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https://mathoverflow.net/questions/176127 | 3 | I suspect this problem is very famous and it must be studied very well. But I searched in Google and I did not find good reference. I will appreciate any answer and reference for any contribution about this question.
Firstly the motivation: We know that if $\gcd(a,b)=1$, then the sequence $an+b$ generates infinite p... | https://mathoverflow.net/users/19885 | irreducible polynomials on the polynomial sequence | If $g\_1(x)$ and $g\_0(x)$ are relatively prime, then there are infinitely many irreducibles of the form $g\_1(x) y + g\_0(x)$. This is due to Kornblum and Landau (1911) and recorded in many places; see Chapter 4 of [Number Theory in Function Fields](http://books.google.com/books?id=vDpa_C5DIbkC&pg=PA193&lpg=PA193&dq=%... | 10 | https://mathoverflow.net/users/297 | 176130 | 88,765 |
https://mathoverflow.net/questions/176111 | 6 | Suppose that $\mathcal{M}$ is a model category which is not combinatorial, does a homotopy limit in $\mathcal{M}$ correspond to a limit in the associated $\left(\infty,1\right)$-category?
How about when $\mathcal{M}$ doesn't have enough limits and colimits, but is otherwise a model category?
How about when $\mathca... | https://mathoverflow.net/users/4528 | Do homotopy limits compute limits in the associated quasicategory in the non-combinatorial model category case? | Homotopy limits in any model category always coincide with limits in the associated $(\infty,1)$-category. To see this, you need to know the following (classical) facts:
1) given a cofibrant object $A$, the mapping space functor $Map(A,-)$ (constructed as in Hovey's book, say, using a Reedy cofibrant resolution of $A... | 19 | https://mathoverflow.net/users/1017 | 176142 | 88,772 |
https://mathoverflow.net/questions/176104 | 2 | Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that $End\_{\overline{L}}(E)\otimes\_{\mathbf{Z}}\mathbf{Q}\simeq K$. One may associate to $E$ a Groessencharacter of $L$
taking values i... | https://mathoverflow.net/users/11765 | On the conductor of the Groessencharacter of a CM elliptic curve | The explicit formula should be: For each prime $p$ of $L$, take the local ring $\mathcal O\_p$ of $p$ and look at the induced map $\mathcal O\_p^{\times} \to \mathbb A\_L^\times \to K^\times$. Let $\pi$ be a uniformizer and take the smallest $n$ such that the image of $1+ \pi^n \mathcal O\_p$ along this map is trivial,... | 3 | https://mathoverflow.net/users/18060 | 176144 | 88,773 |
https://mathoverflow.net/questions/175885 | 6 | Let $M$ be a category equipped with a class of weak equivalences $W$. Is there a name for a morphism $f$ such that every pushout of $f$ (including, of course, $f$ itself) is a weak equivalence?
For example, if $(M,W)$ underlies a model structure, then any acyclic cofibration in the model structure is such a map. But ... | https://mathoverflow.net/users/49 | Morphisms every pushout of which is a weak equivalence | Maps $f$ such that every pushout along $f$ is a weak equivalence were called *couniversal weak equivalences* in the preprint [Homotopy theory for algebras over polynomial monads](http://arxiv.org/abs/1305.0086) by Michael Batanin and Clemens Berger. In left proper model categories such maps are characterized in Lemmas ... | 3 | https://mathoverflow.net/users/11540 | 176147 | 88,775 |
https://mathoverflow.net/questions/158272 | 5 | How to find out examples over elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$ $?$
| https://mathoverflow.net/users/30999 | Elliptic curves over $\mathbb{Q}$ with no rational torsion and $\mu$-invariant equal to 1 at $p=3$ | EDIT: There was a problem with my original answer. Details below the bottom line.
---
If you have any elliptic curve $E/{\mathbb Q}$ with a point of order 3, then we have an exact sequence of Galois modules:
$$
0 \to {\mathbb Z}/3{\mathbb Z} \to E[3] \to \mu\_3 \to 0.
$$
Here the ${\mathbb Z}/3{\mathbb Z}$ term a... | 9 | https://mathoverflow.net/users/86179 | 176154 | 88,778 |
https://mathoverflow.net/questions/164712 | 13 | I've been studying number theoretic methods of classifying hyperbolic 3-manifolds for over a year now. In particular, there is are the trace field, invariant trace field, quaternion algebra, and invariant quaternion algebra of a hyperbolic 3-manifold, or equivalently, of the associated Kleinian group. There are various... | https://mathoverflow.net/users/14835 | Origin of number theoretic invariants associated to hyperbolic 3-manifolds | The original extension of these invariants to general hyperbolic 3-manifolds was by [Neumann and Reid](http://www.math.columbia.edu/~neumann/preprints/nrarith.pdf).
I think they were motivated by the associated classification of arithmetic hyperbolic 3-manifolds. The general theory of arithmetic lattices was due to [... | 4 | https://mathoverflow.net/users/1345 | 176155 | 88,779 |
https://mathoverflow.net/questions/176145 | 2 | I realize there are a few different ways of going about proving this, depending on one's background, but there's a particular number theoretic aspect that I am just blanking on, and can't seem to find the info on. (BTW I'm aware of the "well known" result that noncompact arithmetic Kleinian groups are commensurable to ... | https://mathoverflow.net/users/14835 | Why do noncocompact arithmetic Kleinian groups have quadratic trace fields? | If $\mathcal{A}$ ramifies at a real place, then the group of units cannot contain parabolic elements, because the corresponding Galois conjugate lies in $SU(2)$ (the unit group of the real quaternions). However, a non-cocompact lattice contains unipotent elements (due to the Margulis Lemma, but really the Godement comp... | 3 | https://mathoverflow.net/users/1345 | 176157 | 88,781 |
https://mathoverflow.net/questions/176150 | 4 | Let $C=Spec \mathbb{C}[t]/(t^{n+1})$. Let $X$ be an algebraic (or complex analytic) scheme, flat over $C$ with the structure morphism $f\colon X\to C$. Assume that the special fiber is isomorphic to $\mathbb{C}\mathbb{P}^1$. **Is it true that there exits an isomorphism $g\colon X\tilde\to \mathbb{C}\mathbb{P}^1\times C... | https://mathoverflow.net/users/16183 | Flat family with special fiber $\mathbb{C}\mathbb{P}^1$ | Yes. $X$ is a Severi-Brauer scheme over $C$ (see this [Bourbaki talk](http://archive.numdam.org/ARCHIVE/SB/SB_1964-1966__9_/SB_1964-1966__9__199_0/SB_1964-1966__9__199_0.pdf) by Grothendieck, Cor. 8.3). Since any finite étale covering of $C$ is trivial, this implies by definition (loc. cit.) that $X$ is $C$-isomorphic ... | 7 | https://mathoverflow.net/users/40297 | 176159 | 88,782 |
https://mathoverflow.net/questions/176125 | 7 | If $f:C\to D$ is an equivalence of categories that is injective on objects, then every pushout of $f$ is also an equivalence. This follows, for instance, because such a functor is an acyclic cofibration in the canonical model structure on **Cat**.
Suppose conversely that every pushout of $f$ is an equivalence of cate... | https://mathoverflow.net/users/49 | Pushouts of equivalences of categories | Every flat functor is injective on objects.
In this [blog post](http://sbseminar.wordpress.com/2012/11/16/the-canonical-model-structure-on-cat/) Chris Schommer-Pries proves that there is a unique model structure on $\mathsf{Cat}$ with categorical equivalences as weak equivalences. The argument of his "Somewhat Less T... | 6 | https://mathoverflow.net/users/12547 | 176163 | 88,785 |
https://mathoverflow.net/questions/176175 | 3 | i was thinking about deformations of hyperkahler manifolds, in particular hilbert schemes of points on K3 surfaces and I think I realized something. I'm here to ask you if I'm right.
Take $X^{[n]}$ the Hilbert scheme of points on the K3 surface $X$. Now take $\pi:\mathcal{Y}\rightarrow Def(X^{[n]})$ a universal defor... | https://mathoverflow.net/users/56026 | Generic deformation of Hilbert scheme of points on K3 surface | I am not completely certain what you are asking. But if you want to see deformations of $\text{Hilb}^n$s "in nature", I recommend that you look at the (ever-growing) literature on the hyper-Kähler fourfold parameterizing lines on a smooth cubic hypersurfaces in $\mathbb{P}^5$. I recommend that you start with the semina... | 0 | https://mathoverflow.net/users/13265 | 176176 | 88,787 |
https://mathoverflow.net/questions/176184 | 10 |
>
> Is it the case that, for any given knot $K$,
> there exists some graph $G$ whose every embedding into $\mathbb{R}^3$
> (or into $\mathbb{S}^3$)
> contains a cycle that realizes $K$?
>
>
>
I know the famous Conway-Gordon result that the complete graph
on seven vertices $K\_7$ is intrinsically knotted in th... | https://mathoverflow.net/users/6094 | Is every knot unavoidable in the embeddings of some graph? | **Yes.** See this [paper](http://www.ams.org/journals/tran/1991-324-02/S0002-9947-1991-1069741-9/S0002-9947-1991-1069741-9.pdf) of Negami. The main result is that for any fixed knot (or link) of type $k$, there is a constant $R(k)$ such that every straight line embedding of $K\_{R(k)}$ in $\mathbb{R}^3$ contains a knot... | 12 | https://mathoverflow.net/users/2233 | 176186 | 88,791 |
https://mathoverflow.net/questions/176117 | 16 |
>
> Given a finite set of fields $k\_1, \ldots, k\_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m\_i$ such that $R/m\_i \cong k\_i$?
>
>
>
To prevent things from being too easy, I require two conditions on $R$:
1. $\operatorname{Spec}(R)$ should be connected (otherwise take $R = k\_1 \... | https://mathoverflow.net/users/44201 | Existence of a ring with specified residue fields | This answer basically fills in details in Oliver Benoist's comments: If $K$ and $L$ are fields with $|L| > |K|^{\aleph\_0}$, then $L$ and $K$ cannot be residue fields of $R$, with $\mathrm{Spec}(R)$ Noetherian and connected.
Proof: Suppose otherwise. Then there is a sequence $K = F\_0$, $F\_1$, $F\_2$, ..., $F\_r = L... | 8 | https://mathoverflow.net/users/297 | 176194 | 88,795 |
https://mathoverflow.net/questions/176190 | 1 | Suppose I have Neron model over some discrete valuation ring.
Is there a result such that the number of components of the fiber over the closed point cannot decrease after some based change?
In particular, in the case when it is the Neron model over discrete valuation ring $R$ of Jacobian of the model over $R$ of H... | https://mathoverflow.net/users/10300 | Neron model: can number of components decrease after based change? | Yes, there is such a result for Neron models of abelian varieties with semiabelian (aka semistable) reduction: "the number of components of the special fiber cannot decrease after base change". This is a special case of Prop. 3 in section 7.4 of Bosch, Lutkebohmert, Raynaud "Neron models".
Beyond the semiabelian case... | 8 | https://mathoverflow.net/users/5498 | 176197 | 88,797 |
https://mathoverflow.net/questions/176192 | 3 | Consider the function
$$f(x)=\prod\limits\_{n=1}^{\infty}(1-x^n)$$
I am interested in an asymptotic formula for $f$ as $x\to 1$. Of course $f\to 0$ but I am interested in how fast.
| https://mathoverflow.net/users/43313 | Growth rate of eta related function | From equation (II) in Newman's [A simple proof of the partition formula](http://projecteuclid.org/euclid.mmj/1028998729):
$$f(z) = \sqrt{\frac{2 \pi}{1-z}} \exp\left(-\frac{\pi^2}{6(1-z)}+\frac{\pi^2}{12} \right) (1+O(1-z))$$
Note that Newman's $f$ is the reciprocal of yours; I rewrote his formula to use your $f$.
... | 7 | https://mathoverflow.net/users/297 | 176198 | 88,798 |
https://mathoverflow.net/questions/176167 | 4 | $\newcommand{\scp}[2]{\langle #1,#2\rangle}\newcommand{\id}{\mathrm{Id}}$
Let $f$ and $g$ be two proper, convex and lower semi-continuous functions (on a Hilbert space $X$ or $X=\mathbb{R}^n$) and let $g$ be continuously differentiable. Consequently, the subdifferential $\partial f$ and the gradient $\nabla g$ are mono... | https://mathoverflow.net/users/9652 | Is the prox-residual monotone? | This inequality is not true. Here is a counterexample:
Let ${\bf x},{\bf y} \in \mathbb{R}^2$ with ${\bf y} = 0$ and ${\bf x} = (1,\epsilon)$ for $\epsilon$ small positive. Let $g(x,y) = Cy^2$ with $C(\epsilon)$ very large, chosen say so that
$${\bf x} - \nabla g({\bf x}) = (1, -M)$$
for some $M$ large.
We now cons... | 4 | https://mathoverflow.net/users/16659 | 176201 | 88,799 |
https://mathoverflow.net/questions/176219 | 3 | Suppose $p,q$ are two primes at even distance $k$. Must there necessarily exist a different pair $p',q'$ composed of entirely different numbers such that $p'$ and $q'$ are also at distance $k$?
Edit: None of the primes need to be consecutive.
| https://mathoverflow.net/users/24478 | If there are two primes at an even distance $k$ is there another disjoint pair of primes also at distance $k$? | It is conjectured that for *every* even number there are *infinitely* many pairs of primes at that distance.
Thus conjecturally the answer is: yes.
However, this is not known though there was huge progress around this problem recently by Zhang, Maynard and others.
The methods used to establish existence of prim... | 7 | https://mathoverflow.net/users/nan | 176220 | 88,807 |
https://mathoverflow.net/questions/176217 | 1 | I got a fun problem.
Define the alphabet $\mathcal{A}=\{0,1,2\}$ and the set $\mathcal{A}^{\leq n}=\{ x\_1x\_2\ldots x\_n: x\_i\in \mathcal{A}\}$ of words of length $n,$ for each $n\in\mathbb{N}.$
Suppose that for each $n\in \mathbb{N}$ somebody selects a set $X\_n$ of $2^n$ points of $\mathcal{A}^{\leq n}.$
Is... | https://mathoverflow.net/users/39115 | A problem in symbolic dynamics | An easy procedure to obtain a somewhat stronger result is the following: Consider those words of length $n$ that have exactly one $2$, and this $2$ is not in position $1$. There are $(n-1)2^{n-1}$ such words, which is already more than we need to form $X\_n$. Make an arbitrary selection of $2^n$ such words to define $X... | 3 | https://mathoverflow.net/users/48839 | 176226 | 88,811 |
https://mathoverflow.net/questions/176003 | 16 | A formal group law over a commutative ring $R$, (by [nLab](http://ncatlab.org/nlab/show/formal+group)) is a sequence of power srires
$$
f\_1,...,f\_n\in R[[x\_1,...,x\_n,y\_1,...,y\_n]]
$$
such that, using the notation
$$
x=(x\_1,...,x\_n),y=(y\_1,...,y\_n),f=(f\_1,...,f\_n)
$$
we have
$$
f(x,f(y,z))=f(f(x,y)... | https://mathoverflow.net/users/50409 | Formal group law is a group object in ...? | A formal group law over a scheme $S$ is a group object in the category of framed formal schemes over $S$. Objects in this category are formal schemes $X$ over $S$ equipped with an $S$-isomorphism $X \to \operatorname{Spf} \mathscr{O}\_S[[t\_1,\ldots,t\_n]]$ for some $n$.
There is a functor from framed formal schemes ... | 15 | https://mathoverflow.net/users/121 | 176233 | 88,814 |
https://mathoverflow.net/questions/122078 | 3 | Let $R$ be a local ring (commutative, Noetherian, over an algebraically closed field; if needed Henselian). Suppose one wants to prove some statement.
Suppose $R$ happens to be the ring of "functions", each of which being defined in a small enough neighborhood of the origin, then one can consider some representative ... | https://mathoverflow.net/users/2900 | working with local rings: "abstract" vs "geometric" proofs | In model theory the notion of categoricity captures the concept I think you are looking for. In particular Lefschetz principle is equivalent to the fact that the theory of algebraically closed fields is categorical. By this we mean that any two algebraically closed fields of the same cardinality (and characteristic, pr... | 1 | https://mathoverflow.net/users/56050 | 176234 | 88,815 |
https://mathoverflow.net/questions/176106 | 6 | In the introduction to the paper "On the Geometry of Holomorphic Flows and Foliations Having Transverse Sections" by Ito and Scardua, one reads the following "a holomorphic codimension one foliation on a compact manifold is not necessarily transverse to some compact Riemann surface. Indeed, the existence of such a comp... | https://mathoverflow.net/users/55998 | Holomorphic Foliations having transverse sections | If a smooth curve $C$ has zero self-intersection and is everywhere transverse to a foliation on a surface then there are also strong restrictions.
If $C$ is rational then the foliation is a Riccati foliation (i.e. is birationally equivalent to the projectivization of a meromorphic flat connection on rank two bundle ... | 7 | https://mathoverflow.net/users/605 | 176236 | 88,816 |
https://mathoverflow.net/questions/176237 | 8 | Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes.
*Goldbach's conjecture*: $\tau(x) = 0$
*Asymptotic Goldbach's conjecture*: $\tau(x) = O(1) $
*Weak asymptotic Goldbach's conjecture*: $\tau(x) = \omicron (\frac{x}{ln(x)}) $
**Question**: Is this weak asymptot... | https://mathoverflow.net/users/34538 | Is this weak asymptotic Goldbach's conjecture open? | The weak asymptotic Goldbach conjecture was proved by Chudakov in 1937 (based on the groundbreaking work of Vinogradov). Better bounds are known, see Lucia's comment. The preprint [arXiv:1804.09084](https://arxiv.org/abs/1804.09084) by Pintz shows that $\tau(x)<x^{0.72}$ for $x$ sufficiently large.
You can find the o... | 15 | https://mathoverflow.net/users/11919 | 176238 | 88,817 |
https://mathoverflow.net/questions/176158 | 29 | Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as $n\rightarrow\infty$? Does it increase, decrease, or converge (or "none of the above")?
| https://mathoverflow.net/users/56018 | Shortest path through $\sqrt{n}$ points out of $n$ | The following is a partial answer.
It is useful to rescale. So think of a Poisson process in the plane and consider the box of side $n$. There will be roughly $n^2$ points there.
Let $A\_{[a,b]}$ be the random variable you described, but in the box $[a,b]^2$. You are really interested in $A\_{[0,n]}/n$. You have ess... | 11 | https://mathoverflow.net/users/35520 | 176241 | 88,818 |
https://mathoverflow.net/questions/176239 | 3 | Consider the Grassmanian $G(k,n)$ ($k\le \frac{n}{2}$) and take its Plucker embedding. Consider now the space of all normal rational curves of degree $k$, contained in the Plucker embedding of the Grassmanian. Finally, take the closure of this space. I would like to know if this closure has some nice description.
Exa... | https://mathoverflow.net/users/13441 | Rational normal curves on Grassmanians | In your example it seems that you are identifying the Hilbert scheme $H$ of conics in $G(2,4)$ with $G(3,6)$. There is a morphism from $H$ to $G(3,6)$ mapping a conic $C$ to the plane spanned by $C$. However, there are planes contained in $G(2,4)$. If $L$ is the locus in $G(3,6)$ parametrizing planes contained in $G(2,... | 1 | https://mathoverflow.net/users/14514 | 176246 | 88,821 |
https://mathoverflow.net/questions/176207 | 6 | Let $L$ be a $k$-dimensional lattice in $\mathbb{R}^n$. The *covolume*
$\hbox{CoVol}(L)$ of $L$ is the $k$-dimensional volume of a
fundamental domain for $L$, i.e., the volume of the parallelopiped
spanned by a $\mathbb{Z}$-basis for $L$. Now let $m<n$, let $A$ be an
$m$-by-$n$ matrix with coordinates in $\mathbb{Z}$ a... | https://mathoverflow.net/users/11926 | Covolume of the row span of a matrix and of the kernel of a matrix | Equality (2) can be proved through the [Smith Normal Form](http://en.wikipedia.org/wiki/Smith_normal_form) of $A$. I am pretty sure there is a simpler proof then the one below.
One can decompose $A$ into: $A = U D V^t$ where $D = ( \hat{D}\,\, \left|\right.\,\, {0}\_{m \times (n-m)})$, $\hat{D}$ is diagonal $m \times... | 2 | https://mathoverflow.net/users/44293 | 176251 | 88,824 |
https://mathoverflow.net/questions/176269 | 5 | There is a discrete distribution where integers, $k$, from $1$ to $n$ occur with probability $p\_{k}$, all $p\_{k}$ are unknown.
Rather than having access to the distribution we have access to $n$ coins, with the $k$th coin having probability $p\_{k}$ of giving a result of heads.
Can we sample *exactly* from the di... | https://mathoverflow.net/users/56076 | Exactly sampling from a distribution with access to the probabilities only | I will assume all $p\_i$ are strictly between $0$ and $1$.
**Lemma** We can use a coin with probability $p$ of coming up heads to emulate a coin with probability $p/(1+p)$ of coming up heads.
**Proof** Flip the coin until it first comes up tails. The emulated coin is declared to be heads if we have flipped an even... | 5 | https://mathoverflow.net/users/297 | 176270 | 88,831 |
https://mathoverflow.net/questions/176274 | 1 | Let $X$ be a (compact) homogeneous space and $V$ be a homogeneous vector bundle on $X$ of rank $n$, and such that $\operatorname{dim}X\ge n$. Suppose $V$ has a section $s$, whose zeros $s=0$ form a sub-variety of $X$ of dimension $\operatorname{dim}X-n$. Is it true, that $V$ is a semi-ample vector bundle?
| https://mathoverflow.net/users/13441 | A semi-ampleness criterion for homogeneous bundles on homogeneous spaces? | Yes, that is true. By "compact homogeneous space", I assume that you mean a smooth projective variety over an algebraically closed field $k$ (presumably $\mathbb{C}$ for you) that is homogeneous under an algebraic action of a group $k$-scheme $G$ (or holomorphic action of a complex Lie group). Let $\mathcal{F}$ be a lo... | 3 | https://mathoverflow.net/users/13265 | 176281 | 88,835 |
https://mathoverflow.net/questions/176290 | 9 | I have been learning about homotopy type theory this summer. I am not a homotopy theorist but I am more comfortable with homotopy theory than I am with type theory, so the way I rationalize many of the constructions used is by thinking in terms of homotopy theory. For example, when someone says type, i think homotopy t... | https://mathoverflow.net/users/4002 | How should I be thinking about object classifiers / universal fibrations / universes? | Yes, the universe is the classifying space for small homotopy types. For various reasons, the universe is not itself a small homotopy type; so it fails to be an object classifier for the trivial reason that it does not classify itself.
Under the univalence axiom, the homotopy type of the universe is well-defined: it ... | 10 | https://mathoverflow.net/users/11640 | 176299 | 88,839 |
https://mathoverflow.net/questions/176297 | 1 | For a divisor $D$ on a smooth complex projective surface $X$, the *stable fixed part* is the maximal effective divisor $E$ which, for every $n \in \mathbb{N}$, is contained in every memeber of the complete linear series $|nD|$.
**Question.** *Is there a surface $X$ possessing nef and big divisors $D$ whose stable fi... | https://mathoverflow.net/users/26522 | Boundedness of the number of curves negative on a varying big divisor | I think the answer to (b) should be "no". The stable base locus you're considering is contained in the augmented base locus $\mathbf B\_+(D) = \bigcap\_{\text{$A$ ample}} \mathbf B(D-A)$. By Nakamaye's theorem, since $D$ is big and nef, we have $\mathbf B\_+(D) = \text{Null}(D) = \bigcup \{ C : D \cdot C = 0 \}$. (This... | 2 | https://mathoverflow.net/users/nan | 176300 | 88,840 |
https://mathoverflow.net/questions/176161 | 5 | Say I have $n$ balls each of $k$ different colors (i.e. $nk$ balls altogether), and I throw these balls independently into $N$ bins. Is there anything that can be said (in expectation, limits with respect to any of the variables, or otherwise) about the minimum number of bins that collectively contain at least one ball... | https://mathoverflow.net/users/56018 | Balls and bins with color | If $S$ is a set of bins, the probability that it contains at least one ball of every colour is $(1 - (1 - |S|/N)^n)^k$. There being $N \choose B$ such sets of cardinality $B$, the expected number of sets of cardinality $B$ that contain at least one ball of every colour is $E(B,N,n,k) = {N \choose B} (1 - (1 - B/N)^n)^k... | 2 | https://mathoverflow.net/users/13650 | 176303 | 88,841 |
https://mathoverflow.net/questions/176288 | 1 | Let $P$ be a pair of pants, $H(P)$ be the space of smooth hyperbolic Riemannian metrics with geodesic boundary on $P$, and $T(P)$ be the Teichmüller space of $P$ (quotient of $H(P)$ under smooth isotopies).
The function $L:H(P) \to (\mathbb{R}^+)^3$ which associates to each metric the length of the three boundary cur... | https://mathoverflow.net/users/7631 | Does the Teichmüller space of the pair of pants admit a continuous global section? | Since the fiber is contractible, there is a global section. The details are in the papers
Earle, Clifford J.; Eells, James,
A fibre bundle description of Teichmüller theory.
J. Differential Geometry 3 (1969) 19–43.
Earle, C. J.; Schatz, A.,
Teichmüller theory for surfaces with boundary.
J. Differential Geometr... | 3 | https://mathoverflow.net/users/21684 | 176306 | 88,843 |
https://mathoverflow.net/questions/176278 | 8 | I am trying to understand better the behaviour and character of modular tensor categories over non-algebraically closed fields. How weird can they be?
The reason I am interested in this is that my collaborators and I are investigating extended 3D tqfts. These are known to be related to Modular Tensor Categories via t... | https://mathoverflow.net/users/184 | How weird can Modular Tensor Categories be over non-algebraically closed fields? | An example of MTC is Drinfeld double of a finite group $G$ (over any field of
characteristic zero). This category contains representation category of $G$ as a subcategory. So all endomorphisms rings that you can find in representations of finite groups, you can also find in MTC. For example the quaternions will show up... | 8 | https://mathoverflow.net/users/4158 | 176318 | 88,849 |
https://mathoverflow.net/questions/173068 | 0 | I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form
$$|u(t)|\_{L^p} \leq f(t)|u\_0|\_{L^q}$$
for the usual type of functions $f$.
The problem I have with adapting the techniques from the sta... | https://mathoverflow.net/users/54557 | Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary | The paper *[Asymptotics of the porous media equation via Sobolev inequalities](http://www.sciencedirect.com/science/article/pii/S0022123605001291)* by Bonforte and Grillo seems to be what you require.
They key is the validity of a logarithmic Sobolev inequality.
| 0 | https://mathoverflow.net/users/49672 | 176321 | 88,850 |
https://mathoverflow.net/questions/168154 | 0 | In <http://www.math.uiuc.edu/K-theory/0357/> Karpenko utters the following:
>
> *Conjecture* 1.6. If an anisotropic quadric $X = Q$ possesses a Rost correspondence,
> then the quadratic form (defining $X$) is a minimal Pfister neighbor.
>
>
>
Since the Paper has been released 15 years ago, I wonder if there h... | https://mathoverflow.net/users/51251 | Rost Correspondence and minimal Pfister-Neighbors | Last week i asked Professor Nikita Semenov and he replied that this is still an open Problem.
| 0 | https://mathoverflow.net/users/51251 | 176323 | 88,851 |
https://mathoverflow.net/questions/176322 | 6 | In the 2-category **Cat** of small categories, for each category $C$ (an object of **Cat**) there is also the dual category (I dare not write "dual object") $C^{op}$.
Is ${op}$ the instance in **Cat** of a more general concept for 2-categories? (In the same sense that monads in **Cat** are a special case of monads-in... | https://mathoverflow.net/users/55915 | opposite category | Note that "taking the opposite" is an operation not just on categories, but also on functors and on natural transformations. Given a functor $F: C \to D$, there is an associated functor $F^{op}: C^{op} \to D^{op}$. (Notice that $(-)^{op}$ preserves the direction -- is covariant -- on functors.) Given a natural transfor... | 14 | https://mathoverflow.net/users/2926 | 176330 | 88,852 |
https://mathoverflow.net/questions/176326 | 2 | For an elliptic operator
$$ Lu = (a^{ij} D\_iD\_j + b^i D\_i + c)u = f,$$
with suitable assumptions on the coefficients, one usually has Schauder estimates of the form
$$ \|u\|\_{C^{2, \alpha}} \leq C(\|f\|\_{C^{0, \alpha}} + \|u\|\_{L^\infty}).$$
I am interested in the case $\alpha =0$. It seems that the corresponding... | https://mathoverflow.net/users/16702 | Estimate for elliptic problem on continuous functions | In the standard reference of Gilbarg and Trudinger there is an example of the function $u(x,y)=(x^2-y^2)\log^{1/2}\left(\frac1{x^2+y^2}\right)$ s.t. $\Delta u$ is continuous in some neighborhood of the origin, but $u$ does not belong to $C^2$ there. Adding some solution of а homogeneous equation would't help. So $f=\De... | 2 | https://mathoverflow.net/users/14551 | 176331 | 88,853 |
https://mathoverflow.net/questions/176319 | 4 | The question is in the title, amenability being understood as the existence of a left-invariant finitely additive probability measure on the group of interest. The case of countable groups is treated here :
[How to construct a continuous finite additive measure on the natural numbers](https://mathoverflow.net/questio... | https://mathoverflow.net/users/56097 | Is it compatible with ZF to assume that every amenable discrete group is finite? | It is consistent with ZF that there is no nonprincipal (i.e., singletons have measure 0) finitely additive probability measure at all. This appears in *Definability of measures and ultrafilters* by Pincus and Solovay [J. Symbolic Logic 42 (1977), no. 2, 179–190, doi: [10.2307/2272118](https://doi.org/10.2307/2272118), ... | 8 | https://mathoverflow.net/users/2000 | 176334 | 88,855 |
https://mathoverflow.net/questions/176305 | 5 | Given a *strong* (not only strict) convex function $f: \mathbb{R}^n\to\mathbb{R}$. On such problems, stochastic gradient decent (SGD) has a convergence rate of $O(1/T)$, where $T$ is the number of iterations [[1]](http://jmlr.org/proceedings/papers/v28/zhang13e.pdf).
How is the convergence rate affected if a constraine... | https://mathoverflow.net/users/42531 | Convergence rate of stochastic gradient decent with projections | See [here](http://arxiv.org/pdf/1212.2002v2.pdf) for a convergence analysis. As mentioned in the comments, the rate is still $O(\frac{1}{T})$ if the additional constraint keeps the feasible set $C$ convex and the algorithm used is projected stochastic subgradient descent with a suitable step size rule.
| 3 | https://mathoverflow.net/users/40595 | 176335 | 88,856 |
https://mathoverflow.net/questions/176333 | 10 | Let $I\_{\lambda},$ $\lambda>0$ be a subset of all irrational numbers $\rho=[a\_{1},a\_{2},...,a\_{n},...]\in(0,1)$ such that $a\_{n}\leq \text{const}\cdot n^{\lambda}.$
Here, $[a\_{1},a\_{2},...,a\_{n},...]$ is the continued fraction with partial quotients $a\_1,a\_2,\dots$.
My question is: under what values of $... | https://mathoverflow.net/users/56099 | Lebesgue measure of a set of irrational numbers | The condition $\lambda>1$ is sufficient and, at least almost, necessary:
To clarify, the space of irrational numbers $(0,1)-\mathbb Q$ is homeomorphic to
$\omega^\omega$ under the map that sends $\frac{1}{a\_1+\frac{1}{a\_2+\cdots}}$ to a function $f$ satisfying
$f(n)=a\_{n+1}-1$. (As is well known.)
This way Lebes... | 10 | https://mathoverflow.net/users/4600 | 176336 | 88,857 |
https://mathoverflow.net/questions/176227 | 18 | **Warmup question:**
Let us say that two continuous functions $f,g:[0,1]\to \mathbb R$ are ***topologically transverse*** if their difference $f-g$ has only finitely many zeros, and each zero separates an interval where $f>g$ from an interval where $f<g$ (and let's say that we also impose $f(0)\not = g(0)$ and $f(1)... | https://mathoverflow.net/users/5690 | Topological transversality | The answer to your first two questions is yes. For the third, I don't think I understand the correct generalization of topological transversality to higher dimensions. I'll prove the second. While the first doesn't strictly follow (one needs to show that transversality can be achieved in the space of graphical curves),... | 6 | https://mathoverflow.net/users/300 | 176337 | 88,858 |
https://mathoverflow.net/questions/176339 | 10 | This is a minor curiosity that came up in a joint project recently.
Consider the sequence $a\_n=3\frac {(2n)!}{(n+2)!(n-1)!}$ (A000245 in OEIS).
It has multiple combinatorial descriptions.
One can write the corresponding generating function, but I will do it with an extra sign.
$$
f(t) = -\sum\_{n=0}^{\infty} a\_n... | https://mathoverflow.net/users/38468 | Combinatorial interpretation of composition of power series? | Basically $f(f(t)) = t$ is related to the fact that $y = f(t)$ and $t$ satisfy an algebraic relation $$y^2 t^2 - 3 y t + y + t = 0$$ which is symmetric in $y$ and $t$ and its graph passes through $(0,0)$ tangent to the line $y=-t$. That relation corresponds to the identity
$$ \sum\_{j=1}^{n-1} a\_j a\_{n-j} + 3 a\_{n+1... | 9 | https://mathoverflow.net/users/13650 | 176342 | 88,860 |
https://mathoverflow.net/questions/176320 | 0 | Let $u\_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u\_n \rVert\_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$.
It follows that for almost all $t$, $u\_n(t)$ is bounded in $L^\infty(\Omega)$, so we can extract a weak-\* convergent subsequence and after a bit of work we can show that
$$\text{fo... | https://mathoverflow.net/users/49672 | Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why? | The explanation is that by the hypotheses you in fact have $\lVert u\_n \rVert\_{\ell^{+\infty}([0,T],L^{+\infty}(\Omega))} \leq C$ for all $n$ as for arbitrarily fixed $t\in[0,T]$ you can take a sequence of $t\_i\to t$ with $\lVert u\_n(t\_i) \rVert\_{L^{+\infty}(\Omega)} \leq C$ and $u\_n(t\_i)\to u\_n(t)$ in $H^{-1}... | 2 | https://mathoverflow.net/users/12643 | 176354 | 88,868 |
https://mathoverflow.net/questions/176315 | 4 | I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra computations. Also, I would like this package to be able to do wedge and hodge dual, and other computation relating to forms.
Ca... | https://mathoverflow.net/users/56095 | Mathematica package for supergravity and string theory | You may try xact
<http://www.xact.es/AVF/index.html>
or see if <http://sagemanifolds.obspm.fr/> for sage works for you.
| 1 | https://mathoverflow.net/users/32185 | 176357 | 88,870 |
https://mathoverflow.net/questions/176349 | 4 | Suppose $(a,3q)=1$ and $a\equiv 1\pmod 3$. Are there infinitely many primes $p\equiv a\pmod {3q}$ such that $3$ is a cubic nonresidue modulo $p$?
Or, an equivalent formulation using quadratic forms: by cubic reciprocity the problem is equivalent to asking whether there are infinitely many primes $p\equiv a\pmod {3q}$... | https://mathoverflow.net/users/54486 | How frequently is 3 a cubic residue mod primes in an arithmetic progression? | The answer is yes; this follows from the Chebotarev density theorem.
To avoid technicalities, I only consider here the case when $3$ does not divide $q$. Let $K\_1 = \mathbf{Q}(\zeta\_q)$ and $K\_2=\mathbf{Q}(\zeta\_3, \sqrt[3]{3})$. These are both Galois extensions of $\mathbf{Q}$. We claim that these are linearly d... | 13 | https://mathoverflow.net/users/16510 | 176358 | 88,871 |
https://mathoverflow.net/questions/176360 | 11 | For an associative algebra $A$ one can define the Hochschild cohomology of $A$ as $ HH^n(A,A):= Hom\_{\mathcal{D}(A^{op} \otimes A)}(A, [n]A)$ (this definition also works for the graded and dg cases as well). There is a lot of information contained in this gadget: when $ n=0$ it is the center and when $n=1$ it is the l... | https://mathoverflow.net/users/38075 | Motivation behind the definition of hochschild cohomology | I think you will have different answers to your questions. As you noticed there is more about Hochschild Homology in n-lab page and you also pointed out the interpretation of (positive) low dimension of Hochschild cohomology. I would say that the Hochschild cohomology is "The" cohomology theory for differential graded ... | 8 | https://mathoverflow.net/users/21369 | 176365 | 88,875 |
https://mathoverflow.net/questions/176348 | 2 | The complex projective plane $\mathbb{C}P^2$ can be thought of as the set of rank one 3-by-3 Hermitian matrices with norm one, i.e., $\mathbb{C}P^2 = \{xx^\* : x \in \mathbb{C}^3, x^\*x=1 \}$. As such, the space $\mathbb{C}P^2$ is a manifold with $4$-real dimensions embedded in the space of all $3$-by-$3$ Hermitian mat... | https://mathoverflow.net/users/54756 | Simple connectedness of $\mathbb{C}P^2$ intersected with an affine subspace | Put $V=\{A\in H\_3(\mathbb{C}):a\_{11}=a\_{33}=1/3\}$. I claim that $V\cap\mathbb{C}P^2$ is the set of matrices of the form
$$ P = \frac{1}{3}\left[\begin{array}{ccc} 1 & z & zw \\ \overline{z} & 1 & w \\ \overline{zw} & \overline{w} & 1 \end{array}\right],
$$
where $|z|=|w|=1$. The proof is not hard if you just expand... | 3 | https://mathoverflow.net/users/10366 | 176377 | 88,878 |
https://mathoverflow.net/questions/176386 | 0 | I stumbled across this "dictionary for noncommutative topology" <http://planetmath.org/noncommutativetopology>
and I would be very interested in learning more on the subject, particularly I'd like to see why these results are true.
What is the reference to the results in section 3 of the above linked page?
| https://mathoverflow.net/users/56120 | Specific Reference? Noncommutative topology and C^* algebras | *Les C$^\*$-algèbres et leurs représentations* (answers most of these).
Find it [here](http://www.google.ca/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&ved=0CDgQFjAF&url=http%3A%2F%2Fiecl.univ-lorraine.fr%2F~Nicolas.Prudhon%2Fgnc%2Fbibliographie%2FOperator%2520Algebras%2FDixmier%2520%5B1977a%2C%2520288pp%5Dh---C-star-a... | 0 | https://mathoverflow.net/users/36886 | 176390 | 88,881 |
https://mathoverflow.net/questions/173682 | 1 | Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T\_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider the unit normal vector $v\_p\in T\_{exp(p)}M$. Is it true that for gereric $\pi$ the set $N(\pi)=\{(exp(p),v\_p): p\in\pi... | https://mathoverflow.net/users/40911 | Ratner theorem and dense geodesic planes in hyperbolic manifolds | I don't see how Ratner's theorem could be used here as I do not see any relation of your setup with **unipotent** flows. On the other hand, a stronger statement actually follows from ergodicity of the **frame flow** (a generalization of the geodesic flows).
In the 3-dimensional case the state space of the frame flow ... | 3 | https://mathoverflow.net/users/8588 | 176391 | 88,882 |
https://mathoverflow.net/questions/172371 | 15 | I thought in asking this question on Math StackExchange, but by my experience I don' t think anyone will notice me. Recently, I started studying deformation of complex manifolds in the sense of Kodaira and Spencer, but some things relating infinitesimal approximations are not clear for me.
Let $\varpi: \mathscr{M} \t... | https://mathoverflow.net/users/40883 | Kodaira-Spencer theory of deformation done right | I am not sure which book you are using . But from Kodaira, Morrow's book $\it complex$ $\it manifolds$ page 149-151, the construction of vector-value $(0,1)-$form $\varphi$ will tell you why the equality holds, also you will see why $t$ is required to be close enough to 0.
| 7 | https://mathoverflow.net/users/40220 | 176397 | 88,883 |
https://mathoverflow.net/questions/176338 | 3 | Let $U$ be a finite set and $\mathcal O$ a set of subsets of $U$, how many subsets $\mathcal S$ of $\mathcal O$ satisfy the union of the elements of $\mathcal S$ is equal to $U$?
I think the problem of finding the smallest of these subsets is called the set cover problem, I can't find any reference to this structure ... | https://mathoverflow.net/users/24478 | Given A set $U$ and a set $\mathcal O$ of subsets of $U$, how many subsets of $\mathcal O$ have union $U$? | EDITED:
Let $F({\cal O}, U)$ be the number of ways to cover $U$ using a **multiset** $\cal O$ of subsets of $U$ (so coverings using different versions of the same set are considered as different). Let $A$ be one element of $\cal O$, which occurs with multiplicity $k$ in $\cal O$. We can either include $A$ or not. If... | 2 | https://mathoverflow.net/users/13650 | 176399 | 88,885 |
https://mathoverflow.net/questions/142501 | 3 | I am reading A. Yekutieli's [original article](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Yekutieli,%20A%2a&s5=dualizing%20complexes&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=9&mx... | https://mathoverflow.net/users/17353 | Balanced dualizing complexes according to A. Yekutieli | I happened to be looking at this very issue a few days ago! Here is my explanation to myself...
Indeed, in the notation above, there is a homomorphism of complexes of graded modules (not derived)
$$ \zeta : \Gamma\_{\mathfrak{m}} (J) \to
\mathrm{Hom}\_{A^{\mathrm{op}}} ( \mathrm{Hom}\_A ( J, I ), \Gamma\_{\mathfra... | 4 | https://mathoverflow.net/users/56127 | 176406 | 88,888 |
https://mathoverflow.net/questions/176410 | 2 | Consider the polynomial ring $S = C[x\_1,\ldots,x\_n]$. Let
$$
0 \rightarrow E\_{n-1} \rightarrow \cdots \rightarrow E\_1 \rightarrow E\_0 \rightarrow I \rightarrow 0 ,
$$
be a minimal free resolution of an ideal $I$ of $S$, where
$$
E\_p = \bigoplus \_j S(-a\_{pj}) .
$$
The regularity of $I$ is $... | https://mathoverflow.net/users/50249 | Castelnuovo Mumford Regularity | By the construction of derived functors, $\mathrm{Tor}(I,A)$ is the homology of the complex
$$0 \to E\_{n-1} \otimes A \to \cdots \to E\_1 \otimes A \to E\_0 \otimes A.$$
(All tensor products are $\otimes\_S$.) In the particular case that $A = S/\mathfrak{m} =:k$, we have
$E\_p \otimes k= \oplus\_j k(-a\_{pj})$, so $\... | 3 | https://mathoverflow.net/users/297 | 176414 | 88,890 |
https://mathoverflow.net/questions/176380 | 5 | Let P denote the pair of pants e.g. a sphere minus three small discs $D\_1$,$D\_2$,$D\_3$ about marked points $x\_1,x\_2,x\_3$. I then consider $P \times S^1$. We have boundary components $T\_1$,$T\_2$,$T\_3$. Fix two homeomorphisms $f\_1: \partial(S^1 \times D^2) \to T\_1 $ and
$f\_2: \partial(S^1 \times D^2) \to T\_... | https://mathoverflow.net/users/36931 | What 3-manifolds can be obtained by gluing $ S^1 \times P $ and two copies of $S^1 \times D^2$ | You get:
- the solid torus,
- all the [Seifert manifolds](http://en.wikipedia.org/wiki/Seifert_fiber_space) fibering over the disc with two exceptional fibers;
- the connected sum $(D\times S^1) \# L(p,q)$ of a solid torus and a lens space $L(p,q)$, for all coprime $(p,q)$.
The proof goes as follows. Consider th... | 9 | https://mathoverflow.net/users/6205 | 176421 | 88,894 |
https://mathoverflow.net/questions/176407 | 4 | Let $X,Y$ be Borel spaces and $A\subseteq X\times Y$ be an analytic set. Let $\pi:X\times Y \to X$ denote the projection map onto $X$. Does there always exist a set $B$ such that $\pi(B) = X\setminus \pi(A)$ and such that the union $A\cup B \subseteq X\times Y$ is still analytic?
In particular, $B$ can't be an analyt... | https://mathoverflow.net/users/11768 | Analytic enlargement of an analytic set | The answer is no, not necessarily. It can happen that there is no such set $B$.
A counterexample is provided whenever $A$ is a Borel subset of the plane, while the projection $\pi(A)$ is analytic, but not co-analytic. In this case, if $A\cup B$ were analytic, it would follow that $B$ itself is analytic, since $B=(A\... | 4 | https://mathoverflow.net/users/1946 | 176427 | 88,899 |
https://mathoverflow.net/questions/176417 | 13 | **Theorem.** The Galois group of a quintic polynomial $f\in\mathbb{Q}[x]$ is $A\_5$ if and only if its discriminant is a rational square and its Weber sextic resolvent has no rational root.
**Question.** What are known infinite families of of quintic polynomials in $\mathbb{Q}[x]$, each with Galois group $A\_5$?
I ... | https://mathoverflow.net/users/31084 | Families of quintics in $\mathbb{Q}[x]$ with Galois group $A_5$ | A result of Mestre (Extensions regulieres de $\mathbb{Q}(T)$ de groupe Galois $\overline{A}\_{n}$, Journal of Algebra, 1990) allows you to take "almost all" quintic $A\_{5}$ polynomials $P(x)$ and extend them into an infinite family of polynomials giving $A\_{5}$ extensions. (Mestre's argument works for $A\_{2n+1}$ wit... | 16 | https://mathoverflow.net/users/48142 | 176431 | 88,902 |
https://mathoverflow.net/questions/176385 | 0 | Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta\_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair $(X,\Delta\_X)$ is klt. Let us take the image $\Delta\_Y=f(\Delta\_X)\subset Y$. Is the pair $(Y,\Delta\_Y)$ still klt or not?
... | https://mathoverflow.net/users/14514 | Small birational maps and singularities of the pair | Here is a very simple example and you can generalize this to get higher dimensional and more complicated ones:
---
Let $X=\mathbb P^2$ and $\Delta=\alpha\cdot C$ where $C$ is a smooth proper (and hence irreducible) curve of degree $>1$ and $0<\alpha<1$ a rational number. Clearly $(X,\Delta)$ is klt. Now let $p\_... | 2 | https://mathoverflow.net/users/10076 | 176438 | 88,905 |
https://mathoverflow.net/questions/176416 | 0 | Can anyone provide a proof of the equidistribution theorem using Weyl's criterion for the case of $c\*a \,\,\, \text{(mod 1)}$ where $c=2^n: \,\,\, \forall n \in N\_0$ for irrational algebraic $a$? The original equidistribution theorem is for the case that $c = n: \,\,\, \forall n \in N\_0$. Much thanks in advance.
| https://mathoverflow.net/users/23772 | Proof of equidistribution theorem for exponential coefficients | You are asking whether a number $a$ is a normal number in base 2 or not. This is an extremely complicated problem, and very far from being solved. Generally speaking, there exist several constructions of normal numbers, but we don't have any tool to determine whether a given number is normal or not. It is (sometimes) c... | 3 | https://mathoverflow.net/users/46852 | 176446 | 88,909 |
https://mathoverflow.net/questions/176393 | 11 | 1. Let $G$ be a reductive algebraic group over ${\mathbb Z}$ (or a finite
localization of a ring of integers $R$ in a number field) acting on an
affine scheme of finite type $M=Spec(A)$ over $R$.
We can take the invariants $A^G$, and then reduce
modulo a prime $p$: $A^G\otimes\_{\mathbb Z}{\mathbb F}\_q$. We can first... | https://mathoverflow.net/users/22180 | GIT over integers | The answer to the first question is affirmative if we assume the generic fiber is normal.
First, some general setup (so it is well-posed to speak of "compatibility with any base change" below). Let $S = {\rm{Spec}}(D)$ be an affine scheme and $X = {\rm{Spec}}(A)$ a finitely presented affine $S$-scheme equipped with ... | 12 | https://mathoverflow.net/users/52824 | 176452 | 88,911 |
https://mathoverflow.net/questions/176459 | 4 | Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation $\sim\_f\subseteq X^2$ given by $x\sim\_fx'$ iff $f(x) = f(x')$. Since $\sim\_f = (f\times f)^{-1}(\Delta\_Y)$ where $\Delta\_Y$ i... | https://mathoverflow.net/users/11768 | Quotients of standard Borel spaces | The answer is no, and this kind of question is part of the subject of the theory of [Borel equivalence relations](http://www.math.ucla.edu/~greg/223b.1.06s/borel.pdf).
The equivalence relations $\sim$ for which there is a Borel function $g:X\to Z$ into a standard Borel space $Z$, with $x\sim y\iff g(x)=g(y)$ are, by... | 3 | https://mathoverflow.net/users/1946 | 176461 | 88,912 |
https://mathoverflow.net/questions/175969 | 3 | Given integers $ n,r,s $, we can define $M(n,r,s)$ to be the maximal size of a family $F$ of $r$-subsets of $\{1,...,n\}$ such that the pairwise intersection between any two subsets is at most $s$. Is there any way to get a (non-trivial) upper bound on $M(n,r,s)$?
If we want the intersections to be at least $s$ (inst... | https://mathoverflow.net/users/17599 | An upper bound on families of subsets with a small pairwise intersection | The maximum size is attained by a Steiner system $S(t+1,r,n)$ when it exists. It consists of $\binom{n}{t+1}/\binom{r}{t+1}$ blocks.
See <http://en.wikipedia.org/wiki/Steiner_system>
| 4 | https://mathoverflow.net/users/7076 | 176462 | 88,913 |
https://mathoverflow.net/questions/176356 | 8 | This can be viewed as a toy version of [this](https://mathoverflow.net/questions/176158/shortest-path-through-sqrtn-points-out-of-n) wonderful question. (I'm removing the TSP component, and I'll zoom in on the statistical part.)
Let $X\_1,\ldots, X\_n$ be iid, with uniform distribution in $[0,1]$. Let $\ell\_n$ be th... | https://mathoverflow.net/users/48839 | Clusters of uniformly distributed random points | As in the $2$-dimensional problem linked, the probability that a particular interval much shorter than $n^{\alpha-1}$ contains $n^\alpha$ points is very small, and we can use the union bound over a small set of such intervals to bound the probability that the shortest interval containing $n^\alpha$ points is small. Thi... | 5 | https://mathoverflow.net/users/2954 | 176465 | 88,914 |
https://mathoverflow.net/questions/176456 | 2 | Let $S$ be a semigroup and $I,J$ be two ideals of $S$. For a semilattice we know that $IJ=I\cap J$. Now the question is there a semigroup with the property $IJ=I\cap J$. thanks for your attention
| https://mathoverflow.net/users/56158 | a question about semigroups | I assume you are asking which semigroups have the property that the intersection of any two ideals is their product.
One condition that will do it is von Neumann regularity. $S$ is regular if, for all $s\in S$, there is $t\in S$ with $sts=s$.
In any semigroup $IJ\subseteq I\cap J$. If $S$ is regular and $s\in I\c... | 3 | https://mathoverflow.net/users/15934 | 176469 | 88,916 |
https://mathoverflow.net/questions/176468 | 1 | Rosenthal introduce the notion of the extendable local reflexivity for Banach spaces as follows: Let $X$ be a Banach space and let $\lambda\geq 1$. $X$ is said to be $\lambda$-extendably locally reflexive
($\lambda$-ELR) if for every finite-dimensional subspaces $E \subset X^{\*\*}$
and $F \subset X^{\*}$, and for ever... | https://mathoverflow.net/users/41619 | A formally weaker form of the extendable local reflexivity for Banach spaces | The answer is yes by a small perturbation argument. For a small $\epsilon$, enlarge $F$ so that $F$ $1+\epsilon$-norms $E$. Get $T$ from the condition. Choose any Auerbach basis $u\_k$ for $E$ and take biorthogonal functions $x\_k^\*$ to $u\_k$ with $x\_k^\*$ in $F$. So $\|u\_k\|=1$ and by the norming condition you can... | 1 | https://mathoverflow.net/users/2554 | 176477 | 88,920 |
https://mathoverflow.net/questions/176432 | 4 | For what $n \geq 3$ does there exist an $n \times n$ matrix such that:
* All entries are in $(0, 1)$.
* Each row and column sums to $1$.
* Aside from the rows and columns, no other subsets of the entries sum to $1$.
EDIT: I had a comment about $n = 3$ likely not being possible, but I removed that now after some hel... | https://mathoverflow.net/users/56140 | Magic squares with specific properties | Such a matrix (integer version) can be constructed from a magic square, say with entries $1,\dots,n^2$, by adding suitable multiples of all $n\times n$ permutation matrices: e.g. take $N=n^2$ and add all $N^iP\_i$, where $P\_1, \dots,P\_{n!}$ are in any order.
This is such a wasteful construction that the question r... | 2 | https://mathoverflow.net/users/29783 | 176480 | 88,922 |
https://mathoverflow.net/questions/176449 | 3 | Let $V$ be a proper singular complex algebraic variety, possibly nonprojective ($dim(V)=n>0$). I would like to know:
1) if its second Betti number is non zero,
2) same question but now $V$ is a compact Moishezon space.
Edit: (as written by Francesco Polizzi) $b\_2\geq 1$ in the case of normal projective varieti... | https://mathoverflow.net/users/27816 | Second betti number of compact analytic spaces | $\def\ZZ{\mathbb{Z}}$ Let $X$ be a complete toric variety. I will show that $H^2(X, \mathbb{Z}) \cong A^1(X, \mathbb{Z}) \cong \mathrm{Pic}(X)$. Since there are examples of complete toric $3$-folds with trivial $\mathrm{Pic}$, this shows that these toric varieties have trivial $H^2$. I rely heavily on the theory of Min... | 8 | https://mathoverflow.net/users/297 | 176494 | 88,931 |
https://mathoverflow.net/questions/176443 | 6 | I believe many such links can be constructed by looking at a foliation similar to the hopf fibration, but the wrapping leaves replaced with $(p,q)$ torus knots. However, I'm interested in particular in whether there are classes of examples that don't fall under that construction.
| https://mathoverflow.net/users/50693 | Is it known which links have Seifert fibered complements? | I think this paper exactly answers your question <http://projecteuclid.org/euclid.dmj/1077378799> .
But you are basically right, the links you get are fibres (possibly degenerate) in some (possibly singular) Seifert fibration of $S^3$ and pretty much it can only be fibred in the ways you mentioned.
| 2 | https://mathoverflow.net/users/27433 | 176497 | 88,934 |
https://mathoverflow.net/questions/176460 | 6 | In their paper "On Three-Dimensional Space Groups", Conway et al. write
>
> Although this paper was inspired by the orbifold concept, we did not need to consider the 219 orbifolds of space groups individually. We hope to discuss their topology in a later paper.
>
>
>
Did this discussion ever materialize? Or di... | https://mathoverflow.net/users/48084 | Orbit spaces of crystallographic groups | The orientable Euclidean orbifolds were described in section 7 of this [paper of Dunbar](http://www.mat.ucm.es/serv/revmat/vol1-123/vol1-123f.html).
To complete the list, note that every crystallographic group has an index 2 orientation-preserving subgroup. The corresponding orientable orbifold quotients will be rea... | 8 | https://mathoverflow.net/users/1345 | 176503 | 88,939 |
https://mathoverflow.net/questions/176519 | 8 | Let $A$ be an alphabet of $k$ symbols,
and $p$ a *pattern*.
An example of a pattern is $p=XX$, where $X$ is any finite
string of symbols from $A^+$.
Avoiding $p$ is avoiding any subword repeated twice in a row.
Such strings are called *[square-free](http://en.wikipedia.org/wiki/Square-free_word)*.
It is well known th... | https://mathoverflow.net/users/6094 | Über theorem on unavoidable patterns? | According to the 2013 paper ["Computing the Partial Word Avoidability Indices of
Ternary Patterns"](https://www.uncg.edu/cmp/research/patterns2/TernaryPatterns.pdf) by Blanchet-Sadri, Lohr, and Scott,
>
> The problem of deciding whether a given pattern is avoidable has been
> solved [1, 14], but the one of decidin... | 7 | https://mathoverflow.net/users/4600 | 176520 | 88,946 |
https://mathoverflow.net/questions/176464 | 2 | Let $V=\mathbf{Z}^N$ be a free $\mathbf{Z}$-module of rank $N$. Let $S\subseteq V$
be a fixed finite subset.
Consider the submodule $M:=\langle S\rangle\leq V$ generated by $S$. We know form the general theory that $M$ is a free $\mathbf{Z}$-module of rank $n\leq N$.
**Q**: Is there a theoritical criterion to dete... | https://mathoverflow.net/users/11765 | General criterion to find a Z-basis in a fixed generating subset | The condition is the following. Let $A$ be the matrix whose rows are
the elements of $S$. Then there is some ordering $v\_1,\dots,v\_n$ of
some $n$ elements of $S$ such that the for all $1\leq k\leq n$, the
gcd of the $k\times k$ minors of the matrix whose rows are
$v\_1,\dots,v\_k$ equals the gcd of the $k\times k$ mi... | 3 | https://mathoverflow.net/users/2807 | 176521 | 88,947 |
https://mathoverflow.net/questions/176511 | 3 | Call two arrows $f$ and $g$ *disjoint* if the pullback of $f$ by $g$ is the initial object. Here's my question: Does there exist a sublocale $j: J\to L$ which is not disjoint with any other (non-initial) sublocale of $L$, but which is disjoint with other non-initial arrows.
| https://mathoverflow.net/users/2884 | Disjoint arrows in the category of locales | No, such thing cannot happen.
(Here was some blunder depending on the assumption that pullback of a nontrivial sublocale along a surjection is nontrivial, which, as Simon Henry points out in the comment below is not true.)
What is actually true is that if a sublocale of $L$ is not disjoint from any nontrivial sublo... | 4 | https://mathoverflow.net/users/41291 | 176529 | 88,950 |
https://mathoverflow.net/questions/176528 | 0 | I've been struggling with proving a conjecture concerning order statistics of Brownian motions for a while. The conjecture I'm looking to prove is the following: (I have run Monte Carlo simulations that confirm the conjecture.)
Let $(q\_1(t), …, q\_n(t))$ be a vector of independent Brownian motions with $q\_i(0) = 0$... | https://mathoverflow.net/users/56182 | Order statistics of Brownian motions | The strategy of the other $n-1$ players can be viewed as a single adversary trying to minimize the payoff of the $n$th player. Picking the highest unpicked object is optimal for that adversary by coupling.
If $t\_{i-1} \le t\_i' \lt t\_i$ then picking at time $t\_i'$ instead of $t\_i$ does an imperfect job of choosi... | 0 | https://mathoverflow.net/users/2954 | 176530 | 88,951 |
https://mathoverflow.net/questions/172027 | 3 | A Galton-Watson tree is the family tree of a Galton-Watson process. Let $T\_n$ denote a Galton-Watson tree conditioned on total population size $n$. The time of extinction is its height $H(T\_n)$ and its diameter $D(T\_n)$ is the length of a longest path in the tree. It is known that for certain offspring distributions... | https://mathoverflow.net/users/53071 | Ratio of expected diameter and height of a conditioned Galton-Watson tree | Yes, it is known. By results of Haas & Miermont [<http://arxiv.org/pdf/1003.3632v3.pdf> - Lemma 33] (see also Addario--Berry, Devroy & Janson [<http://arxiv.org/pdf/1011.4121v1.pdf> - Theorem 1.2 - for stronger bounds], $ H(T\_n)/\sqrt{n}$ is bounded in $L^2$, and so is $D(T\_n)/\sqrt{n}$ because $D(T\_n) \leq 2 H(T\_n... | 2 | https://mathoverflow.net/users/56188 | 176537 | 88,954 |
https://mathoverflow.net/questions/176545 | 3 | Let $G$ be a finite group possessing a generating set of order $n \in \mathbb{N}$. Let $H \leq G$ and $x\_1, \dots, x\_n \in G$ for which $\langle H, x\_1, \dots, x\_n \rangle = G$. Must there be $h\_1, \dots, h\_n \in H$ such that $\langle x\_1h\_1, \dots, x\_nh\_n \rangle = G$?
| https://mathoverflow.net/users/38889 | Generating finite groups | Here's an example. Let $S$ be a simple nonabelian group, $s$ an element of order 2 in $S$ and $c\in S$ such that $S$ is generated by $s$ and $c$ (there are plenty of examples).
Let $G=S^3$; it can be generated by 2 elements. Define $H=\{(1,1,1),(1,s,s),(s,1,s),(s,s,1)\}$. It is easy to check that the subgroup generat... | 6 | https://mathoverflow.net/users/14094 | 176549 | 88,961 |
https://mathoverflow.net/questions/176374 | 4 | In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious embedding from finite partial orders into Priestley spaces is the pro-completion of finite partial orders.
**Main question... | https://mathoverflow.net/users/54548 | Profinite completion of a partial order | Let $\mathbf{Po}$ be the category of posets with monotone maps, $\mathbf{PrSp}$ the category of Priestley spaces. Then let $U:\mathbf{PrSp}\to \mathbf{Po}$ be the forgetful functor. The profinite completion functor $P:\mathbf{Po}\to\mathbf{PrSp}$ would be the left adjoint of $U$. Let $\mathbf{DLat}$ be the category of ... | 3 | https://mathoverflow.net/users/54548 | 176551 | 88,962 |
https://mathoverflow.net/questions/176471 | 4 | I just posted this question as a comment to the question [Hypersurfaces and Elliptic Points](https://mathoverflow.net/questions/52554/hypersurfaces-and-elliptic-points)
but I don't know how many people will see it.
It's well known and easy to prove that a point on a closed hypersurface in $\mathbb{R}^n$ that has maxi... | https://mathoverflow.net/users/613 | Point of maximal distance on a non-positively curved PL surface | It is easy to see that point of maximal distance has to be a vertex.
Further, all the edges starting from this vertex point in the directions of open half-space.
That is it — any vertex with the described property can appear as a point of locally maximal distance. (For global maximum, in addition it has to be a verte... | 3 | https://mathoverflow.net/users/1441 | 176557 | 88,964 |
https://mathoverflow.net/questions/176466 | 6 | I have been learning about Brieskorn homology 3-spheres $\Sigma(a\_1,...,a\_n)$ and Seifert manifolds. My reference is the first few pages of Saveliev's "Invariants of Homology 3-spheres."
If I understand correctly, it is explained that one can realize the homology three spheres by considering a trivial bundle $S^2 ... | https://mathoverflow.net/users/36931 | A question about Dehn surgery and Brieskorn homology 3-spheres | Seifert manifolds that are homology spheres can be determined as follows. Let
$$ M =(S^2, (p\_1,q\_1), \ldots, (p\_h,q\_h)) $$
denote the Seifert manifold obtained by filling the $h$ tori in $D\_h \times S^1$ (where $D\_h$ is the 2-sphere minus $h$ discs) with coprime parameters $(p\_i, q\_i)$. Since we consider Seife... | 7 | https://mathoverflow.net/users/6205 | 176561 | 88,965 |
https://mathoverflow.net/questions/176538 | 10 | [This](http://arxiv.org/abs/1310.0989v4) arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but after some point I couldn't due to the lack of explanation, or my ability to understa... | https://mathoverflow.net/users/56187 | Is the Manickam-Miklós-Singhi Conjecture solved? | I am aware of the paper, but I am not sure that MO is the right forum for this sort of question. Nonetheless, let me try to provide some information in as neutral a manner as possible.
Note that there has been a flurry of recent activity concerning the MMS conjecture. Indeed, a paper of Huang and Sudakov *just* appe... | 6 | https://mathoverflow.net/users/2233 | 176563 | 88,967 |
https://mathoverflow.net/questions/176566 | 2 | Let $X$ and $Y$ be standard Borel spaces, and let $f:X\to Y$ be a surjective Borel map. Does there exist a Borel inverse of $f$, that is a Borel map $g:Y\to X$ such that $f\circ g = \mathrm{id}\_Y$.
In case $f$ is injective, such map does exist, so I am interested in the case when $f$ is not injective. The fact that ... | https://mathoverflow.net/users/11768 | Inverse of a Borel surjection | Your question is really about the [uniformization problem](http://www.math.ucla.edu/~ynm/lectures/dst2009/dst2009.pdf#page=16), a major focus of descriptive set theory. A set $B\subset X\times Y$ is *uniformized* by a set $C\subset B$ if $C$ is the graph of a function with the same projection as $B$. In other words, th... | 6 | https://mathoverflow.net/users/1946 | 176568 | 88,969 |
https://mathoverflow.net/questions/176571 | 2 | Let $V\cong\mathbb C^n$ be a complex vector space of dimension $n$. Let $\lambda\in\mathbb Z^r$ be a generalized integer partition $\lambda\_1\ge\cdots\ge\lambda\_r$ with $r\le n$. Denote by $\mathbb S\_\lambda(V)$ the Schur module with respect to $\lambda$, i.e. the irreducible $\operatorname{GL}(V)$ representation of... | https://mathoverflow.net/users/9947 | Symmetric invariants of a Schur Module | The symmetric group action you have described is exactly the action of the weyl group on the weight spaces. Let $v$ be a highest weight vector. Then $$ \sum\_{\sigma \in S\_n} \sigma v $$ is non zero and invariant under the symmetric group action. The reason it is non zero is because the component in the highest weight... | 4 | https://mathoverflow.net/users/4002 | 176575 | 88,973 |
https://mathoverflow.net/questions/176553 | 1 | How can we find the proof of the following statement:
An undirected graph is edge transitive if and only if its line graph is vertex transitive.
| https://mathoverflow.net/users/31179 | Vertex transitive and edge transitive and line graph | G. Sabidussi, Math. Zeitschr. 76, 385-401 (1961) proved that the automorphism group of a (simple) connected graph is isomorphic to the automorphism group of its linegraph, with the obvious isomorphism, unless the graph is one of four exceptions: the complete graphs $K\_2$ and $K\_4$, a triangle with an extra edge attac... | 11 | https://mathoverflow.net/users/9025 | 176579 | 88,974 |
https://mathoverflow.net/questions/107269 | 1 | Hello !
It's well know that any sublocale of regular locale is the intersection of a familly of open sublocale. Hence if $X$ is a regular locale, the map which to a sublocal $Y \subset X$ associate $ \lbrace o \in \mathcal{O}(X), y \subset o \rbrace $ is injective.
my question is : do we know it's image, at least whe... | https://mathoverflow.net/users/22131 | Intersection of open sublocale of a compact regular locale ? | Here is some sort of characterization; it is not really satisfactory but may be useful for further improvements.
Let us talk in terms of nuclei on frames. For a frame $A$ let $\mathbf NA$ be the frame of its nuclei. Then the question is how to characterize those subsets of $A$ which are equal to $\varphi\_\*j:=\{\ a\... | 1 | https://mathoverflow.net/users/41291 | 176582 | 88,975 |
https://mathoverflow.net/questions/176588 | 11 | Let $\text{Cat}$ be the category of (small) categories and functors. There is a ["categorical"](http://ncatlab.org/nlab/show/canonical+model+structure+on+Cat) (also called "canonical" or "folk") model structure on $\text{Cat}$ in which the weak equivalences are the usual equivalences of categories (and as explained [he... | https://mathoverflow.net/users/50409 | Constructing a "geometric" model structure on Cat by localizing the "categorical" model structure | No, there are far too many "canonical cofibrations" for that. For example, let $A$ be a category with two objects and two parallel arrows between them. Then there is a unique functor $A \to [1]$ that is injective on objects and hence a "canonical cofibration". The homotopy type of $A$ is $S^1$ so the homotopy type of t... | 13 | https://mathoverflow.net/users/12547 | 176590 | 88,979 |
https://mathoverflow.net/questions/175816 | 4 | **Question**: Consider the complex simple Lie group $E\_6$. Let $\lambda\_1$ and $\lambda\_6$ be the fundamental weights defining the $27$-dimensional representation $V$ and $V^\*$, resp. Consider the complex span $\mathfrak a^\*=\langle \lambda\_1,\lambda\_6\rangle$. The subquotient of the Weyl group $W=W(E\_6)$ that ... | https://mathoverflow.net/users/15155 | Calculation with weights of $E_6$ | The answer is yes. Let $\lambda,\mu\in{\mathfrak a}^\*$ and $w\in W$ such that $w(\lambda)=\mu$. Then we want to show that there exists $w'\in Z\_W(\mu)$ such that $w'w({\mathfrak a}^\*)={\mathfrak a}^\*$.
We can think of ${\mathfrak a}^\*$ as a subalgebra of ${\mathfrak g}$ (via the Killing form restricted to the Ca... | 3 | https://mathoverflow.net/users/26635 | 176595 | 88,980 |
https://mathoverflow.net/questions/176584 | 6 | Let $\mathbb{S}$ denote Sacks forcing. This is forcing with perfect trees or equivalently forcing with uncountable Borel subsets of ${}^\omega 2$ with the relation $\subseteq$.
Let $G \subseteq \mathbb{S}$ be a generic filter over $V$. Suppose $B$ is a Borel set coded in $V[G]$. This means $B$ is a subset of $({}^\o... | https://mathoverflow.net/users/43354 | Borel Sets in Sacks Generic Extension | No. For example, let $B$ be the set of reals Turing above $G$. Then $B$ has no nonempty Borel subset with a Borel code in $V$.
| 9 | https://mathoverflow.net/users/14340 | 176596 | 88,981 |
https://mathoverflow.net/questions/176598 | 2 | The motivation to this question can be found in:
[About equivalent statements of the Birch and Swinnerton-Dyer Conjecture](https://mathoverflow.net/questions/125050/about-equivalent-statements-of-the-birch-and-swinnerton-dyer-conjecture?rq=1)
My question is about the last equivalences:
$\mathrm{ord}\_{s=1} L(E/K... | https://mathoverflow.net/users/25947 | How the equality in the first case is equivalent to the inequality in the last case? | This follows (for $A/K$ an Abelian variety over a function field or even for $K/\mathbf{F}\_q$ finitely generated) since one always has $r\_{\text{an}} \geq r$ ($r\_{\text{an}}$ the analytic order $\mathrm{ord}\_{s=1}L(A/K,s)$, and $r$ the rank of $A(K)$).
I proved this in my PhD thesis, which will be available in a ... | 1 | https://mathoverflow.net/users/nan | 176599 | 88,982 |
https://mathoverflow.net/questions/176600 | 1 | Let $C$ be a smooth projective curve over a field $K$ of characteristic $0$ (but not necessarily algebraically closed). Let $\mathcal{L}$ be a line bundle on $C$ of degree $0$. Fix an integer $r>1$. Does there exist a line bundle $\mathcal{L}'$ on $C$ such that $\mathcal{L}'^{\otimes r} \cong \mathcal{L}$? If not true ... | https://mathoverflow.net/users/54369 | Perfectness of the Jacobian of a curve | No in general, for instance if $K$ is a number field : assuming that $C$ has a rational point, the group $\mathrm{Pic}(C)$ is naturally isomorphic to $JC(K)$, the group of rational points of the Jacobian of $C$, which is finitely generated by the Mordell-Weil theorem. So $\mathrm{Pic}(C)$ may very well contain points w... | 5 | https://mathoverflow.net/users/40297 | 176604 | 88,983 |
https://mathoverflow.net/questions/176602 | 3 | For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric powers are stable. As Mumford observed, such vector bundles give rise to ruled complex surfaces ($\mathbb{P}^1$-bundles ov... | https://mathoverflow.net/users/26522 | Mumford-Ramanujam examples in characteristic p [and in Arakelov geometry] | The Mumford construction has been extended in any characteristic by Mehta and Subramanian, see [this paper](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0219&DMDID=DMDLOG_0024).
| 3 | https://mathoverflow.net/users/40297 | 176606 | 88,985 |
https://mathoverflow.net/questions/176587 | 1 | Is there a criterion for a based map $f:G\to H$ between deloopable pointed spaces to be deloopable, i.e., that there is a based map $g:X\to Y$ between path-connected pointed spaces such that $G\simeq \Omega X$, $H\simeq \Omega Y$ and $f=\Omega g$?
You may assume that you know $G$ and $H$ are already deloopable, i.e.... | https://mathoverflow.net/users/nan | Criterion for deloopable based map | As Mark Grant says, $f$ has to be an $A\_\infty$-map. Depending on your situation, it may also be useful to look at it in the following way:
The classifying space $BG$ of the loop space $G$ has a skeletal filtration $B^{(n)}G$ coming, for instance, from the $n$-skeleton of the simplicial space $NG$, the nerve of $G$.... | 2 | https://mathoverflow.net/users/4183 | 176608 | 88,986 |
https://mathoverflow.net/questions/176371 | 7 | Let $ M $ be a countable, transitive model for $ \mathsf{ZFC}^\* $, i.e. for a sufficiently large finite fragment of $ \mathsf{ZFC} $. Suppose that $ \mathbb{P} := (P, {\leq\_P}, \mathbb{1}\_P) \in M $ and $ \mathbb{Q} := (Q, {\leq\_Q}, \mathbb{1}\_Q) \in M $ are forcing notions.
---
Reminder
--------
**Definit... | https://mathoverflow.net/users/48586 | Embeddings of forcing notions - preserve properness? | **The following implications are either trivial or well known:**
*(D$ n $)* implies *(C$ n $)*: Trivial.
*(C3)* implies *(C1)*: Use $ G := i^{-1}[H] $. (See [Kun80, VII.7.5].)
*(C3)* implies *(C4)*: The composition of two complete embeddings is a complete embedding.
*(D3)* implies *(D1)*: Use $ G := i^{-1}[H] $... | 2 | https://mathoverflow.net/users/48586 | 176611 | 88,987 |
https://mathoverflow.net/questions/176610 | 2 | I am currently doing some investigations on *Sylvester's 4 Point Problem* [Probability of 4 Points being in Convex Configuration](https://mathoverflow.net/questions/166228/probability-of-4-points-being-in-convex-configuration)
and repeatedly face the problem of solving equations between sums of euclidean distances betw... | https://mathoverflow.net/users/31310 | Tools for Removing Radicals from Equations | I don't know about "free", but the general solution is to introduce a variable for each radical, together with an equation $y^2 = f(x),$ and then solve the resulting system. Since this is easy to do in Mathematica, I assume WolframAlpha will do it for you for free, but I never use WA, so can't promise.
| 5 | https://mathoverflow.net/users/11142 | 176614 | 88,988 |
https://mathoverflow.net/questions/176607 | 8 | Let $G$ be a compact, connected Lie group and $\rho$ any of its irreducible, unitary representations. If $\gamma:S^1\to G$ is an injective homomorphism (a periodic geodesic passing through the identity), what can be said about the matrix valued integral
$$
I(\rho,\gamma)=\int\_{S^1}\rho(\gamma(t))dt?
$$
More explicitly... | https://mathoverflow.net/users/55893 | Integrals of representations over geodesics | I had made a silly error in the first version of this answer.
To try to answer your first question, notice that the image of $\gamma$ is a circle subgroup $T$, say, of $G$ and that the integral $I(\rho,\gamma)$ is (with your normalisation of the measure on the circle) the projector onto the subspace $V^T$ of $T$-inva... | 3 | https://mathoverflow.net/users/394 | 176616 | 88,989 |
https://mathoverflow.net/questions/176556 | 15 | A Witten-Reshetikhin-Turaev (WRT) Invariant $\tau\_{M,L}^G(\xi)\in\mathbb{C}$ is an invariant of closed oriented 3-manifold $M$ containing a framed link $L$, where $G$ is a simple Lie group, and $\xi$ is a root of unity. Components of $L$ are coloured by finite dimensional $G$-modules.
Since Murakami in 1995, people ... | https://mathoverflow.net/users/2051 | Why are Witten-Reshetikhin-Turaev invariants expected to be integral? | My humble point of view is that the Witten-Reshetikin-Turaev invariant (at least for $G=SU(2)$ or $SO(3)$) at the root $\xi$ is (by definition) a rational function on $\xi$ which looks very much like a polynomial with integer coefficients. (Note that the function depends on $\xi$.)
When $M=S^3$, the rational function... | 10 | https://mathoverflow.net/users/6205 | 176623 | 88,991 |
https://mathoverflow.net/questions/176585 | 2 | **Background/Setup**
For any connected scheme $S$, let $\text{FEt}\_S$ denote the category of finite etale $S$-schemes. Let $f : X\rightarrow Y$ be a morphism of connected schemes, then for any finite etale cover $C\rightarrow Y$, we can pull it back to a finite etale cover of $X$, so after making a choice of pullbac... | https://mathoverflow.net/users/15242 | question about the induced homomorphism of etale fundamental groups | (This was meant to be a comment elaborating on Dan Petersen's answer, but got too long.) The error is that you cooked up an isomorphism between $F\_x \circ f^{\ast}$ and $F\_x$ and *by means of that* simply declared the two functors to be "equal" yet never tried to calculate how this identification interacts with *auto... | 6 | https://mathoverflow.net/users/52824 | 176624 | 88,992 |
https://mathoverflow.net/questions/176629 | 18 | 1. Let $M$ be a complete finite volume Riemannian manifold and $\gamma : \mathbb{R}^{\geq 0} \to M$ a geodesic. Suppose that $\mathrm{im}(\gamma)$ is dense. Is it equidistributed in the Riemannian measure? That is, does
$$
\lim\_{T \to +\infty} \frac{1}{T} \int\_0^T f(\gamma(t)) \, dt = \frac{1}{\mathrm{vol}(M)} \int\_... | https://mathoverflow.net/users/26522 | If there is a dense geodesic, are almost all geodesics equidistributed? Dense? | The first question is false as stated.
By Artin's encoding, geodesics on $SL\_{2}(\mathbb{R})/SL\_{2}(\mathbb{Z})$ corresponding to continued fractions, and the geodesic flow corresponds to the shift.
It's easy to find one fraction where you'll see any given prefix (hence dense), but you won't be equidistributed (say t... | 26 | https://mathoverflow.net/users/8857 | 176631 | 88,994 |
https://mathoverflow.net/questions/176627 | -4 | Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature?
For example, can one make sense of
$$\frac{\partial^{\infty}f(x\_1,x\_2,\cdots)}{\partial x\_1 \partial x\_2 \cdots}?$$
I have a feeling that we can do some sort of an analysis here, that might use eventually the ax... | https://mathoverflow.net/users/13904 | Derivatives of infinite order | In the case of a single variable, see for example [this](http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pjm/1103051257&page=record) article concerning the limit $\lim\_{n\to\infty}f^{(n)}(x)$ for a smooth function $f:\mathbb R\to\mathbb R$.
Also, if $f:\mathbb R\to\mathbb R$ is analy... | 4 | https://mathoverflow.net/users/55893 | 176633 | 88,995 |
https://mathoverflow.net/questions/176639 | 16 | I am stuck on one step that occurs without explanation in several Algebraic geometry books.
Starting from the exact sequence
$$0\rightarrow \Omega\_{\mathbb{P}^n}\rightarrow \mathcal{O}\_{\mathbb{P}^n}(-1)^{\oplus n+1}\rightarrow \mathcal{O}\_{\mathbb{P}^n}\rightarrow 0$$
it is concluded that $$\omega\_{\mathbb{... | https://mathoverflow.net/users/49117 | Canonical Sheaf of Projective Space | det of the middle term of a short exact sequence is the tensor product of the dets of the left and right terms (det = top wedge).
The canonical bundle is det of \Omega, det of O is O.
| 18 | https://mathoverflow.net/users/56234 | 176641 | 88,996 |
https://mathoverflow.net/questions/176638 | 16 | I saw the following question from the "Problem Section" in Schoen and Yau, page 281, problem 12:
Let $M\_1, M\_2$ each have negative curvature. If $\pi\_1 (M\_1)=\pi\_1 (M\_2)$, prove that $M\_1$ is differeomorphic to $M\_2$.
The authors then commented:
"
There is some progress due to Cheeger, Gromov, Farrell and ... | https://mathoverflow.net/users/18850 | Is there any progress on Problem 12 (from Schoen and Yau)? | Counter-example was constructed by T.Farrell and L.Jones, in "Negatively curved manifolds with exotic smooth structures", JAMS 1989, volume 2, number 4.
| 17 | https://mathoverflow.net/users/21684 | 176651 | 89,002 |
https://mathoverflow.net/questions/176636 | 4 | can anyone help me with the following statement (it is part of a bigger proof where it is not explained).
Let $B$ be a finite type commutative $A$-algebra (where $A$ is a commutative ring), and consider the kernel $I$ of the diagonal homomorphism $B\otimes\_A B\to B$ (defined by $b\otimes b'\mapsto bb'$). Then $I$ i... | https://mathoverflow.net/users/56228 | commutative algebra, diagonal morphism | Let $I'\subset B\otimes\_A B$ be the ideal generated by the elements $b\_i\otimes 1-1\otimes b\_i$, and define
$$
R=\{b\in B:b\otimes 1-1\otimes b\in I'\}.
$$
It’s not hard to check that $R$ is an $A$-subalgebra of $B$, so that $R=B$ (because the generators $b\_i$ are in $R$ by construction). Now $b\otimes 1-1\otimes b... | 6 | https://mathoverflow.net/users/5263 | 176653 | 89,004 |
https://mathoverflow.net/questions/176558 | 6 | I have a concern about the "Basic Lemma" which Valdimir Vapnik states and proves in his 1998 book *Statistical Learning Theory* (ch. 14.3, pp. 574–76): It seems like a certain coefficient should have been 2 instead of 1. I'm hoping I'm just wrong, but I'd appreciate any comments or clarifications.
First, some context... | https://mathoverflow.net/users/56198 | Is there a mistake in Vapnik's "Basic Lemma"? | You're right, it seems.
Suppose
1. $X=\{1, -1\}$;
2. $\mu(\{1\})=\mu(\{-1\})=1/2$;
3. $S$ is the power set of $X$;
4. our data sets have just one element each, so that $f(\{1\})$ is either 0 or 1; and
5. $\epsilon=\frac12$.
Then
$$\Pr\left(||f - \mu|| >\frac{\varepsilon}{2}\right)\ge \Pr\left(|f(\{1\})-\mu(\{1\})|>... | 4 | https://mathoverflow.net/users/4600 | 176666 | 89,007 |
https://mathoverflow.net/questions/161587 | 2 | I am studying the the forward-backward algorithm used in Hidden Markov Models. I understand that you are trying to propagate through a sequence (and the available states) to find the most probable state in the model at any given time.
I am however struggling with the following notation (for a forward variable) where... | https://mathoverflow.net/users/48807 | forward algorithm Hidden Markov Model | I'm not sure this is exactly what you're looking for, but let me see if I can make these things a bit clearer by translating them into a more conventional probability theory notation. Let me know if I've misunderstood your question.
Suppose first that the model didn't have any state transitions, but just a single unk... | 4 | https://mathoverflow.net/users/56198 | 176683 | 89,012 |
https://mathoverflow.net/questions/176675 | 2 | Let $G$ be a finite group, $p$ a prime, $P\in{\rm Syl}\_p(G)$, and $x\in P$.
Let $Z^\*\_p(G)$ denote the full preimage in $G$ of $Z(G/O\_{p'}(G))$ under the canonical epimorphism $G\to G/O\_{p'}(G)$.
**Question.** Is it true that if $x^G\cap P =\{x\}$ then $x\in Z^\*\_p(G)$?
*Remark.* For $|x|=p=2$ it is Glauberm... | https://mathoverflow.net/users/12961 | Isolated elements of primary order ($Z^*$-theorem revisited) | The result for $x$ of prime power order is an easy consequence of the case when $x$ has prime order, and proceeds by induction: We may suppose $x$ has order greater than $p,$ that $O\_{p^{\prime}}(G) = 1$ and that $x^{p} \in Z(G).$ Then the image of $x$ is central in $G/\langle x^{p} \rangle$, so $x \in O\_{p}(G).$ Sin... | 3 | https://mathoverflow.net/users/14450 | 176688 | 89,016 |
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