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https://mathoverflow.net/questions/345091 | 2 | I consider the property for a group $G$, that every time you take an element $g$ of prime order in $G$, then there is a complement $H$ to $\langle g\rangle$ in $G$ in the sense that $G=\langle g\rangle H$ and $\langle g\rangle\cap H=1$.
Let $G$ be the cartesian product (= unrestricted direct product) of infinitely ma... | https://mathoverflow.net/users/45296 | A question about complements in a group | It's true (I initially claimed the contrary): the case of elements of order $2$ is easy and the case of elements of order $3$ is a little more tricky.
Write $G=S\_3^I$. Let $G\_3=C\_3^I$ be the 3-Sylow subgroup in $G$. Fix a 2-Sylow subgroup $C\_2$ in $G\_3$ and write $G\_2=C\_2^I$. We consider a quotient $Q=G/N$ of ... | 4 | https://mathoverflow.net/users/14094 | 345106 | 146,298 |
https://mathoverflow.net/questions/344962 | 5 | Let $(S,\eta,s)$ be spectrum of a discrete valuation ring $R$. Let $E$ be an elliptic curve over $\eta$. Let $\mathcal{E}$ be the Neron model of $E$.
Is there a concrete example of an $E$-torsor (smooth genus one curve over $K$) that does not extends to an $\mathcal{E}$-torsor over $S$?
(I am not even sure in the c... | https://mathoverflow.net/users/nan | Does torsor of an elliptic curve extend to torsor of its Neron model? | Any non-trivial $E$-torsor over $\mathbb{Q}\_p$ will give you an example. Let me be more precise.
Let $E$ be an elliptic curve over $K=\mathrm{Frac}(R)$. Assume that $E$ has good reduction over $R$. Let $\mathcal{E}$ be its Neron model. If $X$ is an $E$-torsor over $K$ which extends to an $\mathcal{E}$-torsor $\mathc... | 4 | https://mathoverflow.net/users/4333 | 345111 | 146,299 |
https://mathoverflow.net/questions/345071 | 5 | Mather's cube theorem for the category of topological spaces says that given a homotopy-commutative cube:
>
> If one pair of opposite faces are homotopy pushouts and the two
> remaining faces adjecent the source vertex are homotopy pullbacks,
> then the final two faces are also homotopy pullbacks.
>
>
>
What... | https://mathoverflow.net/users/69037 | Geometric intuition for Mather's cube theorem | I'm not sure what counts as an intuitive explanation, but this is sort of how I think about it.
Say that $B=B\_1\cup B\_2$ and $B\_0=B\_1\cap B\_2$. This is the second pushout square.
Now let $E\_1$ be a bundle over $B\_1$ and let $E\_2$ be a bundle over $B\_2$, and suppose that the restriction to $B\_0$ is the sam... | 4 | https://mathoverflow.net/users/6666 | 345113 | 146,300 |
https://mathoverflow.net/questions/345119 | 12 | There seems to be a general sentiment that triangulated categories are not the "correct" notion to use because mapping cones of morphisms are unique, but only up to non-unique isomorphism.
Does anyone know a concrete example of a "proof" or application that we would like to make with them, but which gets hindered bec... | https://mathoverflow.net/users/126543 | A concrete example of the deficiency of triangulated categories? | Since I have already given a similar answer recently, I don't want to be branded as the "anti-triangular" guy: the formalism of triangulated categories can be useful in certain settings. That said they do have some definite shortcomings, and I would like to list a few in this answer. As a homotopy theorist, most of my ... | 15 | https://mathoverflow.net/users/43054 | 345128 | 146,304 |
https://mathoverflow.net/questions/345130 | 2 | It is a well-known fact that closed convex sets in Banach spaces are weakly closed. The common proof is based on the Hahn-Banach theorem that uses the axiom of choice. Is there any proof of this fact that avoids the axiom of choice and uses only ZF?
| https://mathoverflow.net/users/147604 | Are closed convex subsets of a Banach space weakly closed without the axiom of choice? | Of course not.
The Hahn–Banach is equivalent to the assertion that $X^\*$ is nontrivial for any nontrivial Banach space (or normed space, if you prefer).
This means that if HB fails, there is a nontrivial Banach space $X$ whose weak topology is indiscrete, and in particular no set (other than $X$ and $\varnothing$)... | 3 | https://mathoverflow.net/users/7206 | 345132 | 146,305 |
https://mathoverflow.net/questions/345104 | 6 | Let $A$ be a commutative ring, $a \subset A$ be an ideal. For $A$-module $M$ let $S \subset A$ be the set of elements, which are invertible in $M$, so $M$ is actually a $S^{-1}A$-module. It is not hard to show, that
>
> If $S\cap a \neq \emptyset$, then $\operatorname{Tor}\_\*^A(A/a, M) = 0$.
>
>
>
Under wha... | https://mathoverflow.net/users/140292 | Tor functor and invertible elements | Without any finiteness assumptions on $M$, the converse fails already for $A=k[x, y]$ ($k$ is a field).
Take $M=k[x,y^{\pm 1}]\oplus k[x^{\pm 1},y]$ and $a=(x,y)$. The module $M$ is flat over $A$ so $M\otimes^{\mathbb{L}}\_A A/a=M\otimes\_{A}A/a=M/aM=0$. However $S=k\setminus\{0\}$ because the sets of elements inver... | 3 | https://mathoverflow.net/users/39304 | 345158 | 146,312 |
https://mathoverflow.net/questions/345131 | 4 | Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$.
>
> Does there always exist a continuous map $\lambda:X\to S\_H$, where $S\_H$ is the unit sphere in the Hilbert space, such that $\lambda(... | https://mathoverflow.net/users/53155 | A map into a Hilbert space with prescribed orthogonality | Local compactness is not required but separability (and metrizability) seem essential for the construction below. Also, the function $L$ itself is more of a red herring: all we really need is the symmetric closed set $S=\{(x,y):L(x,y)=0\}$ in $X\times X$ disjoint with the diagonal. The condition is that $\lambda(x)\per... | 6 | https://mathoverflow.net/users/1131 | 345168 | 146,319 |
https://mathoverflow.net/questions/344996 | 2 | I am studying the paper M. Ram Murty, V. Kumar Murty: Mean values of derivatives of modular $L$-series, Ann. of Math. (2) 133 (1991), no. 3, 447-475.
Let $L(s)=\sum\_{n=1}^{\infty} \frac{a(m)}{m^s}$ be the $L$-function of a modular elliptic curve with conductor $N$. We decompose $m=m\_1m\_2$ such that $m\_1$ is the p... | https://mathoverflow.net/users/123157 | Upper bound of summation $\sum_{m < \frac{1}{2}X} \frac{|a(m_1m_2^2)|}{m_1m_2^2} \log\frac{X}{m}$ | Here is a proof for non CM elliptic curves. In this case, the Dirichlet series
$$\sum\_{n=1}^\infty\frac{a(n^2)^2}{n^{2+s}}$$
is holomorphic in the closed half-plane $\Re(s)\geq 1$ with a simple pole at $s=1$. This follows, for example, from Lemma 3 in Moreno-Shahidi: The fourth moment of Ramanujan $\tau$-function (198... | 3 | https://mathoverflow.net/users/11919 | 345178 | 146,321 |
https://mathoverflow.net/questions/345120 | 8 | Let $M$ be a closed, smooth manifold and let $PM$ be the space of unbased piecewise smooth paths $[0,1] \to M$. Then restricting a path to its boundary gives a map
$$
PM \to M \times M .
$$
**Question** is this map a fiber bundle?
Andrew Stacey showed that a related map, the free smooth loop fibration $LM \to M$, i... | https://mathoverflow.net/users/8032 | The free smooth path space on a manifold | Yes.
The technical details are in *[Yet More Smooth Mapping Spaces and Their Smoothly Local Properties](https://arxiv.org/abs/1301.5493)*, specifically in Section 5 which establishes that smooth manifolds are *smoothly locally deformable* which means that there are lots of diffeomorphisms flying around.
Interestingly... | 8 | https://mathoverflow.net/users/45 | 345179 | 146,322 |
https://mathoverflow.net/questions/345180 | 2 | Given a fundamental representation $V(\nu\_k)$ of a semisimple Lie algebra $\frak{g}$, and a general irreducible finite-dimensional representation $V$, is it ever possible that the tensor product $V \otimes V(\nu\_k)$ can contain a copy of $V$?
| https://mathoverflow.net/users/147728 | Tensoring $\frak{g}$-modules by fundamental representations | Yes. First of all, if we take a fixed irrep $ V(\mu) $, then for big enough $ \lambda$, we know that $ V(\lambda + \nu) $ will appear in $ V(\lambda) \otimes V(\mu) $ with multiplicity equal to the weight multiplicity of $ \nu $ in $V(\mu) $.
So your question is equivalent to: is $ 0 $ a weight of a fundamental repre... | 6 | https://mathoverflow.net/users/438 | 345181 | 146,323 |
https://mathoverflow.net/questions/345045 | 7 | Let $W \in \mathbb{C}[x\_1, \dots, x\_n]=R$ be a polynomial with an isolated critical point at the origin. A Matrix Factorizations for $W$ consists a $\mathbb{Z}/2\mathbb{Z}$-graded finite free $R$-module $E$ with an odd differential $d:E \to E$ satisfying $d^2 = W \cdot Id$ instead of the usual square zero condition. ... | https://mathoverflow.net/users/184 | Derived Category of the derived critical locus, is it the category of Matrix Factorizations? | These are indeed related. The first thing to know is that they both ``live'' (i.e., sheafify) over the critical locus (this is not saying much if you assume $W$ has isolated critical points, but interesting in more general cases). This is clear for $\operatorname{Coh}(\operatorname{Crit}(W))$. For matrix factorizations... | 14 | https://mathoverflow.net/users/145919 | 345183 | 146,324 |
https://mathoverflow.net/questions/222326 | 25 | It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas Weil cohomology theories should take values in characteristic $0$. It turns out that de Rham cohomology picks up torsion fr... | https://mathoverflow.net/users/82179 | Reference for de Rham cohomology in positive characteristic | Prompted by user Yai0Phah's comment, let me answer my question: I did indeed find a reference. It turns out the correct place to look for these things is any book on crystalline cohomology, because
$$H^i\_{\operatorname{dR}}(X) = H^i\_{\operatorname{cris}}(X/k).$$
One canonical reference for Poincaré duality for smooth... | 4 | https://mathoverflow.net/users/82179 | 345189 | 146,326 |
https://mathoverflow.net/questions/345028 | 10 | Do colimits in the category of (not necessarily locally convex) topological vector spaces (over R, C, respectively) exist in general?
If no, is there a well-known condition of when they exist?
If yes, how can I describe the topology of the colimit?
(The example in my mind has following properties. First, it is a ... | https://mathoverflow.net/users/78655 | Colimits in the category of (not necessarily locally convex) topological vector spaces | The Springer Lecture Notes 639 *Topological Vector Spaces* of Adasch, Ernst, and Keim contain in § 4 a more or less explicit construction of inductive (=co-) limits in the category of topological vector spaces based on the notion of a *string*: A sequence $(U\_n)\_{n\in\mathbb N}$ of balanced and absorbing sets such th... | 9 | https://mathoverflow.net/users/21051 | 345202 | 146,330 |
https://mathoverflow.net/questions/344946 | 4 | I am wondering if there is any computation of stable homotopy groups of $\mathbb{R}P^{\infty}\wedge \mathbb{R}P^{\infty}$ in low dimensions? I would be very grateful for any reference.
| https://mathoverflow.net/users/51223 | Tables for stable homotopy groups of $\mathbb{R}P^{\infty} \wedge \mathbb{R}P^{\infty}$ | Bob Bruner and Christian Nassau both have code that can compute the Adams charts efficiently. For example, Bob has a chart for $\mathbb{R}P^2\wedge\mathbb{R}P^2$ at <http://www.rrb.wayne.edu/cohom/index.html>, and he might be able to do $\mathbb{R}P^\infty\wedge\mathbb{R}P^\infty$ with little effort if you asked him ni... | 6 | https://mathoverflow.net/users/10366 | 345207 | 146,333 |
https://mathoverflow.net/questions/345112 | 2 | Let $B$ be a Brownian motion. Definining a *pathwise* stochastic integral $I(f):=\int f~dB$ for certain classes of deterministic functions is straightforward: For instance if $f=\sum\_ic\_i1\{[t\_i,t\_{i+1})\}$ is a compactly supported simple function, then
$$I(f):=\sum\_ic\_i(B(t\_{i+1})-B(t\_i)).$$
More generally if ... | https://mathoverflow.net/users/50406 | Pathwise stochastic integral as a linear operator on continuous functions | You can do the same for less regular functions, for example for functions with finite $p$-variation for some fixed $p \in [1,2)$ or functions in $\mathcal{C}^\alpha$ for some fixed $\alpha > 1/2$. You cannot go much below that in the sense that there exists *no* Banach space $\mathcal{B}$ with the following properties:... | 4 | https://mathoverflow.net/users/38566 | 345209 | 146,334 |
https://mathoverflow.net/questions/344950 | 2 | Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $n$ simple modules.
Let $d\_1<d\_2<...<d\_r$ be the sequence of projective dimension of simple $A$-modules in increasing order. Define $\phi\_A:= max \{ d\_{i+1}-d\_i | 1 \leq i \leq r-1 \}$.
>
> Question: Is there a class of examples whe... | https://mathoverflow.net/users/61949 | Gaps in the projective dimensions of simple modules | For $n>0$, let $A\_n=KQ\_n/I\_n$ as follows: $Q\_n$ is the quiver with vertex set $\{1,2\}$ and arrows $\alpha\_i\colon 1\to 2$ and $\beta\_i\colon 2\to 1$ for $1\leq i\leq n$, and $I\_n$ is generated by $\beta\_j\alpha\_i$ for $i\leq j$ and $\alpha\_j\beta\_i$ for $i\leq j-1$ (reading composition from right-to-left). ... | 4 | https://mathoverflow.net/users/21483 | 345221 | 146,337 |
https://mathoverflow.net/questions/345109 | 9 | In his ICM Adress at Nice (Proceedings of the International Congress of Mathematicians Nice, September, 1970, Gauthier-Villars, editeur, Paris 6 e ,1971, Volume 2, pp. 133-163.),
Robion Kirby adresses the problem, whether the fourth topological Spin bordism group is $\mathbb{Z}$ or $\mathbb{Z}\oplus \mathbb{Z}/2$.
... | https://mathoverflow.net/users/21985 | Topological Spin manifolds in dimension 4 | The map $\Omega\_4^{\text{Spin}} \to \Omega\_4^{\text{SpinTop}}$ is taken isomorphically by the signature to the inclusion $16\Bbb Z \hookrightarrow 8\Bbb Z$, so that the groups are abstractly isomorphic but that the natural map is not an isomorphism. This is obtained as Theorem 13.1 on page 325 of Kirby and Siebenmann... | 8 | https://mathoverflow.net/users/40804 | 345226 | 146,338 |
https://mathoverflow.net/questions/344864 | 5 | Let $M^3$ be a compact orientable 3-manifold. Then $TM$ is trivial and let's go ahead and fix a trivialization $\tau : M \times \mathbb{R}^3 \to TM$. Then given a map $g : (M, \partial M) \to (SO(3), 1)$ we can consider the new trivialization $g \cdot \tau$ that is given by $g \cdot \tau (p,v) = \tau(p,g(p)(v))$.
The... | https://mathoverflow.net/users/99414 | Group of parallelizations of $M^3$ finitely generated? | I believe that for any finite $n$-complex $X$, the group $$[X, SO(n)]\_\*$$ is finitely generated. I will follow Igor Belegradek's approach. In fact I only think $H^\*(X;\Bbb Z)$ finitely generated and maybe $\pi\_1$ finitely generated is necessary.
---
There is a fibration $B\text{Spin}(n) \to BSO(n) \to B^2(\Bb... | 2 | https://mathoverflow.net/users/40804 | 345242 | 146,341 |
https://mathoverflow.net/questions/345198 | 4 | Let $P\_{\alpha}$ be the principal series representation of $GL\_n(\mathbb{F}\_q)$, where $\alpha = ( \alpha\_1, \alpha\_2, \cdots, \alpha\_n)$ and $\alpha\_i : \mathbb{F}\_q^\* \rightarrow \mathbb{C}^\*$.
The character value of $P\_{\alpha}$ on the unipotent conjugacy class ( Jordan form has eigenvalues $x$) are of... | https://mathoverflow.net/users/140407 | Character values of principal series representations of $GL_n(\mathbb{F}_q)$ | By the standard formula for the character of an induced representation, the value f(q)/|B| is equal to the number of g for which gug-1∈B (here u is my unipotent element).
Rewriting this condition as u∈g-1Bg, we see that f(q) is equal to the number of Borel subgroups containing u, i.e. the number of points in the Spri... | 3 | https://mathoverflow.net/users/425 | 345262 | 146,351 |
https://mathoverflow.net/questions/345274 | 6 | I would like to know the integral cohomology of $SU(\infty)/SO(\infty)$ (to degree 5 or 6, say.)
[Mimura-Toda](https://mathscinet.ams.org/mathscinet-getitem?mr=1122592) says $H^\*(SU/SO,\mathbb{Z}/2\mathbb{Z})=\wedge[w\_2,w\_3,\ldots]$ where $w\_i$ is a pullback of Stiefel-Whitney classes via $SU/SO\to BSO$.
I'd l... | https://mathoverflow.net/users/5420 | Integral (co)homology of $SU/SO$ | First, I'll mostly talk about $U/O$ rather than $SU/SO$ because $U/O$ can be descibed as $B(\mathbb{Z}\times BO)$ or as the $8k-1$'th space in the $\Omega$-spectrum for $KO$. This gives $\pi\_0(U/O)=0$ and $\pi\_1(U/O)=\pi\_0(KO)=\mathbb{Z}$. From the Hurewicz and universal coefficient theorems this gives $H^1(U/O)=\te... | 7 | https://mathoverflow.net/users/10366 | 345290 | 146,359 |
https://mathoverflow.net/questions/345294 | 3 | Let $E \to X$ be a homomorphic vector bundle over a projective variety $X$. Does $\mathbb{P}(E)$ always have holomorphic sections? If not what is the obstruction?
| https://mathoverflow.net/users/15197 | Does the projectivization of a vector bundle have sections? | I'll assume you're asking about sections of $PE$. The bundle $E$ has Chern classes $c\_i(E)\in H^{2i}(X)$ and thus a Chern polynomial $f\_E(t)=\sum\_{i=0}^nc\_{n-i}(E)t^i$, where $n=\dim(E)$. A section of $PE$ corresponds to a line bundle $L\leq E$ and thus a factorisation $f\_E(t)=f\_L(t)f\_{E/L}(t)$, and thus a root ... | 8 | https://mathoverflow.net/users/10366 | 345295 | 146,360 |
https://mathoverflow.net/questions/345244 | 10 | This question is suggested by some results in a paper I am writing. I would like to write it down there but want to make sure that it is not known or at least MO-hard.
Freiman's inequality states that for a set $A$ of vectors that span $\mathbb R^d$, we have:
$$|A+A| \geq (d+1)|A| -\binom{d+1}{2} $$
My question is... | https://mathoverflow.net/users/2083 | Freiman inequality for projective space? | As stated, the answer is no, Freiman's inequality no longer holds. The counter example is $A$ being the vertices of an equilateral triangle on the unit circle. I found this by looking at the proof of Freiman's inequality (given as Lemma 5.13 of Tao-Vu book on Additive Combinatorics) and see where the proof fails.
In... | 5 | https://mathoverflow.net/users/2083 | 345297 | 146,362 |
https://mathoverflow.net/questions/345299 | 4 | In the correspondence between projective and Kaehler geometry an ample line bundle corresponds to a positive line bundle, where the latter requires that the curvature of the Chern connection is a positive $(1,1)$-form. A very ample is a strengthening of ample (no need to take powers). Is there a corresponding Kaehler n... | https://mathoverflow.net/users/126606 | Kaehler analogue of very ample line bundle | J. P. Demailly, Multiplier ideal sheaves and analytic methods, p. 26, (3.15) A holomorphic line bundle $F$ over a compact complex manifold $X$ is
a) *very ample* if the map $\Phi\_{|F|}\colon X \to \mathbb{P}^{N−1}$ associated to the complete linear system $|F| = \mathbb{P}(H^0(X,F))$ is a regular embedding (by this... | 4 | https://mathoverflow.net/users/13268 | 345301 | 146,363 |
https://mathoverflow.net/questions/345300 | 2 | Let $K$ be a compact neighborhood in $\mathbb R^n$, $Z=X+Y$ are (non-vanishing if necessary) smooth vector fields on $K$. Denote by $e^{sZ}p$ an integral curve of $Z$ with initial point $p=e^{0Z}p\in K$ and a terminal point $q=e^{tZ}p\in K$. Then for any $\varepsilon>0$ there exists a large enough natural number $N$ su... | https://mathoverflow.net/users/13842 | Reference request: Lie product formula for vector fields | Abraham and Marsden, **Foundations of Mechanics**, p. 78, Corollary 2.1.27; Let $X, Y$ be vector fields on a manifold $M$ with flows $F\_t$, $G\_t$. Let $H\_t$ be the flow of $X+Y$. Then
$$
H\_t(x) =\lim\_{n \to \infty} (F\_{t/n} \circ G\_{t/n})^n
$$
Each side is defined if and only the other is.
| 4 | https://mathoverflow.net/users/13268 | 345302 | 146,364 |
https://mathoverflow.net/questions/345292 | 7 | This question is about the Remark on the top of page 22 of Serre's Topics in Galois Theory, available here :
<http://www.ms.uky.edu/~sohum/ma561/notes/workspace/books/serre_galois_theory.pdf>
My question is as follows.
>
>
> >
> > Let $L/K$ be a finite separable extension of fields. Assume that $L$ is Hilber... | https://mathoverflow.net/users/135215 | Subfields of Hilbertian fields | The maximal (pro-)solvable extension $L$ of $\mathbb{Q}$ is not Hilbertian, but every proper finite extension of $L$ is. See Fried-Jarden (third edition, 2008), Example 13.9.5.
| 9 | https://mathoverflow.net/users/7666 | 345303 | 146,365 |
https://mathoverflow.net/questions/339661 | 4 | I am reposting [this question](https://math.stackexchange.com/questions/3331534/are-intrinsic-volumes-defined-for-non-polyconvex-non-compact-sets) I asked and bountied on Math SE, which has been upvoted but not answered or commented on.
The [intrinsic volumes](https://en.wikipedia.org/wiki/Mixed_volume#Intrinsic_vol... | https://mathoverflow.net/users/140709 | Intrinsic volumes of non-polyconvex, non-compact sets | The largest level of generality for which intrinsic volumes can be defined is a difficult question. One large class of compact sets for which they can be defined is the class of compact sets admitting a normal cycle. How large this class is is not very well understood, but it contains smooth (or more generally positive... | 3 | https://mathoverflow.net/users/112954 | 345325 | 146,371 |
https://mathoverflow.net/questions/345330 | 0 | Let me first explain the problem using an analogy.
Let's say you have $N$ doors and $M$ keys. Each door can be opened with a combination of keys, each combination is also unique (i.e. there aren't be two doors that are opened with same combination).
For example, to open Door $1$ you need $3$ keys: $A, B$ and $C$. ... | https://mathoverflow.net/users/148182 | Im looking for an algorithm that can solve or approximate the solution to a problem | You can solve this problem via integer linear programming. Let binary decision variable $x\_i$ indicate whether you can open door $i$, and let binary decision variable $y\_j$ indicate whether you select key $j$. The problem is to maximize $\sum\_{i=1}^N x\_i$ subject to
\begin{align}
\sum\_{j=1}^M y\_j &= k \\
x\_i &\l... | 3 | https://mathoverflow.net/users/141766 | 345331 | 146,373 |
https://mathoverflow.net/questions/345275 | 51 | Lebesgue published his celebrated integral in 1901-1902. Poincaré passed away in 1912, at full mathematical power.
Of course, Lebesgue and Poincaré knew each other, they even met on several occasions and shared a common close friend, Émile Borel.
However, it seems Lebesgue never wrote to Poincaré and, according to ... | https://mathoverflow.net/users/88057 | A historical mystery : Poincaré’s silence on Lebesgue integral and measure theory? | It has nothing to do with the conflict with Borel which developed later, and one can find a pretty explicit answer in the aforementioned letters of Lebesgue to Borel.
(These letters were first [published](https://mathscinet.ams.org/mathscinet-getitem?mr=1110360) in 1991 in *Cahiers du séminaire d’histoire des mathéma... | 36 | https://mathoverflow.net/users/8588 | 345333 | 146,374 |
https://mathoverflow.net/questions/345267 | 3 | Let $G$ be a graph of order $n$ with $m$ edges. Color each vertices uniformly at random with $q$ colors. It is easy to see that expected number of monochromatic edges (edge whose end vertices are of same color) equal to $\frac{m}{q}$. Let $t \leq \frac{m}{q}$, is it possible to show (assuming $q= o(m)$)
$$
\mathbb{P}(\... | https://mathoverflow.net/users/141538 | Concentration of monochromatic edges in a graph | Such an estimate cannot hold in general. Here are two different counterexamples:
1. Consider a complete bipartite graph with $n/2$ nodes on each side and $m=n^2/4$ edges.
Take $q$ bounded, e.g. $q=2$. Then for $q=2$, with probability $2^{-n}$ all left nodes are blue and all right nodes are red, so there are no monoch... | 2 | https://mathoverflow.net/users/7691 | 345340 | 146,376 |
https://mathoverflow.net/questions/345337 | 2 | Consider a Gromov-hyperbolic group $\Gamma$ and let $\mu$ be a finitely supported probability measure on $\Gamma$. Assume that the support of $\mu$ generates $\Gamma$ as a semi-group, in other words, the random walk $X\_n$ driven by $\mu$ can visit the whole group $\Gamma$.
Fact : the random walk $X\_n$ almost surely... | https://mathoverflow.net/users/111917 | Equivalence of harmonic measures on hyperbolic groups | No - you are completely off. There is no reason whatsoever for the harmonic measures of the original and of the reflected random walks to be equivalent (unless these random walks coincide, i.e., unless the step distribution of the random walk is symmetric). The fact that these harmonic measures have the same Hausdorff ... | 5 | https://mathoverflow.net/users/8588 | 345342 | 146,378 |
https://mathoverflow.net/questions/345323 | 4 | I'm reading through Complex Multiplication by Reinhard Schertz, and I'm stuck at Theorem 3.1.8.
Let $\mathfrak{O}\_t$ be the order of conductor $t$ in an imaginary quadratic field $K$.
He defines the groups:
$\mathfrak{I}\_t$ is the group generated by invrtible ideals of $\mathfrak{O}\_t$ (i.e., the group of fract... | https://mathoverflow.net/users/103423 | Proof in Schertz's Complex Multiplication | The statement that seems odd to you is indeed false, what (I believe) he wants to say is that $\mathfrak{D}\_t$ is a Noetherian ring in which every *prime* ideal $\mathfrak{p}\neq 0$ is maximal.
I believe that all you need is in Atiyah-Macdonald's *Introduction to Commutative Algebra*, to which I will refer as [AM69]... | 5 | https://mathoverflow.net/users/18238 | 345350 | 146,383 |
https://mathoverflow.net/questions/345310 | 3 | This is computed based on the following recursive formula $$w\_n=\frac{\lambda\_nw\_{n+1}+\mu\_nw\_{n-1}+1}{\lambda\_n+\mu\_n}$$ where: $n$ is the inital state, State $0$ is absorbing, $\lambda\_n$ and $\mu\_n$ are the up and down rates respectively and $$\sum\_{n=0}^\infty\prod\_{j=1}^n\frac{\mu\_j}{\lambda\_j}$$diver... | https://mathoverflow.net/users/141969 | Expected time till extinction in a B&D process | Clearly, the expected time $w\_n$ till extinction from the initial state $n$ is nondecreasing in $n$. So, if $w\_1=\infty$, then $w\_n=\infty$ for all natural $n$, so that the difference $w\_{n+1}-w\_n$ makes no sense, and hence
the desired conclusion
\begin{equation}
\lim\_{n\to\infty}\prod\_{j=1}^n\frac{\lambda\_j}... | 2 | https://mathoverflow.net/users/36721 | 345351 | 146,384 |
https://mathoverflow.net/questions/345343 | 5 | The question that I shall ask here has arisen in the context of Diophantine approximation. I find it rather interesting, and I have no idea how to answer it. Any help, advice, or suggestions for references would be much appreciated.
Let $S$ be a countably infinite subset of $\mathbb{R}\_{>0}$ that satisfies the foll... | https://mathoverflow.net/users/94701 | What does the image of the integer lattice under a norm look like? | I doubt that there is a simple characterization.
In any case conditions (1,2,3) are not sufficient.
For example, if $n=2$ then $S$ cannot be ${\bf Z}\_{\geq 100}$,
and there are similar counterexamples for every $n \geq 2$,
as a consequence of the following observation.
**Proposition.** *If $\eta$ is a norm on ${\bf ... | 6 | https://mathoverflow.net/users/14830 | 345356 | 146,386 |
https://mathoverflow.net/questions/345354 | 2 | I have a collection of closed half-spaces $H\_1, \dots, H\_n \subseteq \mathbb{R}^d$, each given as $H\_i = \{x \in \mathbb{R}^d : a\_i \cdot x \geq c\_i\}$ for some $a\_i \in \mathbb{R}^d$ and $c\_i \in \mathbb{R}$. [This PDF](http://jeffe.cs.illinois.edu/pubs/pdf/Thesis7.pdf) indicates that it's possible to efficient... | https://mathoverflow.net/users/54637 | Algorithm to determine if a union of half-spaces is all of $\mathbb{R}^d$ | The intersection of open half-spaces is empty iff the intersection of their closures does not have full dimension, specifically iff it is included in one of the hyperplanes that bound the half-spaces.
That is, in your notation: $\bigcup\_{i=1}^nH\_i=\mathbb R^d$ iff there exists $i=1,\dots,n$ such that the linear pro... | 3 | https://mathoverflow.net/users/12705 | 345382 | 146,389 |
https://mathoverflow.net/questions/345157 | 10 | So far I have studied fundamental part of derived category theory, for example, the existence of derived functors, the "composition of derived functors", and so on.
Now I came up with some questions about derived functors in the sense of derived category theory.
-
(1) Are the derived categorical derived functors ... | https://mathoverflow.net/users/128235 | Derived categories and classical theorems in homological algebra | Your question might be compacted to someting like: *Do I need derived categories to study cohomology of sheaves?* Of course, the answer depends on your particular interests. Let me anyway give you some starting points to help you to make up your mind.
>
> **(1) Are the derived categorical derived functors universal... | 15 | https://mathoverflow.net/users/6348 | 345390 | 146,394 |
https://mathoverflow.net/questions/345370 | -4 | Is there a systematic procedure to construct a [model](https://en.wikipedia.org/wiki/Model_theory) of an [axiomatic system](https://en.wikipedia.org/wiki/Axiomatic_system) from the system itself?
For example given the abstract postulates of a ring we can show that the integers satisfies them and hence this is a mode... | https://mathoverflow.net/users/7113 | Is there a procedure to derive models from axiomatic systems? | A simple example for a restricted class is free algebras. Given a set of equations in a functional type (no relation symbols other than standard equality), one can construct the term algebra using the function symbols and variable symbols, and then construct congruence relations induced by identifying the left hand sid... | 1 | https://mathoverflow.net/users/3402 | 345401 | 146,399 |
https://mathoverflow.net/questions/345165 | 11 | Consider the nerve functor $N : \mathbf{Cat} \to \mathbf{sSet}$; it is fully faithful, preserves finite limits, products and exponentials, etc. I am wondering if it additionally preserves whatever dependent products exist in $\mathbf{Cat}$.
For instance, even though $\mathbf{Cat}$ is not locally Cartesian closed, we ... | https://mathoverflow.net/users/51336 | Does the nerve functor preserve dependent products when they exist? | I have (I think) a proof under the assumptions I mentioned in my question, that the nerve is built from a dense subcategory and that there is a corresponding realization functor as a left adjoint. I would very much appreciate any readers of this answer to comment or edit if they find errors in my reasoning; and if you ... | 2 | https://mathoverflow.net/users/51336 | 345404 | 146,401 |
https://mathoverflow.net/questions/345388 | 33 | In computability theory, what are examples of decision problems of which it is not known whether they are [decidable](https://en.wikipedia.org/wiki/Undecidable_problem)?
| https://mathoverflow.net/users/8628 | Decision problems for which it is unknown whether they are decidable | An *integer linear recurrence sequence* is a sequence $x\_0, x\_1, x\_2, \ldots$ of integers that obeys a linear recurrence relation
$$x\_n = a\_1 x\_{n-1} + a\_2 x\_{n-2} + \cdots + a\_d x\_{n-d}$$
for some integer $d\ge 1$, some integer coefficients $a\_1, \ldots, a\_d$, and all $n\ge d$. The following problem is som... | 44 | https://mathoverflow.net/users/3106 | 345411 | 146,404 |
https://mathoverflow.net/questions/345410 | 10 | Do there exist some results on the theory of $\mathbb{C}$ in the language $\{0,1+, \times, \overline{\cdot}\}$, where $\overline{\cdot}$ is the conjugation map $\overline{a+ib} = a - ib$?
I'm wondering in particular if it has QE (or if a sensible expansion of the language has QE), and if it's decidable. But propertie... | https://mathoverflow.net/users/126815 | Model theory of the complex numbers with conjugation | It is a decidable theory, because it is interpretable in the real-closed field $\langle\mathbb{R},+,\cdot,0,1\rangle$, which has a decidable theory. We can interpret complex numbers $a+bi$ as pairs of real numbers $(a,b)$, and the complex structure, including conjugation, is definable in the reals. (Indeed, this is eas... | 16 | https://mathoverflow.net/users/1946 | 345420 | 146,406 |
https://mathoverflow.net/questions/345261 | 5 | Are they equivalent?
That is, given a sheaf of sets $\mathscr{F}$ defined on the small etale site on $X$, is there an essentially unique way to extend it to a sheaf on the big etale site on $X$? If not, what is an example of a sheaf which cannot be extended?
What about for sheaves of abelian groups?
| https://mathoverflow.net/users/126543 | Big etale topos vs small etale topos | There is a site morphism
$$i: X\_{\rm \acute{e}t}\to {\rm \acute{E}t}(X),$$
giving an adjunction (indeed, a geometric morphism of topoi—abstract nonsense)
$${\rm EXT}=i^\*: \mathsf{Shv}(X\_{\rm \acute{e}t})\rightleftarrows\mathsf{Shv}({\rm \acute{E}t}(X)): i\_\*={\rm Res},$$
where ${\rm EXT}=i^\*$ is given by some Kan ... | 4 | https://mathoverflow.net/users/42571 | 345422 | 146,407 |
https://mathoverflow.net/questions/344924 | 5 | Let $\phi:X \to Y$ be a projective/proper, birational morphism between complex algebraic varieties, with connected fibers and $\phi\_\*\mathcal{O}\_X \cong \mathcal{O}\_Y$. Suppose further that $X$ is a non-singular. Let $F$ be a subsheaf of a *free* $\mathcal{O}\_X$-module (not just locally-free) such that the quotien... | https://mathoverflow.net/users/58203 | Injectivity of pullback composed with pushforward | The answer to both questions (injectivity ans surjectivity) is no without further hypothesis.
**Surjectivity :** Let $Y$ be a smooth projective variety and let $\phi : X \longrightarrow Y$ be the blow-up of $Y$ along a smooth subvariety. Denote by $L = \mathcal{O}\_{X}(-E) \otimes \phi^\*\mathcal{O}\_{Y}(1)$, where $... | 3 | https://mathoverflow.net/users/37214 | 345425 | 146,409 |
https://mathoverflow.net/questions/345437 | 4 | I'm a little uncertain about the definitions for
1. Chern roots
2. [Chern classes](https://en.m.wikipedia.org/wiki/Chern_class)
3. Chern characters
From perusing several discussions, I gather that if one correlates the nomenclature with that of [symmetric polynomials](https://en.m.wikipedia.org/wiki/Symmetric_polyn... | https://mathoverflow.net/users/12178 | Canonical reference for Chern characteristic classes | It sounds like, in addition to the references, it would be helpful to disentangle the definitions of Chern roots,
Chern classes, and Chern characters. Different mathematicians will have different perspectives; this is mine.
The first thing one defines are **Chern classes**. Given a complex vector bundle $E\to X$, its... | 13 | https://mathoverflow.net/users/97265 | 345443 | 146,416 |
https://mathoverflow.net/questions/345428 | 1 | Let $N\_n:=\{1,2,\cdots,n\}$. Given two finite states Markov chains $\big(X^{(j)}\_i\in N\_n\}\big)\_{i=0}^\infty$ for $j\in\{1,2\}$, both of which have one absorbing state $1$.
Pr$(X^{(1)}\_{i+1}=1|X\_i=1)=$Pr$(X^{(2)}\_{i+1}=1|X\_i=1)=1, \,\forall a\in N\_n$. $$\text{Pr}\big(X^{(1)}\_{i+1}=b|X\_i=a\big)>\text{Pr}\bi... | https://mathoverflow.net/users/32660 | Comparison of hitting probability of two Markov chains both with only one absorbing state | Let $P=(p\_{ij})$ and $Q=(q\_{ij})$ be the $n\times n$ transition matrices for the two respective Markov chains, such that
\begin{gather}p\_{11}=q\_{11}=1,\\ p\_{ij}q\_{ij}>0\text{ if }i>1,\\ p\_{ij}>q\_{ij}\text{ if }1<i<j,\\
p\_{ij}\le q\_{ij}\text{ if }i>1\text{ and }1\le j\le i.
\end{gather}
The conjecture was th... | 1 | https://mathoverflow.net/users/36721 | 345444 | 146,417 |
https://mathoverflow.net/questions/345419 | 3 | Given the size of the cube(L) and the size of a stick (r), and supposing the first head of the stick fell on some random point of the cube, what is the probability that the second head will also be inside of the cube.
I know that if first head of the stick is inside the box and farther than r away from the border, th... | https://mathoverflow.net/users/148227 | What is the probability that a stick will fall completely inside a cube? | I assume one end of the stick is dropped uniformly in the cube and the other end is chosen uniformly in a sphere of radius $r$ around the first. If $r < L$ then conditionally on the orientation ${\bf x}=(x\_1,x\_2,x\_3)$ with $|{\bf x}| = r$ of the stick the probability that it is inside the cube is $\left(1-\frac{|x\_... | 4 | https://mathoverflow.net/users/47484 | 345457 | 146,424 |
https://mathoverflow.net/questions/345450 | 1 | Let $N\_n:=\{1,2,\cdots,n\}$. Given two finite states Markov chains $\big(X^{(j)}\_i\in N\_n\}\big)\_{i=0}^\infty$ for $j\in\{1,2\}$, both of which have two absorbing states at $1$ and $n$.
$\text{Pr}\big(X^{(1)}\_{i+1}=1|X^{(1)}\_i=1\big)=\text{Pr}\big(X^{(1)}\_{i+1}=n|X^{(1)}\_i=n\big)=\text{Pr}\big(X^{(2)}\_{i+1}=1... | https://mathoverflow.net/users/32660 | Comparison of hitting probability of two Markov chains both with only one absorbing state version 2 under stronger condition | Let $P=(p\_{ij})$ and $Q=(q\_{ij})$ be the $n\times n$ transition matrices for the two respective Markov chains, where $n\ge2$. Your conditions imply the following:
\begin{gather}p\_{nn}=q\_{nn},\\
p\_{nj}<q\_{nj}\text{ if }1\le j\le n-1.
\end{gather}
Hence, $1=\sum\_{j=1}^n p\_{nj}<\sum\_{j=1}^n q\_{nj}=1$, which ... | 1 | https://mathoverflow.net/users/36721 | 345458 | 146,425 |
https://mathoverflow.net/questions/345441 | 4 | Let $\mathbf Z\_K$ be the ring of integers of an algebraic number field $K$. It is well known that $\mathbf Z\_K$ has infinitely many non-associated atoms (and hence is not a [Cohen-Kaplansky domain](https://www.sciencedirect.com/science/article/pii/002186939290234D)).
>
> **Q.** Is there a slick proof of this res... | https://mathoverflow.net/users/16537 | A slick proof of "The ring of integers of a number field has infinitely many non-associated atoms"? | **Fact 1.** The ring $\textbf{Z}\_K$ has infinitely many prime ideals and has Krull dimension $1$.
**Proof.** We have a surjective morphism $\mathrm{Spec}\,\textbf{Z}\_K\rightarrow \mathrm{Spec}\mathbb{Z}$. This follows from the fact that the inclusion $\mathbb{Z}\subseteq \textbf{Z}\_K$ is an integral extension of r... | 6 | https://mathoverflow.net/users/147687 | 345476 | 146,430 |
https://mathoverflow.net/questions/345477 | 0 | For any positive integer $n$, let $[n]:=\{1,\ldots,n\}$. Let $S\_n$ denote the set of permutations (bijections) $\pi:[n]\to [n]$. For any $n>1$ and $\pi\in S\_n$ we let the *maximal neighbor distance* be defined by $$\text{mnd}(\pi) = \max \big(\{
|\pi(k) - \pi(k+1)|: k\in [n-1]\}\cup \{|\pi(n)-\pi(1)|\}\big).$$
For $n... | https://mathoverflow.net/users/8628 | Expected value of maximal distance between neighbors in permutations | It is 1.
We will assume $n$ is large enough for this proof. Take all events $A\_{k,i,j}=\{\pi(k)=i,\pi(k+1)=j\}$ for $k\le n$, $i< n^{2/3}$, $j>n-n^{2/3}$. Clearly if any of these events occurs, then $E\_n>n-2n^{2/3}$. Thus we define the random variable
$$
B=\sum\_{k,i,j} 1\_{A\_{k,i,j}}
$$
and we want to show that ... | 5 | https://mathoverflow.net/users/47135 | 345480 | 146,432 |
https://mathoverflow.net/questions/299068 | 6 | The Drinfeld associator $\Phi(x\_0, x\_1)$ encodes the parallel transport of the Knizhnik-Zamolodchikov (KZ) connection $\nabla$ on the bundle $\mathbb{C}\langle\langle x\_0, x\_1\rangle\rangle$ of formal power series in noncommutating variables $x\_0, x\_1$
over $X:=\mathbb{P}^1(\mathbb{C})\backslash\left\{0,1,\infty\... | https://mathoverflow.net/users/60535 | Is there a ''simple'' formula for the inverse of the Drinfeld associator? | Try
\begin{equation}
\Phi(x\_1,x\_2)=\sum\_w \left(-1\right)^{\textrm{weight}(w)} \zeta(w) \tilde{w}
\end{equation}
where $\tilde{w}$ is the word with reversed order. Because the $\zeta(w)$ obey the shuffle product, that should give you a proper inverse. Try the first few orders!
| 2 | https://mathoverflow.net/users/148268 | 345486 | 146,434 |
https://mathoverflow.net/questions/334335 | 6 | This question was asked [here](https://math.stackexchange.com/questions/1622523/lipschitz-function-of-independent-sub-gaussian-random-variables), but I have reason to believe that it's a serious research question appropriate for this forum (also, the answers given at the link aren't satisfactory).
If $X\in\mathbb{R}... | https://mathoverflow.net/users/12518 | Lipschitz function of independent subgaussian random variables | A counterexample to the first part is obtained by considering the uniform distribution on $\{0,1\}^n$, for $n$ even, defining $A\subset\{0,1\}^n$ by $A=\{x:\sum\_{i=1}^n x\_i\le n/2\}$, and defining $f(x)=\inf\_{y\in A}||x-y||$. Then $f$ is a Lipschitz function of iid subgaussian random variables yet fails to satisfy s... | 6 | https://mathoverflow.net/users/12518 | 345509 | 146,439 |
https://mathoverflow.net/questions/345525 | 10 | Consider the action of $\operatorname{GL}(\mathbb{R}^4)$ on the Grassmannian of 2-dimensional subspaces of $\mathbb{R}^4$. In experiments, I observe that four randomly drawn points in this space are simultaneously stabilized by *nontrivial* members of $\operatorname{GL}(\mathbb{R}^4)$. This violates my naive attempts a... | https://mathoverflow.net/users/29873 | What is the hidden symmetry behind four generic planes in $\mathbb{R}^4$? | Work over any field $k$.
Taking two generic planes in 4-dimensions, we can get them to our favourites by linear transformation $k^4=k^2\oplus k^2$, reducing $GL\_4$ to $GL\_2 \times GL\_2$.
A third plane, generically, is a graph of a unique linear map from one to the other $y=Ax$.
The group action is by matrix similari... | 12 | https://mathoverflow.net/users/13268 | 345529 | 146,444 |
https://mathoverflow.net/questions/345528 | 3 | Let $V\_t$ and $W\_t$ be independent standard Wiener processes ($t\ge 0$, $W\_t,V\_t\in\mathbb R$).
Let $C$ be the event that there is a continuous function $f$ such that for all $s$, $t$,
$$
W\_t=W\_s\iff V\_{f(t)}=V\_{f(s)}.
$$
Does $C$ have probability 0?
(The question arose in connection with a [question by Noa... | https://mathoverflow.net/users/4600 | Brownian level sets and continuous functions | Call two sequences $(a\_n)$ and $(b\_n)$ tail-equivalent if there are $p$ and $q$ such that $a\_{p+n} = b\_{q+n}$ for every $n \geqslant 0$. Write $W(t)$ rather than $W\_t$.
Suppose that $f$ with the desired property exists, that both $W(t)$ and $V(t)$ take every real value, and that $V(t)$ is not monotone on any int... | 1 | https://mathoverflow.net/users/108637 | 345532 | 146,445 |
https://mathoverflow.net/questions/345523 | 3 | Let $X,Y$ be a subvarieties of a smooth complex projective variety $Z$ and let $i:X\hookrightarrow Z$ denote the embedding. Lets further assume that the intersection $X\cap Y$ is transverse in $Z$.
Is it true that the cohomology classes $i^\*([Y])$ and $[X\cap Y]$ in $H^\*(X)$ are equal?
Here we are viewing $X\ca... | https://mathoverflow.net/users/148290 | Pullbacks of cohomology classes along embeddings vs classes of intersections | This is true in the smooth case, assuming $j:Y\hookrightarrow Z$ is proper (inverse images of compact sets are compact; this holds for example if $Y$ is compact). It is a consequence of a more general fact, that pushforwards (or Gysin, or Umkehr homomorphisms) for proper maps commute with pullbacks. That is, given a tr... | 2 | https://mathoverflow.net/users/8103 | 345534 | 146,447 |
https://mathoverflow.net/questions/345473 | 3 | Let $X\subset \mathbb{R}^2$ be open connected (and let's say bounded), let $x\in X$ and $y\in\partial X$ be such that there is a Jordan curve $\gamma:[0,1]\to X\cup\{y\}$ such that $\gamma(0)=x$ and $\gamma(1)=y$.
>
> Does there always exist a Jordan curve $\delta:[0,1]\to X\backslash\gamma(0,1)\cup\{y\}$ such that... | https://mathoverflow.net/users/53155 | Two paths to the boundary with no holes in between | Yes, this is true because every topological copy of $[0,1]$ in the plane is unknotted and can be transformed by a homeomorphism of the plane into the straight arc $[0,1]\times\{0\}$. In the latter case the existence of the curve $\delta$ is more-or-less obvious.
| 2 | https://mathoverflow.net/users/61536 | 345536 | 146,448 |
https://mathoverflow.net/questions/345487 | 1 | Let $a,b,c$ be three pairwise coprime positive integers, and $\Gamma=\langle a,b,c\rangle$ be the corresponding numerical semigroup. Consider the linear equations:
$n\_1a=m\_{12}b+m\_{13}c$
$n\_2b=m\_{21}a+m\_{23}c$
$n\_3c=m\_{31}a+m\_{32}b$
Where $n\_i,m\_{ij}$ are positive integers. Also, let $n\_1=\min\{x\in... | https://mathoverflow.net/users/122085 | Positive integer solutions of linear equations under the constraint of Frobenius number | Let $a=30$, $b=31$ and $c=37$. Then
\begin{array}{ll}
7a=2b+4c=210 \\
7b=6a+c=217 \\
5c=a+5b=185 \\
\end{array}
and the Frobenius number of $\langle a,b,c \rangle$ is 267.
| 2 | https://mathoverflow.net/users/135745 | 345538 | 146,449 |
https://mathoverflow.net/questions/345531 | 5 | I am currently discovering descriptive set theory—with much pleasure! It is something of a surprise to me that, while the Borel hierarchy is indexed by $\omega\_1$, the projective hierarchy is only indexed by $\mathbf{N}$, and that classical descriptive set theory stops there…
In particular, it would seem natural to ... | https://mathoverflow.net/users/118629 | Can it be that universal measurability is preserved by projections? | **I'm not an expert, so please correct me if I'm wrong:**
---
We can indeed continue the projective hierarchy beyond its finite levels. And like the Borel hierarchy, we can do this "from below" as follows:
* $\bf\Sigma^1\_1$ is the class of analytic sets.
* $X$ is ${\bf \Pi^1\_\alpha}$ iff $\omega^\omega\setmin... | 7 | https://mathoverflow.net/users/8133 | 345551 | 146,453 |
https://mathoverflow.net/questions/345544 | 7 | Given a random vector
\begin{equation}
x=(x\_1, \ldots, x\_n)
\end{equation}
with independent and identically distributed entries $x\_i \sim \mathcal{N}(0,\sigma^2)$, I would like to find a lower bound $f(n)$
\begin{equation}
\mathbb{E}[||x||^2\_{\infty}] \geq f(n)
\end{equation}
which is reasonably tight. I ... | https://mathoverflow.net/users/130152 | expected value of squared infinity norm of vector of iid gaussians | Let $Z\_i:=x\_i$ and $M:=M\_n:=\|x\|\_\infty=\max\_1^n|Z\_i|$.
By rescaling, without loss of generality $\sigma=1$. So, for real $u>0$
\begin{multline}
P(M^2>u)=P(M>\sqrt u)=1-P(\max\_1^n|Z\_i|\le\sqrt u)=1-P(|Z\_1|\le\sqrt u)^n \\
=1-(1-2G(\sqrt u))^n=1-e^{-ng(u)}, \tag{1}
\end{multline}
where
$$G(x):=P(Z\_1>x)\si... | 4 | https://mathoverflow.net/users/36721 | 345555 | 146,455 |
https://mathoverflow.net/questions/345559 | 7 | Consider the partial order $(P(\omega),\subseteq)$. Let $L$ be a dense linear suborder. Does $L$ have a countable dense subset?
(Note that it contains a copy of $\mathbb R$, via Dedekind cuts of $\mathbb Q$.)
| https://mathoverflow.net/users/11145 | Linear suborders of $(P(\omega),\subseteq)$ | For any $a\neq b\in\omega$ pick a set $A\_{ab}\in L$ such that $a\in A\_{ab},b\not\in A\_{ab}$, if such exists, and let $D$ be the set of all $A\_{ab}$ we have picked. We claim $D$ is dense in $L$.
Let $X,Y\in L$ with $X\subsetneq Y$. Pick any $Z$ such that $X\subsetneq Z\subsetneq Y$ and take $a\in Z\setminus X,b\in... | 14 | https://mathoverflow.net/users/30186 | 345564 | 146,459 |
https://mathoverflow.net/questions/345515 | 23 | Let $X$ be a real manifold (for simplicity). The standard construction of the universal cover $\varphi: \widetilde{X} \longrightarrow X$ involves fixing a basepoint $p \in X$ and considering homotopy classes of paths from $p$ to $x \in X$.
Is there an alternative construction of $\varphi$ that avoids choosing a basep... | https://mathoverflow.net/users/126543 | Does anyone know a basepoint-free construction of universal covers? | I think that homotopy-theorists often fall into the habit of working mainly with based spaces, even when they don't need to. It can be instructive to notice when the use of a basepoint is unnecessary, even artificial. But it's also important to notice the parts of the subject where the use of a basepoint is necessary. ... | 25 | https://mathoverflow.net/users/6666 | 345568 | 146,462 |
https://mathoverflow.net/questions/345587 | 2 | I'm curious if there is a general strategy for solving the following kind of counting problem.
Fix a positive integer $n$, and let $[n] = \{1, \dots, n\}$.
---
**Preliminaries**
*Definition* An $n$-grid of size $i \times j$ is a function $[i] \times [j] \to [n]$. Equivalently, this is an $i \times j$ matrix w... | https://mathoverflow.net/users/54637 | Counting the number of grids with certain disallowed dominoes | The case of $n=2$ and $\mathcal{S}=\lbrace[2,2],[2,2]^T\rbrace$ is the much studied "hard square entropy" problem. No simple formula or recurrence is known, though a recurrence with very many variables (basically, dynamic programming) allows computation of small values. It is known in this case that $a\_{i,i}^{1/i^2}$ ... | 3 | https://mathoverflow.net/users/9025 | 345588 | 146,470 |
https://mathoverflow.net/questions/76422 | 6 | Let $G$ be a group of automorphisms of the countable atomless Boolean algebra $B$. Suppose that every orbit of $G$ on $B$ is an antichain. Does it follow that $G$ preserves a non-zero (probability) measure on $B$?
Does the answer change if we extend $B$ to some complete or $\sigma$-complete algebra, and the action of... | https://mathoverflow.net/users/4053 | Antichains and measure-preserving actions on Boolean algebras | Let $G$ be a torsion group. Then every action of $G$ on every Boolean algebra has only antichains as orbits (indeed $gA<A$ implies $g^nA<A$ for every $n\ge 1$ and this contradicts $g$ being torsion).
Let now $G$ be a non-amenable torsion group (there are several such groups, such as Golod-Shafarevich groups, some of ... | 1 | https://mathoverflow.net/users/14094 | 345593 | 146,473 |
https://mathoverflow.net/questions/345335 | 3 | Let $k\subset K$ be a separable field extension. As a particular case of M. André [Localisation de la lissité formelle](https://link.springer.com/article/10.1007%2FBF01168230) one obtains that the natural inclusion of power series rings $k[[X\_1,\ldots,X\_n]]\subset K[[X\_1,\ldots,X\_n]]$ is regular. Is there an altern... | https://mathoverflow.net/users/15235 | Proving that a morphism between power series rings is regular | I believe the following works, as long as by "separable" you mean "separable algebraic." I will denote $K$ by $L$ to make $k$ and $K$ more distinct.
*Proof.* Consider the factorization
$$k[[X\_1,\ldots,X\_n]] \subseteq L \otimes\_k k[[X\_1,\ldots,X\_n]] \subseteq L[[X\_1,\ldots,X\_n]].\tag{1}\label{eq:factor}$$
Since... | 2 | https://mathoverflow.net/users/33088 | 345618 | 146,481 |
https://mathoverflow.net/questions/345567 | -1 | I have a non-split extension $2^8\mathbin.(2^7\mathbin:\operatorname{Sp}(6,2))$ of $2^8$ by $2^7\mathbin:\operatorname{Sp}(6,2)$. The question is how does $2^7\mathbin:\operatorname{Sp}(6,2)$ act on $2^8$. This group sits maximally inside the unique nonsplit extension $2^8\mathbin.\operatorname{Sp}(8,2)$.
| https://mathoverflow.net/users/148317 | Group action w.r.t. non-split extension group of the form $2^8\mathbin.(2^7\mathbin:\operatorname{Sp}(6,2))$ | OK, your question makes sense now. The group $2^7:{\rm Sp}(6,2)$ is a maximal subgroup of ${\rm Sp}(8,2)$, and is the stabilizer of a vector in the action of ${\rm Sp}(8,2)$ on its natural $8$-dimensional module. This immediately defines its action on that module. This action is the same in the split and nonsplit exten... | 4 | https://mathoverflow.net/users/35840 | 345623 | 146,484 |
https://mathoverflow.net/questions/345590 | 10 | Several authors (also of standard RiemGeo books) write that the sectional curvature of a plane $\pi$ contained in $T\_pM$, where $(M,g)$ is a Riemannian manifold of any dimension, is the "Gaussian curvature" in $p$ of the surface $S$ generated by the geodesics starting at $p$, with tangent velocity belonging to $\pi$. ... | https://mathoverflow.net/users/24152 | Sectional curvature and Gauss curvature | There is no standard / classical definition of Gaussian curvature except for surfaces embedded in $\mathbb{R}^3$. I think the pattern of exposition that the OP is asking about is really just an allusion to an unjustified assumption made by Riemann as he was inventing what we now know as intrinsic geometry. This assumpt... | 10 | https://mathoverflow.net/users/4362 | 345626 | 146,485 |
https://mathoverflow.net/questions/345374 | 8 | Let us consider real vector bundles, and denote by $V\_k$ the tautological bundle $V\_k\to BO(k)$. From
$$
Thom(V\oplus 1\_{\mathbb{R}}\to X)=\Sigma Thom(V\to X)
$$
and from $j^\*V\_{k+1}=V\_k\oplus1\_{\mathbb{R}}$, where $j\colon BO(k) \to BO(k+1)$ is the canonical embedding, one sees that writing $MO\_k=\Sigma^{-k}Th... | https://mathoverflow.net/users/8320 | A Thom spectrum from "doubled" tautological bundles? | I think that both of your examples, $M\_2Spin$ and $M\_2O$, arise naturally in the context of Thom spectra induced by $(B,f)$-structures. Given a $(B,f)$-structure $\mathcal{B}= \{f\_n: B\_n \to BO(n)\}$, the associated Thom spectrum $M\mathcal{B}$ is defined componentwise as:
$$
M\mathcal{B}\_k = Thom(f\_k^\*V\_k\to B... | 3 | https://mathoverflow.net/users/125244 | 345627 | 146,486 |
https://mathoverflow.net/questions/345569 | 19 | It is common for the first or second degree of various cohomologies to classify extensions of various sorts. Here are some examples of what I mean:
1) Derived functor of hom, $\text{Ext}^1\_R(M, N)$. Let $R$ be a ring (not necessarily commutative, with a $1$). As is its namesake, $\text{Ext}^1\_R (M, N)$ [classifies ... | https://mathoverflow.net/users/30211 | Unifying "cohomology groups classify extensions" theorems | $\newcommand{\cA}{\mathcal{A}}\newcommand{\Ext}{\mathrm{Ext}}\newcommand{\Hom}{\mathrm{Hom}}$Let $\cA$ be an abelian category; then, $\Ext\_\cA^i(A,B)$ is literally $\Hom\_{D(\cA)}(A, B[i])$, where $B[i]$ denotes the shift. A good way of thinking about this is as the cohomology of the derived Hom $\mathrm{R}\Hom\_{D(\c... | 17 | https://mathoverflow.net/users/102390 | 345629 | 146,488 |
https://mathoverflow.net/questions/345636 | 0 | The other day I was thinking about mathematicians in history who made fundamental contributions to both pure and applied mathematics. The examples I can think of are Newton, Gauss, Euler, Archimedes and von Neumann (I suppose you could include John Nash).
I was wondering if there were any other examples of mathematic... | https://mathoverflow.net/users/119114 | Examples of Mathematicians who excelled in Pure and Applied Mathematics | Very partial list:
Fourier, Turing, Peter Lax, Noga Alon, Cathleen Morawetz,
<https://en.m.wikipedia.org/wiki/Olga_Ladyzhenskaya>, Jurgen Moser,
| 1 | https://mathoverflow.net/users/7691 | 345639 | 146,491 |
https://mathoverflow.net/questions/345649 | 2 | Let $\frak{g}$ be a simple Lie algebra (over $\mathbb{R}$ or $\mathbb{C}$) and $V\_{\lambda\_i}$ a fundamental representation. What happens if I take the sum, in the dual of the/a Cartan subalgebra $\frak{h} \subseteq \frak{g}$, of the weights of all the non-trivial weight spaces of $V\_{\lambda\_i}$? For $\frak{sl}\_2... | https://mathoverflow.net/users/126606 | The sum of the weights of an irreducible simple Lie algebra module | The set weights occurring in a representation of $\mathfrak{g}$ is invariant under the action of the Weyl group.
Thus the sum you are asking about is fixed by the action of the Weyl group, which means that it must be zero.
| 5 | https://mathoverflow.net/users/38068 | 345650 | 146,494 |
https://mathoverflow.net/questions/343615 | 8 | $\newcommand{\SO}[1]{\text{SO}(#1)}$
$\newcommand{\dist}{\operatorname{dist}}$
Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.
Set
$$X=\text{GL}^+\_n \cup \{ A \in M\_n \, | \text{ the singular values of } \, A \text{ are distinct }\}$$
Here $M\_n$ i... | https://mathoverflow.net/users/46290 | Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have distinct singular values? | Here is a positive answer for $n=2.$ The argument doesn't seem to generalize easily to higher $n.$ The idea is to write $df\in X$ as $\star df\_1 + df\_2\neq 0$ and make use of Hodge decomposition.
The set $X$ consists of the matrices not of the form $(\begin{smallmatrix}a&b\\b&-a\end{smallmatrix}),$ in the $(dx,dy)$... | 3 | https://mathoverflow.net/users/112284 | 345652 | 146,496 |
https://mathoverflow.net/questions/345643 | 14 | I stumbled upon the following claim online: $\lfloor \frac{3^n}{2^n} \rfloor = \lfloor \frac{3^n-1}{2^n-1} \rfloor$ for all integers $n\in \mathbb{N}$, $n\geq2$. Checking with the computer, the claim seems to be true at least for integers $n$ such that $2 \leq n \leq 10000$. How could I prove this claim is true for all... | https://mathoverflow.net/users/148369 | $\lfloor \frac{3^n}{2^n} \rfloor = \lfloor \frac{3^n-1}{2^n-1} \rfloor$ for $n\geq2$ | Mahler proved in 1957 (see [here](https://www.cambridge.org/core/journals/mathematika/article/on-the-fractional-parts-of-the-powers-of-a-rational-number-ii/151ECDAE3CA50DA5F9CE7EF68477D126)) that if $q$ is a positive rational number which is not an integer, then the distance of $q^n$ to the nearest integer is $(1-o(1))... | 26 | https://mathoverflow.net/users/11919 | 345661 | 146,500 |
https://mathoverflow.net/questions/345631 | 6 | Let $(M,\omega)$ be a symplectic manifold and $H \in C^\infty(M \times \mathbb{S}^1)$. Furthermore, let $J$ be an $\omega$-compatible almost complex structure on $M$. The **Floer equation** is the following partial differential equation:
$$\partial\_s u + J(\partial\_tu - X\_{H\_t} \circ u)=0$$
where $X\_{H\_t}$ is... | https://mathoverflow.net/users/98139 | The Floer Equation is Elliptic | First of all, ellipticity is defined in terms of the principal symbol of an operator, and the Hamiltonian term is zeroth order in the derivatives of u, so let's assume wlog that we're taking about J-holomorphic maps. Moreover, in local coordinates, the difference between J and your favourite constant-coefficient comple... | 4 | https://mathoverflow.net/users/10839 | 345662 | 146,501 |
https://mathoverflow.net/questions/345658 | 17 | Let $(U\_n)\_n$ be an arbitrary sequence of open subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum\_{n=0}^\infty \lambda(U\_n)=\infty$ (where $\lambda$ is the Lebesgue measure). **Does there exist a sequence $(q\_n)\_n$ in $\mathbb{R}^2$ s.t. $D(0,1) \subseteq \bigcup\_{n=0}^\infty (q\_n+U\_n)$?**
Wi... | https://mathoverflow.net/users/33927 | Covering the disk with a family of infinite total measure | No even in dimension 1 (and multiplying the example for $\mathbb{R}$ by the small segment you get a counterexample in $\mathbb{R}^2$).
Take the set $A\_n\subset \mathbb{R}$ defined as $\bigcup\_{k\in \mathbb{Z}} (2k\cdot 10^{-n},(2k+1)\cdot 10^{-n})$. I claim that there exists no finite family of translates $\bigcup\... | 14 | https://mathoverflow.net/users/4312 | 345668 | 146,506 |
https://mathoverflow.net/questions/345669 | 11 | This is a jargon-like question.
The fact that this is posted here rather in a physics forum indicates two things
1. I know too little physics.
2. An explanation with more mathematics flavors will be appreciated more..
### Background
I should first explain what a gravity theory is in my imagination: it seems to ... | https://mathoverflow.net/users/124549 | What do physicists mean by a topological quantum gravity theory | Physicists here. The input for a physical theory is always some topological space and some structure (such as a metric) that depends on the specific context. The dynamics are invariant under the isometries thereof. For example, the theory of Special Relativity deals with a manifold of the form $\mathbb R^n$, and with a... | 20 | https://mathoverflow.net/users/106114 | 345670 | 146,507 |
https://mathoverflow.net/questions/345671 | 3 | I am reading the book: Geometric Group Theory by
Cornelia Druţu and Michael Kapovich With an Appendix by Bogdan Nica. [https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf](https://www.math.ucdavis.edu/%7Ekapovich/EPR/ggt.pdf). On page 125 we have:
\*Def: We say that two groups $G\_{1}$ and $G\_{2}$ are virtually isomo... | https://mathoverflow.net/users/147032 | virtual isomorphism of groups is an equivalence relation | OK, here is a proof (I just fixed it). Let $F\_i\triangleleft H\_i<\_{f.i.} G\_i$, $i=1,2$, $F'\_i\triangleleft H\_i'<\_{f.i.} G\_i, i=2,3$ and $f\colon H\_1\to H\_2/F\_2, g\colon H\_2' \to H\_3'/F\_3'$ be surjective homomorphisms with kernels $F\_1, F\_2'$. Let
$H=H\_2\cap H\_2' <\_{f.i} G\_2$. Then $H/(H\cap F\_2')... | 4 | https://mathoverflow.net/users/nan | 345683 | 146,509 |
https://mathoverflow.net/questions/345692 | 10 | Let $A$ be a finite dimensional commutative algebra. We can assume that it is local.
>
> Question: Which such $A$ have the property that every finite dimensional $A$-module has complexity at most 1? (This should be equivalent to the simple module having complexity equal to one or equivalently bounded Betti numbers... | https://mathoverflow.net/users/61949 | Commutative algebras with modules of small complexity | There are no other examples. This property is equivalent to $A$ being a hypersurface (see Avramov's note "Infinite Free Resolutions"). By Cohen Structure Theorem, an Artinian local hypersurface (which is automatically complete) that contains a field must be isomorphic to $k[[x]]/(f)$, which is equal to $k[[x]]/(x^n) = ... | 11 | https://mathoverflow.net/users/2083 | 345694 | 146,510 |
https://mathoverflow.net/questions/345686 | 5 | I'm trying to work through what the $(-1)$-truncated morphisms are in $\def\Catinf{\mathcal{C}\!at\_\infty} \Catinf$.
>
> **BLUF:** The correct characterization is that $F : C \to D$ is a (-1)-truncated map of $\infty$-categories iff, on hom spaces, $C(X,Y) \to D(FX, FY)$ is a (-1)-truncated map of spaces whose ess... | https://mathoverflow.net/users/148397 | Monomorphisms in $\mathcal{C}\!at_\infty$ | The statement at nLab is indeed incorrect, but your condition is also too strong. The first part should be replaced with a weaker condition that the map $C(X,Y) \to D(FX,FY)$ is a $(-1)$-truncated map of spaces.
| 4 | https://mathoverflow.net/users/62782 | 345700 | 146,511 |
https://mathoverflow.net/questions/345703 | 1 | Thomson's principle for electrical networks states that if $G$ is a network (a weighted graph), $a$, $b$ are vertices of $G$, then the effective resistance between vertices $a$ and $b$ in $G$ is given by:
$\mathcal{R}\_{eff}(a\leftrightarrow{b})=\inf\{\mathcal{E}(\theta): \theta \text{ is a flow from $a$ to $b$}, ||... | https://mathoverflow.net/users/80052 | Finding good flows to upper bound effective resistance | Well, if you graph is an approximation to something continuous, then you can cook up the flow from the corresponding solution in the continuum.
For example, Lyons' paper where he proves his criterion of transience contains such a construction for $\mathbb{Z}^3$. He considers the unit cubes with centers at vertices of... | 2 | https://mathoverflow.net/users/56624 | 345717 | 146,515 |
https://mathoverflow.net/questions/345698 | 18 | Let $(U\_n)\_{n \in \mathbb{N}}$ be a Lucas sequences given by
$$U\_0 = 0,\quad U\_1 = 1,\quad U\_n = P U\_{n - 1} - Q U\_{n-2},$$
where $P,Q$ are integers with $P^2 - 4Q \neq 0$. It is well known that the following product formula holds
$$U\_n = \prod\_{d \mid n} \Phi\_d(\alpha, \beta) ,$$
where $\Phi\_d(\alpha, \beta... | https://mathoverflow.net/users/148402 | Counterpart of cyclotomic polynomials for elliptic divisibility sequences | The counterpart of the cyclotomic polynomials are *elliptic division polynomials*, which can be defined recursively by a non-linear recursion (usually presented as a pair of recursions, one for odd indices and one for even). They are classical, dating back to the 19th century, and you can find them in many sources, inc... | 16 | https://mathoverflow.net/users/11926 | 345718 | 146,516 |
https://mathoverflow.net/questions/345501 | 8 | My question has to do with the chain homotopy that appears in Lee's *Introduction to Topological Manifols* and Rotman's *Introduction to Algebraic Topology* proofs that the inclusion
$$C\_\bullet^\mathcal{U}(X)\hookrightarrow C\_\bullet(X)$$
induces an isomorphism in singular homology
$$H\_p^\mathcal{U}(X)\cong H... | https://mathoverflow.net/users/123694 | Geometric intuition behind this chain homotopy | When thinking about chain homotopies in a setting involving simplices it can be helpful to consider the product $\Delta^p\times I$ where $\Delta^p$ is a $p$-simplex and $I=[0,1]$. The formula $h\sigma=\sigma\_{\sharp} b\_p \ast(i\_p-si\_p-h\partial i\_p)$ corresponds to a certain inductively defined subdivision of $\De... | 14 | https://mathoverflow.net/users/23571 | 345730 | 146,519 |
https://mathoverflow.net/questions/345736 | 1 | If $X\_t$ is a mean zero, square integrable process with covariance kernel $k(s,t),$ Mercer's theorem states that there exists an orthogonal basis $\{\phi\_i\}$ in $L^2$ and eigenvalues satisfying $$\int k(s,t)\phi\_n(s)ds=\lambda\_n \phi\_n(t)$$ and $$k(s,t)=\sum\_{i=1}^{\infty} \lambda\_n\phi\_n(s)\phi\_n(t).$$ Is th... | https://mathoverflow.net/users/51480 | A generalized Mercer's Theorem? | The abstract of the following paper <https://arxiv.org/abs/1110.4017> states:
We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space $X$ into $\mathbb{C}^n$. Given a finite measure $\mu$ on $X,$ we represent the reproducing kernel $K$ as conver... | 1 | https://mathoverflow.net/users/17773 | 345737 | 146,521 |
https://mathoverflow.net/questions/345645 | 6 | Let $E, F$ be two smooth vector bundles over a smooth manifold $M$. [Peetre's theorem](https://en.wikipedia.org/wiki/Peetre_theorem) states that any $\mathbb{R}$-linear morphism $D: \mathcal{E} \to \mathcal{F}$ of the sheaves of sections of $E$ and $F$ is a differential operator, i.e. there exist trivializing charts $U... | https://mathoverflow.net/users/123448 | Does Peetre's theorem hold in complex analysis? | Yes, for both the second question.
**Q2**: Since it's a local question, it suffices to consider $M$ open in $\mathbb{C}^n$ and $\mathcal{E},\mathcal{F}$ one dimensional trivial complex bundles. So let $\mathcal{O}$ denote the ring of holomorphic functions on $M$, let $z\_1,\ldots, z\_n \in \mathcal{O}$ denote the sta... | 1 | https://mathoverflow.net/users/745 | 345748 | 146,524 |
https://mathoverflow.net/questions/345729 | 1 | I just know very little about currents but I need vexedly. Thanks for your help.
Let $M$ be a closed orientable surface and $I=(f\_t)\_{t\in[0,1]}$ be an isotopies from identity to $f$. Suppose that $\mu$ is a probability measure on $M$ and $f$ preserves the measure $\mu$. Let $I.z$ be the oriented trajectory of $z$... | https://mathoverflow.net/users/148420 | The currents homology of closed orientable surfaces and Birkhoff Ergodic theorem? |
>
> My question is that the space of 1-currents is finite
> dimension?
>
>
>
There are several versions of currents. The most standard definition is
the dual space of the space of k-forms of certain regularity. It is
definitely infinite-dimensional, except in some very degenerate cases (e.g. dimension of the ... | 0 | https://mathoverflow.net/users/148443 | 345756 | 146,527 |
https://mathoverflow.net/questions/345421 | 3 | Rowbottom's Theorem states that if $\kappa$ is measurable and $U$ is a normal measure on $\kappa$, then for every $f:[\kappa]^{<\omega}\to \tau$, $\tau<\kappa$, there is an $H\in U$ such that for each $n$, $f\upharpoonright [H]^n$ is constant.
I'm hoping to extend Rowbottom's theorem to multiple measures simultaneous... | https://mathoverflow.net/users/62185 | Extending Rowbottom's Theorem to Multiple Measures Simultaneously? | The conjecture as stated is false, since the function $\alpha \mapsto o^{\vec{U}}(\alpha)$ is a counterexample. Yet, sometimes it is the only counterexample.
**Theorem:** Let $f\colon \kappa^{<\omega} \to \tau$ and let $\langle U\_\alpha \mid \alpha < \lambda\rangle$ be a sequence of normal measures on $\kappa$, $\l... | 3 | https://mathoverflow.net/users/41953 | 345771 | 146,533 |
https://mathoverflow.net/questions/345751 | 2 | Having two closed exact Lagrangian submanifolds $L\_1$ and $L\_2$ that intersect cleanly inside a Liouville manifold $M$ with $c\_1(M)=0,$ is there (with possibly some other conditions) any relation between $$H^\*(L\_1 \cap L\_2)$$ and $$HF^\*(L\_1,L\_2)?$$ In particular, (when) are they necessarily isomorphic?
| https://mathoverflow.net/users/114985 | Two Lagrangian submanifolds with clean intersections | There is a spectral sequence from the cohomology of the intersection to the Floer cohomology. This is explained (and used) in Seidel's paper "Lagrangian 2-spheres can be symplectically knotted"
<https://arxiv.org/abs/math/9803083>
and is based on the thesis of Pozniak, available here:
<http://www.math.ethz.ch/~sa... | 3 | https://mathoverflow.net/users/10839 | 345784 | 146,539 |
https://mathoverflow.net/questions/345780 | 2 | I have a question on a harmonic function and the boundary behavior.
Let $\mathbb{U} \subset \mathbb{C}$ be a unit disk. We denote by $\overline{\mathbb{U}}$ the closure of $\mathbb{U}$ in $\mathbb{C}$.
We have a reflected Brownian motion $X=(\{X\_t\}\_{t \ge0}, \{P\_x\}\_{x \in \overline{U}})$ on $\overline{\mathbb... | https://mathoverflow.net/users/68463 | Harmonic functions with boundary condition | Roughly: $u(x) \approx \operatorname{dist}(x,C)$ (in the sense that the ratio is bounded), except near the corners, where $u(x) \approx |x - x\_0|^{(2\alpha / \pi) - 1} \operatorname{dist}(x,C)$, where $x\_0$ is a corner point and $\alpha$ is the interior angle at $x\_0$.
The easiest way to see this is to map your do... | 1 | https://mathoverflow.net/users/108637 | 345787 | 146,540 |
https://mathoverflow.net/questions/345768 | 2 | I think the following should be in the literature but couldn't find it.
Recall that around the 1970's Bousfield described the $R$-localization $EX$ of any space $X$, for $R$ a fixed ring. The construction comes with a natural map $X \to EX$, which induces an isomorphism $H\_\*(X;R) \cong H\_\*(EX;R)$ and satisfies a... | https://mathoverflow.net/users/5450 | Model structure for fiberwise Bousfield localization | Here are two variants on this.
Strictly, there is no such model structure. If $X$ is any space, then the map $X\_+ \to CX\_+$ is an acyclic cofibration under the definitions given, where $CX$ is the cone on $X$: the definition of equivalence does not see components away from the basepoint. Taking the pushout of the d... | 2 | https://mathoverflow.net/users/360 | 345795 | 146,543 |
https://mathoverflow.net/questions/345788 | 2 | In Spivak's "Calculus on Manifolds", his proof is *almost* coordinate free. I think his proof could be altered (as well as preceding results that he uses) basically by using a different metric to produce bounds. I'm pretty sure this is doable, and I'm going to write it up (so no spoilers, please!). But I'd like to make... | https://mathoverflow.net/users/145755 | Any reference including a coordinate free proof of the inverse function theorem? | A coordinate-free proof of the inverse function
theorem in the finite-dimensional case
is provided by Theorem 19.6 in "Topological Geometry"
by Ian R. Porteous.
In general, the cited book is an exposition
of multivariable calculus in a coordinate-free manner.
| 2 | https://mathoverflow.net/users/402 | 345796 | 146,544 |
https://mathoverflow.net/questions/267771 | 8 | Let $sSet = Set^{\Delta^{op}}$, $cSet = Set^{\square^{op}}$, $ccSet = Set^{\square\_c^{op}}$ be the categories of simplicial sets, cubical sets, and cubical sets with connections, respectively. Then there is a left adjoint functor $cSet \to sSet$ such that a representable cubical set $\square\_n$ is mapped to $\Delta[1... | https://mathoverflow.net/users/62782 | Functors between simplicial sets and cubical sets with connections | Such an adjunction between simplicial sets and cubical sets with connections is constructed in a paper by Krzysztof Kapulkin, Zachery Lindsey and Liang Ze Wong, [A co-reflection of cubical sets into simplicial sets with applications to model structures](https://arxiv.org/abs/1906.09203). Moreover, it is shown there tha... | 2 | https://mathoverflow.net/users/62782 | 345813 | 146,549 |
https://mathoverflow.net/questions/345591 | 2 | Suppose I have 2 Markov processes with transition kernels Q\_1(y|x) and Q\_2(y|x). Suppose i also have Lyapunov functions V\_1, V\_2 for these processes w.r.t. a common set, i.e. there exists a compact set S such that for x outside S, the drift of the respective Lyapunov function is bounded above by a negative number f... | https://mathoverflow.net/users/57291 | Is there any foster-Lyapunov criterion for time varying Markov processes? | The answer is negative- the combined process obtained by alternating the kernels need not be stable.
On the state space
$\Lambda=\{(x,y) \in {\bf Z}^2 : x,y\ge 0\}$ (The non-negative quadrant in the square lattice) consider the following two kernels. Along the two axes both kernels will send the particle toward the... | 1 | https://mathoverflow.net/users/7691 | 345821 | 146,551 |
https://mathoverflow.net/questions/345814 | 3 | This question is about compact (but not necessarily effectively compact) $\Pi^0\_1$-classes in Baire space. If I am given an index for a $\Pi^0\_1$-class, and assured that it is compact, is determining whether the class is nonempty a $\Sigma^1\_1$ question of maximal difficulty?
To be more precise, let $A$ be the set... | https://mathoverflow.net/users/32178 | Is checking whether a compact $\Pi^0_1$-class is nonempty $\Sigma^1_1$-hard? | I think that every compact $\Pi^0\_1$-class has a hyperarithmetic infinite path.
>
> **Proof**: Given a compact $\Pi^0\_1$-class $P$. So there is a recursive tree
> $T\subseteq \omega^{<\omega}$ so that $[T]=P$. Since $P$ is compact,
> there is a function $f$ dominating all of the members of $P$. We will
> find... | 5 | https://mathoverflow.net/users/14340 | 345827 | 146,554 |
https://mathoverflow.net/questions/345786 | 3 | This question is moved from math stackexchange, seems like it is a more advanced question. Here the link from the original question: <https://math.stackexchange.com/questions/3415338/maximal-subgroups-of-order-pq2-in-finite-simple-groups>
It is a well-known result in elementary group theory that if $q^{2}\mid (p-1)$,... | https://mathoverflow.net/users/97247 | Maximal subgroups of order $pq^2$ in finite simple groups | No there is no such simple group.
Suppose that $G$ is simple and has $M$ as a maximal subgroup. Let $P \in {\rm Syl}\_q(M)$ and $Z=Z(M)$. So $|P|=q^2$ and $|Z|=q$ with $Z < P$. Since $M$ is maximal in $G$ and $M$ is not normal in $G$, we have $M = N\_G(Z)$.
Now $N\_G(Z)$ contains the centre of a Sylow $q$-subgroup ... | 10 | https://mathoverflow.net/users/35840 | 345833 | 146,556 |
https://mathoverflow.net/questions/345843 | 1 | Consider the function $f$ on $\mathbb{R}^{l}$ given by \begin{eqnarray}\left(x\_{1},...,x\_{l}\right)\mapsto\left(\sum\_{i=1}^{l}\frac{1}{\left(1+x\_{i}\right)^{k\_{i}}}-\left(l-1\right)\right)^{-1} \end{eqnarray} where $l\geq1$ and $k\_{1},...,k\_{l}\in\mathbb{N}$. I claim that the corresponding (multivariate) Taylor ... | https://mathoverflow.net/users/64444 | Radius of convergence of multivariate Taylor series | Let $x = (x\_1,\ldots,x\_l)$. We can write the function $f(x)$ defined in the question as $f(x) = P(x)/(1 - Q(x))$, where $P$ and $Q$ are polynomials and $Q$ has *non-negative* coefficients. Therefore, for $x \in \mathbb{R}\_+^l$ such that $Q(x) < 1$, we can expand and rearrange the terms of the series $\sum\_{n = 0}^\... | 0 | https://mathoverflow.net/users/108637 | 345846 | 146,559 |
https://mathoverflow.net/questions/345839 | 17 | Can monoids of endomorphisms of nonisomorphic groups be isomorphic ?
| https://mathoverflow.net/users/148161 | Monoids of endomorphisms of nonisomorphic groups | For any prime $p$, the endomorphism monoid of $\mathbb{Z}[\frac{1}{p}]$ is a commutative monoid with zero whose submonoid of nonzero elements is the direct sum of a cyclic group of order two (generated by multiplication by $-1$), an infinite cyclic group (generated by multiplication by $p$), and a free commutative mono... | 17 | https://mathoverflow.net/users/22989 | 345850 | 146,561 |
https://mathoverflow.net/questions/345849 | 5 | Let $f:[0,1] \to \mathbb R$ be a uniformly continuous function such that each value of $f(x)$ is greater than zero. Is its infimum greater than zero in BISH?
I believe that it is indeed the case if one assumes the Fan Theorem. But independent of it, I'm not sure.
[edit]
Note: It's possible to get around this prob... | https://mathoverflow.net/users/75761 | BISH: If a function is pointwise positive, is its infimum positive? | In BISH the follwoing two statements are equivalent:
(i) If $f:[0,1] \to \{y\in\mathbb R\, | \,y>0\}$ is uniformly continuous, then there is $n\in\mathbb N$ such that $\forall x \in [0,1]\ [f(x)>\frac{1}{n}]$
(ii) The Fan Theorem **FT**
This was already proved in Julian, W.H., and Richman, F., 1984, “A uniformly ... | 10 | https://mathoverflow.net/users/101577 | 345852 | 146,562 |
https://mathoverflow.net/questions/259418 | 2 | **Introduction**
I am analyzing the average complexity of an algorithm and it boils down to this question:
**Question**
What is the expected substring length which two randomly generated strings will most likely have?
I found a lot of papers covering this topic on subsequences but couldn't find any for substrin... | https://mathoverflow.net/users/103501 | Mean length of the longest substring (not sequence) of a random string | I don't know if the answer still interest you, anyway:
First of all, you need to define clearly what you mean by longest substring:
-you only look at the longest substring at the beginning of the sequences
-your longest substring can be anywhere but needs to be at the same position in both sequences
-your longe... | 3 | https://mathoverflow.net/users/148495 | 345858 | 146,563 |
https://mathoverflow.net/questions/54660 | 10 | I just had a student come to my office hours and ask me a ton of questions, the answer to all of which was "that's a slight variant to the implicit function theorem, which is proved by formal manipulation from the implicit function theorem". For example:
$\def\RR{\mathbb{R}}$ Let $f : \RR^n \to \RR$ be a smooth func... | https://mathoverflow.net/users/297 | Reference for working with the implicit function theorem | Shameless plug: In [my thesis](https://arxiv.org/abs/1909.00744), I introduced the following notion of an (abstract) normal form. It consists of a tuple $(X, Y, \hat{f}, f\_s)$ where:
* $X$ and $Y$ finite-dimensional vector spaces with compositions $X = Ker \oplus Coim$ and $Y = Coker \oplus Im$,
* $\hat{f}: Coim \to... | 1 | https://mathoverflow.net/users/17047 | 345859 | 146,564 |
https://mathoverflow.net/questions/345379 | 7 | Let $L\_1,L\_2$ be two $\mathbb{R}$-linear subspaces of $\mathbb{C}^n$ that are both totally real, that is, $$L\_j \cap \bigl(i\cdot L\_j\bigr) = \{0\}$$ and $$\dim\_{\mathbb{R}} L\_j = \dim\_{\mathbb{C}} \mathbb{C}^n = n$$ or equivalently $$L\_j \oplus (i\cdot L\_j) = \mathbb{C}^n . $$
I would like to know under wha... | https://mathoverflow.net/users/67031 | Symplectic structure vanishing simultaneously on two totally real subspaces | I'll give a positive answer for two generic totally real planes in $\mathbb C^2$. I believe this generalises to larger $n$, though I don't prove it - just give a possible plan of a proof with one step completed.
**Lemma 1.** Suppose $L\_1$ and $L\_2$ are two *generic* totally real $2$-planes in $\mathbb C^2$ then the... | 2 | https://mathoverflow.net/users/943 | 345861 | 146,565 |
https://mathoverflow.net/questions/345816 | 1 | Let $N\_n:=\{1,2,\cdots,n\}$. Given two finite states Markov chains $\big(X^{(j)}\_t\in N\_n\}\big)\_{t=0}^\infty$ for $j\in\{1,2\}$, both of which have two absorbing states at $1$ and $n$. Define
$p\_{i,j}(t):=\text{Pr}\big(X^{(1)}\_{t+1}$ and $q\_{i,j}(t):=\text{Pr}\big(X^{(2)}\_{t+1}=j|X^{(2)}\_t=i\big), \,\forall i... | https://mathoverflow.net/users/32660 | Comparison of hitting probability of two Markov chains both with only one absorbing state version 3 | Let $P=(p\_{ij})$ and $Q=(q\_{ij})$ be the $n\times n$ transition matrices for the two respective Markov chains. Take $n=5$,
$$P=\frac1{1000}\left(
\begin{array}{ccccc}
1000 & 0 & 0 & 0 & 0 \\
1 & 241 & 260 & 38 & 460 \\
22 & 75 & 283 & 448 & 172 \\
389 & 67 & 103 & 158 & 283 \\
0 & 0 & 0 & 0 & 1000 \\
\end{array... | 1 | https://mathoverflow.net/users/36721 | 345865 | 146,567 |
https://mathoverflow.net/questions/345870 | 5 | Does anybody know what is the biggest $r$ such that $\mathbb{Z}^r$ is isomorphic to a subgroup of $\mathrm{SL}\_n(\mathbb{Z})$?
It cannot be bigger that the virtual cohomological dimension of $\mathrm{SL}\_n(\mathbb{Z})$, which is $\frac{n(n-1)}{2}$, since the cohomological dimension respects inclusions. But I suspec... | https://mathoverflow.net/users/113290 | Free abelian subgroups of $\mathrm{SL}_n(\mathbb{Z})$ | The answer is $\lfloor n^2/4\rfloor$, namely $m^2=n^2/4$ for even $n=2m$ and $m(m+1)=(n^2-1)/4$ for odd $n=2m+1$.
Indeed, one has a free abelian subgroup of rank $\lfloor n^2/4\rfloor$, consisting of upper unipotent matrices with two blocks of size $\lfloor n/2\rfloor=m$ and $\lceil n/2\rceil\in\{m,m+1\}$ in $\mathrm... | 8 | https://mathoverflow.net/users/14094 | 345874 | 146,571 |
https://mathoverflow.net/questions/345856 | 3 | The 'usual' Hirzebruch-Riemann-Roch theorem gives a topological expression for $\chi(E)$, where $E$ is a coherent sheaf on a smooth projective variety. Is there a generalisation of this giving a topological expression for $$\chi(E,F)=\sum\_i (-1)^i \mathrm{Ext}^i(E,F)?$$
By definition $\chi(E)=\chi(\mathcal{O},E)$, and... | https://mathoverflow.net/users/113763 | Generalized Riemann Roch theorem | If the variety $X$ is smooth, one has
$$
\chi(E,F) = \deg\Big( \mathrm{ch}(E)^\vee \cdot \mathrm{ch}(F) \cdot \mathrm{td}\_X \Big),
$$
where $(-)^\vee$ is the involution of $\bigoplus H^{2i}(X,\mathbb{Q})$ that acts by $(-1)^i$ on the summand $H^{2i}(X,\mathbb{Q})$.
| 7 | https://mathoverflow.net/users/4428 | 345879 | 146,573 |
https://mathoverflow.net/questions/345857 | 2 | Let $A$ be a *flat* connection on a principal $G$-bundle $G\hookrightarrow P\to M$.
Consider an *homotopically trivial* loop $\gamma \subset M$. For simplicity, suppose $\gamma = \partial D$ is the boundary of an smoothly embedded disk.
I was trying to prove that the holonomy of $A$ along $\gamma$ is trivial (must be b... | https://mathoverflow.net/users/99042 | Flat connections, curvature and holonomy | The Stokes theorem must be modified first to deal with the nonabelian case.
See <http://arxiv.org/abs/0802.0663>,
Section 3.2, Theorem 3.4 and the displayed formula on top of page 48
for an appropriate formulation.
| 4 | https://mathoverflow.net/users/402 | 345883 | 146,576 |
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