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https://mathoverflow.net/questions/222432 | 8 | Let $X$ be a smooth projective variety over an algebraically closed field $k$.
Can it happen that $q(X) := \dim H^1(X,\mathcal O\_X) =0$ *and* $\textrm{Pic} \,X$ is not finitely generated?
Certainly, if $q(X) =0$, then $\textrm{Pic}^0 \,X =0$. But is that enough to conclude that the abelian group $\textrm{Pic} \, X... | https://mathoverflow.net/users/82230 | Are there varieties with non finitely generated Picard group and vanishing irregularity? | As pointed out by Jason Starr, the answer to your last question is *yes*, so the answer to the question in your title is *no*. Let me give a quick (and straightforward) proof in the case $k= \mathbb{C}$.
First of all, any projective variety over $\mathbb{C}$ has the homotopy type of a finite CW-complex, hence all (c... | 10 | https://mathoverflow.net/users/7460 | 222441 | 104,226 |
https://mathoverflow.net/questions/101100 | 5 | Siegel upper half-space, $\mathfrak{h}\_g$, consists of symmetric $g\times g$ complex matrices with positive-definite imaginary part. From an element $Z\in \mathfrak{h}\_g$ we can construct a theta function $\theta(Z, \cdot): \mathbb{C}^g\rightarrow \mathbb{C}$ which may be thought of as a section of a polarizing line ... | https://mathoverflow.net/users/24525 | Topology of theta nulls | For $g>2$ the locus $M\_g^1$ of "vanishing theta nulls" = set of $X$ in $M\_g$ with a spin bundle such that $h^0(L) > 1$ is shown to be a connected divisor in:
MONTSERRAT TEIXIDOR I BIGAS
The divisor of curves with a vanishing theta-null
Compositio Mathematica, tome 66, no 1 (1988), p. 15-22.
| 5 | https://mathoverflow.net/users/34170 | 222449 | 104,229 |
https://mathoverflow.net/questions/222444 | 8 | Is there a linear differential equation on $\mathbb{C}$ with singularities at $0$ and $1$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$? If so, can someone give a specific example?
| https://mathoverflow.net/users/51203 | A specific linear differential equation on $\mathbb{C}-\{0,1\}$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$ | Yes. they exist. The problem breaks up into two steps:
(1) Embed the free group on two generators into $GL\_n(\mathbb{C})$; see [here](https://sbseminar.wordpress.com/2007/09/17/that-trick-where-you-embed-the-free-group-into-a-lie-group/) or [here](https://math.stackexchange.com/questions/1492799/there-is-a-free-grou... | 12 | https://mathoverflow.net/users/297 | 222456 | 104,231 |
https://mathoverflow.net/questions/222260 | 10 | Let $\mathbf{G}$ be a reductive connected linear algebraic group over a totally real global number field, say $\mathbb{Q}$. Let $\mathbb{A}=\mathbb{R}\times\mathbb{A}\_f$ be the ring of rational adele.
Fix a maximal compact subgroup $K\_\infty$ of $\mathbf{G}(\mathbb{R})$ and an open compact subgroup $K\_f$ of $\mathb... | https://mathoverflow.net/users/74964 | Matsushima-Murakami Isomorphism for $L^2$-cohomology | Morally yes, by
**Franke, Jens** Harmonic analysis in weighted $L\_{2}$-spaces. *Ann. Sci. École Norm. Sup.* (4) 31 (1998), no. 2, 181--279
See especially theorem 4 (and note that if you want an equivariant isomorphism, you need to restrict to the twisted action by some character).
| 4 | https://mathoverflow.net/users/2284 | 222463 | 104,235 |
https://mathoverflow.net/questions/222448 | 2 | Is there a way of defining a geodesic on a Banach Manifold $M$ which is not itself a Hilbert Manifold?
| https://mathoverflow.net/users/36886 | Geodesic on Banach Manifold | Sorry not a real answer but too long for a comment:
You can put a "weak Riemannian metric" on a Banach manifold. Weak in this context means that the Riemannian metric does not describe the topology of the tangent space but induces a coarser topology.
Example: The $L^2$-metric on the space $C^\infty ([0,1], \mathbb... | 1 | https://mathoverflow.net/users/46510 | 222471 | 104,237 |
https://mathoverflow.net/questions/222491 | 13 | Given a finite subgroup of $G$ sitting inside the Morava stabilizer group $S\_n$, we can form the homotopy fixed point spectrum $E\_n^{hG}$. There is a spectral sequence with $E\_2^{s,t} = H^s(G;\pi\_t(E\_n)) \Rightarrow \pi\_{t-s}(E\_n^{hG})$.
For resolving further differentials in this spcetral sequence I have see... | https://mathoverflow.net/users/17440 | Computing Homotopy Fixed Point Spectral Sequences related to Morava E thories | In Ravenel's [paper](http://www.math.rochester.edu/people/faculty/doug/mypapers/arf_odd.pdf) on the Arf invariant, he shows (p. 439) that there is a composite of maps
$$
\mathrm{Ext}\_{BP\_\*BP}(BP\_\*,BP\_\*) \to H^\*\_c(\mathbb{S}\_n,E\_\*) \to H^\*(C\_p,E\_\*/\frak{m}),
$$
under which the images of $\alpha\_1,\beta\... | 7 | https://mathoverflow.net/users/16785 | 222512 | 104,246 |
https://mathoverflow.net/questions/222505 | 5 | For the [permutation group isomorphism](https://cstheory.stackexchange.com/questions/32783/quasi-polynomial-time-algorithm-for-permutation-group-isomorphism) problem, a permutational isomorphism between two permutation groups is sought. A closely related but seemingly easier problem arises, if an isomorphism between th... | https://mathoverflow.net/users/20781 | Group acting on two different sets, can isomorphisms be computed efficiently? | I'm still not sure in what form your input is given. I'll assume you have permutations $\lbrace g\_1,\ldots,g\_k\rbrace$ generating $G$, and permutations $\lbrace h\_1,\ldots,h\_k\rbrace$ generating $H$, and you know there is an isomorphism $\phi:G\to H$ defined by $\phi(g\_j)=h\_j$ for each $j$. Now you want to find a... | 2 | https://mathoverflow.net/users/9025 | 222521 | 104,251 |
https://mathoverflow.net/questions/222516 | 55 | Consider a non-empty set $X$ and its complete lattice of topologies
(see also [this thread](https://mathoverflow.net/questions/15841/how-do-the-compact-hausdorff-topologies-sit-in-the-lattice-of-all-topologies-on)).
The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also Hau... | https://mathoverflow.net/users/58682 | Duality between compactness and Hausdorffness | There are several ways I think of expressing this 'duality'. But before describing this, maybe it would help to explain a sense in which 'existence' (at least one element, or totality of a relation) is dual to 'uniqueness' (at most one element, or well-definedness of a relation).
So let's consider the category whose... | 62 | https://mathoverflow.net/users/2926 | 222524 | 104,252 |
https://mathoverflow.net/questions/222459 | 5 | In order to determine the geodesics between two points, one must solve the geodesic differential equations, which are as following
\begin{align}
p &= u'(s)\\
q &= v'(s)\\
p' + \Gamma^0\_{00}p^2 + 2\Gamma^0\_{01}pq + \Gamma^0\_{11}q^2 &= 0\\
q' + \Gamma^1\_{00}p^2 + 2\Gamma^1\_{01}pq + \Gamma^1\_{11}q^2 &= 0
\end{align... | https://mathoverflow.net/users/80965 | Determining geodesics between two points in curved space | I assume that [imranal](https://mathoverflow.net/users/80965/imranal) asks how to find numerically a geodesic connecting two given points if the connection is given.
One way to do it is to implement the solution of the ODE system he wrote in his question numerically (there are many effective ways for it) and then us... | 6 | https://mathoverflow.net/users/14515 | 222535 | 104,257 |
https://mathoverflow.net/questions/222529 | 6 | I got a little bit confused about the definition of geodesic for $\rm SO(3)$ as
* a compact Lie group
* a Riemannian symmetric space
In the former case, it is given by the usual matrix exponential:
$$
\exp\_{g}tX=ge^{tX}\quad g\in{\rm SO(3)}, X\in\mathfrak{so}(3)
$$
In the latter case, the geodesic is given by the ... | https://mathoverflow.net/users/37006 | geodesic of $\rm SO(3)$ as a compact Lie group vs as a Riemannian symmetric space | I think the second formula is wrong. The map $\tau$ should be given as $$\tau\colon (SO(3)\times SO(3))/SO(3)\to SO(3)\;;\quad [(g,h)]\mapsto gh^{-1}\;.$$
A geodesic in this picture is then given by
$$\tau(ge^{tX/2},he^{-tX/2})=ge^{tX}h^{-1}=gh^{-1}e^{t\mathrm{Ad}\_hX}\;.$$
This also proves the equivalence of both cons... | 8 | https://mathoverflow.net/users/70808 | 222537 | 104,258 |
https://mathoverflow.net/questions/222539 | 10 | Zeeman's conjecture in topological combinatorics states that if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision.
What is the status of this conjecture as of 2015?
| https://mathoverflow.net/users/21985 | Status of Zeeman's collapsability Conjecture | These two very recent surveys state the conjecture as open:
* Robert Kropholler, [Zeeman’s collapsibility conjecture](https://web.archive.org/web/20150918223637/http://people.maths.ox.ac.uk/kar/Zeeman.pdf) (2013)
* A. P. M. Kupers, [Zeeman's conjecture](http://math.harvard.edu/~kupers/notes/zeemansconjecture.pdf) (20... | 15 | https://mathoverflow.net/users/43108 | 222543 | 104,261 |
https://mathoverflow.net/questions/222545 | 5 | I am looking for an explicit description of the unipotent radical of a minimal parabolic subgroup of a unitary group, i.e. the group of isometries of a hermitian form, over an arbitrary field.
In his notes "Linear algebraic groups" (Boulder, 1966), Section 6.6 "Examples", Borel gives such a description for an orthogo... | https://mathoverflow.net/users/12858 | Unipotent radical of minimal parabolic subgroup of a unitary group over an arbitrary field | I deal with the simpler case: $G=U(h)$ where $E/F$ is a separable quadratic extension of fields and $h: E^n\times E^n\rightarrow E$ is Hermitian with respect to $E/F$. Suppose $W$ is a maximal isotropic subspace of $E^n$ with respect to the Hermitian form $h$. WE have the partial flag (where $W^{\perp}$ is the orthogon... | 2 | https://mathoverflow.net/users/23291 | 222549 | 104,262 |
https://mathoverflow.net/questions/222552 | 1 | I have the following question: does the Lorentz spaces $\Lambda(w,1)$ have nontrivial cotype and admit an unconditional basis(or even a symmetric basis)?Thank you.
| https://mathoverflow.net/users/41619 | Does the Lorentz spaces $\Lambda(w,1)$ have nontrivial cotype? | I think that your question about cotype is answered in S. Reisner, Ann. Inst. Fourier, Grenoble 31 (1981), 239-255 (see Theorem 2).
| 1 | https://mathoverflow.net/users/37822 | 222558 | 104,266 |
https://mathoverflow.net/questions/222538 | 8 | Let $k$ be a field of characteristic $0$ and let
$$E: y^2 = f(x)$$
be an elliptic curve over $k$, with $\mathrm{deg}(f) = 3$. Kummer theory yields a map
$$\varphi:\mathrm{H}^1(k, E[2]) \to \mathrm{H}^1(k, E)[2].$$ In particular, to each element $c \in \mathrm{H}^1(k, E[2])$ we may associate a principal homogeneous spa... | https://mathoverflow.net/users/5101 | Explicit $2$-descent on elliptic curves | It is perhaps easier to use the isomorphism of $E[2]$ with the kernel of the
`norm' map $R\_{K/k} \mu\_2 \to \mu\_2$, which translates into
$H^1(k, E[2]) \cong \ker(N \colon K^\times/K^{\times 2} \to k^\times/k^{\times 2})$. In this case you start with $c \in K^\times$
whose norm is a square: $N(c) = a^2$. The descent... | 13 | https://mathoverflow.net/users/21146 | 222572 | 104,270 |
https://mathoverflow.net/questions/222576 | 10 | I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand this better, and have two precise questions.
1. Using the definition $p\_i(E)=(-1)^{i}c\_{2i}(E\_\mathbb{C})$, the fact that... | https://mathoverflow.net/users/8103 | Whitney sum formula for Pontryagin classes I |
>
> I do not see why the difference $c\_1(E\_\mathbb{C})c\_1(F\_\mathbb{C})$ between these two expressions is necessarily 2-torsion. What am I missing?
>
>
>
Real line bundles are classified by $H^1(X, \mathbb{Z}\_2)$, which is a 2-torsion group (geometrically, because real line bundles are self-dual, their tens... | 15 | https://mathoverflow.net/users/290 | 222577 | 104,274 |
https://mathoverflow.net/questions/222522 | 5 | I want to ask on the estimates of the sum
$$ \sum\_{n=1}^{\infty} \mu(n)M\Big(\frac{x}{n}\Big)=\frac{1}{2\pi i }\int\_{\gamma-i\infty}^{\gamma+i\infty}\frac{x^s}{s\zeta(s)^2}ds.$$
It seems that the estimate of the sum be $x\exp(-c\sqrt{\log x})$ for some positive constant $c$, but it is little known to the sign-changes... | https://mathoverflow.net/users/49625 | Estimates of a sum involving both the Möbius function and Mertens function | One can treat this sum with the Dirichlet hyperbola method:
$$\begin{align} \sum\_{ab\leq x}\mu(a)\mu(b) &= \sum\_{\substack{a\leq\sqrt x\\b\leq x/a}}\mu(a)\mu(b)+\sum\_{\substack{b\leq\sqrt x\\a\leq x/b}}\mu(a)\mu(b)-\sum\_{a,b\leq\sqrt x}\mu(a)\mu(b)\\
&= 2\sum\_{n\leq\sqrt x}\mu(n)M\left(\frac{x}{n}\right)-M(\sqrt x... | 9 | https://mathoverflow.net/users/11919 | 222579 | 104,275 |
https://mathoverflow.net/questions/222586 | 6 | Consider two doubly-connected open subsets $A$ and $A'$ of the Riemann sphere. We assume these two domains to be of same modulus (the moduli space being one real parameter), i.e. we assume that there exists a holomorphic bijection $\phi:A\rightarrow A'$. Note that the map $\phi$ is then unique up to precomposition by t... | https://mathoverflow.net/users/82298 | Factorization of conformal maps between annuli | There is such a factorization, with $n=1$.
Let us suppose first that the annuli are non-degenerate, and the boundaries are sufficiently smooth, and $\phi$ sends the outer boundary of $A$ to the outer boundary of $A'$.
Then $\phi$ has a quasiconformal extenson, say $\Phi$ which is a homeomorphism between the spheres. ... | 7 | https://mathoverflow.net/users/25510 | 222591 | 104,281 |
https://mathoverflow.net/questions/222544 | 2 | Let $M$ be an $R$-module of finite length and $N$ a maximal submodule of $M.$
>
> Is there an element $m$ in $M$ such that $m(N:M)=0$?
>
>
>
It is a generalization of this result:
In a Notherian ring $R,$ all minimal prime ideals have non-zero annihilator.
| https://mathoverflow.net/users/82283 | Annihilator of minimal prime ideal in a commutative Noetherian ring | Put $P=(N:M)$.
Because $N$ is a maximal submodule, $P$ is a maximal ideal.
Because $M$ has finite length, there is some minimal $k$ such that $MP^k=MP^{k+1}$. By Nakayama, there exists $s\in 1+P$ such that $MsP^k=0$.
Put $T=MsP^{k-1}$. If $T=0$, then $MP^{k-1}\subset MP^k$, contradicting minimality of $k$. So ... | 4 | https://mathoverflow.net/users/10503 | 222592 | 104,282 |
https://mathoverflow.net/questions/222527 | 15 | In his [student guide](http://www.cambridge.org/de/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-students-guide) on page 154, Adams gives a construction of products for cohomology using "pairings" of spectra (now known as maps from $E\wedge E\to E$). But then he says
>
> However, G. W. Whit... | https://mathoverflow.net/users/70808 | Multiplicative cohomology theories and smash products | I do not claim that I know what Adams meant; for all I know his definition of "product" may include products which are not bilinear. Take the following with a grain of salt.
What I suspect is the following. Adams says, just before your quote:
"In order to construct such products, G.W. Whitehead introduced the notio... | 11 | https://mathoverflow.net/users/360 | 222610 | 104,291 |
https://mathoverflow.net/questions/222593 | 21 | Suppose that $A\_1,\dots, A\_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A\_i'$ and $A\_j'$ such that $\|A\_i-A\_i'\|<\varepsilon$, $\|A\_j-A\_j'\|<\varepsilon$ and $A\_i'A\_j'=A\_j'A\_i'$. Can we find unitary... | https://mathoverflow.net/users/18785 | Almost commuting unitary matrices | **Edit** Now this answers the first question for the operator norm and the normalized Hilbert-Schmidt norm.
The answer depends on the norm you are considering. The answer is no for the operator norm, but is yes for the normalized Hilbert-Schmidt norm (at least if you replace $O(\varepsilon)$ by $o(1)$, see the answer... | 14 | https://mathoverflow.net/users/10265 | 222615 | 104,294 |
https://mathoverflow.net/questions/222601 | 3 | On the wikipedia page "Generalizations of derivative" the author mentions: *" in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some of the features of a functional derivative and the covariant derivative."*
My question is what does this derivative lo... | https://mathoverflow.net/users/36886 | Reference: Finsler Derivative? | Chapter 2 in Bao, Chern, Shen: *An Introduction to Riemann-Finsler Geometry* may give you the answer.
| 2 | https://mathoverflow.net/users/60435 | 222618 | 104,295 |
https://mathoverflow.net/questions/222616 | 6 | I have read in several places that the total Pontryagin classes of real vector bundles satisfy a Whitney sum formula $p(E\oplus F) = p(E)\cdot p(F)$ modulo 2-torsion. I would like to understand the 2-torsion part better.
>
> Is there a reference which describes the difference between $p(E\oplus F)$ and $p(E)\cdot p... | https://mathoverflow.net/users/8103 | Whitney sum formula for Pontryagin classes II | Brown, Edgar H., Jr.
The cohomology of BSOn and BOn with integer coefficients.
Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.
Theorem 1.6, last sentence:
Under Whitney sum,
$p\_q\mapsto \sum\_j r\_{2q-j}\otimes r\_j$, where
$r\_{2s} = p\_s$ and $r\_{2s+1} = (\delta w\_{2s})^2+ p\_s\delta w\_1$.
| 9 | https://mathoverflow.net/users/78588 | 222620 | 104,297 |
https://mathoverflow.net/questions/222653 | 2 | We have encountered the following problem that we think that should be true. Let $\{X\_n\}\_{n\geq 0}$ a sequence of random variables which we know that $\mathbb{E}[X\_n]$ tends to infinity.
The question is the following: can we assure that the sequence does *NOT* converge in distribution to a Poisson?
| https://mathoverflow.net/users/46573 | Convergence in distribution to a Poisson | No. It's easy to construct a sequence $Y\_n$ with $Y\_n \to 0$ a.s. but $E Y\_n \to +\infty$. (You can even have $E Y\_n \equiv +\infty$ if you wish.) Now let $X$ be a fixed Poisson random variable and $X\_n = X + Y\_n$.
| 4 | https://mathoverflow.net/users/4832 | 222654 | 104,305 |
https://mathoverflow.net/questions/222642 | 0 | Let $G$ be a simple Lie group of dimension $n$ (connected or even simply connected). Let $T$ be a maximal torus of dimension $d$. Notice that $\frac{n}{d}$ is an integer which I will denote by $m$. Let $g\_{1},\dots g\_{m}$ be $m$ elements of $G$ such that for $i\neq j$ we have
$$ g\_{i}T\cap g\_{j}T=\emptyset$$
Not... | https://mathoverflow.net/users/21369 | Local diffeomorphism from a torus to a Lie group | No. Take $g\_2=1$, $g\_1\notin T$, then $g\_1 T\cap g\_2 T=g\_1T\cap T=\emptyset$.
The differential of the map
$$\phi\colon T\times T\to G,\quad (t\_1,t\_2)\mapsto g\_1 t\_1 g\_2 t\_2=g\_1t\_1t\_2$$
at the point $(1,1)\in T\times T$ (or at any other point of $T\times T$) is clearly of rank $d=\mathrm{dim}\, T$, and not... | 3 | https://mathoverflow.net/users/4149 | 222655 | 104,306 |
https://mathoverflow.net/questions/222621 | 0 | I need to calculate this expression:
$$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $$
I know that I can express this as:
$$ Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5})=-4i(g^{\mu\nu}\epsilon^{\rho\sigma\alpha\bet... | https://mathoverflow.net/users/82323 | Trace of six gamma matrices | To get from your 15 term expression to your 6 term expression, you can use what is sometimes called the Schouten identity, $g^{\mu [ \nu} \epsilon^{\rho \sigma \alpha \beta]} = 0$. The square brackets denote antisymmetrization, and an antisymmetrization of 5 indices in 4 dimensions must vanish. There are 5 terms when e... | 7 | https://mathoverflow.net/users/39284 | 222656 | 104,307 |
https://mathoverflow.net/questions/222494 | 67 | According to Solomon Feferman, in his [slide presentation](https://math.stanford.edu/~feferman/papers/BernaysLecture1.pdf) "Three Problems for Mathematics", Hilbert wrote (in regards to Gödel's second incompleteness theorem):
>
> ...the end goal [is] to establish as consistent all our usual methods
> of mathematic... | https://mathoverflow.net/users/20597 | What was Hilbert's view of Gödel's Incompleteness Theorems? | Some related information :
1) Volume 2 of Hilbert & Bernays, [Grundlagen der Mathematik](https://en.wikipedia.org/wiki/Grundlagen_der_Mathematik) (1939) include full proofs of Gödel's 1st and 2nd Theorems (for the 2nd one, it was the first published complete proof), as well as Gentzen's concistency proof, with detail... | 16 | https://mathoverflow.net/users/42676 | 222657 | 104,308 |
https://mathoverflow.net/questions/191475 | 2 | Let $G$ be a finite group. A map $\rho:G\rightarrow U\_n$ is called an $\epsilon$-homomorphism if and only if for any $g,h\in G$, we have $||\rho(g)\rho(h)-\rho(gh)||\leq \epsilon$ where the $||$ norm is the usual operator norm. Kazhdan proves in his [paper](http://link.springer.com/article/10.1007/BF02761236) that for... | https://mathoverflow.net/users/18785 | Approximating eps-homomorphisms | I just wanted to add that the answer to above question is yes:
<http://arxiv.org/abs/1510.04085>
| 1 | https://mathoverflow.net/users/18785 | 222669 | 104,312 |
https://mathoverflow.net/questions/222651 | 2 | We start with product manifold $ X := S\_g \times (0,1)$, where $S\_g$ is a closed, orientable surface of genus $g \geq 1$. Since this has $\mathbb R^3$ as its universal cover, $X$ is irreducible. Now $X$ can also be seen as the interior of $\bar{X} = S\_g \times [0,1]$. We want to manipulate $X$ a bit in order to crea... | https://mathoverflow.net/users/78554 | Is the following 3-manifold irreducible? | Yes, $Y$ is still irreducible, and this holds by a simple connectivity argument.
Suppose that $\Sigma \subset Y \subset X$ is a smoothly embedded 2-sphere. Since $X$ is irreducible, $\Sigma$ bounds a smoothly embedded ball $B \subset X$. Note that $\bar X - \Sigma$ has two components: the inside of $\Sigma$ which eq... | 3 | https://mathoverflow.net/users/20787 | 222676 | 104,314 |
https://mathoverflow.net/questions/222504 | 7 | In his 1965 paper *Omitting Classes of Elements* (found in *The Theory of Models: Proceedings of the 1963 International Symposium at Berkeley*, published by North-Holland Publ. Co., Amsterdam (1965)), Morley proved the following omitting types theorem:
For a complete countable theory $T$ and a 1-type $\Sigma$, if for... | https://mathoverflow.net/users/82270 | How to extend Morley's omitting type theorem to uncountable languages? | The key step in the proof is to show that given a sequence $(a\_i : i < \beth\_{(2^{|T|})^+})$, there is a sequence $(b\_i : i < \omega)$ of indiscernibles such that for every $m$ there are $i\_0 < i\_1 < ... < i\_{m-1}$ with
$$tp(b\_0...b\_{m-1})=tp(a\_{i\_0}...a\_{i\_{m-1}}).$$
The proof of this assertion (in this ... | 5 | https://mathoverflow.net/users/57712 | 222691 | 104,321 |
https://mathoverflow.net/questions/220827 | 10 | Consider Lemma 1 from Beilinson's paper "[Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra](http://www.ams.org/mathscinet-getitem?mr=509388)", as follows.
>
> Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, $F: \mathcal{C} \to \mathcal{D}$ an exact functor, $\{X\_i\}$ be a family of... | https://mathoverflow.net/users/nan | Lemma 1 from Beilinson's "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", intuition? | I think some of the commenters have forgotten the time when they found vector space linear algebra understandable, but triangulated categories confusing.
For someone in such a state, a useful tool to help understand statements about triangulated categories is passage to the Grothendieck group. Recall that this is do... | 23 | https://mathoverflow.net/users/4707 | 222697 | 104,323 |
https://mathoverflow.net/questions/222696 | 8 | Let $P$ be the Levy-collapse of the ordinals, so $P$ is a class forcing notion that makes every ordinal countable.
Note that since $P$ is weakly homogeneous, for any formula $\phi(\overline{a})$ with parameters from $\mathbf{V}$, $P$ decides $\phi(\hat{\overline{a}})$. Hence it makes sense to ask whether $(HC, \in)$... | https://mathoverflow.net/users/26705 | Consistency Strength of "HC is elementary in V[G]" | $\newcommand\HC{\text{HC}}$Allow me to denote your theory $\Phi$ by "$\HC\prec V[G]$". As you mention, this is not a single statement,
since we have no truth predicate able to express truth in the
extension $V[G]$, but rather it is expressed as a scheme, asserting of each formula $\varphi$ with parameters
$z\in\HC$ th... | 8 | https://mathoverflow.net/users/1946 | 222702 | 104,325 |
https://mathoverflow.net/questions/222180 | 8 | This is a problem about the practicalities of removing singularities in multivariable complex functions.
In trying to derive the generating function (in two variables) for a certain problem in combinatorics, we have obtained an expression of the form $S(x,u) = N(x,u)/D(x,u)$ where
$$
D(x,u) =
(1 + u - u^2) x^2 + (3 ... | https://mathoverflow.net/users/1907 | Removing singularities in generating functions | I have a solution inspired by Fan Zheng's comment. There is a generalization of Study's lemma [1, $\S$6.13] (which is itself a special case of the Nullstellensatz), that says if $D(x,u)$ is irreducible and if $D(x,u)=0$ implies $N(x,u)=0$ then $D(x,u)$ divides $N(x,u)$.
The zero condition ended up being not easy to ... | 6 | https://mathoverflow.net/users/36497 | 222703 | 104,326 |
https://mathoverflow.net/questions/222699 | 1 | Let $X\subset\mathbb{P}(a\_0,...,a\_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a\_0,...,a\_N)$.
Is it true that if $q\geq 1$ then $H^0(X,\Omega\_X^q(q))=0$ ?
| https://mathoverflow.net/users/nan | Sections of a sheaf of differentials on a weighted complete intersection | Way no.
Just take a degree $d$ hypersurface in ordinary projective space: $X\subset \mathbb P^n$. Then $$\Omega\_X^{\dim X}(\dim X)=\omega\_X(n-1)\simeq \mathscr O\_X(d-2),$$ so as soon as $d\geq 2$, i.e., it's not a projective space itself, then this sheaf will have non-zero global sections. If you increase the deg... | 5 | https://mathoverflow.net/users/10076 | 222704 | 104,327 |
https://mathoverflow.net/questions/222708 | 9 | The following was asked on stackexchange but I think it also belongs here:
<https://math.stackexchange.com/questions/1513446/transitive-models-and-ch>
Suppose $M, N$ are two countable transitive models of ZFC which have same ordinals, cofinalities and reals (but not necessarily same sets of reals!). Suppose $M$ mod... | https://mathoverflow.net/users/2689 | Transitive models and CH | This strange situation can indeed happen.
Start with a countable transitive model of set theory $W$,
satisfying CH, and let $G$ be $W$-generic for the forcing
$\newcommand\Add{\text{Add}}\Add(\omega,\omega\_1)^W$. So the model
$M=W[G]$ continues to satisfy CH. Now, let $g$ be $W[G]$-generic
for the collapse of $\omeg... | 12 | https://mathoverflow.net/users/1946 | 222710 | 104,329 |
https://mathoverflow.net/questions/222713 | 3 | Does anyone know where I might find an online version (for free or purchase, translated or in french) of this paper by Henri Cartan from 1953? I know it was published in *Colloque sur les fonctions de plusieurs variables tenu a Bruxelles* but I can't seem to find a conveniently available copy anywhere. Probably there d... | https://mathoverflow.net/users/82370 | H. Cartan's "Variétés analytiques complexes et cohomologie"? | The paper is included on volume II of Springer's edition of Cartan's collected works
* Henri Cartan, [Oeuvres - Collected Works II](http://www.springer.com/gp/book/9783662469088) (1979)
As far as I can tell it hasn't been digitized so far.
| 4 | https://mathoverflow.net/users/43108 | 222717 | 104,331 |
https://mathoverflow.net/questions/222698 | 0 | Question 1. Does the Hardy space $H^{1}$ have an unconditional basis? This problem appeared in S.Kwapien and A.Pelczynski's paper: Some linear topological properties of the hardy spaces $H^{p}$, Compositio Mathematica.33(1976),261-288. I do not know whether this problem was solved.
Question 2. Does the Lorentz functi... | https://mathoverflow.net/users/41619 | On the unconditional basis on the Hardy space $H^{1}$ and the Lorentz function space $L_{w,1}$ | Maurey proved the existenc of an unconditional basis as in question 1 (Acta Math. 1980) and then Wojtaszczyk (Ark. f. Math. 1982) gave an explicit example---the Franklin system.
| 2 | https://mathoverflow.net/users/81335 | 222722 | 104,334 |
https://mathoverflow.net/questions/222721 | 8 | Given a permutation $\sigma\in S\_n$, let $P\_\sigma$ denote the corresponding $n\times n$ permutation matrix. It is easy to see that for $n=3$, there is only one linear relation up to scaling given by $\sum\_\sigma sign(\sigma)P\_\sigma = 0$. For larger $n$, what are all linear relations $\sum\_\sigma c\_\sigma P\_\si... | https://mathoverflow.net/users/82376 | Linear relations among permutation matrices | The standard permutation representation $\mathbb{C}^n$ of $S\_n$ breaks up as a direct sum of a trivial representation $1$ and an irreducible representation $V$ of dimension $n - 1$. Hence the natural map $\mathbb{C}[S\_n] \to \text{End}(\mathbb{C}^n)$ has image in $\text{End}(1) \oplus \text{End}(V)$. In fact, in term... | 12 | https://mathoverflow.net/users/290 | 222727 | 104,336 |
https://mathoverflow.net/questions/222308 | 6 | Let $S$ be a closed oriented surface of genus $\geq 2$ and $\mathcal{F}\_n(S)$ be the space of flat metrics with conic singularities on $S$ whose cone angles are of the form $2k\pi/n$ ($k\in\mathbb{N}$). Let
$\mathcal{F}\_n^1(S)$ denote metics $g\in\mathcal{F}\_n(S)$ of total area $1$.
Each $g\in\mathcal{F}\_n(S)$ y... | https://mathoverflow.net/users/17294 | The hyperbolic metric on a flat surface | The answer for the case $n=2$ of your question is described in my comments above.
In summary, the lower bound can be derived from Linch's paper on the comparison of Teichmuller and Weil-Petersson distances, while an essentially sharp upper bound involving the inverse of the square root of the systole function can be... | 2 | https://mathoverflow.net/users/1568 | 222736 | 104,339 |
https://mathoverflow.net/questions/222740 | 6 | Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}\_r = \{x \in \mathbb{R}^n : \Vert x\Vert = r\}$ within time $t$ (obviously such an expression, if known, will be in terms of $r$ and... | https://mathoverflow.net/users/82390 | Brownian motion in $n$ dimensions | The process $\|B(t)\|$ is called $n$-dimensional Bessel process (or Bessel process with parameter $\nu=\frac{n}{2}-1$). I think formula $\bf 4$.1.1.4 of Borodin-Salminen "Handbook of Brownian Motion -Facts and Formulae" is what you're looking for.
| 9 | https://mathoverflow.net/users/81488 | 222742 | 104,342 |
https://mathoverflow.net/questions/222636 | 2 | **EDIT:** Let $X$ be an $n$-dimensional Alexandrov space with curvature bounded below. Let $f\_1,\dots, f\_n\colon X\to \mathbb{R}$ be $\lambda$-concave functions. Assume that at a fixed point $p$ there exist directions $\xi\_1^\pm,\dots,\xi\_n^\pm\in \Sigma\_p$ such that $f'\_i(\xi\_i^+)>1,\, f'\_i(\xi^-\_i)<-1$ and $... | https://mathoverflow.net/users/16183 | A property of concave functions on Alexandrov spaces | The base of Perelman's induction for Main Theorem 1.4 (is this what you're talking about?) is in fact a collection of trivial statements about admissible maps $X^n \to \mathbb{R}^{n+1}$.
To move to the next step, which you're looking for, requires an application of Property 1.3(c). The complementing function $g\_{n+1... | 2 | https://mathoverflow.net/users/68708 | 222750 | 104,345 |
https://mathoverflow.net/questions/222755 | 6 | Recently during my work, I encountered the following family of sextic polynomials
$$\displaystyle x^6 - 3x^5 + cx^4 + (5 - 2c)x^3 + cx^2 - 3x + 1.$$
By plugging in various values of $c$, I noticed that this sextic always has all real roots or all non-real roots. When the roots are all complex, I noticed the following... | https://mathoverflow.net/users/10898 | Seeking an explanation for a peculiar factorization | Ah, I have used something like this as a math problem in a math competition!
Note that you have inversion symmetry in your polynomial, meaning that $p(x)$ and $x^6p(1/x)$ have the same roots. This means that if $x$ is a root, then so is $1/x$. You can sort of "fold" these roots together, which is the reason for the c... | 10 | https://mathoverflow.net/users/1056 | 222757 | 104,350 |
https://mathoverflow.net/questions/222748 | 5 | Given a set $S$ of $n$ elements. Let $T$ be the set of all subsets of $S$, with size $\frac{n}{2}$ ($n$ is even). We want to select a subset $T'$ of $T$, with the property that for any pair of the elements $x,y \in S$, there exists a subset $Q \in T'$, such that $x \in Q$ and $y \in Q$. What is the minimum size of $T'$... | https://mathoverflow.net/users/51176 | Selecting subsets with size $\frac{n}{2}$ covering every pair of the elements | The answer is $6$ for $n \ge 4$. This is problem 1.3 on IMC 2013:
<http://www.imc-math.org.uk/imc2013/IMC2013-day1-solutions.pdf>
---
It takes $6$ subsets.
Lower bound: For any element $a$, to cover all $n-1 \gt 2(n/2 -1)$ pairs of elements including $a$, there must be at least $3$ subsets containing $a$. Th... | 7 | https://mathoverflow.net/users/4312 | 222759 | 104,351 |
https://mathoverflow.net/questions/222769 | 2 | Let $p<1$ be a constant. Consider two sets $A,B$, each with $n$ vertices. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the remaining edges. Is it true that as $n\rightarrow\infty$, the probability that there exists a matching between $A$ and $B$ approac... | https://mathoverflow.net/users/79906 | Matching with probabilistic edges | A reference is Erdős and Rényi's 1964 paper [On Random Matrices](https://www.renyi.hu/~p_erdos/1964-14.pdf), where they show that $p=\frac{\log n}{n}$ is a sharp threshold for the existence of a matching. The idea behind the proof is essentially combining the union bound with careful asymptotics to show that the expect... | 8 | https://mathoverflow.net/users/405 | 222775 | 104,358 |
https://mathoverflow.net/questions/222767 | 3 | Let us have positive irrational numbers $a$ and $b$ represented by functions $f\_a,f\_b\colon\mathbb{N}\to\mathbb{N}$ respectively such that $f\_a(0)=\left \lfloor{a}\right \rfloor$ and $f\_a(i)$, $i>0$ is the ith digit of $a$, and similarly for $b$.
Is there some "standard" way to compute $f\_{a^b}$ from $f\_a$ and ... | https://mathoverflow.net/users/nan | Computing digits of irrational exponentiation | I do not think there is an algorithm (or you have to change something
to allow algorithms that never end with some data). Consider the two numbers
$$a=(3/2)^\sqrt{2}=1.77431468418218794421950\dots\quad
b=\frac{1}{\sqrt{2}}=0.70710678118654752440084436210\dots$$
We have
$$a^b=\frac{3}{2}=1.5$$
Then slightly changing ... | 8 | https://mathoverflow.net/users/7402 | 222776 | 104,359 |
https://mathoverflow.net/questions/222788 | 2 | Are there sources that treat questions like the following ones?
1. Suppose that $f\colon\mathbb{C}\to\mathbb{C}$ is an entire function such that $f(x)$ is real for all real $x$ and $f(x)\sim1/x$ as $x\to\infty$. Does it then follow that $f^{(k)}(x)=O(1/x^{k+1})$ for $x>x\_0$, where $x\_0$ is a nonnegative real numbe... | https://mathoverflow.net/users/36721 | Asymptotics of the derivatives of analytic functions | I think the answer to both is negative. By a result by Carleman, entire real functions are dense in $C(\mathbb{R},\mathbb{R})$ with the Withney topology, so there is an entire $f$ such that $$1/x + \sin(e^x) /x^2 < f(x) < 1/x + \sin(e^x)/x^2 +1/x^3$$ for all $x>1$; this clearly forces $f'(x)$ to be unbounded for $x\to+... | 3 | https://mathoverflow.net/users/6101 | 222791 | 104,365 |
https://mathoverflow.net/questions/222786 | 3 | Suppose $F(x,y)$ is a polynomial in two variables over a field $K$, and $F(x,y)$ is not a square. When is $F(x,a)$ a square for $a\in K$? I would guess that Hilbert's Irreducibility Theorem might help (if $K$ is hilbertian), but I am not sure how.
| https://mathoverflow.net/users/74453 | If $F(x,y)$ is a polynomial which is not a square, then how often is the specialization $F(x,a)$ a square? | Write $F(x,y)=G(x,y)^2H(x,y)$ with polynomials $G,H$, such that $H(x,y)$, considered as a polynomial in $x$ over $K(y)$, is squarefree. If $H(x,y)$ does not involve $x$, so $H(x,y)=h(y)$, then you are reduced to an arithmetical question depending on $K$. For instance if $K$ is Hilbertian, and $h(y)$ is a not a square, ... | 4 | https://mathoverflow.net/users/18739 | 222794 | 104,367 |
https://mathoverflow.net/questions/222793 | 2 | Consider the set $\{1,2,\ldots,N\}$. Let $LCM(a,a')$ denote the lowest common multiple of the integers $a,a'.$
We say that $A\subset \{1,2,\ldots,N\}$ is $M-$**good** if $LCM(a,a')\geq M,$ for all $a\neq a' \in A.$
How large can the size $|A|$ be for an $M-$good set, as a function of $M,N.$ Specifically let $N\righ... | https://mathoverflow.net/users/17773 | How large can a subset of $\{1,\ldots,N\}$ be if all pairwise LCMs of its elements are lower bounded? | You're asking for the size of a maximum independent set in the graph
with vertices $\{1,2,\ldots,N\}$ and edges $(i,j)$ whenever $\text{lcm}(i,j) < M$.
I tried it in the case $M=N+1$ for $N$ up to $65$. It appears that the maximum
$|A| = \lceil N/2 \rceil$, which is attained when
$A = \{x: \lceil (N+1)/2 \rceil \le... | 4 | https://mathoverflow.net/users/13650 | 222800 | 104,368 |
https://mathoverflow.net/questions/222765 | 3 | Let $X$ be the Jacobian of a genus 2 curve over $\mathbb{C}$. Let $L=\mathcal{O}(nC)$, where n is an even number. Is it possible to find a smooth curve from $|L|$ which is fixed by the involution $x\mapsto -x$ and which passes through the sixteen 2-torsion points? I have the following ideas:
1) if we take $n$ to be s... | https://mathoverflow.net/users/70211 | How do I find a smooth curve in $J(C)$ through the 2-torsion points? | We have $X=\text{Jac}(C)$, for $C$ some hyperelliptic curve. By appropriate choice of basepoint, we can arrange that the embedding $C\to X$ passes through $0\in X$ and is that $[-1]$ on $X$ restricts to the hyperelliptic involution on $C$.
Let $C'=[2]^{-1}(C)$. I claim that this curve satisfies all of the desired pro... | 4 | https://mathoverflow.net/users/6950 | 222808 | 104,372 |
https://mathoverflow.net/questions/218982 | 15 | Are there simple examples of $n+1$D TQFT that assign 1-dimensional Hilbert spaces to both $n$-torus and $n$-sphere but higher dimensional Hilbert spaces to some other $n$-manifolds? Here I am assuming that the manifolds under consideration are all orientable.
For $n=2$, the requirement of 1-dimensional Hilbert spaces... | https://mathoverflow.net/users/61911 | Examples of n+1D TQFT with 1 dimensional Hilbert spaces on n-torus and n-sphere but higher dimensional Hilbert spaces on other n-manifolds | If your topological field theory is at least once-extended, by which I mean it assigns values to $(n+1)$-manifolds, $n$-manifolds, and also to $(n-1)$-manifolds, than this cannot happen.
More precisely we have the following result:
>
> **Theorem**: Suppose that $ Z: Bord\_{n+1} \to C$ is an $(n+1)$-dimensional o... | 7 | https://mathoverflow.net/users/184 | 222812 | 104,374 |
https://mathoverflow.net/questions/222816 | 3 | Please someone suggest me some reference on the topic "Complex Dynamics". I want a brief geometric treatment from the root level. I have graduate level background on complex analysis, riemannian geometry and topology.
| https://mathoverflow.net/users/65972 | reference on complex dynamics | *Dynamics in One Complex Variable*. (AM-160): Third edition (Annals of Mathematics Studies) Jan 22, 2006 by John Milnor
*Complex Dynamics* (Universitext / Universitext: Tracts in Mathematics) Feb 2, 1996
by Lennart Carleson and Theodore Gamelin
*Iteration of Rational Functions*: Complex Analytic Dynamical Systems (... | 4 | https://mathoverflow.net/users/11926 | 222826 | 104,376 |
https://mathoverflow.net/questions/222817 | 1 | Consider a two dimensional square lattice ($n$ by $n$), which is our space $S$ (each point labelled by an index $1\to n^2$), containing two types of particles, distinguished here by either an index $1$ or $2$. There are $4$ of them on the lattice, $A\_1$, $A\_2$, $B\_1$ and $B\_2$. (Note that $A$'s are distinguishable ... | https://mathoverflow.net/users/nan | Two types of random walkers on square lattice | Since the particles interact with each other (they cannot jump simultaneously to the same point or to a point occupied by another particle), I think you cannot construct the global transition matrix from the transition matrix of a single particle taken in isolation.
By the way, aren't $4$ components enough to describ... | 1 | https://mathoverflow.net/users/13388 | 222830 | 104,378 |
https://mathoverflow.net/questions/222831 | 0 | I am writing a paper(physics) where I am using the fact that the **irreducible's of the regular representations** of the **permutation group** are **absolutely irreducible** in the following sense.
If $V$ is an irreducible of the regular representation of $S\_n$ over $R$ (real numbers) then it remain irreducible unde... | https://mathoverflow.net/users/56778 | Reference Request: Irreducibles of the regular representation of the permutation group is absolutely irreducible | Actually, all of these representations are defined over $\mathbb{Q}$ (i.e., they are absolutely indecomposable). This is stated in the second sentence of the Encyclopedia of Math article on [Representation of the symmetric groups](https://www.encyclopediaofmath.org/index.php/Representation_of_the_symmetric_groups). Thi... | 6 | https://mathoverflow.net/users/66 | 222835 | 104,380 |
https://mathoverflow.net/questions/222744 | 5 | Let $\Phi$ be an irreducible root system and $\Delta$ a simple system (base). Let $W$ be the Weyl group of $\Phi$. Let $\theta$ be the highest root and $h^\vee$ be the dual Coxeter number. Choose the shortest $w \in W$ such that $w(\alpha\_i)=\theta$ for some simple root $\alpha\_i$. I got the result that the length of... | https://mathoverflow.net/users/50437 | Length of Weyl group element mapping highest root to a simple root | Apparently this isn't discussed in any of the published literature, even in the numerous exercises for Bourbaki's Chapter VI on root systems in *Lie Groups and Lie Algebras*. I'm not sure how strong the evidence is for the assertion here that the minimal length is always $h^\vee -2$. Even though it's true for some smal... | 4 | https://mathoverflow.net/users/4231 | 222837 | 104,381 |
https://mathoverflow.net/questions/222809 | 5 | Assume that $P\_n$ is the $n$'th prime: Please help me solve the following $$\lim\_{k\to\infty} {k}\prod\_{n=1}^k \frac{P\_{2n-1}}{P\_{2n}}$$
I am not really sure quite where to start here as I am dealing with primes, one of my weaknesses I want to make the important point: I am not a professional mathematician, in f... | https://mathoverflow.net/users/nan | The limit of the following product? What is the closed form of the value? | *Clarification:* This is not a full answer to the original question (which might be a very hard one), but rather an heuristic argument.
Let $F\_k = \prod\_{n=1}^{k} \frac{p\_{2n-1}}{p\_{2n}}, G\_k = \prod\_{n=1}^{k} \frac{p\_{2n}}{p\_{2n+1}}$. You are interested in $F\_k$, but heuristically at least, they should be c... | 4 | https://mathoverflow.net/users/31469 | 222845 | 104,382 |
https://mathoverflow.net/questions/222844 | 6 | Fix an algebraically closed field $F$. Are there only finitely many symmetric algebras with unit over $F$ of a given finite dimension (up to isomorphism)? By symmetric I mean a Frobenius algebra where the form is symmetric. For a general associative algebra I know it is false:
[Are there only finitely many associativ... | https://mathoverflow.net/users/82435 | Symmetric algebras of given dimension | The answer is certainly "no", but I can't immediately think of an example where it's easy to prove all the details.
In Karin Erdmann's (almost) determination of the structure of tame blocks, there are one-parameter families of algebras, where it's not known which parameters give algebras that actually occur as blocks... | 8 | https://mathoverflow.net/users/22989 | 222853 | 104,383 |
https://mathoverflow.net/questions/222852 | 3 | Let $p<1$ be a constant. Consider two sets $A,B$ with $n$ and $nf(n)$ vertices, respectively, where $f(n)$ is an integer. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the remaining edges. Is it true that as $n\rightarrow\infty$, the probability that the... | https://mathoverflow.net/users/79906 | Probabilistic many-to-one matching | The probability that a single vertex in $B$ has zero neighbours in $A$ is $(1-p)^n > 0$, and these events are independent for distinct elements of $B$. So if $f(n)$ is sufficiently large you expect to have an isolated vertex, which rules out having a matching of the desired form. Thus the failure of the union bound is ... | 4 | https://mathoverflow.net/users/25485 | 222854 | 104,384 |
https://mathoverflow.net/questions/222849 | 4 | Let $p, \ell\_1, \ell\_2$ be distinct prime numbers, and $x\_1, x\_2 \in \overline{\mathbf{Q}}^\times$.
If
$$ \frac{\log\_p x\_1}{\log\_p \ell\_1} = \frac{\log\_p x\_2}{\log\_p \ell\_2}, $$
does it follow that both ratios $\log\_p x\_i / \log\_p \ell\_i$ must be in $\mathbf{Q}$? (I know that both ratios must be ei... | https://mathoverflow.net/users/2481 | Transcendence of a ratio of p-adic logarithms | This is a typical case of the $p$-adic Four Exponentials conjecture. It is surely true, but the proof is beyond reach. If you add in a third prime (equality of $\log\_p{x\_i} / \log\_p{\ell\_i}$ for $i = 1,2,3$), then this becomes a case of the $p$-adic Six Exponentials theorem, a proof of which is available: [Serre J.... | 6 | https://mathoverflow.net/users/26522 | 222864 | 104,389 |
https://mathoverflow.net/questions/222851 | 3 | Are there any results about the rigidity of singular curves of rank 1 affine distributions on a connected compact Lie group?
Specifically the case of a right invariant affine distribution: $D\_{U} = \{ aU + \lambda bU | a,b \in \mathfrak{su}(n), \lambda \in \mathbb{R} \}$ on $SU(n)$ is of interest. I am aware of the ... | https://mathoverflow.net/users/41654 | Singular curves of affine distributions on a Lie group | **Corrected Answer**
This is a special case of a more general fact: Suppose that one has two linearly independent vector fields $A$ and $B$ on a manifold $M^n$ and that one wants to study the regularity of the curves $\gamma:[a,b]\to M$ that satisfy
$$
\gamma'(t) = A\bigl(\gamma(t)\bigr) + u(t)\,B\bigl(\gamma(t)\bigr... | 3 | https://mathoverflow.net/users/13972 | 222876 | 104,397 |
https://mathoverflow.net/questions/222819 | 1 | I have an exponentially bounded sequence $m\_n = \lambda^n + c\_n$ (i.e. the $c\_n$ are quadratic in $n$) and would like to know if this sequence of moments defines a distribution. I considered applying the [Hamburger Moment Problem](https://en.wikipedia.org/wiki/Hamburger_moment_problem), which means I would have to s... | https://mathoverflow.net/users/nan | Exponentially Bounded Sequence of Moments defining Distribution? | No. Because $c\_n$ is quadratic, the values of $m\_0, m\_1, m\_2$ can be arbitrary (subject to $m\_0 = 1$ since presumably we are looking at a probability distribution). The first condition that needs to be satisfied is that $m\_0 m\_2 - m\_1^2 \ge 0$, and by tweaking $c\_n$ appropriately you can easily arrange for thi... | 0 | https://mathoverflow.net/users/290 | 222886 | 104,404 |
https://mathoverflow.net/questions/222631 | 5 | I'm reading [Algebraic Geometry over $C^\infty$-rings](http://arxiv.org/abs/1001.0023).
It is written that "If $\mathfrak{C}$ is not finitely generated then $\Phi\_{\mathfrak{C}}:\mathfrak{C}\rightarrow \Gamma(\text{Spec}\mathfrak{C})$ need not surjective, so $\Gamma(\text{Spec} \mathfrak{C})$ can be larger than $\math... | https://mathoverflow.net/users/nan | global section of affine $C^\infty$-scheme | I think the error is in Joyce's definition 4.12 of $\mathop{\mathrm{Spec}}\mathfrak{C}$! Consider the case when $\mathfrak{C}$ has no real points (example below): then for all $c \in \mathfrak{C}$, $U\_c = \emptyset$ (as they are subsets of $X\_\mathfrak{C} = \emptyset$), but definition 4.12 attempts to set $O\_{X\_\ma... | 4 | https://mathoverflow.net/users/644 | 222887 | 104,405 |
https://mathoverflow.net/questions/222881 | 5 | In the proof of Theorem 1.5.7 (in which it is shown the the nuclei on a locale, when ordered by pointwise partial order, themselves form a locale) in the computation at the bottom of p.35, there is used an inequality
$$ j\Bigl( \bigl( j(b)\Rightarrow k(b)\bigr)\wedge b\Bigr) \Rightarrow k(b) ~~\leq~~ j(b)\Rightarrow ... | https://mathoverflow.net/users/82450 | Not sure how to fix an error in the Handbook of Categorical Algebra (vol 3) | You have a point that the last step is unclear. It might have been clearer if the last inequality were written as an equality instead, since that would essentially force the reader to verify that $b = (j(b) \Rightarrow k(b)) \wedge b$, or simply that $b \leq j(b) \Rightarrow k(b)$, which is true since it is equivalent ... | 11 | https://mathoverflow.net/users/2926 | 222891 | 104,407 |
https://mathoverflow.net/questions/222847 | 4 | Let $p=2^n-1$ be a Mersenne prime. We know that $H=PSL(2,p^4)$ has an irreducible character of degree $ p^4$.
Is there any solvable group of order $H$ with an irreducible character of degree $ p^4$. This is equivalent to: if we consider a solvable group $G$ of order $p^4 (p+1)(p^2+1)(p^4+1)$, can we say that $ G $ ha... | https://mathoverflow.net/users/31045 | Existence of some character degree in a solvable group | Given that $p$ is a Mersenne prime, it is easy to check that ${\rm gcd}(p+1,p^{2}+1) = {\rm gcd}(p+1,p^{4}+1)= {\rm gcd}(p^{2}+1,p^{4}+1) = 2$, and we recall that $p+1$ is a power of $2$. Let $G$ be a solvable group of order $p^{4}(p+1)(p^{2}+1)(p^{4}+1)$. Suppose that $G$ has an irreducible character of degree $p^{4}$... | 7 | https://mathoverflow.net/users/14450 | 222894 | 104,408 |
https://mathoverflow.net/questions/222880 | 1 | Let $V=\sum\_{i=1}^{n}a\_i(z\_1,\ldots z\_n)\frac{\partial}{\partial z\_i}$ be a holomorphic vector field defined on a neighborhood $U\subset \mathbb{C}^n$ of the origin, such that the common zero point of $a\_i$ are the origin, i.e. $V$ has isolated singularity. We know that if $V$ is a radial vector field $\sum\_{i=1... | https://mathoverflow.net/users/62735 | Holomorphic vector field with infinite separatrix | No, a lot more vector fields have infinitely many separatrices. They are called «dicritical». For instance, you can change locally the analytic coordinates $(z\_1,\ldots,z\_n)$, the vector field will not be radial anymore but still it will admit infinitely many separatrices (all oft its trajectories, actually). Aside t... | 1 | https://mathoverflow.net/users/24309 | 222900 | 104,410 |
https://mathoverflow.net/questions/222909 | 6 | I find that the following two types of functions are useful to my research.
(i) We know that a function $f: \mathbb{R}\_+^m\rightarrow \mathbb{R}$ is called convex if for all ${\bf x,y}\in \mathbb{R}\_+^m$ and all $\lambda\_1,\lambda\_2\geq 0$ s.t. $\lambda\_1+\lambda\_2=1$, it holds that \begin{equation}f(\lambda\_1... | https://mathoverflow.net/users/69874 | Two (new?) variants of convex functions | 1. If you throw in positive homogeneity, then the first class of functions is what is called *sublinear*, see for instance [Proposition 1.1.4 ("Fundamentals of Convex Analysis"; Hiriart-Urruty, Claude Lemaréchal)](https://books.google.com/books?id=hIYKBwAAQBAJ&lpg=PA123&pg=PA123#v=onepage&q&f=false).
2. The second clas... | 5 | https://mathoverflow.net/users/8430 | 222910 | 104,412 |
https://mathoverflow.net/questions/222872 | 10 | In Chang and Keisler's *Model Theory* I came across the following theorem (Theorem 7.2.13):
**Theorem** There exists a (first-order) sentence $\sigma$ such that for all infinite cardinals $\alpha$, $\sigma$ admits $(\alpha^{++},\alpha)$ iff there exists an $\alpha$-Kurepa tree.
**Definitions**
(1) Fix a predicat... | https://mathoverflow.net/users/13694 | Is there a (first-order) sentence which admits $(\aleph_2,\aleph_0)$ iff a Kurepa tree exists? | The problem, as I see it, is to build into the sentence $\sigma$ something that makes a certain definable set $A$ (in this case a set indexing the levels of your tree) have cardinality $\aleph\_1$. You've already ensured that $A$ isn't any bigger than $\aleph\_1$ by having surjections from $P$ to all of the initial seg... | 5 | https://mathoverflow.net/users/6794 | 222916 | 104,415 |
https://mathoverflow.net/questions/222896 | 5 | Let $X,Y$ be two compact, smooth, orientable 3 manifolds, each with an incompressible boundary component diffeomorphic to some genus $g $ surface $S\_g$. Under an orientation-reversig diffeomorphism $f:S\_g \to S\_g$, those two manifolds can be glued together to obtain a new smooth, orientable manifold $X \cup\_f Y$. I... | https://mathoverflow.net/users/78554 | Gluing two 3 manifolds along their boundary | The original question:
>
> Using a collar, one can show that if $f$ and $g$ are two isotopic diffeomorphisms of $S\_g$, then the corresponding gluings are diffeomorphic. Is this also a necessary condition?
>
>
>
No, it is not. Suppose we have glued to obtain $M = X \cup\_f Y$. Suppose that $X$ admits a self-... | 4 | https://mathoverflow.net/users/1650 | 222919 | 104,417 |
https://mathoverflow.net/questions/222813 | 5 | Suppose you have a finite group and you consider its Cayley graph with respect to some fixed generating set of nonidentity elements closed under inversion. Are there any results known to the effect that structural information about the group can be recovered from the spectrum of the adjacency matrix of the Cayley graph... | https://mathoverflow.net/users/15482 | recovering information about a group from the spectrum of its Cayley graph | The eigenvalues of the adjacency matrix of the graph are images of the generating set $S$ under the left regular representation of the group $G$. One would think that this should give some information about the representations of $G$ and thus information about $G$. However, in any many cases, a graph which may be viewe... | 0 | https://mathoverflow.net/users/11124 | 222930 | 104,420 |
https://mathoverflow.net/questions/222662 | 9 | Have you seen the following statement proven anywhere?
Let $G$ be a strongly regular graph with parameters $(n,k,\lambda,\mu)$ with $\lambda,\mu>0$. Then there is no set $A$ of at least $n/4$ vertices in $G$ such that the neighborhood of $A$ is of size less than $3n/4$.
A vertex in $A$ in this case would be include... | https://mathoverflow.net/users/82343 | Expansion in strongly regular graphs | Consider the complete tripartite graph $G = \langle X\cup Y\cup Z,E\rangle$ with $|X|=|Y|=|Z|=\frac{n}{3}$, and $E = X\times Y\cup X\times Z \cup Y\times Z$. This graph is strongly (n,2n/3,n/3,2n/3)-regular, and e.g. $X$ has more than $\frac{n}{4}$ vertices and less than $\frac{3}{4}n$ neighbors, so this is a counterex... | 5 | https://mathoverflow.net/users/17599 | 222932 | 104,422 |
https://mathoverflow.net/questions/222637 | 12 | This is a problem that has bugged me for quite some time, and I have not been able to find any documentation about it online. It is well known that the NN algorithm can yield the worst possible route - for small cases it would seem any (undirected) complete weighted graph with this property forces the FN algorithm to p... | https://mathoverflow.net/users/nan | Travelling salesman: can the furthest-neighbour algorithm beat the nearest-neighbour? | Label the vertices of the NN path by $0,1,2,\dots,n=0$. Let $x\_0=0,x\_1,\dots,x\_n=0$ be the FN path. Define $A=\{i\in \{1,\dots,n-1\}| \exists j>i \text{ with } x\_j>x\_i\}$. Then for any $i\in A$ we have by construction $d(x\_i,x\_i+1)\leq d(x\_i,x\_j)\leq d(x\_i,x\_{i+1})$. Let $b\_0=0< b\_1\dots < b\_k=n-1$ be the... | 9 | https://mathoverflow.net/users/35593 | 222933 | 104,423 |
https://mathoverflow.net/questions/222821 | 5 | Is there an example of a flat proper relative curve $X/S$ with geometrically connected fibres and with $\mathrm{dim} S > 1$ and $S$ regular and connected with $0$-dimensional locus of bad reduction $S\_{\mathrm{bad}} = \{s \in S: X\_s/s \text{ not smooth}\}$?
| https://mathoverflow.net/users/nan | curve over higher dimensional basis with 0-dimensional locus of bad reduction | The answer is no if $f:X\to S$ is locally projective and the genus $g$ of the general fiber is $\geq1$. (About these restrictions, see remarks at the end).
Assuming this, put $U:=S\smallsetminus S\_\mathrm{bad}$. By assumption, $S$ is regular and $U$ contains all points of codimension $\leq1$ of $S$. I do not assume... | 8 | https://mathoverflow.net/users/7666 | 222935 | 104,424 |
https://mathoverflow.net/questions/222906 | 0 | I am considering the following minimizing problem:
$$
\min\_{u\in BV(\Omega)}\{\frac12\|u-u\_0\|\_{L^2}^2 + |u|\_{TV(\Omega)}\}
$$
where $u\_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, smooth boundary.
The above problem surely has a unique minimizer $\bar u\in BV$.
Then I could write the correspo... | https://mathoverflow.net/users/62560 | The monotone operator in $BV$ space | $\varphi:=|.|\_{TV}$ is a convex and (I think) lower semicontinuous function from $L^2$ to $[0,\infty]$. Its subdifferential $\partial\varphi$ is what you need to write the Euler-Lagrange equation, in the form $\bar{u}-u\_0\in\partial\varphi(\bar{u})$.
| 0 | https://mathoverflow.net/users/75422 | 222942 | 104,426 |
https://mathoverflow.net/questions/222903 | 7 | Are there any known examples of oriented integer homology 3-spheres $Y$ (besides $S^3$) which have exactly one Stein-fillable contact structure up to isotopy? Failing that, what are the known examples with the smallest number of Stein-fillable contact structures where one exists? Note here I am only counting Stein-fill... | https://mathoverflow.net/users/50754 | Homology 3-sphere with a unique Stein-fillable contact structure | One example you might want to start with is the Poincaré homology sphere. It is a Seifert fibred space over $S^2$ with three singular fibres $M(\frac{1}{2}, - \frac{1}{3}, -\frac{1}{5})$ supporting exactly one Stein fillable contact structure. Note that the the Poincaré homology sphere with reversed orientation $\overl... | 4 | https://mathoverflow.net/users/82484 | 222944 | 104,428 |
https://mathoverflow.net/questions/222925 | 2 | A set $A$ is *completely productive* if there exists a computable function $f$ such that for every $e$, $f(e)\in (A-W\_e)\cup (W\_e-A)$. A set is *effectively non-recursive* if it is r.e. and its complement is completely productive (see Odifereddi's *Classical recursion theory*, p.304).
The set $K=\{n:~n\in W\_... | https://mathoverflow.net/users/65878 | Effectively non-recursiveness of some sets | It's not. Let $A$ be the complement of this set, and suppose that $f$ witnesses that $A$ is completely productive. Let $S$ be an infinite, computable set of indices that we control (via the Recursion Theorem), and let $i \notin S$ be another index that we control.
Define $W\_e$ as follows. If $f(e)=\langle m,n \rangl... | 8 | https://mathoverflow.net/users/47312 | 222947 | 104,429 |
https://mathoverflow.net/questions/222918 | 1 | I am encountering the following problem concerning existence of a limiting random variable (in distribution): assume a sequence of positive random variables $\{X\_n\}\_{n\geq 0}$ from which we know their moments ($\mathbb{E}[X\_n^s]$), and also the limit of these moments (which we denote by $\{m\_s\}\_{\geq 0}$). Let m... | https://mathoverflow.net/users/46573 | moment sequence which does not define a random variable vs convergence in distribution | $\newcommand{\bE}{\mathbb{E}}$ As Christian Remling pointed out in his comment, the Stiltjes condition is preserved under limits. Hence there will always exist a at least one positive measure finite measure $\mu$ on $[0,\infty)$ such that
$$ m\_s=\int\_0^\infty x^s \mu(dx),\;\;\forall s=0,1,2,\dotsc. $$
The measure... | 3 | https://mathoverflow.net/users/20302 | 222954 | 104,433 |
https://mathoverflow.net/questions/222938 | 3 | This is to understand better Example 3.9 on page 221 of *[Group actions on stacks and applications](https://perso.univ-rennes1.fr/matthieu.romagny/articles/group_actions.pdf)* by M.Romagny.
For an action of an algebraic group (scheme) $G$ on an algebraic stack $\mathcal{M}$, the author defines a fixed point stack $\m... | https://mathoverflow.net/users/4721 | On an example by Romagny about fixed point stack not commuting with coarse moduli space | I didn't check every detail, but I think it should work like this:
the point is that the existence of a $G$-invariant $H$-torsor (i.e. an object of $(BH)^G$) would give you a section $G\to H$, which does not exist (as Matthieu specifies in the paper).
To see that, assume you have your $H$-torsor $P\to T$, and pull ... | 1 | https://mathoverflow.net/users/5516 | 222956 | 104,434 |
https://mathoverflow.net/questions/222963 | 5 | How do I see that the 1st Chern class is invariant under choice of section? I know metric invariance follows from how two metrics on line bundle have to be conformally equivalent, but how do we show invariance under choice of section? Thanks in advance.
| https://mathoverflow.net/users/nan | 1st Chern class is invariant under choice of section? | There are many definitions of the $1$-Chern class of a complex line bundle $L\to M$, $M$ compact $CW$-complex. The topological one goes as follows. The line bundle $L$ is an oriented rank $2$ real vector bundle over $M$. As such it has a Thom class $\tau\_L\in H^2(D(L), S(L))$, where $D(L)$ and $S(L)$ are the disk and ... | 6 | https://mathoverflow.net/users/20302 | 222974 | 104,446 |
https://mathoverflow.net/questions/222693 | 3 | Consider an augmented commutative ring $R$, with augmentation ideal $\varpi$. Let $\delta$ be a derivation of $R$. The example I have in mind is $R=\mathbb F\_p[x]/(x^{p^i})$ and $\delta=d/dx$, though I would like statements as general as possible.
I would like to know whether $\varpi^m f=0$ implies $\varpi^{m+1}\del... | https://mathoverflow.net/users/10481 | Derivations annihilated by powers of the augmentation ideal | For the sake of completeness, here is the proof I suggested in the comments,
in some more detail.
**Lemma 1.** Let $\mathbf{k}$ be a commutative ring. Let $A$ be a $\mathbf{k}
$-algebra. Let $I$ be a two-sided ideal of $A$. Let $f:A\rightarrow A$ be a
derivation. Then, $f\left( I^{n+1}\right) \subseteq I^{n}$ for eve... | 1 | https://mathoverflow.net/users/2530 | 222998 | 104,453 |
https://mathoverflow.net/questions/222914 | 14 | If $A$ is an abelian group, we have
$Aut\left(K\left(A,n\right)\right)=Aut(A) \ltimes K\left(A,n\right),$
where the left hand side is the space of self-homotopy equivalences. Is there an easy way to see this abstractly rather than by computation?
Also, it seems there is an analogy with the following: if $\mathbb{... | https://mathoverflow.net/users/4528 | Automorphisms of Eilenberg-Mac Lane spaces and semidirect products (and the odd line) | As per Qiaochu Yuan's comment we need to only understand the space of based maps between $K(A,n)$ with a chosen base point.
The loop-deloop pair of functors establish an equivalence between the categories of $A^\infty$-groups and connected spaces with a base point:
$$\Omega: \mathrm{Top}\_{\*,\ \pi\_0=0} \simeq \math... | 10 | https://mathoverflow.net/users/10605 | 223004 | 104,454 |
https://mathoverflow.net/questions/222986 | 3 | Let $C$ be a genus 2 curve over $\mathbb{C}$. Let $X=J(C)$. Consider the involution $i$ on $X$, $x\mapsto -x$. Let $Y=\frac{X}{(i)}$. This is a singular surface with 16 points of singularity - these are the images of the 16 2-torsion points of $X$. Let $f:X\longrightarrow Y$ be the quotient morphism, which is finite of... | https://mathoverflow.net/users/70211 | Pullback of line bundles and divisors from $Kum(C)$ to $Jac(C)$ | 1) is correct, 2) is not.
Indeed, if $i(C)=C$, then the map $C \rightarrow C'$ is a double cover, and $f^\*{\mathcal O}\_Y(C')={\mathcal O}\_X(C)$ since in a neighbourhood of a general point of $C$ the map $X \rightarrow Y$ is biregular.
For a double cover $X\rightarrow Y$, in an analogous situation, you get $f^\*{... | 2 | https://mathoverflow.net/users/46104 | 223005 | 104,455 |
https://mathoverflow.net/questions/222212 | 3 | Let $(X\_1,X\_2,\ldots,X\_k)$ be distributed according to a multinomial distribution with parameters $(n;p\_1,p\_2,\ldots, p\_k),$ i.e.
$$P(X\_1=n\_1,\ldots,X\_k=n\_k) = {n\choose n\_1,n\_2,\ldots,n\_k} p\_1^{n\_1}\ldots p\_k^{n\_k}~,$$
for $n\_i\geq 0, \sum\_{i=1}^k n\_i = n.$
The multivariate central limit the... | https://mathoverflow.net/users/7576 | Uniform convergence of 2-norm of a multinomial vector | Let $(e\_1,\dots,e\_k)$ denote the standard basis in $\mathbb{R}^k$.
Then
$$\frac1{\sqrt n}\,(X\_1-np\_1,\dots,X\_k-np\_k)\overset{D}=\frac1{\sqrt n}\,\sum\_{j=1}^nV\_j=:V,
$$
where $\overset{D}=$ stands for the equality in distribution,
$V\_j:=U\_j-\mathbb{E}U\_j$, and the $U\_j$'s are iid random vectors in $\math... | 4 | https://mathoverflow.net/users/36721 | 223021 | 104,459 |
https://mathoverflow.net/questions/222985 | 4 | I am researching different generalizations of the familiar open mapping theorem from functional analysis. Every "proof" I attempt while simply assuming positive-homogeneity, even in the finite-dim case, has hit a giant brick wall, making me think it is not true, but I cannot seem to write down a counter-example.
Is t... | https://mathoverflow.net/users/82505 | An open mapping theorem for homogeneous functions? | Here is an example. In complex notation, $f(x+iy):=(x^2+iy^2)^4$ defines a homogeneous polynomial map from $\mathbb{R}^2$ to $\mathbb{R}^2$, which is clearly surjective, but not open (for instance, the image of the open set $\{(x,y)\in\mathbb{R}^2: |y|<|x|\}$ is the upper open half-plane plus the half-line $\{(x,0)\in\... | 5 | https://mathoverflow.net/users/6101 | 223030 | 104,462 |
https://mathoverflow.net/questions/222959 | 9 | **Background/Motivation**. We know that some of the usefulness of Martin's Axiom lies in giving certain "smallness" properties to sets of size less than continuum, e.g. we have
that for all infinite cardinals $ \lambda < 2^{\aleph\_0}, 2^{\lambda} = 2^{\aleph\_0}$; also all sets of reals of cardinality less than $2^{\a... | https://mathoverflow.net/users/81309 | Is there a modification of Martin's Axiom which admits non-measurable sets of size less than continuum? | Adding on to Andreas's comment above:
In Section 3 of Chapter XVIII of "Proper and Improper forcing", Shelah proves a very general preservation theorem. "Application 3.8" on page 912 deals with preserving positive outer measure, and in Claim 3.8C(2) he pins down the (very technical) condition he needs to prove that a... | 5 | https://mathoverflow.net/users/18128 | 223040 | 104,466 |
https://mathoverflow.net/questions/223053 | 4 | A crucial step in the "purely algebraic" proof of Weyl's semisimplicity theorem is that the Casimir element $C\in U\mathfrak{g}$ acts by nonzero scalars on a nontrivial irrep $V$. However, at least two sources I have consulted assert that $\text{tr}\_V(C)=\text{tr}(C)=\dim \mathfrak{g}$, i.e. using the fact that $C$ ca... | https://mathoverflow.net/users/58688 | Why is the trace of the Casimir on the irrep of a semisimple algebra nonzero? | The Casimir operator lies in the center of the universal enveloping algebra of $\mathfrak{g}$. By Schur's lemma, it must act by a scalar multiple of identity on any irreducible representation. For a highest weight representation you can actually directly compute this scalar by considering action of the Casimir on the h... | 6 | https://mathoverflow.net/users/6818 | 223055 | 104,473 |
https://mathoverflow.net/questions/223063 | 3 | Let $G$ be an amenable group acting on a space $X$.
Amenability means there is a $G$-invariant mean on $L^\infty(G,{\mathbf R})$.
Given a bounded function $f\colon X\to {\mathbf R}$ one can use the mean to define a $G$-invariant function $\overline{f}\colon X\to {\mathbf R}$: one just uses the mean to average over t... | https://mathoverflow.net/users/39082 | Averaging measurable functions over amenable group actions | I imagine that in your setting the group $G$ is assumed to preserve some measure $\mu$ on $X$, or at least the measure class of $\mu$ (you are not being very precise about what "space" means, or in what sense are things measurable).
The answer is "yes" (at least for discrete groups), and here is one way to see this. ... | 3 | https://mathoverflow.net/users/75274 | 223066 | 104,476 |
https://mathoverflow.net/questions/222719 | 1 | (*Much revised for clarity*.) I was considering the system of equations,
$$-a+nb+c = -d+ne+f\tag1$$
$$a+b+c = d+e+f\tag2$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag3$$
$$a^6+b^6+c^6 = d^6+e^6+f^6\tag4$$
**Question 1.** Is it true that, for a fixed integer $n$, if the system has an infinite number of co-prime integer solution... | https://mathoverflow.net/users/12905 | For what integer $n$ are there infinitely many $-a+nb+c = -d+ne+f$ where $a^6+b^6+c^6 = d^6+e^6+f^6$? | I am sorry to say that the answer to Question 1 is NO.
As you state, $q=(3-n)p/(2n)$ gives a solution to the problem. Thus the quartic must be birationally equivalent to an elliptic curve.
Using standard methods, it is possible to show that this elliptic curve is of the form
\begin{equation\*}
G^2=H^3+25( h\_1 H + ... | 6 | https://mathoverflow.net/users/56617 | 223077 | 104,479 |
https://mathoverflow.net/questions/223089 | 1 | There is at least 3 model structures on the category of topological spaces, the Quillen Model structure, the Storm model structure and the Mixed model structure.
In the Mixed model structure $\mathsf{MixTop}$ ( [mixed model structure](http://ncatlab.org/nlab/show/model+structure+on+topological+spaces)), weak equivalen... | https://mathoverflow.net/users/21369 | Different model structures on Top | No. Let $X$ be an uncountable set and consider $I^X$ with the product topology. Then the inclusion $\{ 0 \} \to I^X$ is a closed embedding between (strongly) contractible spaces which are therefore mixed cofibrant. However, this map is not a Hurewicz cofibration since it would follow that $\{ 0 \}$ is the zero set of a... | 5 | https://mathoverflow.net/users/12547 | 223090 | 104,483 |
https://mathoverflow.net/questions/223050 | 5 | Is it true that every quasi-projective rational irreducible algebraic complex variety is simply connected for the Euclidean topology?
Of course, this is false if we replace "complex" with "real" or if we forget "rational".
| https://mathoverflow.net/users/23758 | Is every complex rational algebraic variety simply connected for the Euclidean topology? | I want to mention the positive direction. Let $X$ be a smooth, projective variety over $\mathbb{C}$, resp. over an algebraically closed field of arbitrary characteristic. Let $Z\subset X$ be a proper closed subset. If $X$ is (separably) rationally connected and if $Z$ has codimension $\geq 2$ in $X$, then $X\setminus Z... | 13 | https://mathoverflow.net/users/13265 | 223095 | 104,485 |
https://mathoverflow.net/questions/223079 | 0 | In his paper "On discrete subgroups of the two by two projective linear group over $\mathfrak{p}$-adic fields", Yasutaka Ihara considers an abstract group $G$ together with a length function $l$ from $G$ into the non-negative integers with the following properties.
For each non-negative integer $l$, denote by $G\_{l}... | https://mathoverflow.net/users/15482 | Existence of the double coset ring on paper of Ihara | Given $x\in G$, then $x\in G\_l$ for some $l$.
Then $U\backslash UxU\subset U\backslash UG\_lU=U\backslash G\_l$. The latter is a finite set.
| 1 | https://mathoverflow.net/users/nan | 223098 | 104,487 |
https://mathoverflow.net/questions/223062 | 5 | Blumenthal's 0-1 law [see theorem 5.8/5.9](http://math.uchicago.edu/~may/REU2012/REUPapers/Leiner.pdf) tells us that an event in the germ $\sigma-$ algebra has either probability zero or one with respect to a measure induced by a Brownian motion starting in some point $x.$ (Theorem 5.8)
Conversely, one can show that ... | https://mathoverflow.net/users/82546 | Blumenthal and Kolmogorov 0-1 law | The issue, informally, is this. Suppose $W\_t$ is a Brownian motion started at some $x \in \mathbb{R}^d$. Let $X\_{t} = t W\_{1/t}$ as in the linked notes. Then $X\_t$ is a Brownian motion, but started at 0, not $x$. (By the strong law of large numbers, we have $\lim\_{t \to 0} X\_t = \lim\_{s \to \infty} \frac{1}{s} W... | 2 | https://mathoverflow.net/users/4832 | 223101 | 104,488 |
https://mathoverflow.net/questions/223001 | 0 | Following the line of [this](https://mathoverflow.net/q/45812/49870) question, I'm in my last year of M.Sc., and I'm looking for a place where I can start my PHD. Since that question has been asked 4 years ago, I thought it may be wise to ask whether something has changed.
That is, there are new research groups/univ... | https://mathoverflow.net/users/49870 | Computational Algebra and Symbolic Computation - Where? | Some particular places have already been mentioned in the answers
to the question you refer to, and I think it is not appropriate to give
in this place advice on where to do your PhD in computational algebra
concretely. That said, some general things I would do in your situation:
* First of all, search on the interne... | 0 | https://mathoverflow.net/users/28104 | 223115 | 104,491 |
https://mathoverflow.net/questions/223136 | 0 | The updated version can be found [here](https://mathoverflow.net/questions/223203/can-we-represent-a-n-1-rectifiable-set-locally-as-a-graph-with-some-price#223203).
Let $S\subset \mathbb R^N$ be a $N-1$ rectifiable set with $\mathcal H^{N-1}(S)<\infty$.
My question, for each $x\_0\in S$, would it be possible to cho... | https://mathoverflow.net/users/62560 | The partition of $N-1$ rectifiable set | The answer is "no".
In fact the answer "yes" to your question would imply that any Lipschitz simple curve the plane $\mathbb{R}^2$ can be presented locally as graph.
The latter does not hold in general.
Say take a logarithmic spiral.
One could also construct a $(1\pm\varepsilon)$-bi-Lipschitz curve which is not... | 1 | https://mathoverflow.net/users/1441 | 223146 | 104,505 |
https://mathoverflow.net/questions/223105 | 5 | The article "[Quasimöbius maps](http://link.springer.com/article/10.1007%2FBF02790198?LI=true)" by Jussi Väisälä states that one always has the implication QM $\implies$ QC. But a proof is only given in for maps of the form
$f:\dot{A} \to \dot{Y}$ where $A \subset \dot{X}$ and where $\dot{X} = X \cup \{\infty\}$ denote... | https://mathoverflow.net/users/82568 | Are quasi-Möbius maps always quasi-conformal? | It holds in general. I can't see exactly what Vaisala writes, since it's behind a paywall, but since quasiconformality is an infinitesimal property and quasi-Mobius is a global one, it is easy to prove the former from the latter. (The general idea to prove quasiconformality at $x$ is to use the cross-ratio condition on... | 3 | https://mathoverflow.net/users/82595 | 223147 | 104,506 |
https://mathoverflow.net/questions/223151 | 6 | Many people find ACC more intuitive than AC ("Pick something from the first set, then something from the second set, then...) and it also doesn't lead to "controversial consequences" (See for eg: [Peculiar examples with Axiom of Countable Choice ?](https://mathoverflow.net/questions/23246/peculiar-examples-with-axiom-o... | https://mathoverflow.net/users/76572 | Replacing Axiom of Choice with Axiom of Countable Choice | First of all, the axiom of countable choice says that given a countable family of non-empty sets, you can choose from each set *simultaneously*. If you want to choose from one, then from another, then from another, and so on you need a strictly stronger form of choice called **Dependent Choice**, abbreviated as $\sf DC... | 17 | https://mathoverflow.net/users/7206 | 223162 | 104,509 |
https://mathoverflow.net/questions/222994 | 8 | Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. Let $B = A^T A$. We know, for example, that $B$ is a square and symmetric $n$ by $n$ matrix with all its diagonal entries equal to $m$ exactly. I am trying to work out how to calculate (or estimate) the expected [Frobenius... | https://mathoverflow.net/users/45564 | How to calculate expected value of matrix norms of $A^TA$? | Expanding a bit on Yemon Choi's comment: concentration is indeed the key. First,
$\|B\|\_F^2$ is simply the sum of the squares of singular values of $A$, and
$E\|B\|\_F^2=m(m-1)n+mn^2$. On the other hand by standard concentration inequalities (using that $\|B\|\_F^2=\sum \sigma\_i^4$ where $\sigma\_i$ are the singular... | 6 | https://mathoverflow.net/users/35520 | 223163 | 104,510 |
https://mathoverflow.net/questions/223160 | 7 | A primary parallelohedron is a polyhedron that can fill space with infinite translated copies.
It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, Tutton, A. E. H. Crystallography and Practical Crystal Measurement, 2nd ed. London: Lubrecht & Cramer, 1964.) that the pr... | https://mathoverflow.net/users/20343 | Are the primary parallelotopes classified? (equivalently, Voronoi cells of lattices) | Voronoi conjectured that every parallelotope is combinatorially equivalent to a Voronoi cell of a lattice. The conjecture was proved for $d\le 4$ by Delone.
Reference: [Handbook of Convex Geometry, P. M. Gruber and J. M. Wills (eds.) page 1005](https://books.google.com/books?id=x2viBQAAQBAJ&lpg=PA1005&ots=Roat2o3I_g&... | 11 | https://mathoverflow.net/users/20186 | 223166 | 104,511 |
https://mathoverflow.net/questions/223168 | 18 | In a series of papers in Compositio Math. entitled On the classification of primitive ideals for complex classical Lie algebras I, II and III, Garfinkle describes an algorithm that allows one to determine the fibers of the Duflo map, and hence the left (or right) Kazhdan-Lusztig cells in the Weyl gorup for a Lie algebr... | https://mathoverflow.net/users/4614 | What happened to the fourth paper in the series "On the classification of primitive ideals for complex classical Lie algebras" by Garfinkle? | It was written, but never published.
[Tyson Gern's 2013 thesis](http://arxiv.org/abs/1304.6074) references it:
* D. Garfinkle. On the classification of primitive ideals for complex classical Lie algebras, IV. **unpublished**.
Fortunately, the same thesis discusses the proof of the type $D$ case (page 41), and giv... | 16 | https://mathoverflow.net/users/43108 | 223171 | 104,512 |
https://mathoverflow.net/questions/223131 | 6 | Let $K\_1$ and $K\_2$ be two $N \times N$ stochastic matrices (hence non-negative and rows adding up to one) with zeros on the diagonal. If $\alpha \in (0,1)$, is it possible to have
$$K\_1 K\_2 = \alpha K\_1 + (1-\alpha)K\_2?$$
| https://mathoverflow.net/users/82584 | Product and convex combination of two stochastic matrices | Let $$A=\pmatrix{0&.5&0&0&.5\cr.5&0&.5&0&0\cr0&.5&0&.5&0\cr0&0&.5&0&.5\cr.5&0&0&.5&0\cr}{\rm\ and\ }B=\pmatrix{0&0&.5&.5&0\cr0&0&0&.5&.5\cr.5&0&0&0&.5\cr.5&.5&0&0&0\cr0&.5&.5&0&0\cr}$$ Then $AB=.5A+.5B$.
| 6 | https://mathoverflow.net/users/3684 | 223181 | 104,516 |
https://mathoverflow.net/questions/223172 | 1 | Suppose $X$ is 1-d stochastically continuous process with $X(0) = 0$, i.e.
$X\_s \to X\_t$ in probability as $s\to t$ for all $t\ge 0$. Let $\tau = \inf\{t>0: |X\_t|>1\}$.
[Q.] Is $\tau>0$ almost surely?
I think the answer shall be yes, because $X$ has a Cadlag version (modification).
Thanks.
| https://mathoverflow.net/users/5656 | Hitting time of a stochastically continuous process | This is not true. Consider some random sequence that converges to $0$ a.s., for example, $Y\_n=(Z\_1+\cdots+ Z\_n)^{-1}$, where $Z$'s are i.i.d. (for instance) Exp(1) random variables. Set
$$
X\_t = \begin{cases}
2, & \text{if $t=Y\_k$ for some $k$},\\
0, &\text{otherwise}.
\end{cases}
$$
Then $X$ is stochastically... | 2 | https://mathoverflow.net/users/81488 | 223185 | 104,519 |
https://mathoverflow.net/questions/223152 | 6 | Let $M$ be a smooth manifold, and $G$ a Lie group with Lie algebra $\mathfrak{g}$. The Lie algebra of the diffeomorphism group of $M$ is the Lie algebra of vector fields on $M$; that is $\text{Lie}(\text{Diff}(M)) = \Gamma(TM)$.
Given a smooth left-action of $G$ on $M$ expressed as a Lie group homomorphism $\lambda ... | https://mathoverflow.net/users/56938 | Relationship between the Lie functor applied to a Lie group action, and the fundamental vector field mapping? | There are some sign issues for Lie algebra actions on manifolds. I will follow the conventions of [[BGV, section 1.1]](http://www.springer.com/us/book/9783540200628).
There is a natural left action on functions by
$$(g.f)(x)=f(\lambda\_{g^-1}x)\;.$$
If you differentiate it, you get
$$(X.f)(x)=\frac d{dt}\Bigr|\_{t=0}f(... | 3 | https://mathoverflow.net/users/70808 | 223197 | 104,524 |
https://mathoverflow.net/questions/223208 | 7 | Consider a vector-valued function $f: [0,1]^n\rightarrow[0,1]^n$. Write $f(x)=\{f\_1(x), ..., f\_n(x)\}$ with $x\in[0,1]^n$, where the $f\_i: [0,1]^n\rightarrow[0,1]$ are continuous functions with the following properties:
1) $f\_i(\{x\_1, x\_2, ..., x\_n\})=0$ if $x\_i=0$.
2) $f\_i(\{x\_1, x\_2, ..., x\_n\})=1$ if... | https://mathoverflow.net/users/44464 | Intermediate value for a vector-valued function | This has its own [Wikipedia entry Poincaré–Miranda theorem](https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Miranda_theorem) and is similar to the Brouwer fixed-point theorem.
| 7 | https://mathoverflow.net/users/78588 | 223217 | 104,532 |
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