.
Is ther... | https://mathoverflow.net/users/nan | Linearly normal but not projectively normal variety | There are many examples! Just take a very ample line bundle $L$ on $X$, and the embedding $\ X\hookrightarrow |L|^\*$ defined by its global sections. It is linearly normal by construction, but there is no reason why for instance the map $\mathrm{Sym}^2H^0(X,L)\rightarrow H^0(X,L^2)$ should be surjective. Here is a conc... | 8 | https://mathoverflow.net/users/40297 | 200036 | 96,637 |
https://mathoverflow.net/questions/200015 | 5 | A theorem of A. Levy says that, if $\kappa$ is an inaccessible cardinal, then $V\_\kappa\prec\_{\Sigma\_1}V$ namely $V\_\kappa$ is an elementary submodel when considering only $\Sigma\_1$ formulas.
Where can I find a proof of this theorem ?
Is this property true also for some other (non inaccessible) cardinals ?
| https://mathoverflow.net/users/69236 | Inaccessible cardinal and $\Sigma_1$ reflection | Here's one way to prove it. Let $H\_\kappa = \{x: |tc(\{x\})|<\kappa\}$. Then we have:
**Theorem 1** If $\kappa$ is an uncountable cardinal, then $H\_\kappa\prec\_1 V$.
*Proof.* Let $\phi$ be $\Delta\_0$ with free variables among $y,x$. Since $\Sigma\_1$ formulas are upward absolute for transitive models, if $H\_\... | 7 | https://mathoverflow.net/users/17968 | 200045 | 96,640 |
https://mathoverflow.net/questions/200051 | 5 | Let $D$ be an infinite UHF algebra, e.g. the infinite tensor product of the matrix algebra $M\_k(\mathbb{C})$. The permutation group $\Sigma\_n$ acts on the $n$-fold tensor product $D^{\otimes n}$ in a canonical way.
>
> What is known about the fixpoint algebra of this action? Is it simple? Is it again a UHF algeb... | https://mathoverflow.net/users/3995 | fixpoint algebras of a permutation action | The answer is more interesting than I thought it would be. First, a clarification: you mean, that $k$ is fixed; the question still makes sense if $k$ is allowed to vary, and the analysis becomes more complicated (but can be done).
**Edit**: I made some serious (computational) errors in the original version, resulting... | 6 | https://mathoverflow.net/users/42278 | 200059 | 96,642 |
https://mathoverflow.net/questions/200068 | 5 | Given a set $D$ of $n$ same radius disks, embedded in the plane, their arrangement induces a number $k$ of connected regions in $\mathbb{R}^2 \setminus \cup\_{d \in D}$ .
I am interested in an upper bound on $k$ as a function of $n$.
Does anybody know (a reference for) a good upperbound on $k$?
Since the Union Co... | https://mathoverflow.net/users/69264 | Upperbounding the number of regions induced by a set of unit disks | If you take a triangular packing of discs and slightly increase the radius of each disc then enclose the packing in a large regular square and remove all discs outside the square. Then inside the square the ratio of discs to regions outside the discs will be two to one since there is a hexagonal tiling with a three col... | 3 | https://mathoverflow.net/users/1098 | 200073 | 96,647 |
https://mathoverflow.net/questions/200077 | 0 | Let $f:X \to Y$ be a proper surjective morphism of quasi-projective noetherian schemes over $\mathbb{C}$. Assume that $Y$ is irreducible and $X$ is reduced, connected with finitely many irreducible components. Suppose further that every *closed* fiber of $f$ is an integral scheme. Is there any known additional conditio... | https://mathoverflow.net/users/58203 | Proper morphism and irreducibility of schemes | This is false. Take for $X$ the union of the two axes in $\mathbf P^1\times\mathbf P^1$ and take for $f$ the first projection to $Y=\mathbf P^1$.
| 1 | https://mathoverflow.net/users/10696 | 200079 | 96,651 |
https://mathoverflow.net/questions/200085 | 3 | Let $G$ be a finite group and let $M,N \lhd G$ be normal subgroups with a trivial intersection. Suppose that $G$ has a subgroup of index $2$. Must $G$ have a subgroup of index $2$ which contains either $M$ or $N$?
| https://mathoverflow.net/users/38889 | Subgroups of index 2 in a fibered product | Then answer is, in general, no. Let $G = {\rm SL}(2,5) \times M$, where $M$ is cyclic of order $2$. Then $V = Z(G)$ is a Klein $4$-group. Let $H = {\rm SL}(2,5)$. Let $N$ be a subgroup of order $2$ of $V$ with $M \neq N$ and $N \not \leq H.$ There is such a subgroup, as $V$ has three subgroups of order $2$. Now we have... | 4 | https://mathoverflow.net/users/14450 | 200089 | 96,652 |
https://mathoverflow.net/questions/199976 | 1 | I am looking for an example in noncommutative ring theory. Namely, I am looking for a left perfect ring with finite left global dimension that is not right coherent. It seems to me that should be possible to get such example, but did not find it yet...
| https://mathoverflow.net/users/69222 | Example of a left perfect ring with finite left global dimension that is not right coherent | It seems the ring $$A=\begin{bmatrix}
\mathbb Q & \mathbb Q & \mathbb R\\
0 & \mathbb Q & \mathbb R\\
0 & 0 & \mathbb Q
\end{bmatrix} /
\begin{bmatrix}
0 & 0 & \mathbb R\\
0 & 0 & 0\\
0 & 0 & 0
\end{bmatrix}$$
is such an example. It is semiprimary, hence perfect on both sides, and the global dimension is $2$.
The ... | 0 | https://mathoverflow.net/users/18756 | 200091 | 96,653 |
https://mathoverflow.net/questions/200078 | 2 | Let $G$ be a finite group. If $M$ is a free $\mathbf{Z}[G]$-module, then $H^1(G',M) = H^2(G',M) = 0$ for all subgroups $G' \subset G$. Are there any other modules, free of finite rank over $\mathbf{Z}$ but not over $\mathbf{Z}[G]$, with this property?
| https://mathoverflow.net/users/1048 | What if the low-degree cohomology of a $G$-module and all its restrictions vanish? | I recommend reading Ken Brown's chapter on "Cohomologically Trivial Modules" from his book *Cohomology of Groups*. In particular, you should note that there is no need for only low-degrees: If $M$ is a free $\mathbb{Z}G$-module then $H^i=0$ for all $i>0$.
Now for general $M$, your conditions on the (low-degree) coho... | 3 | https://mathoverflow.net/users/12310 | 200092 | 96,654 |
https://mathoverflow.net/questions/200099 | 3 | Can some help me prove or disprove the following assertion which I encountered in research? Thanks!
Let $f:\mathbb R\to\mathbb R$ be an analytic function. If for $\forall c > 0$, we can find some $t'>0$ such that
$$\int\_{t'}^{t' + 1} {{f^2}(\tau )d\tau } \le c $$
then
$$\mathop {\lim }\limits\_{t \to \infty } ... | https://mathoverflow.net/users/69272 | A conjecture regarding the integral of the square of an entire function | It's false. Take for example
$$
f(x) =\sum\_{n\in {\Bbb Z}} e^{-n(x-n)^2}.
$$
Clearly $f(n) \ge 1$ for all integers $n$. Since in intervals of length $1$ the function $f$ is large only in a small neighborhood of an integer, it is easy to see that $\int\_x^{x+1} f(t)^2 dt$ tends to zero as $|x| \to \infty$.
| 7 | https://mathoverflow.net/users/38624 | 200101 | 96,656 |
https://mathoverflow.net/questions/200080 | 2 | The braid group on 3 strands has the presentation $\langle x,y \;|\; xyx=yxy\rangle$. A group $G$ is called right-orderable if there is a total order $<$ on the set $G$ such that if $a<b$ then $ac<bc$ for all $c\in G$.
It is known that braid groups are right-orderable.
Is there a non-right-orderable torsion-free quot... | https://mathoverflow.net/users/19075 | Is there a non-right-orderable torsion-free quotient group of the braid group on 3 strands? | Yes, there are many such examples. The braid group is isomorphic to the fundamental group of the trefoil knot complement. The trefoil knot $T$ admits many Dehn fillings, parameterized by $r\in \mathbb{Q} \cup \{\infty\}$. If $|r|\geq 1$, then the Dehn filling $S^3\_r(T)$ is an L-space. Moreover, it is usually Seifert-f... | 7 | https://mathoverflow.net/users/1345 | 200104 | 96,658 |
https://mathoverflow.net/questions/199831 | 5 | We have the integral :
$$\int\_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$$
Where s is a complex parameter, and n is a positive integer. The integral converges by virtue of the exponential factor. I tried to deform the path of integration such that we avoid the branch cut(s... | https://mathoverflow.net/users/20782 | Help with the integral $\int_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$ | $$I\_n(s)=\int\_{0}^{\infty}\log\left(1+\frac{s^{2}}{4\pi^{2}} \log^{2}(1+ix)\right ) e^{-2\pi nx}dx$$
a closed-form evaluation of this integral does not look promising, but small and large-$|s|$ asymptotics is doable:
* small $|s|$ (with $\gamma$ Euler's constant and $\_3F\_3$ the [generalized hypergeometric funct... | 8 | https://mathoverflow.net/users/11260 | 200106 | 96,659 |
https://mathoverflow.net/questions/200109 | -1 | Let $G=(V,E)$ be a finite graph. We write $\nu(G)$ for the matching number of $G$. Is there $\varepsilon > 0$ such that we have $$\frac{\nu(G)+\Delta(G)}{V(G)} \geq \varepsilon$$ for all finite graphs $G=(V,E)$?
| https://mathoverflow.net/users/8628 | Maximum degree and matching number | The answer is no (even for connected graphs: if you consider general graph, simply take $E$ empty for a counter-example).
Let $n,d>2$ be any positive integers, define $G\_1$ as a path of length $\ell$, and let $G$ be the tree obtained by adding $d-2$ leaves adjacent to each vertex of $G\_1$. Then we have $\Delta(G)=d... | 2 | https://mathoverflow.net/users/4961 | 200117 | 96,664 |
https://mathoverflow.net/questions/200121 | 2 | I is well known that there is no explicit formula for the eigenvalues of a general matrix (see *e.g.* [Wikipedia](http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Eigenvalues)). This result is a consequence of (1) Abel's theorem, stating that there is no explicit formula for the roots of a general polynomial o... | https://mathoverflow.net/users/15981 | the impossibility of exactly computing eigenvalues | [A real symmetric tridiagonal matrix with a given characteristic polynomial](http://www-users.york.ac.uk/~slow500/reprints/1993SchmeisserGivenCharactersticPolynomial.pdf), G. Schmeisser (1993)
earlier work (behind a paywall):
[Expressing a polynomial as the characteristics polynomial of a symmetric matrix](http://w... | 6 | https://mathoverflow.net/users/11260 | 200122 | 96,665 |
https://mathoverflow.net/questions/200129 | 5 | Let us work over a ground field of characteristic zero. As is well-known, a K3 surface is a smooth projective geometrically integral surface $X$ whose canonical class $\omega\_X$ is trivial and for which $\operatorname{H}^1(X,\mathscr{O}\_X)$ vanishes.
A bit of folklore (proven e.g. in Beauville's *Complex Algebraic ... | https://mathoverflow.net/users/17907 | Singular models of K3 surfaces | Yes, this is true: the smooth minimal model is a $K3$ surface.
In fact, let $\bar{X}$ be the resolution of the singularities of $X$. Then the following holds.
**(1)** Rational double points impose no adjunction conditions to canonical forms, hence $\omega\_{\bar{X}}$ is trivial.
**(2)** Rational double points hav... | 8 | https://mathoverflow.net/users/7460 | 200133 | 96,670 |
https://mathoverflow.net/questions/200125 | 1 | Let $\mathcal{C}$, $\mathcal{D}$ be two small categories. Let $f\: : \: \mathcal{C}\to \mathcal{D}$ be a functor. Then it induces a functor
$$
f^{\*}\: : \: sPsh(\mathcal{D})\to sPsh(\mathcal{C})
$$
via $f^{\*}(X)=X\circ f$. Now consider $sPsh(\mathcal{D}), sPsh(\mathcal{C})$ equipped with the projective model structur... | https://mathoverflow.net/users/41970 | Does the kan extension preserves contractible presheaves? | $f\_!(X)$ is a cofibrant replacement of $\*$ if and only if the comma category $f/d$ has a weakly contractible classifying space, for all $d\in D$.
Indeed, the value of $f\_!(X)$ on $d$ is weakly equivalent to the homotopy colimit of the constant diagram with value $\*$ on the comma category $f/d$, because the restri... | 6 | https://mathoverflow.net/users/20233 | 200136 | 96,672 |
https://mathoverflow.net/questions/200053 | 2 | Let $p$ be an odd prime (large if it matters) and let $G= Aff(\mathbb{F}\_{p^2}) \cong \mathbb{F}\_{p^2} \rtimes \mathbb{F}\_{p^2}^\*$ be the affine linear group acting on $\mathbb{F}\_{p^2}$ by $x\mapsto ax+b$, $a\in \mathbb{F}\_{p^2}^\*$ and $b\in\mathbb{F}\_{p^2}$.
Consider the $\mathbb{F}\_2$-representation $V=\{f\... | https://mathoverflow.net/users/2042 | Invariant subspaces of an $F_2$-representation of the affine linear group of dimension 1 | There are no other invariant subspaces. As a consequence of Klemm's Satz 8(b) in [here](http://ams.math.uni-bielefeld.de/leavingmsn?url=http://dx.doi.org/10.1007/BF01187052), we obtain the following: Let $G$ be a finite sharply $2$-transitive permutation group on $n\ge2$ letters. Then the permutation module for this ac... | 3 | https://mathoverflow.net/users/18739 | 200142 | 96,675 |
https://mathoverflow.net/questions/199889 | 12 | What are applications of the theory of Berkovich analytic spaces? The analytification $X \mapsto X^{\mathrm{an}}$
| https://mathoverflow.net/users/nan | applications of Berkovich spaces | I would first recommend the paper of Antoine Ducros ([Espaces analytiques $p$-adiques au sens de Berkovich](http://webusers.imj-prg.fr/%7Eantoine.ducros/asterisque.pdf), Séminaire Bourbaki, exposé 958, 2006) for a general survey of the theory, with applications.
Here is a list of applications which I find striking, s... | 18 | https://mathoverflow.net/users/10696 | 200146 | 96,677 |
https://mathoverflow.net/questions/200155 | 0 | In the paper "A new cohomology theory for orbifold" by Chen/Ruan, they define the *orbifold cohomology group* of an orbifold $X$ by
$H^d\_{orb}(X)=\bigoplus\_{(g) \in T} H^{d-2\iota\_{(g)}}(X\_{(g)})$
where $H^\ast(X\_{(g)})$ is the singular cohomology of $X\_{(g)}$ with real coefficients and $\iota(g)$ is a ration... | https://mathoverflow.net/users/67042 | Rationally graded singular cohomology | It is my understanding that $H^d$ is taken to be trivial unless $d$ is an integer. Hence this is just notation for a degree shift by a rational number.
Similarly, in the other formula, the square bracket notation is also just a degree-shift.
| 0 | https://mathoverflow.net/users/14901 | 200159 | 96,682 |
https://mathoverflow.net/questions/200167 | 1 | I am trying to feed information about the solution when solving an inverse problem given by a Fredholm integral of the form
$$
g(t)=\int\_{a}^{b}K(t,s)f(s)ds.
$$
Say I know $g(t)$ and $K(t,s)$, and want to know $f(s)$. But I have some additional knowledge: $g(t)$ and $f(s)$ are both constrained to the interval $[0;1]$.... | https://mathoverflow.net/users/69295 | Fredholm integral with functions constrained to [0;1] | There is not enough information for a thorough answer. An a priori bound on the solution may indeed help theoretically and practically. As usual with measured data you may not want to solve the equation $Af=g$ (where $A$ denotes the integral operator, i.e $Af(t) = \int k(t,s) f(s) ds$) but a "least squares" type proble... | 1 | https://mathoverflow.net/users/9652 | 200171 | 96,686 |
https://mathoverflow.net/questions/199943 | 3 | Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is expressed as a product of the form $\mathbb R\times M^{n-1}$ i.e. the manifold splits off a factor $\mathbb{R}$. Both authors... | https://mathoverflow.net/users/46495 | Is there a characterization of Riemannian manifolds that split off two factors? | Ralf Ponge and Helmut Reckziegel have show in their paper: "Twisted products in pseudo-Riemannian geometry" Geometriae Dedicata, October 1993, Volume 48, Issue 1, pp 15-25, the following result
Let $(M,g)$ be a simply connected psudo-Riemannian manifold with two complementary
foliations $L$ and $K,$ whose leaves int... | 2 | https://mathoverflow.net/users/51420 | 200177 | 96,689 |
https://mathoverflow.net/questions/196243 | 4 | Let $X$ be a smooth, projective variety over $\mathbf{C}$ (to keep things simple) and let $\mathcal{F}$ be a vector bundle on $X$.
If $\mathcal{F}$ has a nowhere vanishing holomorphic section, the top Chern class of $\mathcal{F}$ vanishes. Are there other (weaker, but "natural") conditions on $X$ and/or $\mathcal{F}... | https://mathoverflow.net/users/61815 | Vanishing of the top Chern class of a vector bundle | Using the *splitting principle* (see [Hartshorne, Appendix A] one may compute Chern classes as if the vector bundle admitted a filtration by subvectorbundles with line bundles as intermediate quotients. (The point is that taking the projectivization of your vector bundle splits a line bundle off the pull-back of your o... | 2 | https://mathoverflow.net/users/10076 | 200180 | 96,691 |
https://mathoverflow.net/questions/200150 | 1 | Let $i:X\rightarrow \mathbb{R}^N$ be an imbedding of a topological space $X$. Assume that there exists an open neighborhood $U$ containing $i(X)$ which also admits a retraction $p:U\rightarrow X$. The question is whether any two such retractions $p\_1$ and $p\_2$ are homotopic? (It is allowed to shrink $U$ is needed.)
... | https://mathoverflow.net/users/43129 | Retractions of ENR | Yes. Consider $H: U \times [0; 1] \to \mathbb{R}^N$, $H\_t=tp\_1+(1-t)p\_2$. Clearly, $H^{-1}(U)$ is an open neighbourhood of $\iota(X) \times [0, 1]$, by compactness of $[0; 1]$ it contains an open set of of the form $\tilde{U} \times [0, 1]$ with $\tilde{U}$ open neighbourhood of $\iota(X)$, so for example $p\_1 \cir... | 1 | https://mathoverflow.net/users/37158 | 200181 | 96,692 |
https://mathoverflow.net/questions/197955 | 3 | Let $(M, \xi = \text{ker}\,\alpha)$ be a compact contact manifold with non-empty boundary. Vaguely asked, is there any natural geometric structure on the boundary $\partial M$ induced from the contact structure on $M$ which is inner to $\partial M$?
---
Let me give an analogous example in the symplectic case: Let... | https://mathoverflow.net/users/24221 | Boundary geometry of a contact manifold | Maybe what you're looking for are convex hypersurfaces (due to Emmanuel Giroux). They are mostly used in dimension 3, but can be used in every dimension (but things are more complex ... and: being convex in high dimension is not a *generic* property that you could automatically obtain by small perturbations).
A hyper... | 1 | https://mathoverflow.net/users/67031 | 200188 | 96,696 |
https://mathoverflow.net/questions/200182 | 2 | Suppose I want to compute $$\sum\_{n=1}^\infty \cos(n x)/n^{2k}.$$ Now, For any fixed $k$ this is, at least in principle, doable (see the excellent answer to [my math.SE question](https://math.stackexchange.com/questions/985778/weighted-sum-of-cosines) a while back), but the question is whether the sequence of polynomi... | https://mathoverflow.net/users/11142 | Summing cosines | This can be written as
$$ F\_{2k}(x) = \dfrac{1}{2} \left(\text{polylog}(2k, e^{ix}) + \text{polylog}(2k, e^{-ix})\right)$$
Note that
$F\_{k}''(x) = -F\_{k-2}(x)$, with $F\_k(0) = \zeta(k)$ and $F\_k'(0) = 0$, so
that $F\_k(x) = \zeta(k) + \int\_0^x dt\; (x-t) F\_{k-2}(t)$
It seems we have
$$F\_{2k}(x) = (-1)^{k+1} \d... | 9 | https://mathoverflow.net/users/13650 | 200194 | 96,697 |
https://mathoverflow.net/questions/199723 | 0 | Description: Given the following parametric cubic polynomials ${E}^{3}
- 15\, {\beta}\_{\pm}\, {E}^{2} - 3 \left({71\, {\beta}\_{\pm}^{2} + 352\, {\beta}\_{\mp}}\right) E
+ 135\, {\beta}\_{\pm} \left({5\, {\beta}\_{\pm}^{2} - 32\, {\beta}\_{\mp}}\right)$ where ${\beta}\_{\pm} = 1 \pm \eta$, how do I find all possible ... | https://mathoverflow.net/users/62471 | Find all possible rational values of the parameter of a parametric cubic such that it is reducible | I just reopened the question so I can write an answer, although my comment should have sufficed but I may not be getting my message across.
Anyway, let's use some sane variables ($x=\beta\_{\pm},y = E$), so the polynomial is
$$y^3-15xy^2-3(71x^2+352x)y+135x(5x^2-32x)$$
Let's now replace $y$ by $xz$ and get
$$(z... | 1 | https://mathoverflow.net/users/2290 | 200199 | 96,700 |
https://mathoverflow.net/questions/200189 | 4 | Let a pair of random variables $(X, Y)$ over a finite product space $\mathcal{X}\times \mathcal{Y}$ be given. The conditional expectation operator is defined as
$$(T\_Yf) (y):=\mathbb{E}[f(X)|Y=y],$$
where $f$ is a real-valued function acting on $\mathcal{X}$.
It is well known the the operator $T$ is contractive in ... | https://mathoverflow.net/users/41666 | Monotonicity of a ratio of conditional expectation operator | What if $X=\{1,2\}$ and $Y=\{1,2,3\}$. Equip both $X$ and $Y$ with the uniform probability measure. Set $f(x,1)=1$, $f(x,2)=x$ and $f(x,3)=2$.
Then $Tf(x,1)=1$; $Tf(x,2)=1.5$ and $Tf(x,3)=2$.
Notice that $\|f\|\_1=\|Tf\|\_1=1.5$ and $\|f\|\_\infty=\|Tf\|\_\infty=2$,
but $\|f\|\_2>\|Tf\|\_2$.
| 3 | https://mathoverflow.net/users/11054 | 200209 | 96,703 |
https://mathoverflow.net/questions/200210 | 0 | Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$.
At a certain part in a proof I am reading there is the condition
$S\_k : L\_q (\mu) \to H $ has a dense image if and only if $id: H \to L\_p(\mu)$ ... | https://mathoverflow.net/users/69323 | Injective inclusion map from RKHS function space to $L_p(\mu)$ | It seems, from comments, that the question is based on a misreading. The text does not assert that the inclusion map is always injective; it only gives a necessary and sufficient condition for the map to be injective. It is easy to find examples where it is, and examples where it isn't.
As an incredibly trivial examp... | 1 | https://mathoverflow.net/users/4832 | 200211 | 96,704 |
https://mathoverflow.net/questions/200197 | 2 | I intend to approach the paper of Wolfgang Ziller: "The Free Loop Space of Globally Symmetric Spaces", but I need the proper background on the foundations of the study of Free Loop Spaces. I obtained from my library the "Lectures on Closed Geodesics" by Klingenberg, but found it to be rather dry and difficult to read. ... | https://mathoverflow.net/users/48745 | References on the Free Loop Space | This survey paper on the Morse theory and closed geodesics: <http://arxiv.org/abs/1406.3107> (Morse theory, closed geodesics, and the homology of free loop spaces, by Alexandru Oancea) provides a lot of relevant references. In particular, it says that "a beautiful reference concerning closed geodesics is the book by Be... | 2 | https://mathoverflow.net/users/32389 | 200215 | 96,705 |
https://mathoverflow.net/questions/200064 | 4 | Using the Chen-Stein method, one can bound the total variation distance between a sum of possibly dependent Bernoulli random variables $W=\sum\_{i=1}^n X\_i$ and a Poisson distribution using only the first and second moments of the $X\_i$. For example, see [Two Moments Suffice for Poisson Approximations, by Arratia et ... | https://mathoverflow.net/users/36452 | Approximating by independent Poisson random variables | The short answer is "yes".
Longer answer: it is enough to show that
the variable
$Z:= W\_1+ W\_2$ converges to Poisson with the correct parameter. Now apply the "Two moments suffice". Note that if the $W\_1$ and $W\_2$ are dependent,
the conditions of the latter may fail - but at least they can be checked using fi... | 3 | https://mathoverflow.net/users/35520 | 200217 | 96,706 |
https://mathoverflow.net/questions/200216 | 2 | Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ such that for every ellipse $\gamma$ in the plane we have:$$\text{The Euclidien perimeter of}\; \gamma=\lambda (g\text{-diameter of}\;\gamma)$$ for a universal constant $\lambda$?
Note that the $g\text{-diameter }$ is the diameter of the interior of the ellipse as ... | https://mathoverflow.net/users/36688 | Perimeter of ellipse: Combination of two geometries | No, because otherwise we will have this property also for degenerate ellipses, which are intervals, which would imply that the euclidean distance between two (sufficiently close)
points is $\lambda(g$-distance$)$ which implies that $g$ generated an euclidean distance and is therefore a flat metric.
| 11 | https://mathoverflow.net/users/14515 | 200226 | 96,708 |
https://mathoverflow.net/questions/200192 | 7 | The question I have is the following:
Let $Y,Y'$ be two integer homology 3-spheres. Can we embed $Y'$ into $Y\times I$ such that $Y'$ separates the two boundary components apart?
Do we know any nontrivial examples of this type? (For example for $Y'=S^{3}$ or Brieskorn spheres?)
Also, we can consider the similar pr... | https://mathoverflow.net/users/44651 | What are known examples of a 3-manifold $Y$ embedded into $Y'\times I$ where $Y'$ is another 3-manifold? | To expand on Ian's answer to the first question: I proved many years ago (Seifert surfaces of knots in $S^4$, Pac. J. Math. 145 (1990), 97–116) that for any 3-manifold Y, there is a hyperbolic 3-manifold $Y'$ embedded in $Y \times I$ separating the boundary components, so that both pieces are homology cobordisms. This ... | 8 | https://mathoverflow.net/users/3460 | 200234 | 96,714 |
https://mathoverflow.net/questions/200238 | 5 | Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model (where real operations, elementary functions, and comparisons are elementary steps) when a first order differential equation $x'(t)=F(x(t),t)$ with a real-valued rational function $F(x,t)$ in real scalar... | https://mathoverflow.net/users/56920 | Are all rational exactly solvable differential equations known? | I don't think a general answer to your question is known. I personnaly doubt that it could be positive.
1. Partial decidable answer: if the variational linear differential system obtained along a given (algebraic) trajectory is not solvable by quadrature (condition expressed as the virtually solvability of its [Pica... | 5 | https://mathoverflow.net/users/24309 | 200248 | 96,717 |
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