parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/165723 | 13 | There are several places on the web where one may find quite intuitively understandable accounts of (im)predicativity; here on MO I found two questions with very good detailed answers ([Predicative definition](https://mathoverflow.net/questions/36972/predicative-definition) and [Impredicativity](https://mathoverflow.ne... | https://mathoverflow.net/users/41291 | Formal/rigorous treatment of (im)predicativity/predicativism | **Solomon Feferman's papers provide formal systems for predicativity**, most recently [here](http://math.stanford.edu/~feferman/papers/pfa%281%29.pdf). Other papers on his website and in his book *In the Light of Logic* have other expositions. These systems are predicative either by virtue of their ordinal analysis, or... | 5 | https://mathoverflow.net/users/nan | 165852 | 86,569 |
https://mathoverflow.net/questions/165830 | 11 | Consider 3-dimensional TQFTs for example. One version of them is the
3-2-1-0 fully extended TQFT. **Do we have another version: 2-1-0 extended "TQFT"?**
If yes, **do we have an example of 2-1-0 extended TQFT that is not
3-2-1-0 fully extended TQFT?**
-- added -------
By 2-1-0 extended 3-dim "TQFT", we mean that... | https://mathoverflow.net/users/17787 | Relation between fully-extended TQFT and a "topless" TQFT | If I understand you correctly, your "2-1-0" TQFTs are what are frequently called "2+$\epsilon$-dimensional TQFTs" in the mathematical literature. (The $\epsilon$ means that very thin 3-manifolds, e.g. the mapping cylinder of a homeomorphism of 2-manifolds, can have their path integral defined.)
If you try to construc... | 17 | https://mathoverflow.net/users/284 | 165854 | 86,570 |
https://mathoverflow.net/questions/165847 | 5 | Given a formal power series $$y(x)=\sum\_{i=0}^{\infty} a\_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$ P(x,y)=p\_n(x)y^n+p\_{n-1}(x)y^{n-1}+\cdots+p\_0(x)=0,p\_j(x)\in F[x]$$the series satisfies and if it exists,how to write it down?
| https://mathoverflow.net/users/14024 | Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down? | One result in this area is Christol's theorem, which asserts that
an element of $\mathbf{F}\_p[[X]]$ is algebraic over $\mathbf{F}\_p(X)$
if and only if its sequence of coefficients is a $p$-automatic sequence,
which means that there is a finite state machine for which
the coefficient of $X^n$ is the output of this mac... | 13 | https://mathoverflow.net/users/30412 | 165857 | 86,573 |
https://mathoverflow.net/questions/165868 | 14 | I originally asked this on math.stackexchange, where I asked if there could exist a closed manifold that could be given different geometric structures of constant curvature (not at the same time, of course). It was pointed out that the Chern-Gauss-Bonnet theorem shows that no such manifold exists in even dimensions. Al... | https://mathoverflow.net/users/50693 | Does there exist a closed manifold that can be given both a Euclidean and a Hyperbolic structure? | Here is another approach to impossibility. Some decades after Bieberbach, [Milnor](https://projecteuclid.org/euclid.jdg/1214501132) showed that the ball of radius $R$ in the universal cover of a compact manifold is basically a bunch of copies of fundamental domain, copies indexed by the ball of a similar radius in the ... | 26 | https://mathoverflow.net/users/4639 | 165874 | 86,578 |
https://mathoverflow.net/questions/165563 | 2 | Let $X,Y,Z$ be connected topological spaces, $f\colon X\to Y$ be a continuous map and $p\colon Z\to Y$ be a covering map. The problem is the existence of a continuous lift of $f$ across $p$. A standard result involving fundamental groups and induced homomorphisms requires that $X$ be path-connected and locally path-con... | https://mathoverflow.net/users/50457 | Lifts across covering maps | Suppose you have basepoints $x\_0\in X$, $z\_0\in Z$ and $p(z\_0)=f(x\_0)$. The lift $\tilde{f}:X\to Z$ such that $p\circ \tilde{f}=f$ exists and is continuous if and only if
1) $f\_{\ast}(\pi\_1(X,x\_0))\subseteq p\_{\ast}(\pi\_1(Z,z\_0))$ (this is equivalent to $\tilde{f}$ being a well-defined function).
2) For ... | 6 | https://mathoverflow.net/users/5801 | 165879 | 86,582 |
https://mathoverflow.net/questions/128314 | 13 | Say I have an "ordinary" TQFT $F$ of dimension $n$, assigning groups or vector spaces to closed $(n-1)$-manifolds and linear maps to cobordisms. Consider the different ways $F$ can be obtained from a TQFT "extended one step" which assigns categories to manifolds of dimension $n-2$ (often derived categories of algebras ... | https://mathoverflow.net/users/8041 | How unique are extensions of TQFTs to lower dimension? | The question of which tqfts extend is a very interesting one. To make the question more mathematically precise, we can fix the target n-categories and ask for the tqfts to extend with respect to those targets. Then I can give precise answers.
In general there are both existence and uniqueness issues, even in the n=2... | 11 | https://mathoverflow.net/users/184 | 165891 | 86,587 |
https://mathoverflow.net/questions/165897 | 4 | Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda\_{E}^{alg} $, $ \mu\_{E}^{alg} $, $ \lambda\_{E}^{an} $ and $ \mu\_{E}^{an} $ at $p$.
The "algebraic" Iwasawa invariants $ \lambda\_{E}^{alg} $ and $ \mu\_{E}^{al... | https://mathoverflow.net/users/30999 | Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny | Yes, the main conjecture is isogeny-invariant. See here:
B. Perrin-Riou, *Variation de la fonction $L$ $p$-adique par isogénie*, Algebraic number theory, Adv. Stud. Pure Math. **17** (1989), pp. 347-358.
| 5 | https://mathoverflow.net/users/2481 | 165898 | 86,590 |
https://mathoverflow.net/questions/165862 | 2 | The Cohen-Lenstra statistics describe how often a prime divides the class number of quadratic number field $\mathbb{Q}[\sqrt{d}]$
$$ \mathbb{P}\big[h(d) \not\equiv 0\; (\mod p) \big] = \prod\_{k \geq 2}\left( 1 - \frac{1}{p^k}\right) $$
As can be seen in [Section 6.3](http://www.dms.umontreal.ca/~andrew/Courses/Rat... | https://mathoverflow.net/users/1358 | Prime Divisors of the $x \mapsto 2x+1$ Recursion | The divisibility properties of the numbers defined by $a\_{n+1}=2a\_n+a\_{n-1}$ (and $a\_1=1$ and $a\_2=2$) are not random. One reason for this is that this sequence has the very non-random property of being eventually periodic mod $N$, for every integer $N$. In particular, since
$$
a\_n = \frac{(1+\sqrt{2})^n-(1-\sqrt... | 7 | https://mathoverflow.net/users/30412 | 165901 | 86,592 |
https://mathoverflow.net/questions/165887 | 2 | Have not been able to get an answer to this on <http://math.stackexchange.com>, so trying here too...
---
**Given the following two sets:**
* $P^-(n) = \{p \leq n : p \equiv -1\pmod 6\}$
* $P^+(n) = \{p \leq n : p \equiv +1\pmod 6\}$
**For example:**
* $P^-(40) = \{5,11,17,23,29\}$
* $P^+(40) = \{7,13,19,31... | https://mathoverflow.net/users/27456 | Number of primes with $-1\pmod 6$ vs. Number of primes with $+1\pmod 6$ | Assume that the Riemann hypothesis for the non-principal $L$-series $\pmod{3}$ is false, say, this series has a zero $\rho=\sigma+i\gamma$ with $\sigma>1/2$. Then Turan and Knapowski have shown that both $C^-(n)-C^+(n)>n^{\sigma-\epsilon}$ and $C^-(n)-C^+(n)<-n^{\sigma-\epsilon}$ happen infinitely often.
If the Riema... | 17 | https://mathoverflow.net/users/37555 | 165902 | 86,593 |
https://mathoverflow.net/questions/165612 | 7 | In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed to seeing. What are the advantages of using such a major arc?
Here is the paper: <http://arxiv.org/pdf/1201.6656.pdf>
... | https://mathoverflow.net/users/40983 | Major arcs in the proof that every odd number is the sum of at most 5 primes | The first reason is that you do not gain much by considering major arcs around rational numbers with denominator $\geq 3$. The reason is that the contribution of the major arcs $\{a/q:(a,q)=1\}$ reflects the inhomogenity of the distribution of the function in question modulo $q$, which cannot be explained by the distri... | 4 | https://mathoverflow.net/users/37555 | 165908 | 86,595 |
https://mathoverflow.net/questions/165911 | 9 | I am trying to compute the probability that after a perfect shuffling of a deck of memory cards (n pairs) none of the pairs end up with the two members next to each other.
I get into a messy inclusion-exclusion which I guess I could work through, but I wonder if there is a simpler argument I am missing.
A numerical... | https://mathoverflow.net/users/7368 | Probability of a pair of memory cards ending up as neighbors | It's unlikely that there is an exact answer simpler than the inclusion-exclusion sum. One can also use much the same calculations to show that the number of matched pairs adjacent to each other is asymptotically a Poisson distribution with mean 1, so the limit probability is indeed $e^{-1}$. To prove the Poisson limit,... | 7 | https://mathoverflow.net/users/9025 | 165916 | 86,600 |
https://mathoverflow.net/questions/164321 | 4 | I'm searching for a suitable (hopefully simple enough) solution to the following form of integral:
$$\int\_0^\infty \mathrm{d}x~x^n J\_\nu(a x) J\_\nu(b x) K\_\mu(c x) $$
Where $n$, $\nu$, and $\mu$ are all integers, and $a$, $b$, and $c$ are all real and positive.
If not generally, a specific case would be quite... | https://mathoverflow.net/users/7606 | Integrals of two Bessel functions of the first kind and a modified bessel function of the second kind | Although I'm sure @Zurab's answer will indeed give me a solution (and I've heard the suggestion before in other situations), I'm not very familiar with the technique.
Following the suggestion by @Johannes in the comments, there is a much more straightforward way that, although not fully general, fits my problem exact... | 4 | https://mathoverflow.net/users/7606 | 165927 | 86,602 |
https://mathoverflow.net/questions/165618 | 1 | Good day!
Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.
Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in L^q(0,T;V')$. $1/p + 1/q = 1$
$
A\colon W \to L^q(0,T;V')
$ - *linear* operator
I can't find the theorem of existence of t... | https://mathoverflow.net/users/48757 | Existence of the solution of a linear parabolic pde | It seems that $A=-\Delta$ is one of the operators satisfying your condition.
For this operator and any fixed $1<p<\infty$, the solution of $y'+Ay=f\in L^p(0,T;V')$ is in $W=\{y\in L^p(0,T;V):dy/dt\in L^p(0,T;V')$}.
Existence of solution of this type is called theory of "maximal $L^p$ regularity", which can be found in ... | 1 | https://mathoverflow.net/users/50718 | 165928 | 86,603 |
https://mathoverflow.net/questions/165617 | 1 | Good day!
Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.
Consider the space
$W = \{ y \in L^2(0,T;V) \colon dy/dt \in L^2(0,T;V') \}$.
It is well-known that $W \subset C([0,T];H)$ where $H = L^2(\Omega)$.
My question: for what $\alpha$ and $\beta$ we may assert that $W \subset L^\alpha(0,T;L^\beta(\Omega)... | https://mathoverflow.net/users/48757 | Embedding to $L^\alpha(0,T;L^\beta(\Omega))$ | This is an interpolation problem.
$$
L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;L^6(\Omega))
\hookrightarrow L^{2/\theta}(0,T;L^{1/[(1-\theta)/2+\theta/6]}(\Omega)) ,
$$
for any $\theta\in(0,1)$. You can check that this space coincides with
$L^2(0,T;L^6(\Omega))$ when $\theta=1$, and this space coincides with
$L^\infty... | 0 | https://mathoverflow.net/users/50718 | 165930 | 86,604 |
https://mathoverflow.net/questions/165924 | 1 | Consider function $f(x)$. I've counted 4 possible notations to write a derivative of $f(x)$ at point $x = a$:
1. $f'(a)$;
2. $\frac{\operatorname{d}{f(a)}}{\operatorname{d}x}$;
3. $\left.\frac{\operatorname{d}{f(x)}}{\operatorname{d}x}\right|\_{x = a}$;
4. $f\_x(a)$ in case of partial derivative.
Currently, I need ... | https://mathoverflow.net/users/50615 | Choosing Notation for Variable Substitution into Derivative Expressed with Differentials | #2 seems ambiguous, since it seems, at first glance, to be the derivative of the constant $f(a)$, i.e., the interpretation of #2 is usually the first interpretation in the second list in the question. #3 is the most common, and is used in almost all calculus textbooks and/or papers to denote the derivative of $f(x)$ ev... | 2 | https://mathoverflow.net/users/nan | 165931 | 86,605 |
https://mathoverflow.net/questions/164095 | 1 | Background of my question is, that I need to calculate Clothoids and I found an AMS article "*Chebyhev Approximations for Fresnel Integrals*" by W.J. Cody from 1968 (<http://www.ams.org/journals/mcom/1968-22-102/S0025-5718-68-99871-2/S0025-5718-68-99871-2.pdf>).
**Question:** is something better available/possible, ... | https://mathoverflow.net/users/31310 | State of the Art in Approximating Fresnel Integrals | If you aim to compute clothoids to interpolate two given points with assigned tangents, you can have a look at the paper cited above by Bertolazzi and Frego. The arxiv version is updated with a published paper [G1 fitting with clothoids](http://onlinelibrary.wiley.com/doi/10.1002/mma.3114/abstract).
That paper is good ... | 2 | https://mathoverflow.net/users/50721 | 165934 | 86,607 |
https://mathoverflow.net/questions/165900 | 6 | Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL\_3(k)$ consisting of elements $g$ such that $g(p) \in k p$. My question: is $G$ some well known algebraic group?
| https://mathoverflow.net/users/50708 | Reference for an algebraic group preserving a cubic form | To elaborate on abx's comment: modding out by scalars, i.e., working in $PGL\_3$ instead of $GL\_3$, by definition the stabilizer of $p$ is the group of projective automorphisms of the curve $p=0$ which preserves the embedding of the curve.
If we assume that the curve is smooth and that there is a rational point, for... | 4 | https://mathoverflow.net/users/321 | 165936 | 86,608 |
https://mathoverflow.net/questions/165939 | 4 | Suppose $X$ is a complex manifold, we have the map $H^0(X,K\_X^\*/O\_X^\*)\to H^1(X,O\_X^\*)$, is the canonical line bundle $\wedge^n{\Omega}$ always in the image of the map?
| https://mathoverflow.net/users/nan | Must a canonical line bundle be associated to a cartier divisor? | The answer is no. The group $H^0(X, \mathcal{K}\_X^\*/\mathcal{O}\_X^\*)$ is isomorphic to the divisor group $\mathrm{Div}(X)$ (see e.g. Huybrechts' *Complex Geometry*, Prop. 2.3.9), so its image in $\mathrm{Pic}(X)$ is the group of line bundles associated to some divisor. Now there are complex compact surfaces which d... | 13 | https://mathoverflow.net/users/40297 | 165949 | 86,614 |
https://mathoverflow.net/questions/165935 | 3 | Consider a planar pair of pants
$$P = \left\lbrace z \in \mathbb{C}: |z| \le 1, |z-x| \ge r\_1, |z+x| \ge r\_2 \right\rbrace$$
where $-1 < -x-r\_2 < -x+r\_2 < 0 < x-r\_1 < x+r\_1 < 1$.
There is a unique conformal hyperbolic metric $e^{2u} (d x^2 + d y^2)$ on $P$ such that all boundary components are geodesics. This ... | https://mathoverflow.net/users/7631 | Conformal invariants of planar pairs of pants | There are several references that consider the relation between extremal length and hyperbolic length. Usually they consider closed surfaces, but you can put yourself in that situation by doubling along the boundary. Here are some relevant papers:
Matsuzaki, Katsuhiko. Bounded and integrable quadratic differentials: ... | 3 | https://mathoverflow.net/users/5010 | 165952 | 86,616 |
https://mathoverflow.net/questions/165953 | 2 | Let U be a unirational variety over a field of characteristic 0. I have read that its canonical divisor cannot be ample, but I don't know why.
Any references will be most helpful.
| https://mathoverflow.net/users/46578 | Unirational and ampleness | **Smooth varieties**
Let $X$ be a smooth uniruled variety. Then there is a free rational curve $f:\mathbb{P}^1\rightarrow X$, that is $H^1(\mathbb{P}^1,f^\*T\_X\otimes\mathcal{O}\_{\mathbb{P}^1}(-1)) = 0$. Now $f^{\*}T\_X$ is a rank $n = dim(X)$ vector bundle on $\mathbb{P}^1$. Therefore we can write
$$f^\*T\_X = \m... | 4 | https://mathoverflow.net/users/14514 | 165955 | 86,617 |
https://mathoverflow.net/questions/165967 | 6 | Geometric realization of $B{\mathbb G}\_{\mathfrak m}({\mathbb C})$ is ${\mathbb C}{\mathbb P}^\infty=\varinjlim\_n~ {\mathbb C}{\mathbb P}^n\_k$; what if one considers a separable field $k\neq {\overline k}$? Sheaf-theoretically, $B{\mathbb G}\_{\mathfrak m}$ represents the simplicial sheaf $B{\mbox{hom}}\_{k}(-,{\mat... | https://mathoverflow.net/users/50735 | infinite grassmannian in algebraic geometry | As far as I understand, yes, you can look at it this way. However, at least to me, this seems like a strange way of doing it, and there are a few problems with it:
* Basically, you are saying
that the cohomology of ${\mathbb P}^\infty$ and $B(\mathbb{Gm})$ coincide because
the fiber (which is ${\mathbb A}^\infty-\{0\... | 4 | https://mathoverflow.net/users/2653 | 165970 | 86,625 |
https://mathoverflow.net/questions/165976 | 2 |
>
> What is $\mathbf{B}\Omega A$, where $A$ is a pointed object of an $(\infty,1)$ category with point $\*\to A$, $\Omega A$ is the loop space of $A$, and $\mathbf{B}X$ is the delooping of $X$?
>
>
>
The closest I have come to finding anything about this is in [this $n$lab entry, titled *looping*](http://ncatlab... | https://mathoverflow.net/users/nan | What is the delooping of a looping? | A simple example should indicate the general phenomenon: Let $A$ be a discrete based set.
The $\Omega A$ is a point, so $B \Omega A$ is a point.
The general phenomenon is this:
$B\Omega A$ is always connected, whereas $A$ needn't be. The statement which is true is that there's a map $B\Omega A \to A$ which is a weak ... | 9 | https://mathoverflow.net/users/8032 | 165977 | 86,628 |
https://mathoverflow.net/questions/165980 | 10 | Recall that the cohomotopy set $\pi^k(\mathcal{M})$ is $[\mathcal{M},S^k]$, i.e., the set of pointed homotopy classes of continuous mappings $\mathcal{M}\to S^k$. Recall also the Whitehead theorem:
>
> **Theorem:** *Suppose $X,Y$ are connected CW complexes. Suppose then that $f:X\to Y$ is a continuous map which ind... | https://mathoverflow.net/users/nan | Whitehead theorem for cohomotopy | No, this is false. According to the Sullivan Conjecture (Miller's Theorem),
$\mathrm{map}\_\*(B\mathbb{Z}/p, S^n) \sim \*$ for all $n$, which means
$$
[\Sigma^n B\mathbb{Z}/p, S^k] = \*
$$
for all $n$. So if we let $f: \Sigma^k \mathbb{Z}/ p \to \*$, the induced map
$$
f^\*:
\pi^k(\*)
\to
\pi^k ( \Sigma^n B\mathbb{... | 21 | https://mathoverflow.net/users/3634 | 165988 | 86,631 |
https://mathoverflow.net/questions/165991 | 5 | I want to know what the critical idea behind Hardy-Littlewood circle method is. It seems that they divide the circle into major arcs and minor arcs to ignore the singularities of generating function to be able to use tools from complex analytics to estimate the coefficients of the generating function. Is that right?
| https://mathoverflow.net/users/14024 | What is the critical idea behind Hardy-Littlewood circle method? | The generating function has no singularities in the modern treatment of the method (it is a finite exponential sum). The idea is that the generating function is small at any point which is not close to any rational number with small denominator. This reflects the expectation that the coefficients, which are number theo... | 13 | https://mathoverflow.net/users/11919 | 165995 | 86,634 |
https://mathoverflow.net/questions/165918 | 5 | In paper arXiv:math/0701247
*"Divisibility of the stable Miller-Morita-Mumford classes"* by Soren Galatius, Ib Madsen, Ulrike Tillmann, it was shown
that the Pontryagin numbers for a 4-dim surface bundles are
$0$ mod $D$, where $D=12$ is the maximal divisor of the Pontryagin numbers.
**If we fix the fiber of the 4-di... | https://mathoverflow.net/users/17787 | Pontryagin number for 4-dim surface bundle | Ulrike Tillmann told me that Endo ( <http://projecteuclid.org/euclid.ojm/1200788349> 1998) proved that:
if the genus of the fibre $g>2$, then
there is a surface bundle over a surface of a genus equal or less than 111, and the surface bundle has a signature $\pm 4$, hence realising the minimal Ponrtryagin number 12 for ... | 2 | https://mathoverflow.net/users/17787 | 165998 | 86,635 |
https://mathoverflow.net/questions/165999 | 5 | This is somewhat related to the question found at [What is the DGLA controlling the deformation theory of a complex submanifold?](https://mathoverflow.net/questions/114090/what-is-the-dgla-controlling-the-deformation-theory-of-a-complex-submanifold), though not exactly the same, so I hope it's not duplicating too much.... | https://mathoverflow.net/users/49247 | Deformations of a pair of compact, complex manifolds | The infinitesimal deformations of
$(X,M)$ are controlled by the sheaf $T\_X\langle M\rangle$ of vector fields on $X$ which
are tangent to $M$: see for instance [this paper](http://math.unice.fr/~beauvill/pubs/Fano.pdf), Prop. 1.1. Thus the obstruction you are looking for lies in $H^2(X,T\_X\langle M\rangle)$. The exact... | 6 | https://mathoverflow.net/users/40297 | 166003 | 86,637 |
https://mathoverflow.net/questions/165561 | 9 | We know that all compact orientable manifolds of dimension 3 are spin.
In 4 dimensions, $CP^2$ is not spin. I would like to ask if
all 4-dimensional compact orientable mapping tori are spin?
See also a related question [Spin structure on mapping torus](https://mathoverflow.net/questions/72766/spin-structure-on-mappi... | https://mathoverflow.net/users/17787 | Are 4-dimensional mapping tori always spin? | The answer is no. As was described in the thread: [Spin structure on mapping torus](https://mathoverflow.net/questions/72766/spin-structure-on-mapping-torus)
a mapping torus has a spin structure if and only if the monodromy of the bundle (over $S^1$) fixes a spin structure. The mapping torus I'm going to describe is ... | 7 | https://mathoverflow.net/users/1465 | 166005 | 86,638 |
https://mathoverflow.net/questions/102914 | 11 | Suppose that $X$ and $Y$ are two categories. Let $\operatorname{Funny}(X,Y)$ denote the category whose objects are functors $X\to Y$ and whose morphisms are *unnatural transformations* $F\to G$, where an unnatural transformation $F\to G$ is given by an $\operatorname{Ob}(X)$-indexed family of arrows $\gamma\_x:F(x)\to ... | https://mathoverflow.net/users/1353 | Do the "funny" tensor product and the cartesian product satisfy any algebraic "laws"? | In a recent article
>
> Mark Weber, *Free Products of Higher Operad Algebras*, Theory and Applications of Categories, Vol. 28, No. 2, 2013, pp. 24–65, [journal](http://www.tac.mta.ca/tac/volumes/28/2/28-02abs.html), arXiv:[0909.4722](https://arxiv.org/abs/0909.4722).
>
>
>
there is a nice description of $□$ in... | 15 | https://mathoverflow.net/users/50759 | 166022 | 86,647 |
https://mathoverflow.net/questions/165917 | 0 | Given a family $\mathcal{F}$ of sets over ground set $X$, let $\tau(\mathcal{F})$ be the **transversal number** (aka blocking number), that is the cardinality of the smallest set of points $E \subseteq X$ such that every set in $\mathcal{F}$ meets $E$.
Lovasz (Problem 13.25 in his Problems & Exercises books) has prov... | https://mathoverflow.net/users/22051 | Generalized Helly theorem for $t$-intersecting families | This is the best possible bound. If $\mathcal F$ consists of every $r$-set of a set with $r+\frac{r-t}{k-1}$ elements, then it is still $k$-wise $t$-intersecting.
| 2 | https://mathoverflow.net/users/955 | 166026 | 86,649 |
https://mathoverflow.net/questions/166004 | 6 | Suppose you are given a smooth quartic surface $X$ in $\mathbb P^3$. I would like to find an upper bound for the number of lines on $X$ in the case that there is no plane intersecting the curves in four lines (so that the possible intersections of the surface and a plane are an irreducible quartic curve, a line and an ... | https://mathoverflow.net/users/43951 | adjacency matrix of a graph and lines on quartic surfaces | Yes, this approach has been tried, and we're about to submit a paper [edit: the paper has now been submitted, see [arXiv:1601.04238](http://arxiv.org/abs/1601.04238)]. Alas, very little is known about hyperbolic (as we call them; those with a single positive eigenvalue) graphs, and currently the proof is heavily comput... | 5 | https://mathoverflow.net/users/44953 | 166039 | 86,651 |
https://mathoverflow.net/questions/165121 | 2 | Let $l=p^r$ a prime power and $\zeta$ a primitive l-th root of unity. It is classical result, that $(1-\zeta)^{\varphi(l)}=p\cdot\epsilon\in\mathbb{Z}[\zeta]$ for a unit $\epsilon$.
It should be a classical problem to calculate the quotients $\mathbb{Z}[\zeta]/(1-\zeta)^i$ for nonnegative integers $i$.
Does anyone know... | https://mathoverflow.net/users/34217 | Quotients of number rings IZ[zeta_l] | At least to me, KConrad's advice somehow makes the situation a bit clearer and as I was only interested in the structure as an Abelian group everything is pretty easy. $\mathbb{Z}[\zeta]$ is the free Abelian group on the generators $(1-\zeta)^k$ for $k=0,...,p^r-p^{r-1}-1$. As a subgroup the ideal $(1-\zeta)^i$ is gene... | 1 | https://mathoverflow.net/users/34217 | 166040 | 86,652 |
https://mathoverflow.net/questions/166023 | 4 | In some recent works, such as [this one](http://arxiv.org/pdf/1402.7364v1.pdf) (3.2, page 15), a definition of "gluing of dg-categories along a dg-bimodule" is given. It is obviously the analogue of the notion of [collage](http://ncatlab.org/nlab/show/cograph+of+a+profunctor) (or cograph) of a profunctor.
My question... | https://mathoverflow.net/users/20883 | Universal property of gluing [collage, cograph] of dg-categories | I think it is more natural to ask for a universal property with respect to quasifunctors. There is one and you can find it in Appendix A of <http://arxiv.org/abs/1212.6170>.
| 4 | https://mathoverflow.net/users/4428 | 166041 | 86,653 |
https://mathoverflow.net/questions/166021 | 3 | Let us consider the basic linear elliptic PDE
$$
\mathrm{div} (A\,\mathrm{grad}\,u) + bu = f,
$$
with $f\in L^p,$ $A,b$ uniformly bounded. Do we have, for a weak solution $u\in W^{1,p}(\Omega')$,
$$
(A\,\mathrm{grad}\,u)\in (W^{1,p})^d(\Omega),\ \mbox{ in }\Omega\subset\subset\Omega'\subset\mathbb{R}^d?
$$
or at least... | https://mathoverflow.net/users/20408 | Let $\mathrm{div}\,(A\,\mathrm{grad}\,u) + b u = f$. Is $(A\,\mathrm{grad}\,u)$ weakly differentiable? | As far as I know, there is no such results if you only assume $A\in L^\infty(\Omega)$.
But if you assume that the domain $\Omega$ is partitioned into two subdomains $\Omega\_1\cup\Omega\_2$ by a smooth interface $\Gamma$ inside $\Omega$, and in each subdomain you assume that $\,A\in W^{1,\infty}(\Omega\_1)\cap W^{1,\... | 4 | https://mathoverflow.net/users/50718 | 166050 | 86,657 |
https://mathoverflow.net/questions/166042 | 8 | Given a set of of $N$ points $\{\mathbf x\_i \in \mathcal{S}^d\}\_{i = 1, \ldots, N}$, where $\mathcal{S}$ is a set of possible values, how can I find the point $\mathbf x^\*$ that maximizes the minimum distance to all data points?
In other words, I want to solve:
$\max\_{\mathbf x^\* \in \mathcal{S}^d} \min\_{i = ... | https://mathoverflow.net/users/50770 | Finding a point maximizing the minimal distance to a set of points | "Do you have any pointers to more recent works on the generalization to higher dimensions?"
>
> Xie, Yulai, Jack Snoeyink, and Jinhui Xu. "Efficient algorithm for approximating maximum inscribed sphere in high-dimensional polytope." *Proceedings 22nd Symposium on Computational Geometry*. [ACM link](http://dl.acm.or... | 4 | https://mathoverflow.net/users/6094 | 166053 | 86,660 |
https://mathoverflow.net/questions/165552 | 1 | I start with an automorphism $f$ of the complex unit disc $S^1 = \{ z \in \mathbb{C} : |z| \leq 1\}$. I assume that such a map is given by a Mobius transform, namely
$$
f(z) = \frac{z - a}{-\overline{a}z + 1},
$$
for some $a \in S^1$ with $|a| < 1$. (I know the automorphisms of $S^1$ are slightly more general but for m... | https://mathoverflow.net/users/14581 | How far do conjugated Mobius transforms move points? | You can assume $a$ is real. Then, $f(z) - z = (z-1)/(-\overline{z} a + 1),$ for $z$ on the unit circle. Since the modulus of this thing is the chordal distance, which is related to the angle in the obvious way, this reduces to a simple calculus optimization problem.
| 1 | https://mathoverflow.net/users/11142 | 166063 | 86,661 |
https://mathoverflow.net/questions/166056 | 6 | *Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite.*
We are interested in the following kernel defined for $0\leq x,y \leq 1$:
$$K(x,y) = (x+y)^{3/2} - |x-y|^{3/2}.$$
Numerical simulations hav... | https://mathoverflow.net/users/50777 | Proving that a specific kernel is positive definite | We show below a slightly more general claim (to simplify notation, I'll write only in terms of matrices).
$\newcommand{\reals}{\mathbb{R}}$
**Def.** We say a kernel $\psi: X \times X \to \reals$ is *negative definite* (nd) if $\sum\_{ij}c\_ic\_j\psi(x\_,x\_j) \le 0$ for all $c$ such that $\sum\_i c\_i = 0$.
>
> *... | 6 | https://mathoverflow.net/users/8430 | 166069 | 86,663 |
https://mathoverflow.net/questions/166009 | 4 | Whenever $\kappa$ is an infinite cardinal number, write $L(\kappa)$ for the powerset of $\kappa$ ordered lexicographically. (Where the "$L$" stands for linear order.) Furthermore, write $B(\kappa)$ for the subchain of $L(\kappa)$ consisting of all $X \subseteq \kappa$ such that $X$ is bounded in $[0,\kappa).$ Finally, ... | https://mathoverflow.net/users/26080 | Is it consistent with ZFC that $\mathrm{dv}(\kappa) = \kappa$ for all infinite cardinal numbers $\kappa$? | One can actually show that dv($\kappa$) = $2^{<\kappa}$ outright. Suppose $D$ is any dense subset of $2^{\kappa}$ (one may take $D \subseteq B(\kappa)$, if preferred). I'll argue that $D$ must have size at least $2^{<\kappa}$, which is sufficient for the claim. For any sequence $r \in 2^{<\kappa}$, let $\alpha\_r$ deno... | 8 | https://mathoverflow.net/users/33768 | 166075 | 86,666 |
https://mathoverflow.net/questions/162916 | 4 | I am curious about the following.
Let $K$ be a number field. For any $a \in \mathcal{O}\_K$ in its ring of integers, let $N(a)$ be zero if there exist elements $b, c \in \mathcal{O}\_K \setminus \mathcal{O}\_K^\*$ such that $a = bc$ and $b$ has at least two different (that is, nonassociate) factorizations into irredu... | https://mathoverflow.net/users/10591 | Distinct primitive factorizations over integers of number fields | The quantity in question can be bounded in terms of the class number only. I will show that it is at most the number of partitions of $h^2$, where $h$ is the class number. (This could be improved in several ways.)
Let $a \in \mathcal{O}\_K$. Suppose $N(a) > 1$. Clearly $a$ is neither irreducible nor invertible.
Let... | 2 | https://mathoverflow.net/users/nan | 166081 | 86,669 |
https://mathoverflow.net/questions/166082 | 3 | I recently came upon a recursive formula for the (ordinary) signatures of torus knots. The formula, which I found in Murasugi's book "Knot Theory and Applications" (Springer, 2007), originally appeared in a paper by Gordon, Litherland and Murasugi, "Signatures of Covering Links" (<http://www.maths.ed.ac.uk/~aar/papers/... | https://mathoverflow.net/users/22431 | Parity of knot signatures | This is exercise 3.4 in [Livingston's book (Knot Theory, Carus math. monographs, vol 24, page 123).](http://books.google.com/books?id=KXAS3KRZGRMC&pg=PA123&lpg=PA123&dq=knot%20signature%20is%20even&source=bl&ots=OxeFQFpqv_&sig=r9zshQSSogYUAPIFEfrFdy8A8Vw&hl=en&sa=X&ei=bapyU_SxI8mRyAT81oDwBg&ved=0CHgQ6AEwCA#v=onepage&q=... | 5 | https://mathoverflow.net/users/11142 | 166084 | 86,671 |
https://mathoverflow.net/questions/166088 | 24 | As part of my Phd thesis on aperiodic Wang tilings, I've discovered I need a bound on the irrationality measure of $\gamma = \log 2/\log 6$. That is, I am looking for an upper bound on the quantity
$\eta = \inf \{\alpha : \left|\frac{\log 2}{\log 6} - \frac{p}{q}\right| < \frac{1}{q^\alpha} \text{ for only finitely m... | https://mathoverflow.net/users/50796 | Irrationality measure of log(2)/log(6) | See Georges Rhin: Approximations de Padé et mesures effectives d'irrationalité. (French) [Padé approximants and effective measures of irrationality] Séminaire de Théorie des Nombres, Paris 1985–86, 155–164, Progr. Math., 71, Birkhäuser Boston, Boston, MA, 1987.
Inequality (8) there shows that if $u\_0$, $u\_1$ and $u... | 34 | https://mathoverflow.net/users/38624 | 166091 | 86,674 |
https://mathoverflow.net/questions/165933 | 1 | Can some one suggests an English text covering that part of the book dealing with elliptic functions.
As i understand from [here](https://mathoverflow.net/questions/126420/functions-of-one-complex-variable-geometric-theory), there is no translation of the full book to English but maybe another text that present the sub... | https://mathoverflow.net/users/41258 | Hurwitz, A. and R. Courant: Funktionentheorie , elliptic functions part | There is no English translation.
One book in English which covers most of the material
is Akhiezer's book MR1054205. Another is Whittaker Watson.
Course of modern analysis.
EDIT. There are very many books covering the basic theory of elliptic functions,
old ones and new ones. But I don't know the book which could re... | 2 | https://mathoverflow.net/users/25510 | 166102 | 86,677 |
https://mathoverflow.net/questions/166107 | 7 | From research completely unrelated to Number Theory I stumbled onto the following equation:
$$
xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3
$$
for $x, y, z$ integers, $x,y,z \neq 0$. Are there nonvanishing integers that satisfy it (there are many solutions if one of them is zero)?
| https://mathoverflow.net/users/nan | $xyz = \frac{7}{16}\left(\frac{2x - y - z}{3}\right)^3$ in nonvanishing integers | No. This can be verified via the following Magma code, which can be used on the [free Magma online calculator](http://magma.maths.usyd.edu.au/calc):
```
P<x,y,z>:=ProjectiveSpace(Rationals(),2);
C:=Curve(P,x*y*z-7/16*((2*x-y-z)/3)^3);
E:=EllipticCurve(C);
MordellWeilGroup(E);
```
which outputs
```
Abelian Group... | 13 | https://mathoverflow.net/users/30412 | 166109 | 86,678 |
https://mathoverflow.net/questions/166077 | 2 | A comment on another question (linked below) states
"The group $PSL\_2((\mathbb{Z}/p^n))$ is the automorphisms group of the $(p+1)$ regular tree of depth $n$, where at level $m$ of the tree you have the points of $\mathbb{P}(\mathbb{Z}/p^m)$."
I was unable to find a reference stating this. Is it true, and if so wh... | https://mathoverflow.net/users/50791 | $PSL_2(\mathbb{Z}/p^n)$ isomorphic to automorphism group of depth-$n$, $(p+1)$-regular tree? | As already noted in the comments, the group $G=PSL\_2(\mathbb{Z}/p^n)$ is not the full automorphism group of the $(p+1)$-regular tree of depth $n$. These particular groups of automorphisms have quite some additional structure.
For example, if one fixes any leaf $x$, then the point-stabilizer $G\_x$ contains a normal ... | 3 | https://mathoverflow.net/users/41178 | 166112 | 86,680 |
https://mathoverflow.net/questions/166111 | 1 | First,every language in Chomsky hierarchy(or c.e.language) corresponds to a generating function,the set of the functions is GF,now,a question : is every generating function with integral coefficient in GF?
Secondly,of course,if the coefficients are randomly extracted from N,the function may not be in GF.Are all gener... | https://mathoverflow.net/users/14024 | Approaches to implicitly defining generating function | There certainly are GFs not corresponding to any language in Chomsky hierarchy, as the latter must be recursively enumerable, a property known not to be closed under taking the complement. Indeed, let $F$ be the GF for a recursively enumerable language $L$ with non-recursively enumerable complement, and $A$ the GF for ... | 1 | https://mathoverflow.net/users/11100 | 166113 | 86,681 |
https://mathoverflow.net/questions/166049 | 2 | i would like to ask you a question i can not answer myself, i hope this is not too trivial and i'm not missing something too basic.
Let's suppose we have $X$ and $Y$ Kahler manifolds and $f:X\rightarrow Y$ a bimeromorphic map such that $f^\*:H^2(Y)\rightarrow H^2(X)$ is an isomorphism and also an Hodge isometry. Then... | https://mathoverflow.net/users/50772 | Hodge isometry sending the Kahler class to its opposite | It is impossible, because the birational (movable) nef cone is mapped to birational nef cone, where birational nef cone is a cone of all classes which are non-negative on all curves which move in families covering the whole manifold. Clearly, $\omega$ belongs to the movable nef cone, and $-\omega$ does not.
This is f... | 4 | https://mathoverflow.net/users/3377 | 166133 | 86,688 |
https://mathoverflow.net/questions/166119 | 8 | First, I give my motivation to ask this question. The generalised Neumann trace can be defined as
$$
{}\_{H^{-1/2}(\partial\Omega)}\langle\frac{\partial u}{\partial{\mathbf{n}}},v\rangle\_{H^{1/2}(\partial\Omega)}
={}\_{H^{-1}(\Omega)}\langle\Delta u,v\rangle\_{H^1(\Omega)}-\int\_{\Omega}\nabla u\cdot\nabla v.
$$
But t... | https://mathoverflow.net/users/20408 | Negative real order Sobolev spaces: density and representation | I don't know the references. It's hard to find the references for such questions, as they are not often used by others. It is better to derive such results based on the well-known results.
For $1\leq p\leq\infty$, $W^{-m,p}(\Omega)$ is usually defined as the dual space of $W^{m,p'}\_0(\Omega)$, the completion of $C^\... | 4 | https://mathoverflow.net/users/50718 | 166141 | 86,692 |
https://mathoverflow.net/questions/166065 | 3 | I'm using a 5-point Triangle Moving Average:
$$S\_j = (Y\_{j-2} + 2Y\_{j-1} + 3Y\_j + 2Y\_{j+1} + Y\_{j+2}) / 9$$
The problem is that I often need to smooth my data more than once, and when I do this too much, it becomes noticeability very slow (and I'm using C++).
Is there a way to optimize this formula?
Like ... | https://mathoverflow.net/users/50784 | Triangular Smoothing Formula Optimization | the "clever" way to improve the efficiency of the moving average filter is to implement it recursively; you will then need only two computations per data point, regardless of the length of the filter.
see page 281 and following of [The Scientist and Engineer's Guide to Digital Signal Processing](http://www.analog.com... | 0 | https://mathoverflow.net/users/11260 | 166151 | 86,693 |
https://mathoverflow.net/questions/166154 | 1 | I'm sure this is something silly but I am trying to understand the following paper <http://www.sciencedirect.com/science/article/pii/S0001870807001636#> and something is not clear to me. The paper starts with "irreducible polynomial representations of $GL(r)$ are indexed by sequences $\lambda=(\lambda\_1\geq\ldots\geq\... | https://mathoverflow.net/users/31261 | Some question about polynomial representations of $GL(V)$ | If you remove the assumption that $\lambda\_r$ is nonnegative, then you are indexing all rational representations of $GL(V)$, so the main point is that the author is focusing on *polynomial* representations, i.e., those whose matrix entries can be defined in terms of polynomials.
$V^\*$ is not polynomial because $GL(... | 8 | https://mathoverflow.net/users/321 | 166161 | 86,697 |
https://mathoverflow.net/questions/166153 | 7 | Let $X,Y$ be compact connected manifolds and $\varphi\colon\pi\_1(X)\to\pi\_1(Y)$ be a homomorphism between their fundamental groups. Under what conditions on $X$, $Y$ and $\varphi$ is it true that $\varphi$ is the homomorphism induced by an appropriate continuous map $f\colon X\to Y$?
| https://mathoverflow.net/users/50457 | Realizing homomorphisms between fundamental groups | In general there is an obstruction living in $H^3(X,\pi\_2Y)$. Choose a CW structure on $X$ and $Y$ with only one 0-cell. Then you can use $\varphi$ to define a map at the level of 1-skeleta (just by sending every 1-cell $e$ to a cellular representative of $\varphi([e])$). Since $\varphi$ is a map of fundamental groups... | 18 | https://mathoverflow.net/users/43054 | 166166 | 86,699 |
https://mathoverflow.net/questions/166164 | 1 | Let $X$ be a compact Kahler manifold of complex dimension $n$ and let $Y\subset X$ be an open subset such that $V:=X\setminus Y$ is of complex codimension 2.
I know that by Hartogs' theorem follows $H^{2,0}(X)\simeq H^{2,0}(Y)$ but i read that it is also true $H^2(X,\mathbb{C})= H^2(Y,\mathbb{C})$ i.e. the homology (... | https://mathoverflow.net/users/50824 | Second cohomology on an open subset with complement of codimension 2 | By Poincaré duality, this is equivalent to say that the natural map $H^{2n-2}\_c(Y)\rightarrow H^{2n-2}(X)$ is bijective. This map appears in a long exact sequence
$$\ldots H^{2n-3}(X-Y)\rightarrow H^{2n-2}\_c(Y)\rightarrow H^{2n-2}(X)\rightarrow H^{2n-2}(X-Y)\ldots $$Since $X-Y$ has real dimension $\leq 2n-4$, the spa... | 2 | https://mathoverflow.net/users/40297 | 166167 | 86,700 |
https://mathoverflow.net/questions/162337 | 5 | Let $C,C'$ be rational polyhedral cones in $\mathbb R^n$ both with non-empty interior. Rational means they are generated by vectors with rational entries. One says that $C,C'$ are *isomorphic* if there is a unimodular integer matrix $A\in Gl(n,\mathbb Z)$ such that $AC=C'$.
>
>
> >
> > **Question.** What is the ... | https://mathoverflow.net/users/15934 | Computational complexity of deciding isomorphism of rational polyhedral cones | According to the abstract of "[On the complexity of polytope isomorphism problems](http://arxiv.org/abs/math/0106093)",
>
> [W]e derive that the problems to decide whether two polytopes, given either by vertex or by facet descriptions, are projectively or affinely isomorphic, are graph isomorphism hard.
>
>
>
... | 3 | https://mathoverflow.net/users/297 | 166171 | 86,701 |
https://mathoverflow.net/questions/166139 | 4 | Let $X$ and $Y$ be smooth projective geometrically connected curves over $k$ of genus $g$ at least two.
If $k$ is an algebraically closed field of characteristic zero, there exists a connected variety $T$ over $k$, points $x,y \in T(k)$ and a family of curves $\mathcal C\to T$ such that $\mathcal C\_{t\_0} = X$ and ... | https://mathoverflow.net/users/50816 | Deforming curves to other curves over the field of rational numbers | Yes. Let $d\geq 3$ be an integer. Define $N$ to be $(2d-1)(g-1)$. Denote by $P(t)$ the Hilbert polynomial $2d(g-1)t + 1-g$. Let $H^{P(t)}\_{\mathbb{P}^N\_k/k}$ denote the Hilbert scheme parameterizing closed subschemes $C$ of $\mathbb{P}^N\_k$ with Hilbert polynomial $P(t)$. By Grothendieck, this exists and is a projec... | 4 | https://mathoverflow.net/users/13265 | 166172 | 86,702 |
https://mathoverflow.net/questions/166169 | 1 | Can there exist a right invariant killing field of a right invariant (but not bi-invariant) Riemannian metric on a Lie group?
I am especially interested in the case of $SU(N)$ with a metric of the form (at the identity):
$g(x,y) = \frac{1}{\lambda} B(x,y) + \frac{1}{\lambda^2} B(x,w)B(y,w)$ where $w$ is an arbitrary... | https://mathoverflow.net/users/41654 | Right invariant Killing fields of Right invariant Riemanian metrics | A right invariant Riemannian metric is invariant und all right translations. A left invariant vector field has a flow consisting of right translations (by $\exp(tX)$).
Thus each left invariant field is a Killing field. A right invariant field $R\_X$ is Killing if and only if $S^2(\text{ad}\_X)^\*g\_{e}=0$.
| 3 | https://mathoverflow.net/users/26935 | 166173 | 86,703 |
https://mathoverflow.net/questions/166174 | 1 | If the systole is defined as the length of the shortest essential simple closed curve are there any known upper bounds for hyperbolic surfaces with punctures?
| https://mathoverflow.net/users/38496 | Upper bounds for systoles on punctured surfaces | Yes, since the injectivity radius (defined as the max of injectivity radii over all points of the surface) is bounded by roughly the log of the area (think "embedded disk"). For interesting papers on this subject, check out:
@article {MR1269424,
AUTHOR = {Buser, P. and Sarnak, P.},
TITLE = {On the period matrix of ... | 4 | https://mathoverflow.net/users/11142 | 166175 | 86,704 |
https://mathoverflow.net/questions/166138 | 2 | Let $A$ be a matrix with entries either 0 or 1, where each column contains at least one 1, to remove trivial degenerations.
Let $P$ be the *convex hull of all integer vectors* $x$ that satisfy $Ax \leq y$, and $x\geq 0$, where $y$ is some non-negative integer vector. Clearly, $P$ is an integral polytope.
For exampl... | https://mathoverflow.net/users/1056 | Integrally closed polytopes from 01-matrices | No.
$$x\_1+x\_2 \leq 1 \quad y\_1 + y\_2 \leq 1 \quad z\_1 + z\_2 \leq 1$$
$$x\_1+y\_1+z\_1 \leq 2 \quad x\_2+y\_2+z\_1 \leq 2 \quad x\_2+y\_1+z\_2 \leq 2 \quad x\_1 + y\_2 + z\_2 \leq 2$$
$$(1,1,1,1,1,1) \in 2 P.$$
Note that the first three inequalities imply $x\_1+x\_2+y\_1+y\_2+z\_1+z\_2 \leq 3$.
If we are to sum... | 2 | https://mathoverflow.net/users/297 | 166181 | 86,707 |
https://mathoverflow.net/questions/166176 | 1 | Is there a result relating sutured manifolds and surfaces of minimal genus? perhaps someone has a very clever point of view of these two notions that can share.
In other matters, do we know how to construct minimal genus surfaces out of (weakly) incompressible surfaces? I have been searching for a construction but no... | https://mathoverflow.net/users/50749 | Sutured Manifolds and minimal genus | Sutured manifolds and sutured manifold hierarchies were defined for the very purpose of studying surfaces of minimal genus within a homology class. See the original papers of Gabai on this topic, starting with
* "Foliations and the topology of 3-manifolds." Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 1, 77–80.
The... | 4 | https://mathoverflow.net/users/20787 | 166182 | 86,708 |
https://mathoverflow.net/questions/166188 | 1 | Let $\Phi:A \rightarrow B$ be a flat morphism of commutative rings. Let $f \in A$, not a unit and $A/fA \cong B/fB$ induced by $\Phi$.
Let $M$ be an $A\_f$-module. Is it true that $M \otimes\_A B = 0 \Rightarrow M=0$?
Here's my way of thought so far: by flatness of $B$ it's enough to show this for $M$ a cyclic $A\_... | https://mathoverflow.net/users/33573 | In this special situation, does $M \otimes B=0$ imply $M=0$? | No, that is not true. Let $A$ be $\mathbb{Z}[x]/\langle x(1-x) \rangle$. Let $B$ be $A/\langle x \rangle$ with the obvious quotient morphism, $\Phi$. Let $f$ be $x$. Then the natural $A$-algebra homomorphism, $$ A[y]/\langle yx-1 \rangle \to A/\langle 1-x \rangle, \ \ y \mapsto 1, $$ is an isomorphism. To see this, obs... | 6 | https://mathoverflow.net/users/13265 | 166189 | 86,710 |
https://mathoverflow.net/questions/166190 | 1 | For each commutative monoid $M$, there exists a "groupification" $\widehat{M}$, i.e. an abelian group that satisfies an obvious universal property.
I tried to prove the following: If in the diagram of Monoids
$$ L \stackrel{j}{\rightarrow} M \stackrel{i\_1, i\_2}{\rightrightarrows} N$$
the morphism of monoids $j$ is ... | https://mathoverflow.net/users/16702 | "Exactness" of groupify functor | Let $A=\mathbb{N}\cup\{\infty\}$, considered as a monoid under addition. Let $M=\mathbb{N}$, $N=A\oplus A$, $i\_1(n)=(n,0)$ and $i\_2(n)=(0,n)$. Then the equalizer of $i\_1$ and $i\_2$ is $0\to\mathbb{N}$. But $\widehat{N}=0$, so $0\to\widehat{M}\to\widehat{N}$ is not exact.
If you want an example without absorbing e... | 6 | https://mathoverflow.net/users/75 | 166195 | 86,711 |
https://mathoverflow.net/questions/165919 | 4 | Let $F(x,y)$ be a squarefree binary form with integer coefficients,
possibly reducible, $\deg(F) \ge 3$.
I am interested in ways of getting infinitely many integer solutions $(x,y,m), m \ne 0$
to $F(x,y)=m$, maximizing $\max(|x|,|y|)$ relative to $m$.
More formally, suppose $F$ is as above, $f$ is an increasing fun... | https://mathoverflow.net/users/12481 | Diophantine equations with infinitely many large solutions | If the degree of the form is $d$, I think you get solutions with $f(|m|) > cm^{1/(d-2)}$ by applying Dirichlet's theorem in diophantine approximation to a root of $F(x,1)=0$. And you cannot do better than $cm^{1/(d-2)+\epsilon}$ by Thue-Siegel-Roth.
| 8 | https://mathoverflow.net/users/2290 | 166198 | 86,713 |
https://mathoverflow.net/questions/166179 | 7 | This is a question about ITTM model introduced by Hamkins et al. In [this](http://arxiv.org/pdf/math/9808093.pdf) paper it is proven that no admissible ordinal is clockable, so it either starts or lies within a gap in clockable ordinals. I seek for reference concerning sort of opposite result - that if an ordinal start... | https://mathoverflow.net/users/30186 | Only admissibles start gaps in clockable ordinals | Let me sketch the argument. Philip Welch is also on
MO, and I would encourage him to post further explanation and details.
The main question left open in the original ITTM paper
* *Joel David Hamkins and Andy Lewis*, [**Infinite time Turing machines**](http://dx.doi.org/10.2307/2586556), *J. Symbolic Logic* **65**... | 9 | https://mathoverflow.net/users/1946 | 166204 | 86,716 |
https://mathoverflow.net/questions/166215 | 2 | Does there exist a forcing $P$ which adds a generic real in the sense that $V[G] = V[x]$ for some $x \in ({}^\omega\omega)^{V[G]}$, and for all reals $y \in ({}^\omega\omega)^{V[G]}$, if $V[y] \neq V$, then there exists some $z \in {}^\omega\omega$ such that $V \subsetneq V[z] \subsetneq V[y]$?
Is there a forcing pos... | https://mathoverflow.net/users/43354 | Intermediate Extensions Determined by Reals | I think the forcing that adds one Cohen real fits the requirements in your first question. Although it's not the case that all the new reals in the extension are Cohen reals, it is true that the submodel generated by any new real is also obtainable by adjoining a single Cohen real. So there will be strictly smaller ext... | 6 | https://mathoverflow.net/users/6794 | 166217 | 86,722 |
https://mathoverflow.net/questions/163603 | 1 | I'm looking for a generalization to the urn-ball matching problem. As a reminder of what I've got in mind, here's the simple version:
Randomly assign (with replacement) $N$ balls to $M$ urns. Afterwards, for any urn with more than one ball assigned to it, extract one at random.
The total number of matches (urn-ball... | https://mathoverflow.net/users/49684 | A generalized urn-ball matching problem; Complicated combinatoric/probabilistic limit | The following solves my problem, but not the question stated above, exactly.
I managed to sort of sidestep the intractable sum above by changing the rules of the generalization a little bit.
Fix the assignment of the balls to the urns. However, instead of pulling out balls from overmatched urns with probability pro... | 0 | https://mathoverflow.net/users/49684 | 166219 | 86,724 |
https://mathoverflow.net/questions/117202 | 2 | Is there any simple graph $\Gamma$ with 16 vertices with full automorphism group $G$ such that $H\cong Q\_8$ be a semiregular normal subgroup of $G$?
| https://mathoverflow.net/users/27831 | simple graphs of degree 16 with a semiregular normal subgroup isomorphic to the quaternion group $Q_8$ | Here is an example.
Adjacency matrix:
$ \left[ \begin{array}{cccccccccccccccc}
0&1&1&0&1&1&1&0&0&0&0&0&0&0&0&0\\1&0&0&1&1&1&0&1&0&0&0&0&0&0&0&0\\1&0&0&0&0&0&1&0&1&1&1&0&0&0&0&0\\0&1&0&0&0&0&0&1&1&1&0&1&0&0&0&0\\1&1&0&0&0&0&0&0&0&0&1&0&1&0&1&0\\1&1&0&0&0&0&0&0&0&0&0&1&0&1&0&1\\1&0&1&0&0&0&0&0&0&1&0&0&1&1&0&0\\0&1&0&... | 3 | https://mathoverflow.net/users/22377 | 166220 | 86,725 |
https://mathoverflow.net/questions/166227 | 2 | let us consider following model
$$y(t)=A\_1 \sin(\omega\_1 t+\phi\_1) + A\_2 \sin(\omega\_2 t+\phi\_2) + A\_3 \sin(\omega\_3 t+\phi\_3)+ \ldots +A\_p \sin(\omega\_p t+\phi\_p)+z(t)$$
we have three parameter fixed,but unknown and also $z(t)$ is simple white noise, before i will ask my question let us consider follow... | https://mathoverflow.net/users/6003 | probabilistic distribution of given data | Maybe the following paper [Link](https://ieeexplore.ieee.org/document/1092281) (Probability Distributions for Noise Plus Several Sine Waves--The Problem of Computation, by S.O. Rice) will be helpful.
| 1 | https://mathoverflow.net/users/32389 | 166229 | 86,731 |
https://mathoverflow.net/questions/166234 | 1 | Let $G$ be a finite group such that $G$ has a normal subgroup $H$ and $H$ is isomorphic to the alternating group $A\_5$. Also we know that $G/H \cong A\_5$.
Can we say that $G \cong A\_5\times A\_5$?
Thanks for your helps
| https://mathoverflow.net/users/31045 | on the extensions of $ A_5$ by $A_5$ | Well, $G$ must be isomorphic to the direct product $A\_{5} \times A\_{5}$ in any case, under your assumptions, since the outer automorphism group of $A\_{5}$ just has order $2$. Note that $H$ is a maximal normal subgroup of $G,$ and that $F(G) = 1$ under your assumptions. If $G \not \cong A\_{5} \times A\_{5},$ then $H... | 4 | https://mathoverflow.net/users/14450 | 166238 | 86,735 |
https://mathoverflow.net/questions/166150 | 0 | Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Let $I$ an ideal of $R$. We have
$$0:\_MI = \cap\_x(0:\_Mx),$$
where $x$ runs a set of generators of $I$.
Now set $S = R[T]$ with $T$ is a variable. We have $M\otimes S = M[T]$ and $IS = I[T]$ and
$$0:\_{M[T]}IS = (0:\_MI)[T]$$
by the... | https://mathoverflow.net/users/17901 | Colon operation after adjoint variables | The question is not true by the example of Neil Epstein as above.
Edit: Let me finish this question.
We give a generalization for the example of Neil Epstein.
Let $(R, \mathfrak{m})$ be a Gorenstein local ring of dimension $0$ and the embedded dimension $\ell(\mathfrak{m}/\mathfrak{m}^2)>1$. We show that $M = R$ ... | 0 | https://mathoverflow.net/users/17901 | 166242 | 86,738 |
https://mathoverflow.net/questions/166184 | 4 | I have been working on the Mahler conjecture for over a year now and have made some progress for certain classes of convex polytopes and I'm now attempting to write up my results specified to $\mathbb{R}^3$ and $\mathbb{R}^4$ for ease of explanation before attempting to generalize to $d \geq 5$. I understand that the M... | https://mathoverflow.net/users/20343 | The Mahler conjecture and non-zonoidal 3-polytopes (4-polytopes) | It is correct that the Mahler conjecture was proven for zonoids. However, your definition of zonoid is not quite right.
A ***zonotope*** is the Minkowski sum of finitely many line segments. And a ***zonoid*** is a compact convex body that is the Hausdorff limit of a sequence of zonotopes. So every zonotope is a zonoi... | 5 | https://mathoverflow.net/users/48084 | 166243 | 86,739 |
https://mathoverflow.net/questions/166149 | 2 | Let $f \in C^{\infty}(\mathbb{R}^2)$ be smooth and compactly supported. Can we approximate $f(x,y)$ by sums of the form $\sum\_{i=1}^m g\_i(x) h\_i (y)$ where $g\_i, h\_i \in C^{\infty}(\mathbb{R})$ are smooth with compact support.
Exact formulation:
Suppose $f \in C^{\infty}(\mathbb{R}^2)$ with $supp(f)\subseteq [a,... | https://mathoverflow.net/users/50820 | Approximation of smooth compactly supported functions on $\mathbb{R}^2$ using sums of products of one variable functions | The situation is even much better, because $C^\infty([a,b]\times [c,d])=C^\infty([a,b])\tilde{\otimes}\_\pi C^\infty([c,d])$ (the completed projective tensor product) and due to a celebrated result of Grothendieck every Element of $X\tilde{\otimes}\_\pi Y$ for two Frechet spaces is even a series $\sum\limits\_{n=0}^\in... | 2 | https://mathoverflow.net/users/21051 | 166250 | 86,742 |
https://mathoverflow.net/questions/166253 | 2 | Given a finite set $A$ on the Riemann sphere and a homeomorphism $f$, may I say there exists a quasiconformal homeomorfism isotopic to $f$ relative to the set $A$?
| https://mathoverflow.net/users/50849 | Quasiconformal deformation | Yes, as long as $f$ is orientation-preserving.
By a theorem of Munkres-Smale-Whitehead (see Corollary 1.18 in Manifolds with Transverse Fields in Euclidean Space, by Whitehead), $f$ can be approximated arbitrarily well by smooth diffeomorphism.
Let $g$ be a diffeomorphism close enough to $f$. We can modify $g$ in... | 6 | https://mathoverflow.net/users/38319 | 166265 | 86,748 |
https://mathoverflow.net/questions/166269 | 8 | Let $A$ be a $n\times n$-matrix. We let $\|A\|\_p$ denote the norm of $A$ when considered as a linear operator on $\ell^p(\{1,2,\ldots,n\})$, that is,
$$
\|A\|\_p = \sup\_{x\neq 0}\frac{\|Ax\|\_p}{\|x\|\_p}.
$$
By the Riesz-Thorin theorem, the $p$-norm of $A$, as a function of $p$, is log-convex, meaning that the fun... | https://mathoverflow.net/users/24916 | Is the p-norm of a matrix strictly log-convex? | The answer is "no".
First, a trivial counterexample: let $A\_n$ be the $n \times n$ matrix with all $1$s on the first row and zeroes elsewhere, then $\frac{1}{p} \mapsto \|A\_n\|\_p = n^{1/p}$ is log-convex but non-constant and not strictly log-convex.
One can deal with this example by replacing "constant" with "lo... | 5 | https://mathoverflow.net/users/766 | 166277 | 86,754 |
https://mathoverflow.net/questions/166013 | 13 | Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete topology. Weibel, in his $K$-theory book, computes the torsion in its coefficient ring. I would like to know the torsion-free ... | https://mathoverflow.net/users/7108 | What is the coefficient ring of algebraic K theory of the discrete $\mathbb{C}$? | I think the answer to this question is not known. All we can say about the K-theory of $\mathbb{C}$ concerns the torsion.
The trouble starts with $K\_1(\mathbb{C})\cong\mathbb{C}^\times$, which is pretty difficult to understand as an abelian group. There is a formula for $K\_2$ of a field due to Matsumoto which is $K\... | 11 | https://mathoverflow.net/users/50846 | 166281 | 86,756 |
https://mathoverflow.net/questions/166055 | 2 | A paper by Kaczorowski & Perelli [arXiv:1207.2312](http://arxiv.org/abs/1207.2312) dealing with the elements of the Selber class with degree two suggests that $S\_2$ coincides with the automorphic l-functions over $GL\_2( \mathbb{Q} )$.
**EDIT.** We know from [[1]](http://www.ams.org/mathscinet-getitem?mr=1253620)[[2... | https://mathoverflow.net/users/43108 | Automorphic L-functions over $GL_n( \mathbb{Q} )$ | I think your question is too broad, at least it is not clear to me what you are really asking. At any rate, if the Ramanujan-Selberg conjecture is true, then the $L$-function of an automorphic representation of $GL\_n$ over $\mathbb{Q}$ (with unitary central character) belongs to the Selberg class. Moreover, it is beli... | 2 | https://mathoverflow.net/users/11919 | 166282 | 86,757 |
https://mathoverflow.net/questions/166280 | 6 | On a four-manifold, there is apparently a relation between the first Pontryagin class modulo 4 and the Pontryagin square of the second Stiefel-Whitney class:
$\mathfrak{P}(w\_2) = p\_1 \; {\rm mod} \; 4$
This fact is for instance mentioned in the comments of [this question](https://mathoverflow.net/questions/61043/... | https://mathoverflow.net/users/2183 | Pontryagin square of Stiefel-Whitney classes and Pontryagin classes | In your first claim, it is a bit unclear what bundle you are considering. It is false for the tangent bundle of $\mathbb{C}P^2$: $\mathfrak{P}(w\_2) = c\_1^2 = 9 \; {\rm mod} \; 4$, while $p\_1 = 3$. The context of the question you link to is arbitrary oriented rank 3 bundles over a 4-dimensional base. Is that what you... | 6 | https://mathoverflow.net/users/13061 | 166291 | 86,759 |
https://mathoverflow.net/questions/166292 | 3 | In Example 2.2.19 of
Lazarsfeld, Positivity in Algebraic Geometry I,
I found the following statement:
Let $D$ be a divisor on an irreducible projective variety $X$. Then $D$ is nef and big if and only if there exists an effective divisor $N$ such $D-\frac{1}{k}N$ is ample for $k\gg 0$.
It is clear to me that $D... | https://mathoverflow.net/users/nan | Big and Nef divisors | $D-\frac{1}{k}N$ ample for any $k\gg 0$, that is $D-\epsilon N$ ample for any $0<\epsilon\ll 1$ ample, implies that $D$ is nef. This is because $Nef(X)$ is the closure of $Amp(X)$. If $C$ is an irreducible effective curve then $(D-\epsilon N)\cdot C = D\cdot C-\epsilon N\cdot C >0$. Therefore $D\cdot C>\epsilon N\cdot ... | 3 | https://mathoverflow.net/users/14514 | 166293 | 86,760 |
https://mathoverflow.net/questions/166288 | 10 | I was reading the paper on "curves of every genus with many points II" by: Elkies, Howe, et al. And some of the terms are not clear to me. Is there any elaborate exposition on these stuffs?
In particular I appreciate an explanation on the following. Let $C$ be a smooth, projective curve over field $\mathbb{F}\_q$ ($q... | https://mathoverflow.net/users/50859 | Quadratic twist of curve defined over finite field | Here is an explanation via explicit equations. First suppose $q$ is odd. Since the function field extension $\mathbf{F}\_q(B)/\mathbf{F}\_q(C)$ has degree $2$, it is the extension gotten by adjoining to $\mathbf{F}\_q(C)$ the square root of some nonsquare element $\,f\in\mathbf{F}\_q(C)$. Geometrically this means that ... | 13 | https://mathoverflow.net/users/30412 | 166301 | 86,764 |
https://mathoverflow.net/questions/166231 | 1 | **A revision:**
**According to the comment of Nate Eldredge, in order to avoid the triviality, we revise the property $P$.**
Assume that $A$ is a commutative unital Banach algebra. Its maximal ideal space is denoted by $\Delta\_{A}$. For $a\in A$, the Gelfand transform of $a$ is denoted by $\hat{a}$. An element $a\... | https://mathoverflow.net/users/36688 | Injective element of a commutative Banach algebra | $A(D)$ does not satisfy P. I will actually work on the upper semi-disk, but of course one could map things back to $D$. Consider $f\_{\epsilon}(z)=1-z^2-i\epsilon z$. The other point with the same image as $z$ is $w=-z-i \epsilon$, which is never in the (closed) semi-disk if $z$ is, so all $f\_{\epsilon}$ are injective... | 2 | https://mathoverflow.net/users/48839 | 166302 | 86,765 |
https://mathoverflow.net/questions/166297 | 27 | Weil's bound for Kloosterman sums states that for $(a,b)\not=(0,0)$,
$$
|K(a,b;q)|:=\left|\sum\_{x\in\mathbb{F}\_q^\*}\chi(ax+bx^{-1})\right|\leq 2\sqrt{q},
$$
where $\chi$ is a non-trivial additive character on $\mathbb{F}\_q$ (the field with $q$ elements).
My question is, is it known to be false that $\sqrt{q}$ can... | https://mathoverflow.net/users/36862 | Is Weil's bound for Kloosterman sums ever attained? | Going a bit beyond $61$, I find that the first counterexample to $|K(a,b;q)| < 2 \sqrt{q-1}$ with prime $q$ has $(q,ab) = (139,38)$, when $K(a,b;q) = -23.51308393\ldots = -2 \sqrt{138.216\ldots}\,$, and there are no further prime counterexamples up to $10^3$.
[*added later*] Extending the search overnight reached a b... | 29 | https://mathoverflow.net/users/14830 | 166305 | 86,767 |
https://mathoverflow.net/questions/165965 | 9 | So the question is that, over a finite field, does there exist an abelian variety $A$ for which there does not exist a generically one-to-one morphism from a hyperelliptic curve $C$ to $A$.
p.s. A result of Oort and de Jong proved that a "generic" abelian variety satisfies this property. However it seems generic mean... | https://mathoverflow.net/users/31327 | Motives over finite field not generated by hyperelliptic curves | I don't know the answer, but I'll note that this is explicitly raised as a question by Bogomolov and Tschinkel in remark 8 of this paper:
<http://www.cims.nyu.edu/~tschinke/princeton/papers/yuri/jacob/jacob6.pdf>
I'll also remark that a version of this question often comes up among people who study ranks of ellipti... | 3 | https://mathoverflow.net/users/431 | 166310 | 86,769 |
https://mathoverflow.net/questions/166298 | 7 | Given a finite group $G$, write $K(G)$ for the complete digraph on the elements of $G$. Label the edge from $g$ to $h$ by element $g^{-1}h$.
**Question**: For what groups does there exist a Hamiltonian path in $K(G)$ whose edge labels exhaust the elements of $G$, apart from the identity?
Some observations:
1. If ... | https://mathoverflow.net/users/10909 | Paths in groups | Groups with this property are known as *sequenceable*, although the standard
definition of a sequenceable group looks a little different: namely, a finite
group $G$ is called sequenceable if its elements can be arranged in a
sequence $(g\_1,\ldots,g\_n)$, where $n=|G|$, so that all partial products
$a\_1:=g\_1,\ a\_2:=... | 12 | https://mathoverflow.net/users/9924 | 166316 | 86,771 |
https://mathoverflow.net/questions/166307 | 4 | Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho\_0,\rho\_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L\_2$ transportation distance between $\rho\_0$ and $\rho\_1$ can be computed as follows:
$$d(\rho\_0,\rho\_1)^2=\inf\_{\rho(x,t),v(x,t)}\int\_{\Omega}\int\_0^1\rho(x,t)|... | https://mathoverflow.net/users/25311 | PDE-Based Triangle Inequality for Optimal Transportation | I first wrote a longer answer, but it got somehow lost. Here is a rewritten shorter one. The idea is to simply construct a curve going from $\rho\_0$ to $\rho\_1$ to $\rho\_2$. The only thing is to check how to make the reparametrization, i.e. decide the time $T$ when you move from the first transport to the other. Thi... | 4 | https://mathoverflow.net/users/11716 | 167321 | 86,773 |
https://mathoverflow.net/questions/66765 | -1 | the title is quite explicit: I would like to know the consequences of the degree conjecture for the Selberg class.
Thank you in advance.
| https://mathoverflow.net/users/13625 | Consequences of the degree conjecture | I think there are no direct consequences of the conjecture, but it seems like the natural first step in solving the bigger orthonormality conjecture (and/or related issues like unique factorization) which in turn imply a major open problems in number theory, the Artin conjecture (Dedekind conjecture, in the case of uni... | 2 | https://mathoverflow.net/users/43108 | 167327 | 86,776 |
https://mathoverflow.net/questions/163044 | 13 | Let $k$ be a field, and denote by $K\_p(k)^{(n)}$ the weight $n$ eigenspace of the Adams operations on the $p$-th $K$-group of $k$.
The Beilinson-Soulé (BS) vanishing conjecture predicts that
$$
K\_{2q-p}(k)^{(q)}=0
$$
for $p \leq 0$ and $q>0$ (cf. Levine, "Tate motives and vanishing conjectures...").
If this conje... | https://mathoverflow.net/users/14349 | Over which fields (of positive characteristic) is the Beilinson-Soulé vanishing conjecture known to hold? | Maybe let me begin with the remark that the Adams eigenspace decomposition concerns the rationalized algebraic K-theory $K\_i(k)\otimes\_{\mathbb{Z}}\mathbb{Q}$. Consequently, the Beilinson-Soulé conjectures as well as the abelian category of motives of Levine are rational objects.
Now for what's known in positive c... | 15 | https://mathoverflow.net/users/50846 | 167333 | 86,778 |
https://mathoverflow.net/questions/167323 | 53 | It is well known that there are functions $f \colon \mathbb{R} \to \mathbb{R}$ that are everywhere continuous but nowhere monotonic (i.e. the restriction of $f$ to any non-trivial interval $[a,b]$ is not monotonic), for example the [Weierstrass function](http://en.wikipedia.org/wiki/Weierstrass_function).
It’s easy t... | https://mathoverflow.net/users/7845 | Everywhere differentiable function that is nowhere monotonic | Everywhere differentiable but nowhere monotonic real functions do exist. It seems that the first correct examples were found by A. Denjoy in [this paper](http://www.numdam.org/numdam-bin/item?id=BSMF_1915__43__161_0). A short existence proof, based on Baire's category theorem, was given by C. E. Weil in [this paper](ht... | 32 | https://mathoverflow.net/users/11919 | 167335 | 86,780 |
https://mathoverflow.net/questions/167322 | 4 | I recently heard about the following problem:
Let $X$ be a projective variety with klt singularities and such that $-K\_X$ is big. Is $X$ a Mori Dream Space ?
Now, $-K\_X$ big if and only if $-K\_X -\epsilon D = -(K\_X+\epsilon D)$ is ample for some effictive divisor $D$ and some positive rational number $\epsilon ... | https://mathoverflow.net/users/nan | Varieties with big anti-canonical divisor | Yes, the issue concerns singularities of the pair $(X,\epsilon D)$.
Assume that $-K\_X$ is big then $-K\_X -\epsilon D = -(K\_X+\epsilon D)$ is ample for some effictive divisor $D$ and some positive rational number $\epsilon >0$. If $(X,\epsilon D)$ is a klt pair (that is $X$ is a log Fano variety) then, by BCHM $X$ ... | 4 | https://mathoverflow.net/users/14514 | 167339 | 86,782 |
https://mathoverflow.net/questions/158535 | 5 | I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch–Gordan coefficients.
I would be very grateful if you could share with me a closed form of the ratio between
$$
\begin{pmatrix}
l\_1 &l\_2... | https://mathoverflow.net/users/47396 | Closed form for 3j-symbol ratios | The following recursion relation
$$
C(m\_2+1,m\_3-1)\begin{pmatrix}l\_1 &l\_2 &l\_3\\m\_1&m\_2+1&m\_3-1\end{pmatrix}+
D(m\_2,m\_3)\begin{pmatrix}l\_1 &l\_2 &l\_3\\m\_1&m\_2&m\_3\end{pmatrix}+$$ $$
C(m\_2,m\_3)\begin{pmatrix}l\_1 &l\_2 &l\_3\\m\_1&m\_2-1&m\_3+1\end{pmatrix}=0,
\tag{1}$$
where $$C(m\_2,m\_3)=\sqrt{(l\_... | 4 | https://mathoverflow.net/users/32389 | 167350 | 86,786 |
https://mathoverflow.net/questions/167348 | 1 | If we have a family of classes $(\mathfrak{M}\_\alpha)\_{\alpha\in D}$ of $\in$-structures with $D$ being a limit ordinal or the class of ordinals, and a family $(\chi\_{\alpha,\beta})\_{\alpha<\beta(<D)}$ being a direct system (see [here][1] for a definition of a direct system) of end extensions (an end extension $\ch... | https://mathoverflow.net/users/43258 | How can one define the direct limit of classes? | You can define it the same way as you define a direct limit of sets, using [Scott's trick](http://en.wikipedia.org/wiki/Scott%27s_trick) to form equivalence classes. Whenever you have an equivalence relation defined on a class $X$, you can form equivalence classes that are sets by sending $x\in X$ to the set of all $y\... | 2 | https://mathoverflow.net/users/75 | 167352 | 86,788 |
https://mathoverflow.net/questions/54027 | 7 | There seems to be general opinion that, for positive integral quadratic forms in at least three variables, spinor genera in the same genus all have the same mass (not representation measures of some number, that is different, indeed some recent authors write of representation mass of numbers and it throws me off). Auth... | https://mathoverflow.net/users/3324 | Mass of spinor genus, positive integral quadratic forms | Better late than never:
In the adelic setup, the mass or measure of a lattice $L$ (the german word Maß translates to measure in English) is given as
$\mu(O(V)\backslash O(V)O\_{\mathbb A}(L))$, where $\mu$ is the (Tamagawa-normalized) Haar measure on the adelic orthogonal group $O\_{\mathbb A}(V)$ of the underlying ... | 5 | https://mathoverflow.net/users/8099 | 167353 | 86,789 |
https://mathoverflow.net/questions/167340 | 8 | Consider $\phi(A)$ a formula of second-order arithmetic with one free variable $A$ of type "set". Suppose $\exists A : \phi(A)$ is a true sentence. Does it follow (not in second order arithmetic itself, but in a stronger theory of your choice, e.g. ZFC) that there is a formula of second-order arithmetic $\psi(n)$ with ... | https://mathoverflow.net/users/11146 | Are there "non-constructive" sets in second-order arithmetic? | This may or may not be true depending on the background set theory.
On the one hand, it is consistent with ZFC that there exists a $\Delta^1\_2$-definable well-ordering $\prec$ of the reals. Then one can take for $\psi(n)$ the formula saying “$n$ is in the $\prec$-minimal set $A$ satisfying $\phi(A)$”, so the answer ... | 15 | https://mathoverflow.net/users/12705 | 167354 | 86,790 |
https://mathoverflow.net/questions/167349 | 28 | I asked the following question on [math.stackexchange](https://math.stackexchange.com/questions/448772/when-is-mathfraks-n-times-mathfraks-m-a-subgroup-of-mathfraks-p) several months ago:
>
> Let $n,m,p>1$ be such that $S\_n \times S\_m \hookrightarrow S\_p$. Does it imply that $p \geq n+m$?
>
>
>
Derek Holt g... | https://mathoverflow.net/users/43559 | When is $S_n \times S_m$ a subgroup of $S_p$? | I think I can do this with an elementary argument, but I have to rush off somewhere soon, so I will answer quickly and hope I get it roughly right!
Assume $p < n+m$, so we can assume also that $n>p/2$. $S\_n$ must have a faithful orbit in its action on $p$ points, and if that orbit has size $k$ then $n \le k \le p < ... | 26 | https://mathoverflow.net/users/35840 | 167359 | 86,792 |
https://mathoverflow.net/questions/166279 | 3 | Let $S$ be a $C^2$-regular hypersurface with $S=\partial V$ for some open set $V \subset R^{N+1}$, and let $\nu(P)$ be the exterior unit normal of $S$ with respect to $V$.
Assume that $S$ satisfies the R-sphere condition, that is for every $P\in S$ the tangent balls $B^\pm(P,R):= \{Q \in R^{N+1}: \ |P\pm R \nu(P) - ... | https://mathoverflow.net/users/50856 | Does the R-sphere condition imply that a surface is locally a graph of function on a ball of radius R? | It is true and not too hard. WLOG, $R=1$
Let $p,q$ be two points on $S$. Let $a$,$b$ be the outer unit normals to $S$ at $p$ and $q$ respectively. Let $v=\overline {pq}$. Since $B(p-a,1)\cap B(q+b,1)=\varnothing$, we have $|a+b+v|^2\ge 4$. Since $B(p+a,1)\cap B(q-b,1)=\varnothing$, we get $|a+b-v|^2\ge 4$. Adding tho... | 3 | https://mathoverflow.net/users/1131 | 167364 | 86,796 |
https://mathoverflow.net/questions/167336 | 7 | A distance function $d: \mathbb{R} \times \mathbb{R} \rightarrow [0,\infty)$ that is defined by a smooth Riemannian metric on the real line satisfies the following properties:
1. $d$ is a length metric (a.k.a. intrinsic metric, inner metric, Menger-convex ...);
2. $d$ is continuous;
3. for every $x,y \in \mathbb{R}$,... | https://mathoverflow.net/users/21123 | Riemannian distance functions on the real line | [Moved from the comments/chat.]
Define $D(x)$ by
$$
\begin{align\*}
D(x) =
\begin{cases}
d(0,x), & \text{ if } x\geq 0;\\
-d(x,0), & \text{ if } x< 0.
\end{cases}
\end{align\*}
$$
Function $D(x)$ is non-decreasing. It defines a measure $\nu$ on $\mathbb R$ such that $\nu[a,b] = D(b) - D(a)$ for every $b > a$.
Appl... | 9 | https://mathoverflow.net/users/26349 | 167369 | 86,798 |
https://mathoverflow.net/questions/167383 | 3 | Let $X$ be the closed subspace of Schwartz space $\mathcal{S}(\mathbb{R}^N)$ defined by
\begin{equation\*}
X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}.
\end{equation\*}
My question: Is $C\_0^\infty(\mathbb{R}^N)\cap X$ dense in $X$, w.r.t. to the topology of $\mathcal{S}(\mathbb{R}^N)$?
-... | https://mathoverflow.net/users/50891 | Can I approximate Schwartz functions which integrate to zero by $C_0^\infty$ functions which integrate to zero? | Yes. Your approximations (or any other reasonable method) will have small integrals $\epsilon\_k=\int \eta\_k f = O(k^{-N})$ for all $N$, so you can fix this by adding $-\epsilon\_k \varphi(x)$, $\varphi\in C\_0^{\infty}$, $\int\varphi = 1$.
| 6 | https://mathoverflow.net/users/48839 | 167384 | 86,803 |
https://mathoverflow.net/questions/38382 | 33 | I've read in the textbooks that the non-trivial generator $\eta\_n$ of $\pi\_{n+1}(S^n)$ is the suspension of the Hopf map $S^3\to S^2$, and the generator $\chi$ of $\pi\_5(S^3)$ is given by $\eta\_3 \circ \eta\_4$. Fine.
My question is, how I can visualize them? Is there a nice explicit way to describe these maps $... | https://mathoverflow.net/users/5420 | How can I visualize the nontrivial element of $\pi_4(S^3)$ and $\pi_5(S^3)$ ? | (This is a bit late, but I hope you find it interesting!)
Here's smooth representation of the generator of $\pi\_4(Sp(1))$ (and so the same homotopy group of $S^3$ and $SU(2)$). Consider $S^4 = \mathbb{HP}^1$, and $Sp(1)$ the unit sphere in $\mathbb{H}$. Then the following function $t\colon \mathbb{HP}^4 \to Sp(1)$ r... | 7 | https://mathoverflow.net/users/4177 | 167389 | 86,805 |
https://mathoverflow.net/questions/167378 | 1 | I wanted to know if there are any computations of cohomology groups $H^n(\Gamma,A^{(\Gamma)})$ in the literature for certain $n\in\mathbb{N}$, Abelian groups $A$, and infinite groups $\Gamma$.
Here $A^{(\Gamma)}$ is the direct *sum* of copies of $A$ (one for each $g\in\Gamma$) and $\Gamma$ acts on $A^{(\Gamma)}$ via... | https://mathoverflow.net/users/46541 | Known computations of certain 2-cohomology groups? | When $A$ is a commutative ring, the module $A^{(\Gamma)}$ you describe coincides with the group ring $A[\Gamma]$ with it's canonical (right) $\Gamma$-action.
In particular, let $A=\mathbb{Z}$. Then $\Gamma$ is a *duality group* if there exists an integer $n$ such that $H^i(\Gamma ; \mathbb{Z}[\Gamma])=0$ for $i\neq n... | 3 | https://mathoverflow.net/users/8103 | 167403 | 86,810 |
https://mathoverflow.net/questions/167404 | 3 | The question I want to ask is inspired by [this](https://mathoverflow.net/questions/109/what-do-epimorphisms-of-commutative-rings-look-like) mathoverflow post about epimorphisms in the category of commutative rings. I found the seminar (by P. Samuel) referenced by David Rydh particularly interesting so I wanted to ask ... | https://mathoverflow.net/users/48273 | What do epimorphisms in noncommutative rings look like? | Chapter XI of B. Stenström: *Rings of Quotients*, Springer Grundlehren vol. 217, 1975; [SpringerLink](http://link.springer.com/book/10.1007/978-3-642-66066-5/page/1); an earlier draft had appeared as LNM 237.
H. H. Storrer: [*Epimorphic Extensions of Non-Commutative Rings*](http://retro.seals.ch/digbib/view?rid=comah... | 4 | https://mathoverflow.net/users/27465 | 167405 | 86,811 |
https://mathoverflow.net/questions/167406 | 3 | I am reading an article and came across this expression and would appreciate some explanation.
"We have a function
$$u(F)=2\Big[\frac{F-L}{L}-ln\frac{F}{L}\Big]$$
Since $u''(F)>0$ a Taylor series expansion with second-order remainder of $u(F\_n)$ about $F\_n=L$ implies
$$u(F\_n)=\int\_0^L\frac{2}{K^2}(K-F\_n)^+dK+... | https://mathoverflow.net/users/50898 | Taylor Series Expansion | you can find this worked out in Appendix 1 of [Towards a Theory of Volatility Trading](http://pricing.online.fr/docs/TradingVolatilityStrat.pdf)
the general identity (for $L>0$) is
$$u(F)=u(L)+u'(L)[(F-L)^+-(L-F)^+]$$
$$\qquad+\int\_0^L u''(K)(K-F)^+\,dK+\int\_L^\infty u''(K)(F-K)^+ dK,$$
where $(K-F)^+$ means $K... | 0 | https://mathoverflow.net/users/11260 | 167413 | 86,813 |
https://mathoverflow.net/questions/167362 | 3 | Let $M\in\mathbb{R}^{n\times n}$ be symmetric positive definite and consider a matrix $Q\in\mathbb{R}^{n\times m}$ ($m<n$) with orthonormal columns ($Q^TQ=I$). I'm interested in finding an exact expression for
$$
K\equiv\max\_{x}\frac{x^Tx}{x^TQ^TM^{-1}Qx}
$$
in terms of the Rayleigh quotient of $M$ (no inverse).
Equiv... | https://mathoverflow.net/users/40734 | Maximising a Rayleigh quotient over a subspace | If $z=M^{-1/2}Qx$, then
$$K=\max\_{z\in S}\frac{z^TMz}{z^Tz},$$
where $S$ is the range of $M^{-1/2}Q$.
| 3 | https://mathoverflow.net/users/9833 | 167420 | 86,815 |
https://mathoverflow.net/questions/167410 | 1 | Let $S\subset\mathbb{P}^n$ be a finite set of $s$ reduced points. Let $\mathcal{I}$ be the ideal sheaf of $S$ in $\mathbb{P}^n$. We consider the sheaf
$$\mathcal{F}\_k:=\mathcal{O}\_{\mathbb{P}^n}(kd)\otimes\mathcal{I}^{km}.$$
Therefore $H^{0}(\mathbb{P}^n,\mathcal{F}\_k)$ is the space of hypersurfaces of degree $kd$ ... | https://mathoverflow.net/users/nan | Higher cohomology of sheaves on a projective space | It seems to me that you are looking for a kind of regularity result. By Remark $1.8.44$ in Lasarsfeld, Positivity in algebraic geometry 1, you have that if $X\subset\mathbb{P}^n$ is a smooth variety of codimension $c$ defined scheme-theoretically by polynomials of degrees $d\_1\geq d\_2\geq ...\geq d\_m$. Then
$$H^i(\... | 0 | https://mathoverflow.net/users/14514 | 167427 | 86,820 |
https://mathoverflow.net/questions/167414 | 2 | Ramanujam's vanishing theorem states for $X$ a Kähler surface and (1)$D$ a 1-connected divisor, (2)$h^0(O\_X(nD))\ge 2$ for some $n\ge 1$, (3) $|nD|$ is not an irrational pencil, we have $h^1(O\_X(-D))=0$.
In the proof given in BPV, it used a topological lemma which is based on Hodge decomposition of a Kähler manifol... | https://mathoverflow.net/users/nan | Algebraic proof of Ramanujam's vanishing theorem | Perhaps this helps: Kyungho Oh, [*Vanishing Theorems for Singular Varieties*](http://www.ams.org/journals/proc/1993-119-01/S0002-9939-1993-1163335-X/S0002-9939-1993-1163335-X.pdf), (Proceedings of the American Mathematical Society).
| 1 | https://mathoverflow.net/users/43108 | 167432 | 86,823 |
https://mathoverflow.net/questions/167338 | 10 | Let $X$ be an irreducible affine variety (integral, reduced scheme of finite type over an algebraically closed field $\Bbbk$ *of characteristic zero*) of dimension $n$. The well-known Noether Normalization Lemma states that there is a finite morphism $\pi:X\to\mathbb A\_\Bbbk^n$.
Assume now that I have an irreducibl... | https://mathoverflow.net/users/9947 | Control ramification in Noether Normalization | Let $y\_0\in Y$ be a point in the smooth locus of $X$. We are going to construct a finite morphism $\pi: X\to \mathbb A^n\_k$ unramified (thus étale) at $y\_0$. As the ramification locus is closed, this will imply that $\pi$ is unramified at the generic point of $Y$. From now on we can merely forget $Y$. I will suppose... | 4 | https://mathoverflow.net/users/39387 | 167436 | 86,825 |
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