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https://mathoverflow.net/questions/172004 | 7 | I am currently reading [this paper](http://projecteuclid.org/euclid.jdg/1214442469) in which we have a map $g:I\rightarrow Gauge(P)$ for some principal bundle $P$ which is differentiated. I am looking for a reference or explanation what the "most common" differentiable structure on $Gauge(P)$ would be.
Thank you very... | https://mathoverflow.net/users/37072 | Differentiable structure on the Gauge group of a principal bundle? | For *compact* base manifold and reasonably well behaved structure groups $G$ (like a finite-dimensinoal or a Banach-Lie group) there is the structure of a Frechet-Lie group on the group $Gauge(P)$. This is worked out in Chapter 1 of [Lie Group Structures on Symmetry Groups of Principal Bundles](http://dx.doi.org/10.101... | 5 | https://mathoverflow.net/users/5937 | 172068 | 87,898 |
https://mathoverflow.net/questions/172061 | 3 | How can I numerically compute the Skorokhod integral of a non-adapted process? If it is adapted, that is easy since the integral is just an Ito integral.
I have found that computing the Malliavin derivative is also easy using finite differences, by bumping the Brownian path at a time t. But I can't figure out how to ... | https://mathoverflow.net/users/53088 | Numerical computation of Skorokhod integral | If you are looking for approximations of the Skorohod integral we can use Wick Riemann sums.
Define for $F \in \mathbb{D}^{1,2}$ and $W$ a Wiener process the Wick product $F \star W\_t$ by
$F \star W\_t=FW\_t -\int\_0^t D\_s F ds$
then the Skorohod integral $\delta (u)$ of a possibly non-adapted process $u$ is th... | 4 | https://mathoverflow.net/users/48356 | 172069 | 87,899 |
https://mathoverflow.net/questions/172073 | 15 | Let $H$ be an infinite dimensional separable complex Hilbert space. Denote by $\mathcal{B}(H)$ the C\*-algebra of bounded operators on $H$, $\mathcal{K}(H)$ the ideal of compact operators on $H$, and $\mathcal{C}(H)=\mathcal{B}(H)/\mathcal{K}(H)$ the Calkin algebra. The collection of projections $\mathcal{P}(\mathcal{C... | https://mathoverflow.net/users/16107 | Why do the projections in the Calkin algebra not form a lattice? | Maybe it's clearer in our original email correspondence!
From me to Ilijas, 12/31/06:
>
> The poset of projections in any von Neumann algebra is a lattice. Is this true of C(H)? The answer seems to be no. I can give a pair of projections that looks like a counterexample, though proving this might take some work.
... | 12 | https://mathoverflow.net/users/23141 | 172077 | 87,903 |
https://mathoverflow.net/questions/171969 | 5 | In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form
$$\cos\left(t\right)\partial\_{x}\mathbf{u}+\sin\left(t\right)\partial\_{y}\mathbf{u}+\mathbf{A}\partial\_{t}\mathbf{u}+\mathbf{B}\left(t\right)\mathbf{u}=0$$
with $\left(x,y,t\right)$-independent mat... | https://mathoverflow.net/users/37254 | Analytic solution of a system of linear, hyperbolic, first order, partial differential equations | For simplicity, consider solutions where $ u$ does not depend on $x, y$:
$A u\_t + B(t) u = 0$. If $y^T A = 0$, that says $y^T B(t) u = 0$, so $u$ is
restricted to belong to a certain (possibly $t$-dependent) subspace.
Thus for your example $$A = \pmatrix{1 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 0 & 0\cr 0 & 0 & 0... | 3 | https://mathoverflow.net/users/13650 | 172087 | 87,908 |
https://mathoverflow.net/questions/171775 | 3 | In trying to follow the proof of Proposition 4.11 in
M. C. White, [Injective modules for uniform algebras](http://www.mas.ncl.ac.uk/~nmcw/papers/ua21.pdf), Proc. London Math. Soc. 73 (1996) 155--184
there is a part which seems unclear.
Let $I$ be a left ideal in a unital Banach algebra $A$.
Assume $I$ is weakly c... | https://mathoverflow.net/users/19593 | Existence of certain bounded approximate identity | Thanks to the hints of Narutaka Ozawa and Yemon Choi I can answer this question by myself.
Recall that $I^{\perp\perp}$ is the weak${}^\*$ closure of $I$ in $A^{\*\*}$, so by Goldstine theorem we can choose a net $(e\_\nu'')\_{\nu\in N''}\subset I$ such that it weak${}^\*$ converges to $e$. Clearly $(1-e\_\nu'')\_{\n... | 1 | https://mathoverflow.net/users/19593 | 172104 | 87,914 |
https://mathoverflow.net/questions/162193 | 8 | I have a question which I already asked on a more specialized site (<http://logicblogfrontend.hoelzl.fr/>), but perhaps M.O. will allow me to reach a wider range of experts.
Suppose that $X$ is Martin-Löf random, and $Y$ is some real. Must there be a sequence $R$ of rationals converging to $Y$ such that $X$ is rando... | https://mathoverflow.net/users/12126 | Martin-Löf randomness relative to a $\Delta^0_2$-representation of a real | Frank came up a quite short negative answer:
Let $x=\Omega\oplus \Omega^{\Omega}$ and $z$ is low for $x$. Then by van-Lambalgen's theorem relativized to $z$, $\Omega^{\Omega}$ is $\Omega\oplus z$-random. So $0''\equiv\_T x\not\leq\_T \Omega\oplus z\equiv\_T 0'\oplus z$.
| 6 | https://mathoverflow.net/users/14340 | 172107 | 87,915 |
https://mathoverflow.net/questions/171982 | 3 | I am not sure whether I should ask for help here or math stackexchange. I got trouble with an inequality involving the Schrodinger operator on manifolds. Any suggestion is appreciated!
Let $(M,g)$ be a closed Riemannian manifold, and two functions $\phi, \psi\in C^{\infty}(M)$ satisfying
$$\phi+\psi>0\ \mathrm{on}\ ... | https://mathoverflow.net/users/51632 | A question on Schrodinger operator | Here is an alternative formulation of a question related to yours.
Consider the case when $M=\Omega\subset\mathbb{R}^{n}$, a (smooth)
open bounded domain. Suppose that $u>0$ and
$$
-\Delta u+\psi u=f \mbox{ in }\Omega.
$$
Integrate against $\frac{w^{2}}{u}$ and rearranging you obtain for
all $w\in H\_{0}^{1}(\Omega)... | 0 | https://mathoverflow.net/users/40120 | 172108 | 87,916 |
https://mathoverflow.net/questions/167546 | 3 | I am reading a paper which refers to a maximally skewed stable distribution $F(x;1,-1,\pi/2,0)$ . Is there an efficient way to sample from this distribution?
If $X$ has distribution $F(x;\alpha,\beta,\gamma,\delta)$, then when $\alpha=1$ its characteristic function $\phi(\theta) = \mathbb{E}(e^{-i\theta X})$, $\theta... | https://mathoverflow.net/users/45564 | Sampling from maximally skewed stable distribution | Table 1 of <http://arxiv.org/abs/0908.3961v2> contains an explicit algorithm for this.
| 1 | https://mathoverflow.net/users/45564 | 172111 | 87,917 |
https://mathoverflow.net/questions/172129 | 4 | In deformation of complex analytic spaces, one usually considers an analytic proper simple surjective map $\varpi: \mathscr{M} \twoheadrightarrow \mathscr{P}$ as an analytic family. However the term simple differs by literature. There are two notions of simple: as a locally trivial map and as a flat map. **Are these de... | https://mathoverflow.net/users/40883 | Simple maps: Flat versus locally trivial | The answer is **no**.
In fact, flat maps are not required to be smooth maps in general. For instance, a flat family of smooth curves degenerating to a nodal curve is clearly *not* locally trivial.
However, the answer is still **no** even if we only consider smooth maps.
In fact, there is the following theorem due... | 3 | https://mathoverflow.net/users/7460 | 172131 | 87,925 |
https://mathoverflow.net/questions/172017 | 0 | Let $X$ be a smooth projective variety such that $-K\_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that:
* $-K\_Y$ is big,
* there exists an effective $\mathbb{Q}$-divisor $D\subset Y$ such that $-(K\_Y+D)$ is ample.
| https://mathoverflow.net/users/nan | Big divisors and small transformations | Let $f:X\dashrightarrow Y$ be a small map. Let $D$ be a big divisor on $D$. Then for $m >>0$ the map $\phi\_{mD}:Y\dashrightarrow Z$, induced by $mD$ is birational. Now, if $\Delta:=f^{\*}D$, then the map induced by $m\Delta$ is $\phi\_{mD}\circ f$, which is birational as well. Therefore $\Delta$ is big. We conclude th... | 0 | https://mathoverflow.net/users/14514 | 172137 | 87,926 |
https://mathoverflow.net/questions/171911 | 3 | H.H. Corson in [C] introduced the following version of Lindelöf property for convex closed subsets of Banach spaces:
>
> A Banach space $X$ has *property (C)* if every family of convex closed subsets of $X$ with empty intersection contains a countable subfamily with empty intersection.
>
>
>
For instance, all ... | https://mathoverflow.net/users/15860 | Productivity of Corson's property (C) | **Yes**, if $X$ and $Y$ have property (C), then so does $X\oplus Y$ because property (C) is a three-space property; this is a result of Pol (Proposition 1 in the paper you are referring to).
However, I am not sure if it is known whether property (C) passes from $C(K)$ to $C(K\times K)$. (If this is true, it should fo... | 2 | https://mathoverflow.net/users/15129 | 172140 | 87,928 |
https://mathoverflow.net/questions/167386 | 3 | The context I am thinking of mainly is GL(2), and accordingly, the degree 2 Artin L-function. But comments about GL(n) in general are also welcome.
In light the local Langlands correspondence, what can we say about Artin's L-function? For example, does it follow that for a fixed automorphic representation $\pi$ and $... | https://mathoverflow.net/users/48554 | GL(2) Local Langlands and Artin's L-function | I am not sure exactly of what is asked in this question, but I wish to correct what has been said in an answer because it is a little bit outdated. (Edited after Kevin's comment)
For $n=2$, Artin's conjecture had been solved in the 80's by Langlands and Tunnel for representations of $G\_{\mathbb Q}$ to $GL\_2(\mathbb... | 12 | https://mathoverflow.net/users/9317 | 172151 | 87,932 |
https://mathoverflow.net/questions/172146 | 9 | I've already asked this question on [math.stack](https://math.stackexchange.com/questions/834312/strange-definition-of-ergodicity) a few days ago and haven't received an answer, so I'm asking here.
In an engineering course, a stationary process was defined to be ergodic if for all $k\in \mathbb{N}$ and for any bounde... | https://mathoverflow.net/users/53127 | Strange definition of ergodicity | Yes. That is the same thing as ergodicity. To explain it, you have probably seen somewhere that the way to understand random variables formally is functions from a (hidden) underlying space $\Omega$ to $\mathbb R$. That is: knowing the point $\omega$ of $\Omega$, one can recover $X\_n(\omega)$ for each $n\in \mathbb Z$... | 14 | https://mathoverflow.net/users/11054 | 172152 | 87,933 |
https://mathoverflow.net/questions/172053 | 12 | I've been reading a preprint by David Hansen (with appendix by James Newton) called *Universal eigenvarieties, trianguline Galois representations and p-adic Langlands functoriality*. In it he talks about using overconvergent cohomology to construct eigenvarieties.
Now I was wondering if I could get a reference/ expl... | https://mathoverflow.net/users/15566 | Overconvergent cohomology and overconvergent modular forms | Unfortunately there *isn't* a simple direct relationship between OC mod forms and OC cohomology! The miracle is that they contain the same finite-slope Hecke data. As you've guessed, one proves this by using classical Eichler-Shimura and Coleman's and Stevens's control theorems, plus $p$-adic interpolation - this is a ... | 6 | https://mathoverflow.net/users/1464 | 172157 | 87,934 |
https://mathoverflow.net/questions/171605 | 4 | This is a (self-contained) followup question to <https://math.stackexchange.com/questions/380672/analogue-of-the-schwartz-zippel-lemma-for-subspaces>.
Let $f : \mathbb{R}^n \to \mathbb{R}$ be a nonzero multivariate polynomial of total degree $d$ over a field $\mathbb{R}$, and $S \subset \mathbb{R}$ be finite. Pick a ... | https://mathoverflow.net/users/22930 | Stronger versions of Schwartz-Zippel for random linear subspaces | For question 1, I think it's enough to take a fixed $z$ that is transcendental over $\mathbb{Q}(S)$, and consider $$g(z,z^2,\dots,z^k)=f(zy\_1+z^2y\_2+\dots+z^k y\_k).$$
Each coordinate of $zy\_1+\dots+z^ky\_k$ is independently uniform on
$$S'=\{zs\_1+\dots+z^k s\_k \, | \, s\_1, \dots, s\_k \in S\},$$
so by Schwarz-... | 5 | https://mathoverflow.net/users/405 | 172162 | 87,936 |
https://mathoverflow.net/questions/172145 | 6 | Let $k$ be an algebraically closed field of positive characteristic, and let $G$ be a reductive algebraic group over $k$ (for instance a classical group).
Let $V$ be a (rational) $G$-module. We say that $V$ admits a good filtration [see for instance Jantzen, Representations of Algebraic Groups, §4.19] if there exists... | https://mathoverflow.net/users/23236 | About G-modules with good filtrations | The answer to both questions is no.
For 1) let $V=H^0(\lambda)$ with $\lambda$ dominant, $U\leq V$ any submodule that doesn't have a good filtration and $\Delta(U)=\{(u,u): u\in U\}\leq V\oplus V$. Take $M\_1=V\oplus0\leq (V\oplus V)/\Delta(U)$, $M\_2=0\oplus V\leq(V\oplus V)/\Delta(U)$, and $M$ an injective hull of ... | 7 | https://mathoverflow.net/users/22989 | 172163 | 87,937 |
https://mathoverflow.net/questions/172171 | 3 | Let $A$ be matrix in $M\_{n}$ (i.e., $n\times n$ complex matrices), and $\|A\|\le 1$, we call it a contraction.
Assume that $A$ and $B$ are contractions such that
$I-AA^\*$ and $I-BB^\*$ are positive-definite.
How to show that
$$\text{Tr}\left(1-AA^\*\right)^{-1}\cdot \text{Tr}\left(1-BB^\*\right)^{-1} \ge \left(\... | https://mathoverflow.net/users/38620 | A similar Cauchy-Schwarz inequality with linear-algebra | Here's one way to prove this.
Assuming $A$ and $B$ are contractions, first recall *Hua's Matrix Identity*
\begin{equation\*}
(I-B^\*B) + (A-B)^\*(I-AA^\*)^{-1}(A-B) = (I-B^\*A)(I-A^\*A)^{-1}(I-A^\*B).
\end{equation\*}
From this it identity immediately follows that the following matrix is (Hermitian) positive definit... | 7 | https://mathoverflow.net/users/8430 | 172174 | 87,939 |
https://mathoverflow.net/questions/172109 | 9 | I found a paper: 'A New Method of Finding the Distribution of Prime Number', saying
>
> We stack discs and annuluses with certain rules then turn on the light to illuminate. The projection of annuluses corresponds prime number,and the projection of discs corresponds composite number.
>
>
>
See: <http://en.cn... | https://mathoverflow.net/users/33646 | Optical methods for number theory? | There's a way to use physics to calculate the digits of $\pi$. I quote from <https://math.stackexchange.com/questions/138289/intuitive-reasoning-behind-pis-appearance-in-bouncing-balls>
Let the mass of two balls be $M$ and $m$ respectively. Assume that $M=16\times100^nm$. Now, we will roll the ball with mass $M$ towa... | 8 | https://mathoverflow.net/users/3684 | 172175 | 87,940 |
https://mathoverflow.net/questions/172156 | 1 | What is the smallest subalgebra of $\mathfrak{su}(4)$ containing the span of the set $A = \{A, B\_1, B\_2\}$ where:
$A = i (J^x \sigma\_x \otimes \sigma\_x + J^y \sigma\_y \otimes \sigma\_y + J^z \sigma\_z \otimes \sigma\_z)$ for some real numbers $J^k$ and
$B\_1 = i\sigma\_z \otimes I$ and
$B\_2 = I \otimes \sigma... | https://mathoverflow.net/users/41654 | Smallest subalgebra of $\mathfrak{su}(4)$ arrising from a control problem on $SU(4)$ | The answer depends on the values of the constants $J^x$, $J^y$, and $J^z$. Here is what direct computation yields:
If $J^x=J^y=J^z=0$, so that $A=0$, then $B\_1$ and $B\_2$ span a $2$-dimensional abelian subalgebra.
If $J^x=J^y=0$, but $J^z\not=0$, then $A$, $B\_1$, and $B\_2$ span a $3$-dimensional abelian subalge... | 3 | https://mathoverflow.net/users/13972 | 172194 | 87,945 |
https://mathoverflow.net/questions/172178 | 4 | Let $X$ be a reasonable smooth scheme over some base $S$. The tangent sheaf $T\_X$ is a Lie algebroid, locally free as a $\mathcal{O}\_X$ module, and its Universal Enveloping Lie Algebroid $\mathfrak{U}(T\_X)$ is the sheaf of differential operators on $X$.
However, when $X$ is not necessarily smooth, for any open $U ... | https://mathoverflow.net/users/50797 | About the construction of the Universal Enveloping Lie Algebroid | Suppose that $X$ is affine.
A natural way to prove that localisation commutes with the universal enveloping Lie algebroid construction is via Rinehart's Theorem 3.1 in "Differential forms on general commutative algebras." Transactions of the American Mathematical Society (1963): 195-222. This says that if the global ... | 3 | https://mathoverflow.net/users/345 | 172200 | 87,947 |
https://mathoverflow.net/questions/172010 | 1 | I am interested in solutions to the Monge-Ampere equation for a smooth function
$h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is:
$$\det[\nabla^2(h(x,y))]=0 $$
Here $\det$ denotes the determinant and $\nabla^2$ denotes the Hessian. Geometrically this says th... | https://mathoverflow.net/users/36931 | Question about extending a solution to Monge-Ampere solution | The answer in the smooth case is 'no', because the differential equations governing the second fundamental form of the graph (which has Gauss curvature $K\equiv0$) show that the entire part of the domain $y>0$ (and $x>R$) will have to belong to the non-planar locus, i.e., the open set where the second fundamental form ... | 5 | https://mathoverflow.net/users/13972 | 172201 | 87,948 |
https://mathoverflow.net/questions/172187 | 4 | In the [paper](http://arxiv.org/abs/1210.5126v2), geometric RSK correspondence is given by
$$
\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \mapsto \left( \begin{matrix} \frac{bc}{b+c} & ab \\ ac & \frac{ad}{b+c} \end{matrix} \right).
$$
How this map relates to the classical RSK correspondence? Thank you v... | https://mathoverflow.net/users/11877 | Geometric RSK correspondece and classical RSK correspondence | I have some notes that hopefully explain some of this: <http://www-users.math.umn.edu/~shopkins/docs/rsk.pdf>.
By the way, I believe you got the map slightly wrong. It should be:
$$
\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \mapsto \left( \begin{matrix} \frac{bc}{b+c} & ab \\ ac & ad(b+c) \end{matrix}... | 5 | https://mathoverflow.net/users/25028 | 172204 | 87,950 |
https://mathoverflow.net/questions/172208 | 3 | The following excerpt is from Connes' [Noncommutative Geometry](http://www.alainconnes.org/docs/book94bigpdf.pdf)
>
> Let $(X, \mathcal{B}, \mu)$ be a standard Borel space equipped with a probability measure $\mu$, and let $\ T$ be a Borel transformation of $(X, \mathcal{B})$ that leaves $\mu$ invariant. Let $M = L... | https://mathoverflow.net/users/44405 | Measure Preserving Transformation Induced by a $*$-automorphism on $L^\infty(X,\mu)$ | 1. Every idempotent on $L^\infty$ is multiplication by (the equivalence class of) the characteristic function of a measurable set
2. Such an automorphism maps an idempotent onto another idempotent.
3. This gives your result at the level of sets, i.e., it provides an isomorphism of the Boolean $\sigma$-algebra of equiva... | 6 | https://mathoverflow.net/users/53122 | 172213 | 87,953 |
https://mathoverflow.net/questions/172215 | 2 | Let me explain the motivation for my question by talking about the finite-dimensional situation. Let's say we have a $d$-dimensional $C^\infty$ manifold $M$ embedded smoothly in $\mathbb R^n$. We fix some subspace $H\subset \mathbb R^n$ that intersects $M$ transversally, so that $H\cap M$ is a smooth submanifold of $M$... | https://mathoverflow.net/users/4129 | When do curves exist in infinite-dimensional submanifolds? | Take $V = L^2(M)$ with its usual Hilbert space topology and $W = C^\infty(M)$. Take $X = \mathrm{span}\{f\}$ for some non-smooth function. Then $X \cap W = \{0\}$, hence the kinematic tangent space of $X \cap W$ is zero as well, and not equal to $X$.
Your situation is quite strange and the example in the beginning of... | 2 | https://mathoverflow.net/users/16702 | 172217 | 87,954 |
https://mathoverflow.net/questions/172172 | 2 | Let us consider the wave equation $$(\partial\_t^2 - \Delta)u = 0$$ on a domain $U$ with coercive homogeneous boundary conditions $$Bu|\_{\partial U} = 0$$ that make $-\Delta$ self-adjoint. My question is, how can we construct such boundary conditions that the wave equation does not have finite speed of propagation any... | https://mathoverflow.net/users/53143 | Finite speed of propagation of wave equation | The propagation speed is still finite because the following standard argument works independently of what happens at the boundary:
Assume that $u\in C^2$ solves the wave equation and $u(t=0,x)=0$ on $B\_r(0)$ for some $r>0$, such that $\overline{B}\_r\subset U$. Then $u(t,x)=0$ on $B\_{r-|t|}$. If we apply this to th... | 6 | https://mathoverflow.net/users/48839 | 172223 | 87,958 |
https://mathoverflow.net/questions/150087 | 15 | Suppose $\kappa$ is a regular cardinal and $P$ is a $\kappa$-c.c. partial order. I want to know when are small sets added by subforcings of size $<\kappa$. The following seems well-known:
>
> **Fact:** If $\kappa$ is weakly compact, and $P$ has size $\kappa$ and is $\kappa$-c.c., then for any $P$-name $\tau$ for a ... | https://mathoverflow.net/users/11145 | capturing small sets in small factors | Assume that a regular cardinal $\kappa > \omega\_1$ has the property that for every $\kappa$.c.c. forcing notion $\mathbb{P}$ and name of set of ordinals $x \in V^{\mathbb{P}}$ smaller than $\kappa$, there is a subforcing $\mathbb{Q}$ of cardinality less than $\kappa$ that already decides the value of $x$.
We will s... | 5 | https://mathoverflow.net/users/41953 | 172224 | 87,959 |
https://mathoverflow.net/questions/172231 | 9 | I'm looking for examples of the following phenomena. Let $X$ be a reasonable space (say, a CW complex) and $G$ be a finite group acting on $X$. For all $k \geq 1$, the projection map $X \rightarrow X/G$ induces a map $H\_k(X;\mathbb{Q}) \rightarrow H\_k(X/G;\mathbb{Q})$ which factors through the $G$-coinvariants $(H\_k... | https://mathoverflow.net/users/53181 | Rational homology and finite group actions | The maps $\psi\_k$ are all isomorphisms; this is a simple application of the transfer ("averaging") construction. See Theorem 2.4, Chapter III, of Bredon's book "Introduction to compact transformation groups".
| 14 | https://mathoverflow.net/users/21684 | 172241 | 87,966 |
https://mathoverflow.net/questions/172081 | 7 | Warning: I am not a differential geometer, so some of the following might not make sense.
Background:
Let $w: (T\Omega)^k \to \mathbb{R}$ be a $k$-tensor on $\Omega$, an open subset of $\mathbb{R}^n$.
We can define the "derivative" of this tensor as follows. $Dw:(T\Omega)^{k+1} \to \mathbb{R}$ is a $k+1$-tensor ... | https://mathoverflow.net/users/1106 | A cohomology group which depends on the connection | Let me try to describe a slightly more general situation and hopefully prove my comment above.
Let $V$ be a vector bundle on a manifold $M$ with connection $D : \Gamma(V) \to \Gamma(V \otimes T^\*M)$. Let $Z(V)$ be the group of sections $\omega \in \Gamma(V \otimes T^\*M)$ where, for an open covering $U\_i$ of $M$, t... | 2 | https://mathoverflow.net/users/947 | 172242 | 87,967 |
https://mathoverflow.net/questions/172097 | 3 | Let $[n]:=\{1,\dots,n\}$ and $0\leq p\_n\leq n$. Fix any subset $A\_n$ of $[n]$ with $p\_n$ elements. The number of subsets $B$ of $[n]$ with $p\_n$ elements that are disjoint from $A$ is $\binom{n-p\_n}{p\_n}$ and so the number of subsets $B$ with $p\_n$ elements that have a non-empty intersection with $A$ is given by... | https://mathoverflow.net/users/43528 | Asymptotics of the number of elements in the intersection of two growing sets | To avoid subscripts, I'll write simply $p$ instead of $p\_n$. We'll assume that $p$ is small compared with $n$; certainly say $3p<n$. Given $A$ of cardinality $p$, the number of sets $B$ of size $p$ intersecting $A$ in exactly $j$ elements is
$$
\binom{p}{j} \binom{n-p}{p-j} \le \frac{p^j}{j!} \Big(\frac{p}{n-2p}\Big... | 4 | https://mathoverflow.net/users/38624 | 172258 | 87,974 |
https://mathoverflow.net/questions/172254 | 4 | Let $R$ be a possibly noncommutative left Noetherian ring and $M$ an $R$-module. I am looking for a reference or a proof for the following fact: $M$ is finitely generated and projective if and only if it is finitely presented and flat. (I am not interested in references that treat only the commutative case.)
| https://mathoverflow.net/users/53197 | Finitely generated projective = finitely presented flat over a noncommutative Noetherian ring | This holds over any ring, noetherian or not. See Bourbaki *Algebra* X, §1, no. 5.
| 11 | https://mathoverflow.net/users/40297 | 172260 | 87,975 |
https://mathoverflow.net/questions/172259 | 5 | It is extremely well-known that $H^\*(BQ\_8;\mathbb{Z})=\mathbb{Z}[\alpha,\beta,\gamma]$ with relations $2\alpha=2\beta=8\gamma=\alpha^2=\beta^2=\alpha\beta-4\gamma=0$, $|\alpha|=|\beta|=2$ and $|\gamma|=4$. I was wondering if anyone can given an explicit reference showing that $c\_2(\sigma)$ generates $H^4(BQ\_8)\cong... | https://mathoverflow.net/users/53199 | Explicit reference on generator of $H^4(BQ_8;\mathbb{Z})\cong \mathbb{Z}_8$ identified with second Chern class of standard representation | It is enough to show that the reduction mod 2 of that class is not zero. However, reducing $c\_2$ mod 2 gives the Stiefel-Whitney class $w\_4$ of the underlying real representation. That this class is non-zero is classical, and explicitly stated and proved for example in Quillen's "The Mod 2 Cohomology Rings of Extra-s... | 2 | https://mathoverflow.net/users/37021 | 172261 | 87,976 |
https://mathoverflow.net/questions/172032 | 7 | Let $\chi'\_f(G)$ be the fractional chromatic index.
Based on limited experiments (up to 8 vertices and few larger graphs),
I suspect:
**Conjecture** For perfect graphs $\lceil \chi'\_f(G) \rceil = \chi'(G)$
**Conjecture 2 (new)** For cubic claw-free perfect graphs $\lceil \chi'\_f(G) \rceil = \chi'(G)$
**Conje... | https://mathoverflow.net/users/12481 | Conjecture: for perfect graphs the fractional chromatic index rounded up equals the chromatic index | A result of Cai and Ellis (see Theorem 5 in <http://www.sciencedirect.com/science/article/pii/0166218X9190010T>) implies that deciding whether a cubic perfect line-graph is $3$-edge-colorable is NP-complete. Counter-examples to Conjecture 2 can be built from their argument as follows:
First, notice that every cubic b... | 6 | https://mathoverflow.net/users/53170 | 172273 | 87,983 |
https://mathoverflow.net/questions/172266 | 1 | This question arises when I am reading Klainerman&Machedon's paper "On the Uniqueness of Solutions to the Gross-Pitaevskii Hierarchy". The author made a comment on page 3, which in effect is as follows:
Let $\gamma(x,y)$ be some complex valued function in $L^{2}(\mathbb{R}^{2})$ such that
$$
\gamma(x,y)=\overline{\ga... | https://mathoverflow.net/users/18850 | Cauchy-Schwarz type formula for positive integral operator | Let me introduce the notation
$$ \langle \psi,\phi\rangle\_\gamma = \int\_{\mathbb{R}^2} \bar{\psi}(y) \gamma(x,y) \phi(x) ~\mathrm{d}x~\mathrm{d}y $$
so that
$$ \langle \phi,\psi\rangle\_\gamma = \overline{\langle \psi,\phi\rangle\_\gamma}$$
---
Firstly, your definition of $S$ I think is not strictly what they h... | 2 | https://mathoverflow.net/users/3948 | 172276 | 87,985 |
https://mathoverflow.net/questions/172277 | 3 | Let $S$ be a closed surface endowed with a Riemannian metric of negative curvature and let $US$ be the unit tangent bundle. Let $\mu$ be the Liouville measure on $US$.
Let $f: US\rightarrow\mathbb{R}$ be a smooth function and $M=\int\_{US}f(v)d\mu(v)/\int\_{US}d\mu$ be its mean value.
For each $v\in US$, let $(v(t)... | https://mathoverflow.net/users/17294 | On the Birkhoff ergodic theorem for geodesic flows | The answer is NO. There is no chance for such a thing.
The reason is the following. The asymptotic behaviour of the ergodic average depends on the asymptotic geometric behaviour of the ... geodesic you are looking at! And the hyperbolicity of the geodesic flow means that starting from everywhere (for example from $x... | 8 | https://mathoverflow.net/users/30691 | 172293 | 87,990 |
https://mathoverflow.net/questions/172279 | 2 | We know that every compactification of locally compact Hausdorff spaces correspond to a unitization of $C^{\*}$ algebras. For example the one point compactification corresponds to the minimal unitization and the Stone Chech compactification correspond to the maximal unitization "Multiplier algebra".
In this question ... | https://mathoverflow.net/users/36688 | Unitization via "End points compactification" | There is [**Noncommutative End Theory**](http://www.msp.org/pjm/1998/185-1/pjm-v185-n1-p03-p.pdf) by Akemann and Eilers.
| 1 | https://mathoverflow.net/users/1291 | 172300 | 87,995 |
https://mathoverflow.net/questions/172290 | 4 | I read the following problem, claimed to be in the IMO shortlist in 1988:
>
> A test consists of four multiple choice problems, each with three options, and the students should give an unique answer to each problem. After the test, one finds that for every three students, there is one problem to which their answers... | https://mathoverflow.net/users/20595 | "Codes" in which a group of words are pairwise different at a certain position | It is called perfect hash families in the design theory and computer science literature.
A *perfect hash family* PHF$(N; k, v, t)$ is an $N \times k$ array on $v$ symbols with $v \geq t$,
where for every $N \times t$ subarray at least one row consists of distinct symbols. The following shows a PHF$(6; 12, 3, 3)$.
$$\... | 7 | https://mathoverflow.net/users/27829 | 172304 | 87,997 |
https://mathoverflow.net/questions/172295 | 0 | Let $X, Y$ be positive real numbers, and let $p\_1, ... p\_n$ be the primes less than $Y$. How many subsets $S$ of the integers from 1 to $n$ are there such that the product of the $p\_i$'s with $i$ in $S$ is less than $X$, as a function of $X,Y$?
Clearly, if $X$ is sufficiently large relative to $Y$, then all choice... | https://mathoverflow.net/users/10898 | Number of products of distinct primes lying in an interval | Distinct sets of primes have distinct products, hence you are counting the number of square-free $Y$-smooth integers below $X$.
The number of $Y$-smooth integers below $X$ is
$$\Psi(X,Y)=X\rho(u)+O(X/\log Y),$$
where $X=Y^u$, and $\rho$ is Dickman’s function, which is $1-\log u$ for $u\in[1,2]$. In the range you are ... | 5 | https://mathoverflow.net/users/12705 | 172307 | 87,999 |
https://mathoverflow.net/questions/172284 | 4 | Let $\mathscr{T}$ be atriangulated category.
The third axiom for triangulated categories, namely,
if in the diagram
$$\begin{array} 0X &\stackrel{u}{\longrightarrow}&Y&\stackrel{v}{\longrightarrow}&Z&\stackrel{w}{\longrightarrow}&\Sigma X\\
\downarrow{f}&&\downarrow{g}&&\downarrow{\exists h}&&\downarrow{\Sigma f}... | https://mathoverflow.net/users/12826 | Proof without using Yoneda's lemma? | Yes, it is possible (compare Balmer, [Triangular Witt groups. Part I: The 12-term localization exact sequence](http://www.math.ucla.edu/~balmer/research/Pubfile/TWG-I.pdf), page 5).
One easily reduces your question to the following situation:
\begin{array}
0
X &\stackrel{u}{\longrightarrow} & Y & \stackrel{v}{\longri... | 10 | https://mathoverflow.net/users/53211 | 172313 | 88,001 |
https://mathoverflow.net/questions/172269 | 5 | Sorry if this question is a dumb one.
I used a special family of matroids in my research. One of them, of rank 3, can be represented by the following matrix over $\mathbb F\_5$ or over $\mathbb R$:
$\left(\begin{array}{cccccccccccc}
1&0&0&1 &1 &0 &-1&2 &-1& 2&0 &0\\
0&1&0&-1&0 &1 &2 &-1&0 & 0&-1&2\\
0&0&1&0 &-1&-1... | https://mathoverflow.net/users/49233 | Does this matroid have a name? | Matrices that contain at most two non-zero entries per column are called *frame matrices*. The matroids representable by frame matrices (over a finite field $\mathbb{F}$) are in fact a fundamental class in the structure theorem of Geelen, Gerards and Whittle for the class of all $\mathbb{F}$-representable matroids. I t... | 8 | https://mathoverflow.net/users/2233 | 172322 | 88,005 |
https://mathoverflow.net/questions/172327 | 4 | I'm looking for a reference, expository in nature, for the proof of the following theorem of Coifman, Lions, Meyer and Semmes.
Theorem:
For all $u\in W^{1,n}(\mathbb{R}^n;\mathbb{R}^n)$, $\operatorname\*{det}Du\in\mathcal{H}^1(\mathbb{R}^n).$
While I have the paper of Coifman, Lions, Meyer and Semmes where the r... | https://mathoverflow.net/users/53221 | Reference: Hardy space regularity of the Jacobian determinant | This is Lion's own exposition: [Jacobians and Hardy spaces](http://repository.cmu.edu/cgi/viewcontent.cgi?article=1301&context=math).
>
> The present report gives some elementary proofs, explained in the
> simple solution of $R^2$, of the embedding results of Jacobian
> determinants into Hardy spaces.
>
>
>
| 1 | https://mathoverflow.net/users/11260 | 172328 | 88,007 |
https://mathoverflow.net/questions/172312 | 1 | I would appreciate any help with the following problem:
Let $(M,g)$ be a 3 dimensional Riemann manifold with boundary. Let $ \Gamma $ be a surface of sufficient regularity dividing M into two connected components $M\_{+}$ and $M\_{-}$. Is it possible to construct a harmonic function such that it is zero on $\Gamma$ a... | https://mathoverflow.net/users/50438 | Harmonic Function with special property | Depends on what you mean by "sufficient regularity". No simple interpretation will work. Think of the disk $|z|<10$ on the plane split by the parabola $y=x^2$. Put $v(z)=u(z+iz^2)$. Then $v$ is still harmonic in the disk $|z|<2$ and vanishes on its real diameter $(-2,2)$. Thus we have the identity $v(\bar z)=-v(z)$ by ... | 2 | https://mathoverflow.net/users/1131 | 172335 | 88,010 |
https://mathoverflow.net/questions/172324 | 8 | Let $M$ be a compact manifold and $\text{diff}(M)$ its diffeomorphism group. Various quotients of $\text{diff}(M)$ appear in the literature, oftentimes with geometric significance. A well-known example is the space $\text{diff}(M)/\text{diff}\_\mu(M)$, where $\mu$ is a measure on $M$ of total volume one and $\text{diff... | https://mathoverflow.net/users/41626 | When are quotients of the diffeomorphism group Fréchet manifolds? | (1) Yes, it is the set of smooth positive densities of the same total volume as $\mu$.
This can be seen by the "Moser trick" directly, and it is a one dimensional affine subspace of a space of smooth post ice sections of a line bundle, that tame.
(2) Yes, it is, see
* MR2670086 (2011m:58006) Reviewed
Molitor, Ma... | 6 | https://mathoverflow.net/users/26935 | 172361 | 88,019 |
https://mathoverflow.net/questions/172349 | 2 | Let $n,m$ be two positive integers. By $r\_n$ we denote the largest prime not exceeding $n$. If $r\_n\leq m\leq n$ and $q$ is the largest prime factor of $n!/m!$ such that $q\geq 17$ and $q\geq n-m+3$, then I would like to find some examples of integers $n$ and $m$ such that $q^4$ divides $n!$.
| https://mathoverflow.net/users/25674 | Finding a suitable number | Given a positive integer $n$, let $P(n)$ denote the largest prime factor of $n$. What you are looking for are integers $n$ such that $q := \max\{P(n-15),P(n-14),\dots,P(n)\} < Cn$ for some constant $C \in ]0,1[$ depending on whether you require $q^2$ to divide $n!$ (as in the first version of your question) or $q^4$ li... | 2 | https://mathoverflow.net/users/28104 | 172382 | 88,029 |
https://mathoverflow.net/questions/172381 | 1 | Consider the (strong) Bruhat order, $\leq\_B$, on the symmetric group $S\_n$. Suppose there are permutations $\pi,\sigma∈S\_n$ such that $\pi\geq\_B \sigma$. Suppose further that they satisfy the following property: if i precedes j in $\sigma$ written in single line notation, then i precedes j in $\pi$ written in singl... | https://mathoverflow.net/users/nan | maximal chain in (strong) Bruhat order satisfying constraint | Yes. The relation you are imposing between $\pi$ and $\sigma$ is called [weak order](http://en.wikipedia.org/wiki/Weak_order_of_permutations#Weak_order_of_permutations) -- specifically, you are saying that $\sigma \leq\_W \pi$ where $\leq\_W$ is weak order. Every maximal weak chain is also a maximal strong chain, so ta... | 3 | https://mathoverflow.net/users/297 | 172383 | 88,030 |
https://mathoverflow.net/questions/172387 | 1 | It is well known that the asymptotic density of an infinite union of disjoint sets of integers may not be the sum of their individual asymptotic densities.
Can this failure of countable additivity occur even if each of the sets being unioned is periodic (calling a set periodic if there is some nonzero integer such th... | https://mathoverflow.net/users/3902 | Additivity of asymptotic density of periodic sets | For each $r\in\mathbb{Z}$, let $a\_{r}$ be a positive real number so that $\sum\_{r\in\mathbb{Z}}a\_{r}<1$. Let $A\_{r}\subseteq\mathbb{Z}$ be a periodic set with $\mu(A\_{r})<a\_{r}$ and $r\in A\_{r}$. Then $\mathbb{Z}=\bigcup\_{r\in\mathbb{Z}}A\_{r}$. Now reenumerate the set $\{A\_{r}|r\in\mathbb{R}\}=\{B\_{n}|n\in\m... | 3 | https://mathoverflow.net/users/22277 | 172389 | 88,032 |
https://mathoverflow.net/questions/172391 | 1 | Are there any good references which explain the injectivity part of Grothendieck's section conjecture in details? I heard that Grothendieck himself gave a proof in the letter to G.Faltings. But I need more kind explanation.
I already read a book `Galois groups and fundamental groups' by T.Szamuely. I do not know whet... | https://mathoverflow.net/users/44006 | Injectivity of Section conjecture | You can find it in <http://www.springer.com/mathematics/algebra/book/978-3-642-30673-0>, Chapter 7 by Jakob Stix.
| 2 | https://mathoverflow.net/users/nan | 172397 | 88,034 |
https://mathoverflow.net/questions/172343 | 18 | Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ the number of primitive $n$th roots of unity with negative real part. Then $\varphi(n)=R(n)+L(n)$ for $n\ne 4$. I am pa... | https://mathoverflow.net/users/23827 | Number of primitive $n$th roots with positive versus negative real parts | As Gerhard Paseman has commented, more general problems were studied by D.H. Lehmer in his paper [The distribution of totatives](http://books.google.com/books?id=k5Y-agKNh68C&pg=PA347&lpg=PA347&dq=lehmer+the+distribution+of+totatives&source=bl&ots=P35LNI-VcC&sig=SvYgiAEnfQDa17xCMRlpPxnzG8I&hl=en&sa=X&ei=zk2mU9fGHoP6oAT... | 11 | https://mathoverflow.net/users/38624 | 172419 | 88,043 |
https://mathoverflow.net/questions/172392 | 5 | Continuing the curiosity of my last couple questions: Is it the case that for every set of primes $F$, the asymptotic density of the integers coprime to all of $F$ is $\displaystyle \prod\_{p \in F} (1 - 1/p)$?
(The case of finite $F$ is obvious. Furthermore, as the product is an upper bound on the density, this hold... | https://mathoverflow.net/users/3902 | Relationship of Euler product to coprimality densities for arbitrary sets of primes | GH from MO is answering a more difficult question where one asks for uniform estimates for the number of integers up to $x$ not divisible by primes in a set $F$ (which may also depend on $x$). This is delicate, and the answer is indeed no.
The problem, as I understand it, keeps $F$ fixed, and asks for the density as... | 11 | https://mathoverflow.net/users/38624 | 172421 | 88,044 |
https://mathoverflow.net/questions/172422 | 10 |
>
> Are there arbitrarily long arithmetic progressions in which all the
> prime factors of all the terms are at most $N$, for some $N$? Assume
> all the terms are positive and the sequence of terms is increasing.
>
>
>
I have proved that no such infinite sequence exists. Note the $N$ may vary from AP to AP.
Fo... | https://mathoverflow.net/users/50068 | Arbitrarily long arithmetic progressions | A simple proof is available as well. Pick p coprime to d and let t be such that td=1 mod p. Then, mod p, t times the arithmetic progression looks like a sequence of consecutive integers. Thus its length has to be less than p to avoid one of the terms being a multiple of p, which means the original progression also has ... | 16 | https://mathoverflow.net/users/35626 | 172428 | 88,047 |
https://mathoverflow.net/questions/172431 | 1 | Assume $X$ to be a Notherian topological space such that any irreducible closed
subset has a unique generic point. Consider $Y\subseteq X$ as a topological space with the induced topology from $X$. Is it true that any irreducible closed
subset of $Y$ has a unique generic point?
What properties on $Y$ make above quest... | https://mathoverflow.net/users/49402 | Sober topological subspace | Surely not: the subspace of $\text{Spec}(\mathbb{Z})$ consisting of nonzero primes, which is homeomorphic to $\mathbb{N}$ with the cofinite topology, is not sober. See also [Wikipedia](http://en.wikipedia.org/wiki/Sober_space).
| 4 | https://mathoverflow.net/users/2926 | 172433 | 88,048 |
https://mathoverflow.net/questions/172427 | 7 | Let us start with the following definition.
Let $1\leqslant k\leqslant n$ and let $\omega\_1,\omega\_2\in\Lambda^k(\mathbb{R}^n)$. We say that $\omega\_1$, $\omega\_2$ are **equivalent**, if there exists $T\in GL(n,\mathbb{R})$ such that
$$
T^\ast\omega\_1=\omega\_2,
$$
where $T^\ast:\Lambda^k(\mathbb{R}^n)\rightarr... | https://mathoverflow.net/users/27832 | Equivalence of exterior forms | Normal forms for exterior forms is a classical subject. It is well-known that when $k=1,2,n{-}2,n{-}1$, or $n$ and when $(n,k)=(6,3),(7,3),(7,4),(8,3),(8,5)$, there are only a finite number of 'normal forms', i.e., orbits in $\Lambda^k(\mathbb{R}^n)$ under the action of $\mathrm{GL}(n,\mathbb{R})$.
In particular, in... | 11 | https://mathoverflow.net/users/13972 | 172435 | 88,049 |
https://mathoverflow.net/questions/172379 | 14 | Verdier localization is one of the more intuitive ways to localize a triangulated category, "killing" a suitable class of objects via a functor which is universal with respect to this property.
I would like to know whether it is possible to reproduce the construction of $\mathcal{T}/\mathcal C$ in a $\infty$-stable s... | https://mathoverflow.net/users/7952 | Verdier localization for stable $\infty$-categories | One good source is Blumberg-Gepner-Tabuada, "A Universal Characterization of Higher Algebraic K-Theory."
See Definition 5.4, which defines the Verdier quotient as the cofiber (In the oo-category of presentable stable oo-categories) of the fully faithful functor $\mathcal{C} \to \mathcal{T}$.
Proposition 5.6 shows ... | 13 | https://mathoverflow.net/users/3593 | 172436 | 88,050 |
https://mathoverflow.net/questions/163217 | 9 | Let $G$ be a reductive group and $\lambda$, $\mu$ and $\nu$ be dominant weights of $G$. Denote by $V\_\lambda$ the irreducible representation of $G$ of highest weight $\lambda$. It seems to be true that
$$
V\_\lambda \otimes V\_\mu^\* \subseteq V\_{\lambda+\nu}\otimes V\_{\mu+\nu}^\*,
$$
where $W^\*$ as usual is the du... | https://mathoverflow.net/users/49530 | Tensor products of two irreducible representations of reductive groups and their inclusions | There is a natural inclusion, defined as follows:
The space of sections of weight $\mu$ on $\mathcal{F} = G/B$ can naturally be identified with $V\_\mu^\*$ (via the map $w \rightarrow [g \rightarrow \langle w, g v\_\mu \rangle ]$ where $v\_\mu$ is a highest-weight vector). Similarly, if we consider $\mathcal{F} = G/B... | 7 | https://mathoverflow.net/users/44191 | 172441 | 88,052 |
https://mathoverflow.net/questions/172440 | 7 | Given rational numbers $a$ and $b$, what is the fastest way to determine whether there are any rational solutions to $a=x^2+by^2$?
I am interested in the case where the numerator and denominator of $a$ have about $4$ digits, and those of $b$ have about $8$ digits. I am willing to restrict attention to solutions where... | https://mathoverflow.net/users/10366 | Fast quadratic norm algorithms | This is a conic, so factoring the discriminant is important, then you can check local solubility at all bad primes. This gives whether it is solvable. With a little extra work (maybe negligible compared to factoring), Simon's algorithm will give you a solution. <http://www.ams.org/journals/mcom/2005-74-251/S0025-5718-0... | 6 | https://mathoverflow.net/users/52997 | 172442 | 88,053 |
https://mathoverflow.net/questions/166248 | 15 | I remember reading somewhere that it takes about a week to convert a page of math into something a proof-assistant like Isabelle or HOL Light would accept.
Is this type of conversion something that requires a lot of search and creativity, or is it near-mechanical?
| https://mathoverflow.net/users/30352 | Does formalizing math require search and creativity, or is it near-mechanical? | Formalisation time per page is far from constant. It depends on the material. Sometimes you race through the pages, and at other times you make no progress at all.
Obstacles to formalisation include
* gaps or errors in the presentation;
* flights into metamathematical arguments, appeals to global symmetries, etc.... | 14 | https://mathoverflow.net/users/44555 | 172445 | 88,055 |
https://mathoverflow.net/questions/172406 | 5 | I know that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). In dimension 2 there are strong restriction on the fundamental group of closed manifolds.
I wanted to know what happen in dimension 3: which are the conditions on a ... | https://mathoverflow.net/users/52926 | On the fundamental group of closed 3-manifolds | Igor's suggestion of the recent [paper by Aschenbrenner, Friedl, and Wilton](http://arxiv.org/abs/1205.0202) is probably the best place to start as it has a good treatment this problem which includes both a summary of recent advances and a litany of open problems.
If you want to work through a classification of geome... | 7 | https://mathoverflow.net/users/27453 | 172454 | 88,059 |
https://mathoverflow.net/questions/172452 | 0 | If $M=G/H$ is a reductive homogeneous space then we can write $\frak{g}=\frak{m}+\frak{h}$
where $[\frak{h}, \frak{m}]\subset \frak{m}$. Here $\frak{g}$ and $\frak{h}$ are the Lie algebras of $G$ and $H$ respectively. Let $T\_o$ be the torsion of the canonical connection (of the second kind) w.r.t. this decomposition a... | https://mathoverflow.net/users/14454 | Decomposition of Lie subspaces | I am starting to think that you can always do this, but perhaps I am missing something at this later hour.
The torsion of the canonical connection at the origin defines a skewsymmetric bilinear map $T\_0: \Lambda^2 \mathfrak{m} \to \mathfrak{m}$ by
$$
T\_0(X,Y) = -[X,Y]\_{\mathfrak{m}}
$$
where the subscript indicate... | 1 | https://mathoverflow.net/users/394 | 172455 | 88,060 |
https://mathoverflow.net/questions/172459 | -3 | Let $A=\{x\ |\ x\in\mathbb Z\_{\ge 0},\ x\ $ with some conditions$\ \}$.
Let $B=\mathbb Z\_{\ge 0}-A$.
Define $\ 2A= \{a+b : a \in A,\ b \in A\}$.
Define $\ 2B=\{a+b : a \in B,\ b \in B\}$.
Then the set $\ \{n,\ n+1k ,\ n+2k, \ ...\}\ \subseteq\ 2A\ $or $\ 2B$ for some positive integers $n,k$?
| https://mathoverflow.net/users/33646 | A general question on nonnegative integer sequence | No. As suggested by S. Carnahan (with the exact numbers tweaked),
$$
A = \bigcup\_{k=0}^\infty [3^{2k},3^{2k+1}) \quad\text{and}\quad B = \Bbb N\setminus A = \bigcup\_{k=0}^\infty [3^{2k+1},3^{2k+2}).
$$
Then
$$
2A \subset \bigcup\_{k=0}^\infty [3^{2k},2\cdot3^{2k+1}) \quad\text{and}\quad 2B \subset \bigcup\_{k=0}^\inf... | 2 | https://mathoverflow.net/users/5091 | 172460 | 88,062 |
https://mathoverflow.net/questions/172025 | 1 | This question is probably easy but I only have "tedious case checking" proof strategy in sight, and I'm sure there should be a reference lying around...
The question concerns the TSP problem (with fixed start- and end-vertex). The graph is the graph induced on the ball in an infinite grid. More precisely, the vertice... | https://mathoverflow.net/users/18974 | Estimate for the travelling salesman problem for balls inside a grid | For question (1), $k$ grows linearly with $n$, so it cannot be chosen independently.
Color the vertices of the grid black and white, so that the vertices $(x,y)$ with $x+y$ even are black and the remaining vertices are white. Clearly, every edge of the graph has one end black and the other end white. Therefore, on ev... | 3 | https://mathoverflow.net/users/24076 | 172462 | 88,063 |
https://mathoverflow.net/questions/172365 | 13 | Let $A\to B$ be a morphism of (unitary commutative) rings such that $B$ is module-finite over $A$ and there exists $f\in A$ which is a nonzerodivisor in $A$ and in $B$, with $A[1/f]\to B[1/f]$ an isomorphism.
My question is: is the diagram $A\to B \rightrightarrows B\otimes\_A B$ exact ?
In geometric terms, this is... | https://mathoverflow.net/users/17988 | Descent of functions along finite birational morphisms | This is not always true. For example, it fails for the inclusion $k[x^3,x^5]\to k[x]$, where $k$ is a field. This is because $x^7$ is in the equalizer: $x^7\otimes 1 = x^2\otimes x^5=x^5\otimes x^2 = 1\otimes x^7$.
Such examples are indeed in the literature. I saw this one in SGA3, exp V.
| 11 | https://mathoverflow.net/users/1114 | 172467 | 88,065 |
https://mathoverflow.net/questions/172476 | 0 | Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $u \in L^2(0,T;H^1(\Omega))$ with $u\_t \in L^2(0,T;H^{-1}(\Omega))$. Define the truncation function$$T\_\epsilon(x) = \begin{cases}
-\epsilon &: x \in (-\infty, -\epsilon]\\
x & x \in (-\epsilon, \epsilon)\\
\epsilon &: x \in [\epsilon, \infty)
\end{cases}.... | https://mathoverflow.net/users/49672 | Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$ | This is a classical trick: the function $\frac{T\_{\epsilon}(.)}{\epsilon}$ is an approximation to the sign function so at least formally the result is clear.
In order to work ou the details, define
$$
S\_{\epsilon}(z)=\int\_0^z\frac{T\_{\epsilon}(s)}{\epsilon}ds.
$$
You can compute explicitly $S\_{\epsilon}(z)$ so I... | 3 | https://mathoverflow.net/users/33741 | 172479 | 88,070 |
https://mathoverflow.net/questions/172481 | 10 | Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
My teacher has conjectured this. I haven't been able to produce a counterexample, neither a solution. Is this claim true? If not a ... | https://mathoverflow.net/users/50068 | Polyhedron not circumscribed about a sphere | This is exercise 21.3, of [Mathematical Omnibus: Thirty Lectures on Classic Mathematics](https://rads.stackoverflow.com/amzn/click/0821843168), by D.B. Fuks and Serge Tabachnikov. The solution is given on page 448:
>
> Assume that $P$ is circumscribed. Consider a face $A\_1$, $A\_2$,...
> $A\_n$ and let $O$ be its... | 18 | https://mathoverflow.net/users/11260 | 172484 | 88,072 |
https://mathoverflow.net/questions/172432 | 7 | Jyrki Lahtonen has suggested I write a blog post relating binary quadratic forms to quadratic field class numbers, <https://math.stackexchange.com/questions/209512/binary-quadratic-forms-over-z-and-class-numbers-of-quadratic-%EF%AC%81elds/209543#comment1727526_209543>
The coincidence part of this is Jyrki bringing up... | https://mathoverflow.net/users/3324 | Class number for binary quadratic forms discriminant $\Delta$ to class number $\mathbb Q(\sqrt \Delta)$ | It seems to me that what Buell says about the narrow class group is not quite right (it's hard for me to say, as I don't have a copy of it). Magma tells me that in $\mathbb{Q}(\sqrt{210})$, the narrow class group is $(\mathbb{Z}/2\mathbb{Z})^{3}$ and the ideal class group is $(\mathbb{Z}/2\mathbb{Z})^{2}$. The squares ... | 7 | https://mathoverflow.net/users/48142 | 172490 | 88,074 |
https://mathoverflow.net/questions/172500 | 2 | During my work with order preserving homeomorphisms, I got interested in the double arrow space and, subsequently, in the lexicographic square.
I would really like to find examples of spaces like these two.
Specifically, I am looking for a space $X$ with the following properties:
1. $X$ is a compact linearly ordered... | https://mathoverflow.net/users/31446 | Compact, densely ordered spaces | One of my favorite spaces has this property: the *extended long ray*.
First, the long ray itself: this is just the space $L$ gotten by pasting together $\omega\_1$-many copies of $[0, 1)$ in the natural way. Formally, $L$ is the lexicographic order on $\omega\_1\times[0, 1)$, with both viewed as linear spaces in the ... | 5 | https://mathoverflow.net/users/8133 | 172501 | 88,076 |
https://mathoverflow.net/questions/167332 | 5 | Let $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}\_n(K)$, where $\textrm{Sym}\_n(K)$ is the vector space of all symmetric $n \times n$ matrices with entries in $K$, such that every matrix in $M$ has a... | https://mathoverflow.net/users/36563 | Vector Spaces of Symmetric Matrices of Low Rank | There are three references you may want to check out:
* R. Meshulam, On two extremal matrix problems, Linear Algebra Appl. 114/115 (1989), 261--271
* R. Loewy, N. Radwan, Spaces of symmetric matrices of bounded rank, Linear Algebra Appl. 197-198 (1994), 189--215
* R. Loewy, Large spaces of symmetric matrices of boun... | 6 | https://mathoverflow.net/users/34951 | 172504 | 88,078 |
https://mathoverflow.net/questions/172505 | 4 | I asked this question already on stackexchange, but I did not get any resonance at all, so maybe anybody here can give me a few hints about my problem.
My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}\_{\ge 0}\rightarrow \mathbb{C}$ $$ i\partial\_t f(x,t) = -\partial\_x^2 f(x,t) + g(t)V(x)f(x,t).$$
I wo... | https://mathoverflow.net/users/nan | Solution to Schrödinger equation | I agree with @Christian Remling that the product structure of your potential $V(x,t)=g(t)V(x)$ is not helpful in general, but it would help if $g(t)$ is a monotonically decreasing function of time, see [this paper](https://web.archive.org/web/20160313020436/http://pdf.easechem.com/pdf/32/d34de190-5a35-4f5e-83d0-51d0303... | 3 | https://mathoverflow.net/users/11260 | 172513 | 88,082 |
https://mathoverflow.net/questions/171711 | 4 | Let $T\_{p,q}$ be the (p,q) torus knot. Could anybody possibly compute either unreduced or reduced Khovanov $\mathfrak{sl}(2)$ homology of the connected sum $T\_{2,3} \sharp T\_{3,4}$ of the (2,3) and (3,4) torus knots? The (3,4) torus knots $T\_{3,4}$ is homologically thick. Is the Khovanov $\mathfrak{sl}(2)$ homology... | https://mathoverflow.net/users/17644 | Khovanov $sl_2$ homology of a connected sum of some torus knots | KnotTheory` gives the following Khovanov polynomial for the knot $K = T\_{2,3}\# T\_{3,4}$ you're looking at:
$$q^7 + q^9 + 2q^{11}t^2 + 2q^{15}t^3 + (q^{13} + 2q^{15})t^4 + (2q^{17} + 2q^{19})t^5 + (q^{17} + q^{21})t^6 + 2q^{21}t^7 + q^{25}t^8$$
The two summands have $s$ equal 2 and 6, therefore $s(K)=8$; the terms $q... | 5 | https://mathoverflow.net/users/13119 | 172525 | 88,087 |
https://mathoverflow.net/questions/172521 | 10 | Let $k$ be a field, let $f \colon X \to Y$ be a morphism of $k$-varieties, and assume $X$ and $Y$ are smooth and projective. Let $H(\\_)$ be a classical Weil cohomology theory (i.e. one of $\ell$-adic étale, Betti, algebraic de Rham, crystalline). *Of course it depends on $k$ which of the theories are applicable, but I... | https://mathoverflow.net/users/21815 | What morphisms induce injective/surjective maps on (Weil) cohomology? | Kleiman, Algebraic Cycles and the Weil Conjectures, Proposition 1.2.4: Let $f: X \to Y$ be surjective. Then $f^\*: H^\*(Y) \to H^\*(X)$ is injective.
Let me recall the proof.
Let $x$ be a closed point of the generic fibre of $f$ and set $Z := \overline{\{z\}} \subseteq X$ and let $z = cl(Z)$ be the cycle class of $... | 12 | https://mathoverflow.net/users/nan | 172527 | 88,088 |
https://mathoverflow.net/questions/172128 | 15 | I noticed that the following is true, and that there is a reasonably elementary proof of it (in particular, the classification of finite simple groups is not needed). Let $G$ be a finite permutation group which contains two $p$-cycles which do not commute (where $p$ is any odd prime other than a Mersenne prime). Then $... | https://mathoverflow.net/users/14450 | Permutation Groups Containing non-commuting $p$-cycles | *Fixed several inaccuracies, many thanks to Frieder Ladisch for spotting them:*
Let $x$ and $y$ be two non-commuting $p$-cycles, $G=\langle x,y\rangle$, and $G$ be considered as a transitive permutation group on the support $\Omega$ of $G$.
We show that either $\text{AGL}\_1(\mathbb F\_q)\le G$ for a Mersenne prime... | 8 | https://mathoverflow.net/users/18739 | 172529 | 88,089 |
https://mathoverflow.net/questions/172526 | 1 | Consider the following system of ODEs
$$
Y^{'}(t) = - \left[ A Y(t) + Y(t) A \right] + B(t)
,
$$
where $Y(t)$,$A$,$B(t)$ are all matrices, with the properties $A=A^T$, $Y=Y^T$. $Y(t)$ is the matrix of functions to be solved. Of course, there's a standard way to solve this system, for example, by rearranging the $n \ti... | https://mathoverflow.net/users/54306 | A Non-homogeneous, Linear (Matrix) System of ODEs: What's Known About it? | Well, there won't be solutions unless $B(t)^T = B(t)$ as well. If you have this, then consider $U(t) = e^{At}Y(t)e^{At}$. This will satisfy
$$
U'(t) = e^{At}B(t)e^{At},
$$
so your solution with given initial value $Y(0)$ is going to be
$$
Y(t) = e^{-At}\left(Y(0) + \int\_0^te^{A\tau}B(\tau)e^{A\tau}\ \mathrm{d}\tau\ri... | 4 | https://mathoverflow.net/users/13972 | 172546 | 88,094 |
https://mathoverflow.net/questions/172545 | 3 | Let $F$ be a field, $k$ and $m$ natural numbers with $k \leq m$, and $c \in F^m$.
Is there some name for the set $\mathcal{B}\_c = \{ B \in F^{m \times k}\, | \,\, \exists x \in F^k $s.t. $ Bx = c\}$ of all matrices whose columns can be linearly combined to form $c$?
Furthermore, is there some known structure for t... | https://mathoverflow.net/users/54315 | Space of matrices B for which there is a solution to Bx=c for a given c | If you are willing to replace the vectors $c$, $x$, and $y$ by the spaces $U\_1 = \text{span}(c)$ and $U\_2 = \text{span}(x,y)$, which in some sense doesn't change the problem, then one generalization is the concept of a [partial flag variety](http://en.wikipedia.org/wiki/Generalized_flag_variety#Partial_flag_varieties... | 2 | https://mathoverflow.net/users/5963 | 172548 | 88,095 |
https://mathoverflow.net/questions/172537 | 10 | Separable algebras in modular tensor categories are interesting algebraic structures, which have received significant attention because of their connection to conformal field theories. My understanding is that it is only the Morita class of the algebra which is important for determining the conformal field theory. Also... | https://mathoverflow.net/users/799 | Is every separable algebra in a modular tensor category Morita equivalent to a commutative one? | The answer to your first question is no. The category of modules over a commutative algebra has a tensor category structure such that the forgetful functor is a tensor functor, but the category of modules over noncommutative algebras need not. Since the category of modules is a Morita invariant this gives a negative an... | 9 | https://mathoverflow.net/users/22 | 172551 | 88,096 |
https://mathoverflow.net/questions/172550 | 6 | Let $k$ be an algebraically closed field. To any del Pezzo surface $S$ over $k$ we may associate its graph of lines, which has one vertex for each line and an edge (with multiplicity if required) between two vertices if and only if the corresponding lines insect. For me a "line" means a $(-1)$-curve. It is well-known t... | https://mathoverflow.net/users/5101 | Graphs of lines on del Pezzo surfaces | (Making my comment into an answer...)
Yes, these are the vertex-edge graphs of the $E\_7$ and $E\_8$ polytopes. This is written up in Section 8.2.5 (in the published version) of Dolgachev, *Classical Algebraic Geometry*.
| 10 | https://mathoverflow.net/users/nan | 172555 | 88,100 |
https://mathoverflow.net/questions/172558 | 13 | Suppose we have a set of $n$ points $\{X\_1,X\_2,\dots,X\_n\}$ in the real plane and $\mathcal{A}$ a family of subsets of $\{1,\dots,n\}$.
By a "set of collinearity conditions for $\mathcal{A}$" we mean the conditions of collinearity for every subset $\{X\_i:i\in A\}$ s.t. $A\in\mathcal{A}$. I'm interested in determi... | https://mathoverflow.net/users/51663 | When does a set of collinearity conditions imply collinearity of all of the points? | Yes, if a collection of colinearity conditions is realizable, then they are realizable using only 3 distinct points.
Suppose that $x\_1$, $x\_2$, ..., $x\_n$ in $\mathbb{R}^2$ is a collection of points realizing your colinearity conditions, and not all on a line. Choose two unequal points $x\_i$ and $x\_j$, and let $... | 10 | https://mathoverflow.net/users/297 | 172562 | 88,102 |
https://mathoverflow.net/questions/172560 | 4 | I would like to ask the community for a reference on the following subject: is there some thing as an equivalence between the definitions of Uniform Visibility manifolds and Gromov $\delta$-hyperbolic manifolds? Or even an implication. By visibility I mean:
A Riemannian manifold $(M,g)$ with riemannian distance $d$ i... | https://mathoverflow.net/users/22339 | Visibility spaces and Gromov hyperbolicity | For Cartan-Hadamard manifolds the equivalence of uniform visibility and Gromov hyperbolicity is proved in the following [paper](http://www.ams.org/mathscinet-getitem?mr=1293874) by Kaimanovich, Theorem 2.9.
| 4 | https://mathoverflow.net/users/8588 | 172567 | 88,105 |
https://mathoverflow.net/questions/172461 | 8 | Consider the example of $\mathfrak{g} = sl\_3$. Then
$$
\mathfrak{g} = \mathfrak{n} \oplus \mathfrak{h} \oplus \mathfrak{n}^{-},
$$
where $\mathfrak{n}$ is generated by $E\_{12}, E\_{13}, E\_{23}$, $\mathfrak{h}$ is generated by $E\_{11}-E\_{22}, E\_{22}-E\_{33}$, $\mathfrak{n}^{-}$ is generated by $E\_{21}, E\_{32}, ... | https://mathoverflow.net/users/11877 | PBW basis and canonical basis | It is possible to obtain the canonical basis from the PBW basis, as long as you are working in the quantum group Uq(n). The canonical basis seems to be inherently a quantum phenomenon, so it shouldn't be surprising that we want to compute in the quantum group.
If Eπ is an indexing of the quantum PBW basis, then what ... | 6 | https://mathoverflow.net/users/425 | 172588 | 88,112 |
https://mathoverflow.net/questions/172604 | 3 | Let $R$ be a commutative ring; does every ideal in the $I$-adic completion of $R$
$$ \varprojlim\_i R/I^i $$
arise as the $I$-adic completion of some ideal inside of the original ring $R$?
| https://mathoverflow.net/users/54347 | Ideals in the I-adic completion of a ring | No: take the ring $k[[t\_1, ..., t\_n, ...]]$ of power series of countably many variables, and let $I$ be the ideal generated by $r:=\sum t\_i^i$. Since there is no invertible $u$ such that $r=uP$, where $P$ is a polynomial, the ideal $I$ does not come from any ideal in the polynomial ring.
| 5 | https://mathoverflow.net/users/3377 | 172607 | 88,119 |
https://mathoverflow.net/questions/172559 | 0 | Let us consider a generalized O-U process $X\_t \in L^2[0, 1]$ defined by the following spde:
$dX\_t = \frac{1}{2}\partial\_x^2X\_t + dW\_t, $
$\partial\_x X\_t(0) = \partial\_x X\_t(1) = 0, $
$X\_0 = 0, $
where $W$ is a space-time white noise. Let $\{\varphi\_j\}$ be a CONS of $L^2[0, 1]$. I want to estimate ... | https://mathoverflow.net/users/44590 | On the superior of generalized Ornstein-Uhlenbeck process | You won't have much luck with $\sum\_j\mathbb{E}[\sup\_T\langle X\_t,\varphi\_j\rangle^2]$, since this is not expected to converge. For simplicity say $\varphi\_j$ are the eigenvectors of the Laplace operator, so that $\langle X\_t,\varphi\_j\rangle$ is your $X^j$. Rescale $Y^j\_t = X^j(t/\lambda\_j)/\sqrt{\lambda\_j}$... | 1 | https://mathoverflow.net/users/8966 | 172615 | 88,123 |
https://mathoverflow.net/questions/172599 | 3 | Let $F$ be a free group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq F$, a subgroup of finite index in $F$. Must there be a basis (free generating set) for $H$ containing $A$?
The existence of some basis is guaranteed since $H$ is a free group (Nielsen-Schreier Theorem). Even ... | https://mathoverflow.net/users/38889 | Bases of free groups | As Derek Holt says in comments, the answer to your first question is 'yes'. You can argue topologically.
There is a rose $R$ corresponding to $X$ with $\pi\_1R\cong F$. The subset $A$ defines a connected subrose $R'\subseteq R$. The subgroup $H$ corresponds to a based, finite-sheeted covering space $S\to R$. The asse... | 4 | https://mathoverflow.net/users/1463 | 172618 | 88,126 |
https://mathoverflow.net/questions/172464 | 4 | I'm studying Pin cobordism groups of a point for some low dimensions. I found a general result by Anderson, Brown, Peterson in Theorem 5.1 of their paper "Pin cobordism and related topics" [http://retro.seals.ch/cntmng?pid=comahe-002:1969:44::42](http://retro.seals.ch/cntmng?pid=comahe-002:1969:44%3a:42). Using the Pon... | https://mathoverflow.net/users/45600 | Lower dimensional Pin cobordisms | The paper "Pin cobordism and related topics" [http://retro.seals.ch/cntmng?pid=comahe-002:1969:44::42](http://retro.seals.ch/cntmng?pid=comahe-002:1969:44%3a:42). only gives the homotopy groups of
$RP^{\infty }\wedge X$ where $X$ is either $\Sigma ^nHZ/2$, $bo\langle 8n\rangle$ and
$bo\langle 8n+2\rangle$, noting that... | 3 | https://mathoverflow.net/users/43326 | 172628 | 88,131 |
https://mathoverflow.net/questions/171974 | 5 | In a finite distributive lattice $L$ one has $height(L) = |J(L)|$ i.e. the size of the largest chain equals the number of join-irreducible elements.
Briefly, this follows by arranging the subposet $J(L) = \{x\_1,\dots,x\_n\} \subset L$ so that $j > i \implies x\_j \nleq x\_i$, and then observing that $0 < x\_1 < x\_1... | https://mathoverflow.net/users/5152 | Finite lattices whose number of join-irreducibles does not exceed its height | Join semi-distributive lattices don't have this property because weak order on $S\_n$ is join semi-distributive and doesn't have this property. (Eg, for $n=3$.)
Lattices satisfying the property you are interested in are called "join-extremal" by George Markowsky, in a paper Primes, irreducibles and extremal lattices.... | 4 | https://mathoverflow.net/users/468 | 172634 | 88,135 |
https://mathoverflow.net/questions/172197 | 3 | Let $X = \mathbb{N}^\mathbb{N}$ and $f: X \to \mathbb{R}$ be a function such that
$$|f|\_{var}= \sum\_{n=1}^{\infty} var\_n f < \infty,$$
where $var\_n f = sup\{|f(x)-f(y)|: x,y \in X , x\_k = y\_k, \forall k = 1,\dots, n\}$.
Also, suppose that
$\sum\_{i =1}^\infty \exp(\sup\_{x \in [i]}f(x)) < \infty$, where $[i]... | https://mathoverflow.net/users/52960 | For a summable function, with summable variation, prove that $\sup_{i \in I} \sup_{x \in [i]}|f(x)| \exp((t-1) sup_{x \in [i]}f(x) )$ is bounded | I have just found out that is very simple.
We have that $\sup f - \inf f \le \text{var}\_1f$ and also that $\sup\_{x \in [i]}f(x) \to - \infty $ as $ i \to \infty$.
Then, for $i$ sufficiently large
\begin{align}
\sup\_{x \in [i]}|f(x)| \exp( (t-1) sup\_{x \in [i]}f(x))&
= \sup\_{x \in [i]}(-f(x)) \exp( (t-1) sup\_{... | 2 | https://mathoverflow.net/users/52960 | 172663 | 88,145 |
https://mathoverflow.net/questions/172650 | 4 | If $X$ and $Y$ are two sets linearily ordered by $<$, $X$ is called cofinal in $Y$ if $X \subseteq Y$ and and for every $y \in Y$, there is a $x \in X$ with $y < x$.
If $M$ is some model and $\varphi(x)$ is a formula in the corresponding language, possibly with parameters from $M$, then $\varphi(M)$ denotes the subse... | https://mathoverflow.net/users/nan | Elementary chains of $\aleph_1$-saturated models | The answer is not necessarily.
Let me describe how to make a counterexample.
First, notice that your requirement that $\varphi(M\_i)$ is cofinal in $\varphi(M\_{i+1})$ is equivalent merely to the assertion that $M\_i$ is cofinal in $M\_{i+1}$, which is a special case if one uses a trivial formula. The reason is t... | 2 | https://mathoverflow.net/users/1946 | 172666 | 88,147 |
https://mathoverflow.net/questions/172508 | 8 | Using standard notation, we refer to $H^s(\mathbb R) = W^{s,2}(\mathbb R)$ to be the Sobolev Hilbert spaces. As is often the case, it's natural to then consider properties of functions in $H^s(\mathbb R)$ by looking at how it behaves in the Fourier domain, more specifically, we have that for any $s\in \mathbb{R}$,
$... | https://mathoverflow.net/users/48539 | Equivalent Norms for the Dual of Sobolev / Bessel Spaces | Q1 has been addressed by Mark in the comments (absolute values are missing in your formula), but let me quickly look at this again: Just take Fourier transforms to see that
$$
\|g\|\_{-s}^\*=\sup\_{\|h\|\_s=1} |\langle g,h\rangle | = \sup \left\{ |\langle \widehat{g}, \widehat{h}\rangle | : \|(1+t^2)^{s/2}\widehat{h}\|... | 6 | https://mathoverflow.net/users/48839 | 172669 | 88,148 |
https://mathoverflow.net/questions/172626 | 13 | A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi\_n(f)\colon\pi\_n(X)\to\pi\_n(Y)$ on stable homotopy groups is zero is called a *ghost map*.
Not every ghost map $f$ is the zero object of the abelian group $\operatorname{Hom}\_{SH}(X,Y)$ (more precisely, its image $\bar f\in... | https://mathoverflow.net/users/46913 | Is there a non-zero ghost map between finite suspension spectra? | Following André's suggestion I will promote this comment to an answer.
The OP's question is Freyd's Generating Hypothesis, which has been an open question for nearly 50 years. It is very hard to test, because there is no example of a nontrivial finite spectrum $X$ where all the groups $\pi\_k(X)$ are known. Devinatz ... | 17 | https://mathoverflow.net/users/10366 | 172674 | 88,151 |
https://mathoverflow.net/questions/172675 | 4 | For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2\_{p,q}=H\_p(B,h\_q(F))\Rightarrow h\_{p+q}(E)\end{equation}For theories $h$ (such as ordinary homology or oriented bordism), are there results about when the... | https://mathoverflow.net/users/51107 | stability results for the Atiyah-Hirzebruch spectral sequence | The path-loop fibration for the sphere $S^n$ is:
$$\Omega S^n \to PS^n \to S^n$$
and $PS^n$ is contractible. In singular homology, the Atiyah-Hirzebruch-Serre spectral sequence (in this case, usually just known as the Serre spectral sequence) collapses at $E^n$, and not before. That is: there is a nontrivial diffe... | 8 | https://mathoverflow.net/users/4649 | 172678 | 88,153 |
https://mathoverflow.net/questions/172640 | 4 | By a rational homology 3-sphere, I mean a compact oriented manifold three-manifold $Y$ with $H\_1(Y)$ finite. My question is whether there exists a reasonable classification of such manifolds such that $\pi\_1(Y)$ has a minimal presentation with two generators? How about three generators?
Here are the examples I kno... | https://mathoverflow.net/users/6986 | Question about the fundamental group of rational homology 3-spheres | As indicated in the comments, the greatest class of rank 2 3-manifolds (including rational homology spheres) are the genus 2 manifolds, which as indicated by Ruberman are double branched covers over links (coming from the hyperelliptic involution of the genus 2 surface which extends over both handlebodies). This includ... | 5 | https://mathoverflow.net/users/1345 | 172680 | 88,154 |
https://mathoverflow.net/questions/172690 | 50 | How and when did the word "normal" acquire [this meaning](http://en.wikipedia.org/wiki/Normal_(geometry))? When I first thought of this, I couldn't really come up with any explanation that wasn't complete speculation -- pretty much all I was able to see was that it isn't any stranger than "right" in "right angle" -- th... | https://mathoverflow.net/users/20803 | How did "normal" come to mean "perpendicular"? | *normalis* already meant *right-angled* in classical Latin; for example, *angulus normalis* appears in the first century text *De institutione oratoria* (volume XI, paragraph 3.141) by [Marcus Fabius Quintilianus.](https://en.wikipedia.org/wiki/Quintilian)
In a [commentary](http://catalog.hathitrust.org/Record/00926... | 78 | https://mathoverflow.net/users/11260 | 172694 | 88,162 |
https://mathoverflow.net/questions/172695 | 2 | The question is in the title, here is my motivation:
$\require{AMScd}$Let $(\mathcal C,\otimes,I)$ be a monoidal symmetric closed category. Then, the tensor product commutes with colimits, and if $\mathcal C$ has infinite coproducts, the object $N:=\displaystyle\coprod\_{n\in\mathbb N}I$ has the following properties ... | https://mathoverflow.net/users/25887 | Example(s) of monoidal symmetric closed category with NNO without infinite coproducts? | The effective topos is an example of a (locally!) cartesian closed category with NNO where infinite coproducts do *not* exist. Indeed, the main feature of the NNO $N$ in the effective topos is that the endomorphisms of $N$ are precisely the computable functions. As such, $N$ cannot be (isomorphic to) $\coprod\_{n \in \... | 7 | https://mathoverflow.net/users/11640 | 172698 | 88,163 |
https://mathoverflow.net/questions/172707 | 7 | Let $(I,<)$ be a directed, partially ordered set. Consider an inverse system $(S\_i)\_{i \in I}$ of finite sets, i. e. a functor $S:I^{op}\to \mathbf{FinSet}$. What is the maximum possible cardinality of the inverse limit $A=\varprojlim S$ of this system? It is obvious that $|A|=2^{\aleph\_0}$ can be achieved (for exam... | https://mathoverflow.net/users/37059 | Maximum cardinality of a filtered limit of finite sets | You can get arbitrarily large cardinalities. For instance, let $X$ be any set and consider the poset $I$ of finite partitions of $X$, ordered by refinement. There is a "tautological" filtered system of finite sets indexed by $I$, and $X$ naturally maps to it. Clearly the map from $X$ to the limit of this system is inje... | 9 | https://mathoverflow.net/users/75 | 172712 | 88,167 |
https://mathoverflow.net/questions/172706 | 10 | From Fermat's little theorem we know that every odd prime $p$ divides $2^a-1$ with $a=p-1$.
>
> Is it possible to prove that there are infinitely many primes not
> dividing $2^a+2^b-1$?
>
>
>
(With $2^a,2^b$ being incoguent modulo $p$)
**1.** Obviously, If $2$ is not a quadratic residue modulo $p$ then we... | https://mathoverflow.net/users/38851 | Primes dividing $2^a+2^b-1$ | This is a heuristic which suggests that the problem is probably quite hard. We have that $p | 2^{a} + 2^{b} - 1$ if and only if there is some integer $k$, $1 \leq k \leq p-1$ with $k \ne \frac{p+1}{2}$ for which $2^{a} \equiv k \pmod{p}$ and $2^{b} \equiv 1-k \pmod{p}$ are both solvable. If $r$ is the order of $2$ modu... | 10 | https://mathoverflow.net/users/48142 | 172714 | 88,168 |
https://mathoverflow.net/questions/172709 | 3 | Let $B$ be a connected pointed CW complex, let $E$ and $E'$ be two CW complexes and let $f\colon E\to B$ and $f'\colon E'\to B$ be two Serre fibrations. Let $g\colon E\to E'$ be a continuous map such that
\begin{eqnarray}
E &\xrightarrow{f}& B\\
\small{g}\downarrow & &\|\\
E'&\xrightarrow{f'}& B
\end{eqnarray}
comm... | https://mathoverflow.net/users/46913 | Equivalence of the total spaces of two Serre fibrations with equivalent fibers | Note that $\pi \_1(B)$ acts on $\pi \_0(F)$. So the ending of the long exact sequence for the fibration
$$\cdots\rightarrow \pi \_1(B)\rightarrow \pi \_0(F)\rightarrow \pi \_0(E)$$
is not just an exact sequence of sets, but it also shows that $\pi \_0(E)$ is the quotient of the set $\pi \_0(F)$ by the group $\pi \_1(B... | 8 | https://mathoverflow.net/users/43326 | 172718 | 88,170 |
https://mathoverflow.net/questions/172719 | 2 | Let $X \subseteq \mathbb{P}^n$ be a projective variety. I would like to have a morphism $f: \tilde{X}\to X \subseteq \mathbb{P}^n$ where $f$ is finite and birational, $f^\* \mathcal{O}\_{\mathbb{P}^n}(1)$ is a very ample line bundle and $\tilde{X}$ is a smooth projective variety. Under which conditions does such a map ... | https://mathoverflow.net/users/36563 | Some Kind of Resolution of Singularites | In other words, you want the normalization of $X$ to be smooth, plus a condition on the embedding: the pull back of $\mathcal{O}\_X(1)$ to $\tilde{X} $ should be very ample. The latter property depends on the embedding and is not always true, even for curves: see [this post](https://mathoverflow.net/questions/163802/do... | 5 | https://mathoverflow.net/users/40297 | 172721 | 88,172 |
https://mathoverflow.net/questions/172716 | 5 | Let $p$ be a prime number, $\mathbb{Q}\_p$ the field of $p$-adic numbers, $G\_p$ the absolute Galois group of $\mathbb{Q}\_p$ and $V$ a finite dimensional vector space over $\mathbb{Q}\_p$. Assume we are given a linear representation $\rho : G\_p \to GL(V)$.
Can we find a closed subgroup of finite index $H$ in $G\_{p... | https://mathoverflow.net/users/38416 | Absolutely irreducible p-adic representation of the absolute Galois group of Q_p | The answer is no. To prove it, it suffices to show that there exists a representation $\rho: G\_p \rightarrow Gl(V)$ of open image. There are several ways to do this, one is to use Chenevier prop. 1.8 in "Quelques courbes de Hecke se plongent dans l'espace de Colmez", which says that for $p=2,3,5,7$ and
$\rho: G\_{\mat... | 5 | https://mathoverflow.net/users/9317 | 172731 | 88,175 |
https://mathoverflow.net/questions/172388 | 11 | Let $X$ be a projective complex manifold of dimension $n$.
Are torsion cohomology classes in $H^{2n-2}(X,\mathbb{Z})$ algebraic?
(We may assume, without loss of generality, that $n=3$, because of the Lefschetz
hyperplane theorem.) I know that torsion classes (of even codimension) aren't always algebraic;
the first co... | https://mathoverflow.net/users/9833 | A special case of the integer Hodge conjecture | This [2013 paper](https://www.math.ucla.edu/~totaro/papers/public_html/integral.pdf) of Totaro says that it is an open question. But the integral Hodge conjecture is known to fail for dimension 1, just not via torsion.
| 3 | https://mathoverflow.net/users/4639 | 172758 | 88,186 |
https://mathoverflow.net/questions/172735 | 7 | If $A \to M$ is a Lie algebroid over a smooth manifold $M$ then a representation of $A$ is a vector bundle $E \to M$ with a flat $A$-connection
$$
\nabla : \Gamma(E) \to \Gamma(E\otimes A^\*).
$$
If $G$ is a Lie groupoid over $M$ then a representation is a vector bundle $E\to M$ and a Lie groupoid homomorphism $G \to G... | https://mathoverflow.net/users/4622 | Integrating representations of Lie algebroids | The usual condition that one uses to ensure that all representations of $A$ integrate to representations of $G$ is that $G$ be "source 1-connected", meaning that all of the fibres of the source map $s : G \to M$ are connected and simply connected.
That the source 1-connected condition is sufficient is a consequence o... | 6 | https://mathoverflow.net/users/39915 | 172769 | 88,189 |
https://mathoverflow.net/questions/172739 | 6 | What is the definition of ambiguity of context-sensitive
grammar?This is relevant to the definition of inherent ambiguity of
context-sensitive language.And any proof for the inherent ambiguity of the
language $L = {a^ib^ic^id^je^jf^j \bigcap a^ib^jc^id^je^if^j} $ or
$a^nb^nc^nd^ne^nf^n$?
Since another question about ... | https://mathoverflow.net/users/14024 | Inherent ambiguity of the context-sensitive language $L = {a^ib^ic^id^je^jf^j \bigcap a^ib^jc^id^je^if^j} $ or $a^nb^nc^nd^ne^nf^n$ | Let me define a context-sensitive language *strongly unambiguous* if it is recognizable by a context-sensitive grammar such that every string in the language has a *unique* derivation. (Note that in the CFL world, this condition is much stronger than plain unambiguity.)
I don’t know if there *is* a sensible well-beha... | 3 | https://mathoverflow.net/users/12705 | 172786 | 88,196 |
https://mathoverflow.net/questions/172764 | 2 | The Riemann mapping theorem (cf e.g. <http://en.wikipedia.org/wiki/Riemann_mapping_theorem>) essentially guarantees the existence of a biholomorphic mapping of a simply connected, open subset of the complex plane onto the unit disk.
**Questions:**
* Are there any results known about length-preserving mappings from... | https://mathoverflow.net/users/31310 | Length-preserving Analogue of Riemann's Mapping Theorem | Any length preserving map from one plane domain to another must be also conformal.
Because the angle of a small triangle can be found if you know the sides.
This observation solves all your questions, because if a conformal map is also
length preserving, than the derivative must have constant absolute value one, thus i... | 5 | https://mathoverflow.net/users/25510 | 172792 | 88,199 |
https://mathoverflow.net/questions/172772 | 7 | Let $\mathcal M$ be a compact connected real-analytic manifold. It is well known that every continuous map $f\colon\mathcal M\to\mathbb S^1$ is homotopic to a smooth map. My question is the following. Are there any sufficient conditions on $\mathcal M$ that would guarantee the existence of an analytic map $\mathcal M\t... | https://mathoverflow.net/users/50457 | Analytic maps in homotopy classes | This is always true. The Morrey-Grauert theorem says that $M$ has a real-analytic embedding in Euclidean space, so real-analytic functions $M\to\mathbb{R}$ separate points, so they are dense in the algebra of all continuous function (by the Stone-Weierstrass theorem). Thus, given a map $f:M\to S^1\subset\mathbb{C}$ we ... | 10 | https://mathoverflow.net/users/10366 | 172794 | 88,200 |
https://mathoverflow.net/questions/172498 | 3 | As far as I can tell, there are a number of major types of quantum field theory. For example,
Constructive QFT, which has two major branches (Algebraic/Axiomatic QFT and Functorial QFT).
Topological QFT, which has three major subsets (Cohomological QFT, Homotopy QFT, and Topological Conformal Field Theory).
Confo... | https://mathoverflow.net/users/51216 | Comparison of Different Types of QFT | There is quite a bit of confusion in the statement "Constructive QFT, which has two major branches (Algebraic/Axiomatic QFT and Functorial QFT)."
Axiomatic QFT provides mathematical definitions for the concept of a QFT, for instance the Garding-Wightman axioms or the Haag-Kastler axioms. Here the word "axiom" can be ... | 3 | https://mathoverflow.net/users/7410 | 172795 | 88,201 |
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