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https://mathoverflow.net/questions/116407 | 2 | Is there a classification of the compact MU or BP modules in any category of spectra? Can the periodicity theorem be finagled to give a MU-module structure on finite spectra?
| https://mathoverflow.net/users/11546 | Compact MU or BP Modules | No nonzero finite spectrum admits an $MU$-module structure. Indeed, suppose $F$ is a finite spectrum with an $MU$-module structure. Then for all $n$, $F$ has a map $v\_n:\Sigma^{2p^n-2}F\to F$, which induces an isomorphism on $K(n)\_\*F$ (there's a subtlety here in that it's not obvious that the $v\_n$ map on $F$ and t... | 11 | https://mathoverflow.net/users/75 | 116415 | 65,812 |
https://mathoverflow.net/questions/116408 | 11 | Given a compact Lie group G acting freely on a topological manifold M, is it true that the orbit space M/G is also a topological manifold? If so, why?
| https://mathoverflow.net/users/29948 | orbit space of a topological manifold | The answer is no. Bing constructed a space $X$ (called [the dogbone space](http://en.wikipedia.org/wiki/Dogbone_space)) so that $X$ is not a manifold, but $X\times R$ is homeomorphic to $R^4$. In particular, $M^4=X\times S^1$ is a $4$-manifold (since its universal cover is $R^4$) and $X$ is the quotient of $M^4$ by fre... | 15 | https://mathoverflow.net/users/21684 | 116419 | 65,813 |
https://mathoverflow.net/questions/116165 | 6 | This is inspired by a recent [question](https://mathoverflow.net/questions/116138). A set $A \subset \mathbb{Z}/n\mathbb{Z}$ with $|A|=m$ is a Sidon set if all the pairwise sums of distinct elements are unequal: $A+A=\{a+a' \mid a,a' \in A, a \ne a'\}$ has $\binom{m}2$ elements. Since $a+b=c+d$ implies $a-c=b-d$ it is ... | https://mathoverflow.net/users/8008 | Recovering Sidon sets from difference sets, part 2. | N0. It is NOT the case that it is true in general. It *is* true for all cases with both $m \le 5$ and prime $n \lt 100$. It is also true for $m=6$ and $n = 31$. But actually it is false ( for $m=6$) over $\mathbb{Z}$ and also over $\mathbb{Z}/{n}\mathbb{Z}$ for any $n \ge 36$, prime or composite.
The sets $$A=\{0, 1,... | 2 | https://mathoverflow.net/users/8008 | 116420 | 65,814 |
https://mathoverflow.net/questions/116413 | 4 | I think my questions relates to this other: "[counterexamples to differentiation under integral sign](https://mathoverflow.net/questions/105769/counterexamples-to-differentiation-under-integral-sign)"
In fact, it provides a counterexample
Consider $f(x,y)=y^3e^{-y^2x}$ and define $F(y) =\int\_0^{\infty}f(x,y)dx$
... | https://mathoverflow.net/users/29954 | Why can't I interchange integration and differentiation here? | You need a convenient locally convex space $E$ of functions in $x$ such that
$g\mapsto \int\_0^\infty g(x)\,dx$ is a bounded linear functional, and such that
$y\mapsto f(\quad,y)$ is a differentiable curve in $E$, at least of Hoelder class $C^{1,\alpha}$. Then the interchange follows by the chain rule.
The space $\ma... | 3 | https://mathoverflow.net/users/26935 | 116434 | 65,824 |
https://mathoverflow.net/questions/116417 | 3 | To be honest, I don't really know, whether or not the following is a research level
question:
Let $M$ be a smooth manifold, $C^\infty(M)$ the smooth function ring on $M$ and
suppose $R\subset C^\infty(M)$ is a subring. What are conditions, such that
$R$ is the smooth function ring of a smooth manifold ?
On a first... | https://mathoverflow.net/users/21302 | Smooth submanifolds defined by Subrings | You have to be careful: let $Y\subset V$ be a submanifold of $V$, you have a restriction map $$C^{\infty}(V)\rightarrow C^{\infty}(Y)$$
whose kernel is an ideal $p\_Y$, and if $Y$ is a closed submanifold:
$$C^{\infty}(Y)\cong C^{\infty}(V)/p\_Y$$
Thus the question is rather what ideals of $C^{\infty}(V)$ are of the ... | 7 | https://mathoverflow.net/users/27816 | 116439 | 65,825 |
https://mathoverflow.net/questions/116017 | 7 | Hello,
I wonder if anyone has a copy of Deligne's letter to Looijenga from 1974 mentioned as reference [26] in Bessis' paper *Finite complex reflection arrangements are $K(\pi,1)$* from 2006, see <http://arxiv.org/abs/math/0610777>, and is willing to share it / make it publicly available.
We were recently proving s... | https://mathoverflow.net/users/21291 | Deligne's letter to Looijenga from 1974 | Eduard Looijenga provided a scanned version of the letter, which now can be found at
<http://homepage.univie.ac.at/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf> (outdated)
<http://homepage.rub.de/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf> (2020-01-24)
Many thanks for making it publicly av... | 13 | https://mathoverflow.net/users/21291 | 116444 | 65,827 |
https://mathoverflow.net/questions/116430 | 7 | One way to construct an Aronszajn tree is to build a sequence of functions $\langle e\_\alpha : \alpha < \omega\_1 \rangle$ such that $e\_\alpha$ is an injection from $\alpha$ to $\omega$, and for any $(\alpha, \beta)$, $e\_\alpha$ agrees with $e\_\beta$ except on a finite set.
Is the following generalization possibl... | https://mathoverflow.net/users/11145 | larger coherent family of functions | If the continuum hypothesis holds, then this is impossible. To see this, consider a family of $\omega\_2$ many pairwise disjoint subsets of $\omega\_2$, and for each $x$ in this family, consider the range of $e\_x$, which is a subset of $\omega$. By the continuum hypothesis, there are only $\omega\_1$ many possible ran... | 5 | https://mathoverflow.net/users/1946 | 116460 | 65,835 |
https://mathoverflow.net/questions/116463 | 8 | It is a classical result that if $0^{\sharp}$ exists, then there is a model of $ZFC$ in which there is a $\Delta^1\_3$ well ordering of reals but no $\Delta^1\_2$-well ordering.
My question is: Is $0^{\sharp}$ necessary to prove this? Or does the statement $\mathbf{CON}(ZFC+\mbox{there is a }\Delta^1\_3\mbox{ well or... | https://mathoverflow.net/users/14340 | $\Delta^1_2$-well ordering vs $\Delta^1_3$ | Hi Yu,
No, your statement is equiconsistent with $\mathsf{ZFC}$. In
>
> Leo Harrington. *Long projective wellorderings*, Annals of Mathematical Logic 12 (1977) 1-21, [MR0465866 (57 #5752)](http://www.ams.org/mathscinet-getitem?mr=465866).
>
>
>
it is shown that it is equiconsistent with $\mathsf{ZFC}$ to ha... | 7 | https://mathoverflow.net/users/6085 | 116465 | 65,837 |
https://mathoverflow.net/questions/116467 | 2 | Is there any counter example for the following statement?
STATEMENT:
Let $0 \to F \to A \to Q \to 0$ be a short exact sequence of abelian groups.
Assume that $F$ is a finite group, and $Q$ is a uniquely divisible abelian group.
Then, this exact sequence splits.
Please give me any advice.
| https://mathoverflow.net/users/39742 | Finite / uniquely divisible abelian groups | By "uniquely divisible", I guess you mean torsion-free and divisible, i.e., $Q$ is a $\mathbb{Q}$-vector space. If so, the answer to your question is that every such exact sequence splits.
It is enough to show $\mathrm{Ext}(Q, \mathbb{Z}/n) = 0$. We have an exact sequence
$$\hom(Q, \mathbb{Z}/n) \to \mathrm{Ext}(... | 8 | https://mathoverflow.net/users/2926 | 116473 | 65,843 |
https://mathoverflow.net/questions/116476 | 8 | I wonder, why we consider the notion of pseudoholomorphic curve: By definition a pseudoholomorphic curve in an almost complex manifold $X$ is a smooth map $f: C \rightarrow X$
from a Riemann surface $C$ into $X$ such that $df \circ J=J \circ df$ for the respective almost complex structure $J$. Why does it make sense to... | https://mathoverflow.net/users/29973 | Where does the notion of pseudoholomorphic curve come from? | As many have pointed out: Gromov introduced it\* wrote a seminal paper utilizing it, and we continue to use it today because it's an incredibly useful tool. I've never spoken to Gromov about why he introduced it (who knows how great mathematicians come up with great ideas) but I can try to give some (probably historica... | 12 | https://mathoverflow.net/users/3593 | 116480 | 65,848 |
https://mathoverflow.net/questions/116455 | 0 | Hi,
I'm reading the following paper: <http://fma2.math.uni-magdeburg.de/~holm/ARTIKEL/holm-hu-23-05.pdf>
I've come across a piece of information, which I don't understand, and wanted to ask, if I overlook something.
Let $A\_0:=k[x,y]/\langle x^2,xy,y^2\rangle$.
Let $A\_0^0=\langle \tilde{1} \rangle$ be the regu... | https://mathoverflow.net/users/12826 | Representation dimension of a special algebra | I don't think (\*) is correctly copied from the paper. The corresponding claim in the paper is that every morphism from an indecomposable summand of $M$ except for the identity morphism from $T$ to $T$ factors through $N\_1$.
I think a precisely correct statement is that, for any direct summand $T$ of $M$, Hom($T$,$... | 1 | https://mathoverflow.net/users/468 | 116482 | 65,849 |
https://mathoverflow.net/questions/116485 | 0 | Hello,
I would like to know if there is some kind of heuristics according to which the sequence $(S\_n)\_{n\in\mathbb{N}}$ would be a divisibility sequence, where $S\_n$ is the $n$-th superperfect number (i.e a number $m$ such that $\sigma(\sigma(m))=2m$).
Thanks in advance.
| https://mathoverflow.net/users/13625 | Is the sequence of superperfect numbers a divisibility sequence? | The sequence of superperfect numbers is a dvisibility sequence if and only if no odd superperfect numbers exist.
To see this just note that the even superperfect numbers are known to be all powers of two, precisely they are $2^{k-1}$ such that $2^k -1$ is a (Mersenne) prime.
Then, if all are even it is clearly a d... | 2 | https://mathoverflow.net/users/nan | 116488 | 65,853 |
https://mathoverflow.net/questions/116362 | 12 | Below I will give some definitions. My question is: do these appear in the literature, and if so, under what name?
Let $G$ and $H$ be groups that may not be commutative. For $y\in G$, define $R\_y:G\to G$ by $R\_y(x)=xy$. Let $f$ be a function from $G$ to $H$. Define $\delta\_n(f):G^n\to H$ by \begin{align\*}
\delta... | https://mathoverflow.net/users/10366 | Polynomial maps between noncommutative groups | Polynomial mappings of groups were [investigated](https://people.math.osu.edu/leibman.1/preprints/pon.pdf) by Leibman in connection with density Ramsey theory.
**Definition.** Let $G$ be a group and $f:\mathbb{Z}\to G$ any map. The *discrete derivative* of $f$ is the map $D\_a f(b) = f(b+a) f(b)^{-1}$. The map is cal... | 5 | https://mathoverflow.net/users/24201 | 116492 | 65,855 |
https://mathoverflow.net/questions/116493 | 1 | Hello,
I would be glad, if someone could answer a question concerning the following:
<http://www.math.uni-bonn.de/people/schroer/preprints/repdim.pdf>
On page 5 they show (3)=>(1). The last step is not clear to me.
I wanted to ask, why (and under which general conditions) the Hom-functor can be omitted.
Thank... | https://mathoverflow.net/users/29976 | Question about an exact sequence | For any abelian category $\mathcal{A}$, an object $M\in\mathcal{A}$ is called a generator if, given an injective nonsurjective morphism $U\to V$ in $\mathcal{A}$, there always exists a morphism $M\to V$ that cannot be factorized as $M\to U\to V$.
In particular, an object $M$ in the abelian category of left $A$-module... | 4 | https://mathoverflow.net/users/2106 | 116498 | 65,860 |
https://mathoverflow.net/questions/116490 | 2 | First recall how the cup product is defined for the cohomology of a group $G$:
Fix a projective resolution $P \to \mathbb{Z}$ over $\mathbb{Z}G$. Then $P \otimes P \to \mathbb{Z} \otimes \mathbb{Z} = \mathbb{Z}$ is a projective resolution of $\mathbb{Z}$ over $\mathbb{Z}G \otimes \mathbb{Z}G=\mathbb{Z}[G \times G]$.... | https://mathoverflow.net/users/18571 | Why do we use the diagonal for diagonal approximations ? | The diagonal map $\Delta$ is "coassociative": the two maps $(\Delta \otimes 1) \circ \Delta$ and $(1 \otimes \Delta) \circ \Delta$ from $\mathbb{Z}G$ to $(\mathbb{Z}G)^{\otimes 3}$ are equal. Therefore $\Delta$ induces an *associative* product on cohomology. Similarly, the map $\varepsilon: \mathbb{Z}G \rightarrow \mat... | 2 | https://mathoverflow.net/users/4194 | 116500 | 65,861 |
https://mathoverflow.net/questions/116445 | 8 | Many examples comes to mind, the most famous being the Gödel's theorems viewed as formalisations of the Liar's paradox. I just realised that the proof of non-calculability of Kolmogorov complexity is a positive rewriting of Berry's paradox. My question (perhaps to be made into collective mode) is a) what are the best e... | https://mathoverflow.net/users/17164 | Positive results coming from paradoxes | Kritchman and Raz [adapted](http://www.ams.org/notices/201011/rtx101101454p.pdf) the surprise examination paradox (a/k/a the unexpected hanging paradox) to prove Godel's second incompleteness theorem (extending the ideas underlying Chaitin's Berry's-paradox-inspired proof of Godel's first incompleteness theorem).
I ... | 9 | https://mathoverflow.net/users/10503 | 116509 | 65,864 |
https://mathoverflow.net/questions/116502 | 2 | Is there a triangle whose vertices, as well as the four classical points, the centroid, the orthocenter, the incenter, and the circumcenter, all have integer coordinates?
| https://mathoverflow.net/users/18207 | Integer triangle | Clearly it is enough to find a triangle in which all seven points are rational, as then you can make them integral by rescaling. But given *any* triangle with rational coordinates, aren't the centroid, orthocenter, and circumcenter all automatically at rational coordinates?
1. The centroid is the arithmetic average o... | 9 | https://mathoverflow.net/users/78 | 116510 | 65,865 |
https://mathoverflow.net/questions/116507 | 1 | If $F \rightarrow E \rightarrow B$ is a Serre fibration, and I know the complex $K$-theory of two of these spaces, what can I learn about the $K$-theory of the third? I wish there was a spectral sequence
$$
K^i(B,K^j(F)) \implies K^{i+j}(E)
$$
but it doesn't look like it. What can I do instead?
| https://mathoverflow.net/users/29980 | Can I compute K theory in Serre fibrations? | Here are a couple possible answers:
(1). You could follow Dylan's suggestion about the Atiyah-Hirzebruch spectral sequence for
$$H^{\ast}(B, K\_{\ast}) \implies K^{\ast}(B)$$
and use the fact that you know the K-theory of $B$ to conclude something about the differentials in this spectral sequence. Then plug thi... | 5 | https://mathoverflow.net/users/4649 | 116517 | 65,867 |
https://mathoverflow.net/questions/67015 | 15 | Hi, everyone!
I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such methods as algebraic extensions and complex analysis. Is there any way to prove this fact with only elementary methods?
| https://mathoverflow.net/users/11072 | Elementary Luroth theorem proof? | See here:
<http://math.berkeley.edu/~gbergman/grad.hndts/Luroth.ps>
| 5 | https://mathoverflow.net/users/29987 | 116522 | 65,869 |
https://mathoverflow.net/questions/116511 | 4 | So I've been thinking about a problem for a little bit and I decided it's time to ask those that know more about the subject than I do. I've been working on some Stochastic Calculus (a new area of math for me), and I've been looking at change of measures, and how they affect different properties of stochastic processes... | https://mathoverflow.net/users/29981 | Continuous Markov Process and Change of Measure | You have exactly the right idea. To make it precise, let $Y$ be a Brownian motion on some probability space $(\Omega,\mathcal{F},P)$, with respect to its natural filtration $\mathbb{F}$. Define a new measure $Q$ by
$\frac{dQ}{dP} = \exp\left(\int\_0^T\int\_0^tY\_sdsdY\_t - \frac{1}{2}\int\_0^T\left|\int\_0^tY\_sds\ri... | 1 | https://mathoverflow.net/users/26459 | 116547 | 65,879 |
https://mathoverflow.net/questions/116556 | 7 | I only ask because I don't know how to look for the answer.
| https://mathoverflow.net/users/1631 | Status of the Isomorphism problem for automatic groups? | It's still an open problem. The isomorphism problem for hyperbolic groups (a much smaller class) was only solved recently by Dahmani and Guirardel (see [here](http://arxiv.org/abs/1002.2590)), following work of Sela.
In the same vein, the conjugacy problem is also still open for automatic groups. It has been solved f... | 7 | https://mathoverflow.net/users/317 | 116557 | 65,884 |
https://mathoverflow.net/questions/116525 | 4 | Hi,
in 1971 M.Auslander showed that the representation dimension of $A$ is $\neq 1$ for every Artin algebra $A$.
Does anybody have a reference paper or book proving this? Is the proof easy and / or does it need many prerequisites?
Thanks for the help.
| https://mathoverflow.net/users/12826 | Why is the representation dimension of an Artin algebra never equal to 1? | First of all, you have to assume that $A$ is non-semi-simple. For a semi-simple Artin algebra, the representation dimension is defined to be 1.
For a non-semi-simple algebra, the representation dimension is, by definition, the smallest $d$ such that there exists $M$ an $A$-module which is both a generator and a co-g... | 6 | https://mathoverflow.net/users/468 | 116558 | 65,885 |
https://mathoverflow.net/questions/116559 | 7 | The [Kähler identities](http://mathworld.wolfram.com/KaehlerIdentities.html) (sometimes known as the Hodge identities) are an important collection of relationships between operators on the exterior algebra of a Kähler manifold. These relationships generalise to hermitian manifolds, sections of hermitian holomorphic vec... | https://mathoverflow.net/users/21564 | Where do the Kähler Identities first appear? | As you hinted yourself, they were indeed discovered by W.V.D. Hodge. They appear explicitly in his 1941 book "The Theory and Applications of Harmonic Integrals", which you can find [here,](http://en.bookfi.org/book/582372) see e.g. the lemmas on pages 172 and following. They were discovered by him some times in the 30s... | 7 | https://mathoverflow.net/users/13168 | 116560 | 65,886 |
https://mathoverflow.net/questions/116519 | 8 | By definition, an eigenform is a simultaneous eigenvector for all the Hecke operators $T\_n$. Suppose I know that $f$ is an eigenvector of a particular Hecke operator, say $T\_N$, does it follow that $f$ is also an eigenvector for all the other Hecke operators? Or are there counter-examples to this?
| https://mathoverflow.net/users/21596 | Is an eigenvector of a Hecke operator automatically an eigenform? | Following the comments, here is perhaps the simplest counterexample (once you know the Breuil-Conrad-Diamond-Taylor modularity theorem). The curves $y^2 + y = x^3$ and $y^2 + y = x^3 + 2x$ both reduce mod 2 to the same smooth curve, so their Hecke eigenforms have equal $T\_2$ eigenvalues.
However, the curves over $\m... | 9 | https://mathoverflow.net/users/121 | 116565 | 65,887 |
https://mathoverflow.net/questions/116548 | 11 | Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$.
For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9), that $\Gamma(E)$ (if not empty) is a (tame) Fréchet manifold with respect to the topology of uniform convergence of... | https://mathoverflow.net/users/17047 | Space of sections of a fibre bundle with non-compact base space | Your definition depends on the choice of the exhaustion and on the choice of the metric on $E$. To get a meaningful theory you have to add many more assumptions (like: a Riemannian metric of bounded geometry on $M$ where the open sets are geodesic balls ...).
For example, if you want to let the diffeomorphism group of ... | 13 | https://mathoverflow.net/users/26935 | 116573 | 65,891 |
https://mathoverflow.net/questions/116562 | 3 | I have encountered a problem in my elec. eng research that I find rather challenging, being a bear of very little brain. The question has been considered under a slightly different aspect here: <http://arxiv.org/abs/math-ph/0607011>
My formulation of the problem is as follows: with all quantities real, is it possible... | https://mathoverflow.net/users/30004 | An interesting equation of some practical interest. | You might be interested in a series solution. After a change of variable we can rewrite the equation in the form
$$ {{\rm e}^{-x}}-b x \left( sx-1 \right) = 0$$
so that for $s=0$ we have a solution at $x = w$ where $w = \text{LambertW}(-1/b)$.
Then we have a series solution in powers of $s$, which should converge for s... | 5 | https://mathoverflow.net/users/13650 | 116578 | 65,894 |
https://mathoverflow.net/questions/116564 | 14 | I'm teaching myself bits and pieces of forcing at the moment, for the purposes of translating them into sheaf-theoretic versions. I'm trying to write down what I feel is a cleaner description of the Easton product of forcing posets, by which I mean a global description rather than one in terms of elements like
$$
\left... | https://mathoverflow.net/users/4177 | I'll admit it: I don't understand the definition of the Easton product. | Here is a general way to think about product forcing, which may be
helpful. One has forcing notions $\mathbb{Q}\_\gamma$ for each
$\gamma$ in a class $D$ of ordinals. (In your example, $D$ is the
class of regular cardinals, and
$\mathbb{Q}\_\gamma=\text{Add}(\gamma,E(\gamma))$ for the Easton
function $E$.) One wants to... | 17 | https://mathoverflow.net/users/1946 | 116579 | 65,895 |
https://mathoverflow.net/questions/116581 | 0 | Consider the following classical construction (which is called Pfaffian representation as Sasha indicates):
Let $ V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space
of alternating two-forms. Then the equation $a\wedge a\wedge a=0$, $a\in V^4$
is homogeneous of degree three and hence define... | https://mathoverflow.net/users/13441 | Real Pfaffian representations of real cubic surfaces | In other words you are asking about Pfaffian representations of a cubic hypersurface. To find a Pfaffian representation of a smooth cubic hypersurface of dimension $d$ is equivalent to constructing a vector bundle $E$ of rank 2 with $c\_1 = 2$ generated by $6$ global sections and with $H^\bullet(S,E(-k)) = 0$ for $1 \l... | 3 | https://mathoverflow.net/users/4428 | 116587 | 65,896 |
https://mathoverflow.net/questions/116583 | 4 | Not sure if this makes sense, but is it possible Fermat's Last Theorem to fail
with a parametrization over some extension of $\mathbb{Z}$, i.e. are there
not all constant $x(t),y(t),z(t) \in K[t]$ where $K$ is an extension of $\mathbb{Z}$
s.t. $$x(t)^p + y(t)^p=z(t)^p, x(t)y(t)z(t) \ne 0, p > 2,\gcd(x(t),y(t),z(t))=1 $... | https://mathoverflow.net/users/12481 | Can FLT fail with a parametrization over some extension of Z? | No; in fact, you can take $K = \mathbb{C}$. This follows from the [Mason-Stothers theorem](http://en.wikipedia.org/wiki/Mason%E2%80%93Stothers_theorem) in the same way that FLT for sufficiently large $n$ follows from the abc conjecture.
As Chandan indicates in the comments, a more geometric reason this is false is t... | 8 | https://mathoverflow.net/users/290 | 116589 | 65,897 |
https://mathoverflow.net/questions/116602 | 1 | Let $X\subset\mathbb{P}^n\subset\mathbb{P}^N$ be a smooth irreducible projective complex variety, $L\subset\mathbb{P}^N\setminus\mathbb{P}^n$ be a linear $(N-n-1)$-dimensional variety and consider the cone $Y=C\_L(X)=\bigcup\_{p\in L \atop x\in X} \langle p,x \rangle\subset\mathbb{P}^N$.
Is it true that $Y$ is locall... | https://mathoverflow.net/users/15606 | Factoriality of cones | Let me describe a situation where your statement is true.
Let $X \subset \mathbb{P}^n$ be any smooth, *projectively normal* subvariety of the projective space and let $C(X)$ be the projecting cone over $X$, with vertex the point $p$. Then $C(X)$ is factorial (i.e. the coordinate ring of $C(X)$ is a UFD) if and only i... | 3 | https://mathoverflow.net/users/7460 | 116610 | 65,908 |
https://mathoverflow.net/questions/116627 | 29 | I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.
My idea is to amend my article with some theories that seemed useless when they are created but found use after some time.
I came with some ideas like the Turing machine but I think I'm not grasping the right exam... | https://mathoverflow.net/users/30026 | Useless math that became useful | Number theory, in particular investigations related to prime numbers, was famously considered useless (e.g., by Hardy) for practical matters. Now, since "everybody" needs some cryptography it is quite useful to know how to generate primes (e.g., for an RSA key) and alike, sometimes involving prior 'useless' number theo... | 58 | https://mathoverflow.net/users/nan | 116629 | 65,919 |
https://mathoverflow.net/questions/116586 | 0 | What's the defination of "quasi-conformal mappings"between Riemannian Manifolds?
Espectially I want to konw how the proof in section 4 of [Ha] works.
Thanks!
[Ha]R.S.Hamilton.Convex hypersurfaces with pinched second fundamental form.
Comm.Anal.Geom.,Vol2(1994),167-172.
| https://mathoverflow.net/users/30012 | What's the defination of "quasi-conformal mappings"between Riemannian Manifolds? | Here is the standard definition: Suppose that $M, N$ are oriented $n$-dimensional Riemannian manifolds, $f: M\to N$ is an orientation-preserving homeomorphism, which locally belongs to $W^{1,n}$. Then $f$ is called quasiconformal if there exists $K<\infty$ so that almost everywhere in $M$ the following inequality holds... | 4 | https://mathoverflow.net/users/21684 | 116631 | 65,920 |
https://mathoverflow.net/questions/116580 | 3 | Hello,
While studying Sobolev spaces, the following question came to my mind. Any help in this direction is appreciated.
**QUESTION**
Let $U\subseteq\mathbb{R}^n$ be open. Does there exist a function $f\in L^1\_{\text{loc}}(U)$ such that
1) the classical derivative $Df$ exists everywhere in $U$.
2) $f$ is w... | https://mathoverflow.net/users/27832 | Classical Derivative, Weak Derivative and Integration by Parts | Suppose $f \in W^{1,1}\_{loc}(U)$. Then no, since for such an $f$, we have that $Df$ exists and the approximate limit
$ap\lim\_{y\to x} \frac{f(x)-f(y)-Df(x)(x-y)}{|x-y|} = 0$
exists for almost every $x$, while from assuming classical differentiability we have
$\lim\_{y\to x} \frac{f(x)-f(y)-\nabla f(x)(x-y)}{|x-... | 3 | https://mathoverflow.net/users/28090 | 116642 | 65,922 |
https://mathoverflow.net/questions/116634 | 1 | Let $G$ be a graph and $e$ be an edge of the graph $G$ such that the subgraph $G\setminus e$
is connected. The subgraph $G\setminus e$ is the subgraph of $G$ obtained by deletion of the edge $e$ of $G$.
Assume that $G$ has $n$ vertices.
Is it true that $\lambda(G)-\lambda(G\setminus e)\geq \frac{1}{n}$?
Here $\lambda(... | https://mathoverflow.net/users/19075 | Difference of the maximum eigenvalue of a graph with the one of one-edge-deleted subgraph | It's always useful to test these questions on actual examples. The largest eigenvalue of the cycle $C\_n$ is 2 and the largest eigenvalue of the path $P\_n$ on $n$ vertices is $2\cos(\pi/(n+1))$. When $n=8$, this is 1.879385 and $2-1.8793852=0.120615 <1/8$.
In fact it is not hard to see that for large $n$ the differe... | 3 | https://mathoverflow.net/users/1266 | 116650 | 65,926 |
https://mathoverflow.net/questions/116582 | 1 | Suppose $F$ is a holomorphic (or polynomial if you prefer) function on $\mathbb C^3$ and $0$ is an isolated singularity of the surface $F=0$. Then on the one hand we can define Milnor number of this singularity, which is equal to the co-dimension of the Jacobian ideal of $F$ (the ideal generated by derivatives of $F$ a... | https://mathoverflow.net/users/13441 | Milnor number in terms of minimal resolution of an isolated singularity. | In principle it is possible, but it you need to know a bit more than the exceptional divisor. Denote by $X\_f$ the Milnor fiber of the singularity, and by $\mu\_f$ the Milnor number. Then
$$ \mu\_f= b\_2(X\_f)= \chi(X\_f)-1. $$
So the computation boils down to computing the Euler characteristic of the Milnor fiber.... | 4 | https://mathoverflow.net/users/20302 | 116654 | 65,930 |
https://mathoverflow.net/questions/116625 | 2 | Let $X$ be an (as nice as you prefer) alg. variety (or alg. stack) defined over $\mathbb{F}\_{q}$ and let $\mathcal{F}$ be an l-adic sheaf on $X\_n = X {\times\_{\mathbb{F}q}} \mathbb{F}\_{q^n}$. Fix an isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}\_l$ (otw replace below $\mathbb{C}$ by the $l$-adic numbe... | https://mathoverflow.net/users/1328 | recover trace of l-adic sheaves defined over an extension | As anon points out, the answer in general is that one cannot recover it up to trace. Indeed, a sheaf on $\mathbb F\_q$ is just a matrix up to conjugation. The corresponding sheaf on $\mathbb F\_{q^n}$ is just that matrix to the $n$th power. There is no relation between the traces.
However, if you have the characteris... | 3 | https://mathoverflow.net/users/18060 | 116660 | 65,935 |
https://mathoverflow.net/questions/116667 | 13 | What is the continuous probability distribution that maximizes entropy, given only the bounds of the random variable [a,b] and the mean mu of the probability distribution?
For example:
* if a=0, b=1, and mu=0.5, it should return a U[0,1].
* if a=10, b=20, and mu=20, it should return Dirac delta at x=20.
* if a=0, b... | https://mathoverflow.net/users/30040 | What's the maximum entropy probability distribution given bounds [a,b] and mean? | You can maximize the entropy using standard calculus of variations; you need to take into account the constraint that the probability distribution is properly normalized and that the mean is known, using Lagrange multipliers. You then find that the probability distribution is of the form:
$$p(x) = \frac{\alpha e^{\al... | 12 | https://mathoverflow.net/users/30041 | 116670 | 65,938 |
https://mathoverflow.net/questions/116669 | 4 | Does anyone has a nontrivial example of Cartier divisor $D$ on a projective variety $X$, such that $D$ is pseudoeffective but $-D$ is nef?
| https://mathoverflow.net/users/14854 | Pseudoeffective but anti-nef divisor | A nef divisor is also pseudo-effective so if $D$ is pseudo-effective and $-D$ is nef, then both $D$ and $-D$ are pseudo-effective and hence $D\equiv 0$ (numerically trivial). I assume this is not considered a non-trivial example.
| 4 | https://mathoverflow.net/users/10076 | 116674 | 65,940 |
https://mathoverflow.net/questions/116663 | 13 | In the proof of the Nilpotence Theorem, or at least in Ravenel's account of it in his Orange Book, a sequence of spectra are used, denoted $X(n)$ with $X(0)=\mathbb{S}$ and and $X(\infty)=MU$ such that $\langle X(n)\rangle\geq\langle X(n+1)\rangle$. These are the Thom spectra associated to the map $\Omega SU(n)\to BU$.... | https://mathoverflow.net/users/11546 | Connection of X(n) spectra to formal group laws | Although I'm not aware an interpretation of $X(n)$ in terms of formal group laws (aside from
the ones in Eric's and Dylan's comments), I would like to point out that the nilpotence theorem is fundamentally a geometric fact and is proved that way. The nilpotence theorem has a number of corollaries to the effect that th... | 20 | https://mathoverflow.net/users/344 | 116678 | 65,941 |
https://mathoverflow.net/questions/116524 | 2 | I'm reading the following paper: <http://math0.bnu.edu.cn/~huwei/paper/Holm-Hu-1.pdf>
On page 795 and 796 there are the definitions (in a diagrammatical way) of some $A\_n$ modules, whereupon $A\_n:=k[x,y]/\langle x^2,xy^{n+1},y^{n+2}\rangle$.
All the $A\_n$ modules defined there should be indecomposable.
>
>
... | https://mathoverflow.net/users/12826 | Proving indecomposability of special modules | Hello.
I believe that the endmorphism ring may not be hard to compute in this case.
Let $R = k[x,y]$ where $k$ is a field, and let $I = (x^2, xy^n, y^{n-1})$. Then $End\_R(R/I) = Hom\_R(R/I,R/I) \cong Hom\_{R/I} (R/I,R/I) \cong R/I$. But $R/I$ is a local ring since $\sqrt{I} = (x,y)$ which is a maximal ideal in $R... | 3 | https://mathoverflow.net/users/22388 | 116688 | 65,945 |
https://mathoverflow.net/questions/115366 | 17 | The following algebraic structure came up when I was thinking about invariants of coloured knots. The elements are all elements of a noncommutative free group $F$, and the operations are:
1. $a^b= b^{-1}ab$, taking the conjugate in $F$.
2. $[a,b]= aba^{-1}b^{-1}$, taking the commutator of two elements in $F$.
And ... | https://mathoverflow.net/users/2051 | What are the relations between conjugates and commutators? | There is a notion of multiplicative Lie algebras introduced here: Ellis, Graham J.
On five well-known commutator identities. J. Austral. Math. Soc. Ser. A 54 (1993), no. 1, 1–19. The signature there does include multiplication, though. The problem of finding axioms was solved there (I think somebody finally proved that... | 10 | https://mathoverflow.net/users/nan | 116693 | 65,947 |
https://mathoverflow.net/questions/116671 | 3 | In some deduction systems there is a rule\* that given $\exists x (\phi(x))$, we can infer $\phi(y)$, where $y$ is a fresh variable (i.e., one we haven't yet mentioned in this context). Call this rule "EI."
(Edit: in the opening sentence I originally said "in natural deduction systems there is typically a rule that..... | https://mathoverflow.net/users/29956 | Existential instantiation in Hilbert-style deduction systems | First, the standard definition of semantic entailment is neither the “simple” one nor the “complicated” one, but the following: $T\models U$ iff for every $M$, if $M,A\models T$ for every $A$, then $M,A\models U$ for every $A$.
First-order Hilbert-style usually employ some form of a generalization rule: the simplest ... | 6 | https://mathoverflow.net/users/12705 | 116696 | 65,948 |
https://mathoverflow.net/questions/116630 | 2 | Background/Motivation
---------------------
I'm working on algorithms for canonical labeling of a certain class of graphs (motivated by biology). The "difficult" instances of this problem can be reduced to graphs of the type mentioned in the title.
*Rigid* refers to graphs having only the trivial automorphism, i.e... | https://mathoverflow.net/users/30027 | Isomorphism of connected, rigid, N-regular graphs with chromatic index N? | Based on the further information, I am addressing the question of canonical labelling of connected, edge-coloured, regular graphs. I don't recall any literature on this. Let $n$ be the number of vertices and $d$ the degree.
As you noted, DFS (or BFS, etc) can be used to make a unique labelling for each starting verte... | 0 | https://mathoverflow.net/users/9025 | 116697 | 65,949 |
https://mathoverflow.net/questions/116592 | 5 | Given $2n$ integral points of $\mathbb Z^2$, is there a polynomial algorithm which gives
a matching consisting of $n$ non-intersecting straight vertical or horizontal segments between pairs of points if such a matching exists? (Not all segments have to be vertical
or horizontal, there can be $a$ vertical and $n-a$ hor... | https://mathoverflow.net/users/4556 | Complexity of a matching problem on the grid $\mathbb Z^2$ | I believe the full details, along the lines of what domotorp posted, are provided in "Reconstructing sets of orthogonal line segments in the plane" by Rendl and Woeginger. From the abstract,
"We show that reconstructing a set of $n$ orthogonal line segments in the plane from the set of their vertices can be done in $... | 6 | https://mathoverflow.net/users/11828 | 116720 | 65,956 |
https://mathoverflow.net/questions/116712 | 6 | Probably this is a trivial question, but I am unable to find an answer: is there a function $v(x)$ such that
$$
\int\_{0}^\infty x^n e^{v(x)} dx =\frac{1}{n!}
$$
for all positiv integer n?
| https://mathoverflow.net/users/3840 | Integral transform and $\frac{1}{n!}$. | Your question is a special case of the *Hamburger moment problem*: given a sequence of positive numbers $(\mu\_n)\_{n\geq 0}$ decides if there exists a positive measure $\mu$ on $\mathbb{R}$ such that $\newcommand{\bR}{\mathbb{R}}$
$$\mu\_n=\int\_{\bR} x^n \mu(|dx|),\;\;\forall n=0,1,2,\dotsc. $$
There exist many n... | 8 | https://mathoverflow.net/users/20302 | 116722 | 65,958 |
https://mathoverflow.net/questions/116707 | 1 | $\def\p{\phantom-}$Call a $\lbrace 0,-1,1\rbrace$-matrix $M$ an even sign configuration if every row of $M$ contains an even number of 1's and every column of $M$ contains an even number of $-1$'s. The matrix
$$\begin{bmatrix} \p1 & -1 & 1 & 0 \\\\ -1 & \p1 & 0 & 1 \\\\ \p0 & -1 & 1 & 1 \\\\ -1 & \p0 & 0 & 0 \end{bmatr... | https://mathoverflow.net/users/4556 | Even sign configurations with prescribed support | The answer to both of your questions is **yes**.
As suggested by the edit, consider the graph $G(A)$ whose vertex set is the set of non-zero entries of $A$, and where two entries are adjacent if they are in the same row or column. Now, as mentioned in the latest edit, if $G(A)$ contains a perfect matching then $A$ i... | 2 | https://mathoverflow.net/users/2233 | 116725 | 65,959 |
https://mathoverflow.net/questions/116724 | 0 | Let suppose that I have a box with $k$ different balls, each one with a different color.
At each time I have to extract a ball and observe the color. Then I put the ball back in the box.
How many extraction I need in order to see all the colors with probability at least $1-\alpha$?
This looks like a multinomial d... | https://mathoverflow.net/users/20067 | Multinomial -- how many trials in order to see all the values with prob 1-\alpha | This is the Coupon collector's problem. Check out <http://en.wikipedia.org/wiki/Coupon_collectors_problem>, in particular, the section on tail estimates.
| 2 | https://mathoverflow.net/users/15695 | 116726 | 65,960 |
https://mathoverflow.net/questions/116649 | 7 | Recently, while undertaking a study of commutative algebra, I learned three concepts: (i) a local ring, (ii) a regular local ring and (iii) a regular ring.
At the end, I found myself asking this seemingly naïve question: Are regular local rings the same objects as local rings that are regular? At first, I thought, "M... | https://mathoverflow.net/users/26077 | On similar concepts in mathematics whose similarity is a non-trivial fact. | 1. $f:\mathbb R^2\to \mathbb R$ is $C^\infty$.
2. $f:\mathbb R^2\to \mathbb R$ is $C^\infty$ along each $C^\infty$-curve
$c:\mathbb R\to \mathbb R^2$; i.e., $f\circ c$ is $C^\infty$ for each such $c$.
Equivalence was proved only in 1979 by Jan Boman.
EDIT: It was 1967, sorry for being careless.
EDIT: Using "gener... | 12 | https://mathoverflow.net/users/26935 | 116729 | 65,962 |
https://mathoverflow.net/questions/116730 | 5 | Hello,
Let's say that an integer $n$ is $k$-primal if $k$ is its smallest primality radius (i.e non negative integer $r$ such that both $n-r$ and $n+r$ are primes).
I think that for every positive integer $m$ and every non negative integer $k$, there exists an arithmetic progression made of $m$ $k$-primal integers. ... | https://mathoverflow.net/users/13625 | About a possible generalization of Green-Tao's theorem | János Pintz considered such questions recently, see his preprints [here](https://arxiv.org/abs/1002.2899) and [here](https://arxiv.org/abs/1004.1067). In particular, under a weak form of the Elliot-Halberstam conjecture there is an integer $d>0$ such that there are arbitrary long arithmetic progressions of primes $p$ s... | 6 | https://mathoverflow.net/users/11919 | 116731 | 65,963 |
https://mathoverflow.net/questions/116709 | 7 | Let $G$ be a semisimple algebraic group, $C$ be a smooth projective curve, and $\omega$ be the canonical line bundle.
The stack $\mathrm{Higgs}\_{\omega}$ is defined as the stack associating to each $S$ the groupoid consisting of $(E, \phi)$, where $E$ is a $G$-torsor over $X \times S$ and $\phi \in \Gamma(C \times S... | https://mathoverflow.net/users/2623 | Identifying $T^* \mathrm{Bun}_G$ with Higgs bundles | The tangent complex to $\operatorname{Bun}\_G(C)$ can be identified with $T\_{\operatorname{Bun}\_G(C)}=\mathbf{R}\pi\_\*{\operatorname{ad} P[1]}$, where $\pi:\operatorname{Bun}\_G(C)\times C\rightarrow \operatorname{Bun}\_G(C)$ is the natural projection and $P$ is the universal bundle.
Then the cotangent stack is $T... | 13 | https://mathoverflow.net/users/18512 | 116733 | 65,965 |
https://mathoverflow.net/questions/115636 | 10 | Background
----------
Many properties of permutations can be stated in terms of *classical patterns*.
For example:
* a permutation is *stack-sortable* if and only if it avoids 231 (Knuth 1975)
* a permutation corresponds to a *smooth* Schubert variety if and only if it
avoids 1324 and 2143 (Lakshmibai and Sandhya 1... | https://mathoverflow.net/users/340 | Properties of permutations with unknown pattern avoidance descriptions | Here's one idea. For every permutation $\pi$ of length $n$, there are $n^2+1$ permutation of length $n+1$ containing $\pi$. However, once you look at permutations of length $n+2$, this quantity depends on $\pi$. In their paper "[Posets of matrices and permutations with forbidden subsequences](https://doi.org/10.1007/s0... | 5 | https://mathoverflow.net/users/2663 | 116742 | 65,972 |
https://mathoverflow.net/questions/116749 | 18 | This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes.
Let $ \mathcal{A} $ be a unital Banach algebra over $ \mathbb{C} $, with $ \mathbf{1}\_{\mathcal{A}} $ denoting the identity ... | https://mathoverflow.net/users/26077 | Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem. | I think Hahn-Banach can be eliminated from the usual proof, but
being a non-expert in set theory, I cannot guarantee that the proof
is completely independent of the axiom of choice.
Here is a sketch of a basic calculus proof. A function $U\to B$
from a region $U\subset C$ to a Banach space $B$ is called analytic if
... | 8 | https://mathoverflow.net/users/25510 | 116751 | 65,977 |
https://mathoverflow.net/questions/116756 | 3 | Consider two $N \times N$ hermitian indefinite matrices $A\_1$ and $A\_2$. Consider their affine combination
\begin{align}
M(t)=(1-t)A\_1+tA\_2
\end{align}
I am interested in the minimum eigenvalue of $M(t)$. I can write this as
\begin{align}
\lambda(t)=\min\_{x ~\in~\mathbb{C}^{N\times 1}}~&x^HM(t)x \\\
&x^{H}x = 1
... | https://mathoverflow.net/users/27249 | Minimum eigenvalue of a Affine Combination of two Hermitian matrices | This is related to so-called *hyperbolic polynomials*, studied by L. Gaarding in the fifties. More generally, let $\lambda(\xi)$ be the least eigenvalue of $A(\xi)=\sum\_\alpha\xi\_\alpha A^\alpha$, where $A^\alpha$ are Hermitian matrices and $\xi$ is a real vector. Then $\lambda$ is a concave function. It is generical... | 6 | https://mathoverflow.net/users/8799 | 116760 | 65,982 |
https://mathoverflow.net/questions/116752 | 2 | I am interested about Social Network Analysis (SNA) with multiple links between pairs of nodes. I'm not aware of works in this area, and I am trying to find any reference in these regards.
In particular, is there also something about SNA with negative edges, possibly multigraphs?
| https://mathoverflow.net/users/13822 | Multigraphs and Social Network Analysis | The ideas of Formal Concept Analysis may be of use to you.
There is at least one paper on the general area
"Understanding social networks using Formal Concept Analysis"
Vaclav Snasel, Zdenek Horak and Ajith Abraham
<http://www.softcomputing.net/wi08.pdf>
| 1 | https://mathoverflow.net/users/3502 | 116761 | 65,983 |
https://mathoverflow.net/questions/116771 | 0 | This may be a stupid question.But I am stuck with it.Is Q\_p(the p-adic) connected under the usual topology?I was confounded with this problem while trying to construct a counter-example related to my master's thesis.
| https://mathoverflow.net/users/30081 | On topology of p-adic numbers. | Any non-empty open can be written as a disjoint union of opens ; for example
$\mathbb{Z}\_p=\cup(a+p\mathbb{Z}\_p)$ where $a$ runs through $\{0...(p-1)\}$. Those spaces are said totally disconnected.
| 1 | https://mathoverflow.net/users/12664 | 116772 | 65,985 |
https://mathoverflow.net/questions/116774 | 1 | In a note I saw this fact that $PSL(3,q)$ where $q=p^n$ does not have any abelian subgroup of order $q^3$. But I could not prove it or find any reference about it, could you please help me about it?
Thanks
| https://mathoverflow.net/users/30082 | Abelian p-subgroups of PSL(3,p^n) | One of the Sylow $p$-subgroups of $PSL(3,q)$ is the subgroup of all unitriangular matrices (i.e. upper triangular matrices with 1 on the diagonal): just compute the order of $PSL(3,q)$ and the order of the unitriangular subgroup which is $q^3$ (or look in Bogopolsky's group theory book). Therefore if $PSL(3,q)$ contain... | 8 | https://mathoverflow.net/users/nan | 116775 | 65,987 |
https://mathoverflow.net/questions/116781 | 3 | The cone $P\_{n}$ of positive semidefinite matrices of order $n$ can be represented in this form: $P\_{n}=\{A|\forall x\geq 0: \langle A,xx^{T}>0 \rangle \}$ with $x$ running over $\mathbf{R}^{n}-0$.
Question: can every convex cone of matrices be represented in this way, *i.e.* if $K$ is a cone of (say, real symmetri... | https://mathoverflow.net/users/22051 | When are cones of matrices "generated" by vectors? | Let $K$ be a closed convex cone in ${\bf Sym}\_n({\mathbb R})$. I assume a generic cone: non void interior, strictly convex. Let
$$K^0=\{ S\in{\bf Sym}\_n({\mathbb R})\quad|\quad{\rm Tr}(SH)\ge0,\quad\forall H\in K\}.$$
be its dual. Then $K=(K^0)^0$. If $K=Z^0$ for some conical set $Z$ (that is, $tZ=Z$ for $t>0$), it... | 6 | https://mathoverflow.net/users/8799 | 116782 | 65,992 |
https://mathoverflow.net/questions/116797 | 12 | For every ring $A$, the structural morphism of schemes $\pi\_A : {\bf P}^n\_{A} \to {\rm Spec}{A}$ is a closed map. The usual proof of this fact is not constructive : given equations of a closed subset $Z$ of ${\bf P}^n\_{A}$, it doesn't produce equations for $\pi\_A(Z)$.
In the case $A$ is a polynomial ring over an ... | https://mathoverflow.net/users/6506 | Constructive proof of "Projective implies proper" | 3, and thus 2 and 1: yes. By checking equality between the two sets at each point, we reduce to the case where the base is a point. But points are always Noetherian schemes, and the statement is obviously true for Noetherian schemes.
Edit: I was just reminded of this question and I realize that I now know the answer.... | 7 | https://mathoverflow.net/users/18060 | 116805 | 66,003 |
https://mathoverflow.net/questions/116808 | 4 | Is there a classification of singularities from $S^2 \to \mathbb{R}^2$ ? The critical points of the map $(x,y) \mapsto (f\_1(x,y),f\_2(x,y))$ where the matrix:
\[ \left[\begin{array}{cc}\frac{\partial f\_1}{\partial x} & \frac{\partial f\_1}{\partial y}\\\\
\frac{\partial f\_2}{\partial x} & \frac{\partial f\_2}{\par... | https://mathoverflow.net/users/1358 | visualizing singularities of maps from sphere to R^2 | I think you want to look at Guillemin and Golubitsky's book *Stable mappings and their singularities*, which has a thorough description of what the singularity types of stable mappings are between surfaces.
Basically, the only stable singularities for smooth maps between surfaces (i.e., $2$-manifolds) are folds and c... | 8 | https://mathoverflow.net/users/13972 | 116811 | 66,006 |
https://mathoverflow.net/questions/116788 | 8 | I'd like to know which of the set theories in SOSOA prove what versions of Cantor-Schroder-Bernstein? For my own purposes I can use arbitrarily high quantifier complexity, but I wonder how little transfinite recursion will suffice.
| https://mathoverflow.net/users/38783 | What is the status of Cantor-Schroder-Bernstein in Reverse Math? | I will show that variants of the following proof work in extremely weak set theories but perhaps not in $\mathsf{B}\_0^{\mathrm{set}}$.
>
> We can always reduce to the case where one of the two injections is an inclusion. Suppose that $B \subseteq A$ and $f:A \to B$ is an injection. Say that $x \in B$ is a $B$-stop... | 6 | https://mathoverflow.net/users/2000 | 116813 | 66,008 |
https://mathoverflow.net/questions/116691 | 1 | Let $\Omega \subset \mathbb{R}^n$ be a compact smooth hypersurface. Suppose $\varphi \in C\_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (so $\varphi(t) \in H^1(\Omega)$ for each $t$ and $\varphi(0) = \varphi(T)= 0$), and $f \in C^1([0,T] \times \Omega)$. Let $w \in L^2(0,T;H^1(\Omega))$ with wea... | https://mathoverflow.net/users/28178 | weak derivative and continuous function | Multiplication of a distribution by a smooth function is defined in the way you indicate. So there is nothing to prove.
| 3 | https://mathoverflow.net/users/12120 | 116816 | 66,010 |
https://mathoverflow.net/questions/115677 | 19 | **Conventions:** A *polytope* in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A *polyhedron* in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be an intersection of finitely many closed halfspaces in $V$ (that is, the set of solutions ... | https://mathoverflow.net/users/2530 | Is the tensor product of polyhedra a polyhedron? | Not always! However, the closure is a polyhedron.
---
Not always: Take $P = \{a | 0 \leq a \leq 1\}$. Take $Q= \{ b,c | b\geq 0, c=1\}$. Then under the map $x=ab$, $y=ac$. Since $P$ is the convex hull of $(0)$ and $(1)$, and $Q$ is the ray starting at $(0,1)$ and going in direction $(1,0)$, $P \otimes Q$ is the c... | 12 | https://mathoverflow.net/users/18060 | 116823 | 66,015 |
https://mathoverflow.net/questions/116824 | 2 | Let $\mathbf C(t)$ be the field of rational functions and let $\overline{\mathbf C(t)}$ be an algebraic closure. Let $G$ be the Galois group of $\overline {\mathbf C(t)}$ over $\mathbf C(t)$.
Let $\rho:G \to GL\_d(\mathbf Q\_\ell)$ be a Galois representation.
Is there a useful notion of "unramified" at $x$, where $... | https://mathoverflow.net/users/29591 | Useful notion of unramified Galois representation | Yes. For $t$ a local coordinate at a pont $P$, choose an embedding $\overline{\mathbb C(t)} \subset \overline {\mathbb C((t))}$ that sends $t$ to $t$. This turns every representation of the absolute Galois group of $\mathbb C(t)$ into a representation of the absolute Galois group of $\mathbb C((t))$. Define the Galois ... | 3 | https://mathoverflow.net/users/18060 | 116826 | 66,017 |
https://mathoverflow.net/questions/111464 | 11 | Let $X/k$ be a surface nonsingular and proper over an algebraically closed field $k$. Let $C \subset X$ be a nonsingular curve. Then it is clear that the self-intersection $(C \cdot C)\_X$ is $\textrm{deg}\_C ( \mathcal{N}\_{X/C} )$ , basically a matter of definition in intersection theory. More generally, if $X/k$ is ... | https://mathoverflow.net/users/25854 | Self-intersection and the normal bundle | I'd like to expand a bit on the excellent comments of Charles Staats and Donu Arapura. They both suggest understanding the self-intersection number of a curve as the number of fixed points of an infinitesimal deformation of the curve, which is manifestly the degree of the normal bundle when such a deformation exists. H... | 16 | https://mathoverflow.net/users/6950 | 116834 | 66,019 |
https://mathoverflow.net/questions/116714 | 2 | Definition: A poset $P$ is called a binomial poset if it satisfy
a. $P$ is locally finite with a $\hat{0}$, and contains a infinite chain.
b. Every interval $[x, y]$ of $P$ is graded. If $l(x,y)$ = n, then we call $[x,y]$ an
n-interval.
c. For all $n \in \mathbb{N}$, any two $n$-intervals contain the same number of ... | https://mathoverflow.net/users/29921 | Are there any binomial poset which has non-isomorphic interval of the same length? | To avoid creating new binomial posets by taking two of them with the
same factorial function (i.e., number of maximal chains in an $n$-interval)
and identifying their least elements, we
should add the extra condition that there exists a maximal chain
$\hat{0}< x \_0< x \_1< \cdots$ such that every element $x$ satisfies... | 5 | https://mathoverflow.net/users/2807 | 116836 | 66,021 |
https://mathoverflow.net/questions/116847 | 28 | This is a tough one, but does anyone know of any images that recall characteristic p geometry (**over algebraically closed fields**) in some sense? It is not enough if it is some picture that can be also understood solely in characteristic 0.
A quick search through the literature has proved fruitless.
I have been t... | https://mathoverflow.net/users/1887 | Intuitive pictures in characteristic p | I don't think you can draw something meaningful - I would be surprised if someone made a good drawing of the Frobenius morphism ;).
That being said, here is an example (possibly misleading or unrelated to your research) I saw in the slides of Benedict Gross's lectures on the arithmetic of hyperelliptic curves. Take a... | 22 | https://mathoverflow.net/users/3847 | 116850 | 66,025 |
https://mathoverflow.net/questions/116848 | 3 | The transfinite subway puzzle (see <http://mathforum.org/kb/message.jspa?messageID=229112>) is one of those clever puzzle only mathematicians can enjoy (other ones being the blue-eyed islanders puzzle (see <http://terrytao.wordpress.com/2011/04/07/the-blue-eyed-islanders-puzzle-repost/> ), or the use of axiom of choice... | https://mathoverflow.net/users/17164 | Aronszajn trees and the transfinite subway | To my of thinking, the subway puzzle does not have to do with the fact that there is an $\aleph\_1$-Aronszajn tree, but rather simply with the fact that $\omega\_1$ has uncountable cofinality. The reason is that the subway will also be empty at $\aleph\_2$, at $\aleph\_3$ and indeed, at any ordinal having uncountable c... | 6 | https://mathoverflow.net/users/1946 | 116858 | 66,028 |
https://mathoverflow.net/questions/116837 | 5 | My question is whether the axiom of extensionality is required to show that the schema of collection follows from the schema of replacement in the usual Zermelo-Fraenkel environment with choice. In other words: Is the schema of collection a theorem schema in Zermelo-Fraenkel set theory with choice minus the axiom schem... | https://mathoverflow.net/users/37385 | Collection from Replacement in ZFC-extensionality | Collection is not provable in ZFC minus extensionality, a simple countermodel is described in <https://mathoverflow.net/questions/54328> . (That the model cannot provably satisfy collection follows from Gödel’s theorem. For a specific instance of collection which fails, let $\bar\omega$ denote one of the many represent... | 4 | https://mathoverflow.net/users/12705 | 116859 | 66,029 |
https://mathoverflow.net/questions/116861 | 3 | Let $E/K$ be an elliptic curve over a number field, and $\mathfrak{p}$ a prime of good supersingular reduction. Let $p$ be the prime below $\mathfrak{p}$. I believe that the following is true, but I can't prove it, hence my asking here:
>
> $E$ does not possess a $K$-rational $p$-isogeny.
>
>
>
I think this is... | https://mathoverflow.net/users/13741 | Supersingular Elliptic Curves with rational isogeny? | You can't prove it because it is untrue.
Let $E$ be an elliptic curve with CM by $\mathbf{Z}[\sqrt{-p}]$ defined over a number field $K$ which
* Contains $\mathbf{Q}(\sqrt{-p})$ so that the action of $\mathbf{Z}[\sqrt{-p}]$ is $K$-rational and
* Over which $E$ has good reduction (in fact, since $E$ has CM, there's ... | 8 | https://mathoverflow.net/users/3384 | 116865 | 66,032 |
https://mathoverflow.net/questions/46827 | 7 | This is a question in elementary geometric topology, of which I know little. It has to do with the result that geometric realizations of simplicial sets and geometric realizations of abstract simplicial complexes coincide up to homeomorphism, and with how many subdivisions are needed to effect the argument.
Here is ... | https://mathoverflow.net/users/2926 | number of subdivisions needed to compare simplicial sets to simplicial complexes? | Let $X$ be a simplicial set. The partially ordered set $(X^{nd}, \leq)$ of non-degenerate simplices of $X$, with $x\le y$ if $x$ is a face of $y$, was considered by Barratt in a 1956 Princeton preprint. I write $B(X) = N(X^{nd}, \le)$ for its nerve, the Barratt nerve.
It is not quite clear to me if, in your definitio... | 10 | https://mathoverflow.net/users/9684 | 116867 | 66,034 |
https://mathoverflow.net/questions/116868 | 4 | I encountered the following statement without a reference many times. For a smooth variety $X$ over a perfect field $k$.
$Hom(H^1\_{et}(X, \mathbb{Z}/n), \mathbb{Z}/n) \cong \pi^{ab}\_1(X)/n$
Is there any reference for this? Why this is true?
| https://mathoverflow.net/users/25696 | Fundamental Group and Etale Cohomology | See Milne's online course notes on Étale Cohomology, Example 11.3, or Lei Fu's Étale Cohomology Theory, Proposition 5.7.20. (By passing to the direct limit over all $n$, you can even prove it for $\mathbf{Q}/\mathbf{Z}$.)
You only need $X$ to be connected Noetherian.
I am interested in alternative proofs not using ... | 5 | https://mathoverflow.net/users/nan | 116871 | 66,035 |
https://mathoverflow.net/questions/116870 | 3 | If $x\_{a+1}$-$x\_{a}$ converges to $0$ and $x\_{2a}$-$2x\_{a}$ converges to $0$ , does that imply $x\_a$ converges to $0$?
| https://mathoverflow.net/users/27712 | Given a sequence of real numbers,do the following conditions suffice to guarantee convergence to 0? | Yes. There's probably a clever proof, but here's a non-clever one. Suppose you had a sequence satisfying your hypotheses but not converging to 0. Multiplying it by a suitable constant (which doesn't affect the hypotheses), you can assume that $x\_a>1$ for infinitely many $a$, in particular for some $a$ so large that $|... | 10 | https://mathoverflow.net/users/6794 | 116875 | 66,039 |
https://mathoverflow.net/questions/116800 | 1 | Hello there, i need to solve this problem:
I have 2 different bi-partite weighted graph, g1 and g2 and i would like to measure their similarity, g1 and g2 may have different number of vertex and edges and they are a result of a clustering algorithm over different data-sets.
Ideas,hints,thoughts are HIGHLY appreciate... | https://mathoverflow.net/users/30098 | Similarity measure between 2 bi-partite graph. | Is it enough to have something which is defined or do you also want it to be relatively easy to compute?
We can say (as you do) that distance is $0$ when and only when the two graphs are identical in the sense that they have equal numbers of vertices and edges and corresponding edges have the same weight. But it can ... | 1 | https://mathoverflow.net/users/8008 | 116887 | 66,044 |
https://mathoverflow.net/questions/116886 | 3 | Let $X$ be a curve or an abelian variety (over a finite field). Then the Galois representation associated to $X$ via the etale cohomology of $X$ (in degree $1$) is integral of weight $1$ and its dimension is determined by the Hilbert polynomial of $X$. This is a theorem of Weil.
Let $X$ be a variety with fixed Hilber... | https://mathoverflow.net/users/29591 | Does the Hilbert polynomial determine the weight of the Galois representation associated to a variety | No. By Hirzebruch-Riemann-Roch, the Hilbert polynomial of a surface embedded in $\mathbb P^1$ with hyperlane class $D$ is determined by the invariants $\chi(O\_X)$, $D \cdot D$, and $D \cdot K$. There is no reason to expect two surfaces with the same arithmetic Euler characteristic, and that each have a divisor with a ... | 3 | https://mathoverflow.net/users/18060 | 116893 | 66,048 |
https://mathoverflow.net/questions/116896 | 45 | Liouville's theorem from complex analysis states that a holomorphic function $f(z)$ on the plane that is bounded in magnitude is constant. The usual proof uses the Cauchy integral formula. But this has always struck me as indirect and unilluminating. There is a proof via harmonic function theory, but this also seems to... | https://mathoverflow.net/users/683 | Liouville's theorem with your bare hands | I think the most illuminating proof of Liouville's theorem uses Riemann surfaces. Let $f : \mathbb{C} \rightarrow \mathbb{C}$ be a bounded holomorphic function, and set $g(z) = f(1/z)$. Then $g : \mathbb{C} \setminus 0 \rightarrow \mathbb{C}$ is a bounded holomorphic function, so Riemann's removable singularities theor... | 36 | https://mathoverflow.net/users/317 | 116902 | 66,051 |
https://mathoverflow.net/questions/116898 | 2 | I am working on a translation from French about, ultimately, the axiom of constructibility. In the opening paragraphs, the author describes how we are going to assume the ZF axioms are consistent to create a stronger theory with this axiom. Then, he goes on to say that it is harmless to also assume that there is a tran... | https://mathoverflow.net/users/nan | Assuming a transitive set model of ZF | Most likely the author is referring to the following fact:
The Reflection Scheme proves each instance of the following scheme: Let $\Delta$ be a finite subset of the axioms of ZF. ZF proves that there is a transitive set which is a model of $\Delta$.
Let $\tau$ be a constant symbol, and assume ZF is consistent. By ... | 5 | https://mathoverflow.net/users/11145 | 116909 | 66,056 |
https://mathoverflow.net/questions/116906 | 1 | Lemma 1 in this paper: <http://ttic.uchicago.edu/~nati/Publications/SrebroShraibmanCOLT05.pdf> claims that
$\|X\|\_{\Sigma} = \min\_{V^TU=X} \frac{1}{2}(\|U\|\_{Fro}^2 + \|V\|\_{Fro}^2),$
where $\|X\|\_{\Sigma}$ denotes the tracenorm and $\|U\|\_{Fro}$ denotes the Forbenius norm. $U$ and $V$ are assumed to be of su... | https://mathoverflow.net/users/25102 | Proof of Tracenorm Equality | Let $X=W\_1^T\Sigma W\_2$ be the singular value decomposition of $X$, where $W\_1, W\_2$ are unitaries and $\Sigma$ diagonal matrix. Taking $V=\sqrt{\Sigma}W\_1$, $U=\sqrt{\Sigma}W\_2$, the minimum is achieved.
I am not sure whether I understand your comment correctly. The other direction is easy, $\|X\|\_1=\|V^TU\|... | 2 | https://mathoverflow.net/users/24492 | 116912 | 66,059 |
https://mathoverflow.net/questions/116895 | 1 | A Bessel Bridge is a Brownian Motion, conditioned such that $B(0) = B(1) = 0$ and $B([0, 1]) \ge 0$. A raised Bessel Bridge is a generalization of this: it's a Brownian Motion conditioned such that $C(0) = a, C(1) = b, C([0, 1]) \ge 0$ for some nonnegative constants $a, b$.
My end goal is to find a density function f... | https://mathoverflow.net/users/21816 | Can we express a one-dimensional raised Bessel Bridge as a function of a single Brownian Motion? | There is no such function, simply because the raised bridge is a nonhomogeneous Markov process (its drift away from zero becomes small as time approaches $1$).
Moreover, there is no such function that also depends on $t$: both the Brownian motion and the bridge have quadratic variation $dt$, so for any process $V(x(t... | 3 | https://mathoverflow.net/users/22758 | 116915 | 66,060 |
https://mathoverflow.net/questions/116913 | 3 | Let $k$ be an algebraically closed field and $G$ an algebraic group over $k$ which is also a $k$-variety (so $G$ is integral, etc). Let $I$ be the ideal defining the identity $e \in G$ and let $\{ t\_1, \ldots, t\_n \} \subseteq I$ be a set of local parameters at $e$. Since $e$ is a smooth point, the $t\_i$ are algebra... | https://mathoverflow.net/users/1528 | On local parameters at the origin in an algebraic group | This *never* happens for a reductive group $G$ of dimension $n>0$. The conditions you give induce a surjective finite group homomorphism $G\to \mathbb{A}^n$, for some mysterious group structure on $\mathbb{A}^n$. In particular, $G$ would act transitively on $\mathbb{A}^n$, which a reductive group cannot do.
On the ot... | 3 | https://mathoverflow.net/users/6950 | 116922 | 66,066 |
https://mathoverflow.net/questions/116923 | 3 | This is probably easy, but I was just wondering if there is a nice and easy formula for the topological Euler characteristic of a K3 surface $X$ with say $k$ nodes. If there is no general formula, is it known what the answer is for $k=1$?
| https://mathoverflow.net/users/13139 | Euler characteristic of nodal K3 surfaces (as in singular) | The nodes do not modify the birational invariants of a surface. So if we blow-up the $k$ nodes of $X$ we obtain a smooth K3 surface $S$, containing $k$ $(-2)$-curves, whose topological Euler number is $24$. Coming back to $X$, we substitute each $(-2)$-curve (which is topologically a sphere, so has Euler number $2$) wi... | 6 | https://mathoverflow.net/users/7460 | 116926 | 66,067 |
https://mathoverflow.net/questions/116901 | 1 | For $\tau > 0$ define $\theta\_{\tau}(x) = e^{\tau(x-x^{2})}$. I am curious about the asymptotics of $\widehat{\theta}\_{\tau}(\tau)$, that is
$\int\_{\mathbb{R}} e^{\tau(x - x^{2})}e^{-2\pi i \tau\cdot x}dx\ \sim\ ?\ \ \ \ \ \ \ \ (\tau \to +\infty)$
But I don't know how to get anything from the oscillation. Bring... | https://mathoverflow.net/users/12968 | Asymptotics of a one-parameter family of Schwartz functions | $$
I(\tau)=\int\_{\mathbb R}e^{-\pi\frac{\tau}{\pi} x^2}e^{-2i\pi \tau x (1-\frac{1}{2i\pi})}dx=
(\frac{\pi}{\tau})^{1/2}e^{-\pi\frac{\pi}{\tau} \tau^2 (1-\frac{1}{2i\pi})^2}=
(\frac{\pi}{\tau})^{1/2}e^{-{\pi^2\tau} (1-\frac{1}{2i\pi})^2},
$$
so that
$$
I(\tau)=(\frac{\pi}{\tau})^{1/2}e^{-{\pi^2\tau} (1-\frac{1}{4\pi^2... | 3 | https://mathoverflow.net/users/21907 | 116930 | 66,070 |
https://mathoverflow.net/questions/116537 | 3 | **Definition** (***Open Manifolds***):An open manifold is a manifold without boundary with no compact component. For a connected manifold, "open" is equivalent to "without boundary and non-compact.
we know that every symplectic manifold admits an almost complex structure but for open manifolds , the inverse is also cor... | https://mathoverflow.net/users/nan | an extended question of Gromov: Every **generalized open almost complex manifold** admits a **generalized symplectic structure**? | In his thesis
<http://arxiv.org/abs/math/0401221>
Marco Gualtieri explains that a generalized almost complex structure on an $n$-manifold $M$ is a reduction of the structure group of $TM \oplus T^\ast M$, which has its canonical hyperbolic quadratic form, from $O(n,n)$ to $U(n,n)$. He points out (p. 48) that since ... | 3 | https://mathoverflow.net/users/2356 | 116932 | 66,071 |
https://mathoverflow.net/questions/116924 | 8 | Given a group $G$ and $G$-modules $M,N$ with $M$ $\mathbb{Z}$-free then it's well known that
$$Ext\_{\mathbb{Z}G}^i(M,N) \cong H^i(G,Hom(M,N))$$
for all $i \ge 0$ (a reference is Brown, Cohomology of Groups, Proposition 2.2).
But what happens if $M$ is not $\mathbb{Z}$-free ? Is it still possible to express $Ext\_{... | https://mathoverflow.net/users/25869 | Can Ext over a group ring always be expressed as group cohomology ? | There is a long exact sequence
>
> $$0 \to H^1(G,Hom(M,N)) \to Ext\_{\mathbb{Z}G}^1(M,N) \to \cdots $$
> $$\begin{array}{lll}
> \cdots & \to & H^i(G,Hom(M,N)) \to Ext\_{\mathbb{Z}G}^i(M,N) \newline
> & \to & H^{i-1}(G,Ext\_{\mathbb{Z}}^1(M,N))\to H^{i+1}(G,Hom(M,N)) \to \cdots
> \end{array}$$
>
>
>
For, as ... | 13 | https://mathoverflow.net/users/10194 | 116937 | 66,074 |
https://mathoverflow.net/questions/116933 | 1 | Let $f:M\_0\rightarrow M\_1$ be a diffeomorphism between two compact $n$-dimensional manifolds. Let $W^{n+1}$ be an $h$-cobordism between $M\_0$ and $M\_1$. Assume that the cobordism has no torsion and its dimension is high enough so that, by the $s$-cobordism theorem, there is a diffeomorphism $F:M\_0\times I\rightarr... | https://mathoverflow.net/users/5069 | Diffeomorphism coming from the s-cobordism theorem | The question is unclear, mainly because it's not clear what "can be taken ... up to isotopy" means. Under the only interpretation that seems at all reasonable, the answer is pretty trivially "no".
I'll assume that the given $F$ is assumed to take $M\_0\times 1$ to $M\_0$ by the identity, and that the question is whet... | 6 | https://mathoverflow.net/users/6666 | 116943 | 66,076 |
https://mathoverflow.net/questions/116938 | 2 | A group $G$ is generated by $1, -1, g\_1, g\_2, \ldots, g\_n$. The relation of its generators is given by a simple undirected graph $G = (V=[n], E)$, where $(i, j) \in E$ means $g\_i g\_j = -g\_j g\_i$. In the group $1$ is the identity such that $1g = g$ for any element $g$; $(-1)\*(-1) = 1$, and $(-1)$ is commuting wi... | https://mathoverflow.net/users/26659 | Unitary representations of a group given generating set | Groups that you described are central extensions of RAAGs (Right Angled Artin Groups). Let's call such groups "almost RAAGs" for lack of a better name (since the name "extended Artin groups" is already taken by Looijenga). The answer to your question (for a general graph $G$) is: "Awfully complicated." For instance, co... | 3 | https://mathoverflow.net/users/21684 | 116949 | 66,079 |
https://mathoverflow.net/questions/116948 | 1 | We know in a semisimple ring R, for every R-module, Noetherian is equivalent to Artinian, my question is:
If for every R-module M Noetherian is equivalent to Artinian, can we prove R is a semisimple ring?
| https://mathoverflow.net/users/30129 | The equivalence of Artinian and Noetherinan for the modules of a semisimple ring | Let $R= k[\epsilon]/\epsilon^2$. Then module is Artinian if and only if it is Noetherian if and only if it is finite-dimensional. This is clear, since if a module is finite-dimensional there can't be an infinite ascending or descending sequence, and if it's infinite dimensional than $M/\epsilon$ is an infinite-dimensio... | 7 | https://mathoverflow.net/users/18060 | 116952 | 66,080 |
https://mathoverflow.net/questions/116934 | 2 | I intend to study mathematical logic , my purpose is to get to Godel's incompleteness theorems
I haven't study any mathematical logic before
so what is the good text which I can use for this purpose ?
I search for a book give me the right picture , and good explanations
I will use it as self-study
| https://mathoverflow.net/users/27947 | which texts do you recommend to study mathematical logic ? | I suggest Cori and Lascar's "Mathematical logic", two small books ; the first book covers propositional calculus, Boole algebras and predicate calculus ; the second recursive functions, Gödel's theorems, set theory and the basics of model theory. I found the proofs precise, the examples nice.
| 1 | https://mathoverflow.net/users/12664 | 116955 | 66,081 |
https://mathoverflow.net/questions/116962 | 2 | Let $G$ and $Cay(A,S)$ be strongly regular graphs with the same parameters. Is it true that $G$ is a cayley graph?
| https://mathoverflow.net/users/8725 | Strongly regular cayley graphs | no, this is certainly not true. IIRC already on 25 vertices there is a family of 15 non-isomorphic s.r.g.'s with the same parameters, some of them Cayley graphs, some not: see <http://www.win.tue.nl/~aeb/graphs/Paulus.html>.
| 6 | https://mathoverflow.net/users/11100 | 116964 | 66,086 |
https://mathoverflow.net/questions/116954 | 3 | I've just stumbled across the following theorem ([here](http://www.math.unt.edu/~moliver/fa04s/JacksonNotes/products.pdf)):
**Theorem** Let $P$ be a partial class order in $M$, a transitive model of ZF (resp. ZFC). Suppose for arbitrarily large cardinals $\kappa$ we have an isomorphism $P \simeq P^- \times P^+$ where... | https://mathoverflow.net/users/4177 | Can we weaken GCH in this class forcing? | It's impossible to accurately answer this since you don't say what forcing you have in mind. However, in the "typical" case, the answer is yes.
The "typical" case is when the forcing poset $P$ comes from a "typical" iteration of set forcings. These are "typically" arranged so that for unboundedly many $\kappa$, the p... | 3 | https://mathoverflow.net/users/2000 | 116972 | 66,089 |
https://mathoverflow.net/questions/116975 | 4 | Hi, I have a question on weak-equivalences of spectra.
More precisely, I wonder whether filtered colimits of weak-equivalences of spectra are again weak-equivalences of spectra. Here, spectra are in the sense of Bousfield-Friedlander, i.e. a sequence of pointed simplicial sets $(E\_0, E\_1, \cdots, )$ with the morphi... | https://mathoverflow.net/users/3168 | Are filtered colimits of weak-equivalences of spectra again weak-equivalences? | The answer is **yes**, but the reason is technical.
The reason is that, if I understand well, you're asking whether weak equivalences are closed under *arbitrary* filtered colimits. These are $\aleph\_0$-filtered colimits, and there is a hierarchy of degrees of filtration parametrized by all infinite regular cardina... | 6 | https://mathoverflow.net/users/12166 | 116987 | 66,099 |
https://mathoverflow.net/questions/116993 | 1 | Let $X$ be a canonically polarized smooth projective geometrically connected variety over $k$ with Hilbert polynomial $h$.
What is the Hilbert polynomial of $X\times\_k \mathbf{P}^1\_k$? How does it depend on $h$?
Example. Let $g\geq 2$ be an integer and let $X$ be a genus $g$ curve. Then the Hilbert polynomial of ... | https://mathoverflow.net/users/29591 | Hilbert polynomial of $X\times P^1$ | Rather than using a canonical bundle, the obvious choice is the Segre embedding $\mathbb P^n \times \mathbb P^1 \to \mathbb P^{2n+1}$. This is equivalent to choosing the ample line bundle that is tensor product of the pullback of the original ample line bundle with the pullback of the line bundle of degree $1$ on $\mat... | 6 | https://mathoverflow.net/users/18060 | 116994 | 66,101 |
https://mathoverflow.net/questions/116304 | 20 | We have a characterization when we want $|X|$ to be a PL-manifold, in particular that the links of all the vertices are themselves (PL) spheres. If we are in the category of PL- spaces then this is a necessary and sufficient condition. If however, we leave the PL category, then we get simplicial complexes that are topo... | https://mathoverflow.net/users/14167 | If $X$ is a simplicial complex, is there a characterization of the links of its vertices that is equivalent to the statement "$|X|$ is a manifold"? | In dimensions $\geq 5$, Theorem 1.5 of [Galewski and Stern](http://www.maths.ed.ac.uk/~aar/papers/galester.pdf) implies that if the links of the vertices of a homology manifold of dimension $\geq 5$ are simply-connected, then it is a manifold.
I'm not sure if this is equivalent to your condition. Certainly your cond... | 11 | https://mathoverflow.net/users/1345 | 116998 | 66,103 |
https://mathoverflow.net/questions/117016 | 2 | I know the notion of the link of a vertex of a 3-manifold. In his article *Geometric structures on low-dimensional manifolds*, Suhyoung Choi first defined the notion of "projective triangulation of an orbifold with a projective structure" which is a cellular decomposition of the underlying space induced by a triangulat... | https://mathoverflow.net/users/25609 | Link of a vertex of a 3-orbifold (link orbifold) | Let $O$ be the 3-orbifold, $\widetilde{O}$ its universal cover (it sounds like we're assuming that $O$ is 'good', ie has a manifold universal cover) and $\pi\_1O$ its fundamental group.
Let $v$ be a vertex covered by a vertex $\tilde{v}$ in the universal cover. Then the link of $v$ is just the 2-orbifold $\mathrm{Lin... | 6 | https://mathoverflow.net/users/1463 | 117017 | 66,113 |
https://mathoverflow.net/questions/117011 | 1 | If B is a Boolean ring is of uncountable cardinality c, does B have 2^c distinct maximal ideals ?
Can you please give me a reference where this question is answered (hopefully) positively ? Thanks
| https://mathoverflow.net/users/30146 | Cardinality of the set of maximal ideals in a Boolean ring/algebra | This particular question is easy to answer. Many related questions about the relationships between various cardinals associated with Boolean algebra (or Boolean rings), such as: cardinality, number of ultrafilters, density, cellularity, distributivity etc, can be found in
* Monk, J. Donald: *Cardinal invariants on B... | 3 | https://mathoverflow.net/users/14915 | 117023 | 66,115 |
https://mathoverflow.net/questions/117026 | 1 | Suppose that $G$ is a connected graph with equitable partition $\pi$. Then the eigenvalues of the *divisor multigraph* $G / \pi$ are all eigenvalues of $G$. (Perhaps excluding some pathological cases) the largest eigenvalue of $G/\pi$ is the Perron value of $G$ and thus simple in the spectrum of $G$.
I would like to... | https://mathoverflow.net/users/22051 | The smallest eigenvalue from an equitable partitions | There are infinitely many counterexamples. Let $\pi$ be the distance partition relative to a vertex in a strongly regular graph. Then $\pi$ is equitable and $G/\pi$ is a path, so its eigenvalues are all simple. But if $G$ is not bipartite, its least eigenvalue is not simple.
| 4 | https://mathoverflow.net/users/1266 | 117030 | 66,117 |
https://mathoverflow.net/questions/117009 | 17 | Hi all! I am trying to understand Specker (1953)'s proof (found [here](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063889/pdf/pnas01594-0080.pdf)) that the axiom of choice is false in New Foundations. I am stuck on the following point. At 3.5 Specker writes:
3.5. The cardinal numbers are well ordered by the relation... | https://mathoverflow.net/users/29956 | Understanding Specker's disproof of the axiom of choice in New Foundations | First notice that one can carry out in NF Zermelo's proof that the axiom of choice implies that all sets can be well-ordered. It follows that the relation on cardinals defined in your quote from Specker satisfies the conditions for a linear ordering. (It's trivially reflexive and transitive. Antisymmetry is the Cantor-... | 20 | https://mathoverflow.net/users/6794 | 117032 | 66,119 |
https://mathoverflow.net/questions/117027 | 1 | Let $G$ be an algebraic group acting on a variety $V$. Which information can be obtained by looking the action of $G$, and subgroups of $G$ that fixes points of $V$?. In other words how we obtain $V$ from the group $G$?
| https://mathoverflow.net/users/25202 | How we obtain information about a variety from an algebraic group acting on it | A lot if $G$ is transitive. Then $V=G/H$ for a subgroup $H$ (if it has a point), or a $G$-torsor mod $H$ (if it doesn't). Then most questions about the geometry of the variety are best answered by studying the group action. For instance, we can study line bundles on a flag variety of a reductive group using the root la... | 3 | https://mathoverflow.net/users/18060 | 117035 | 66,120 |
https://mathoverflow.net/questions/117033 | 4 | This must be an easy question but I don't have a good argument for it and have not found a counterexample: Let $G$ be a connected semisimple algebraic group over $\mathbb{Q}$ such that the center of $G$, $Z(G)$ is trivial, i.e. $Z(G)=\{1\}$ is it true that the center of $G\_{\mathbb{R}}$, denoted $Z(G\_{\mathbb{R}})$ i... | https://mathoverflow.net/users/18000 | Center of the algebraic group $G_{\mathbb{R}}$ for a centerless $G$ | The short answer is that the center Z(G) of a semisimple algebraic group is a well-defined (finite) algebraic subgroup which commutes with extension of the base field, so if it's trivial over $\mathbb{Q}$, then its trivial over every field. But presumably, that is not what you meant to ask. Perhaps you mean: does $Z(\m... | 4 | https://mathoverflow.net/users/30149 | 117040 | 66,122 |
https://mathoverflow.net/questions/117038 | 3 | All topological spaces considered here are Hausdorff.
It is a well-known consequence of the minimality of a compact topology that an injective continuous map
$f\colon X\to Y$
where $X$ is compact, must be automatically a homeomorphism onto its range. I am interested in possibly non-compact spaces which share this... | https://mathoverflow.net/users/29433 | Automatic continuity of the inverse map | The spaces that you are looking for are precisely the minimal Hausdorff spaces. i.e. A Hausdorff space $X$ is a minimal Hausdorff if every injective continuous map from $X$ to a Hausdorff space is an embedding. See the Book General Topology by Stephen Willard problems 17M for more information about these spaces. The Mi... | 8 | https://mathoverflow.net/users/22277 | 117049 | 66,128 |
https://mathoverflow.net/questions/117062 | 0 | Suppose you are given a convex function $f: R^d \rightarrow R$. Let us say you are given $x,x' \in R^d$ and $x\_1, x\_2, \ldots, x\_n \in R^d$ such that
$$\sum\_{i=1}^n (x\_i - x') = x - x'.$$
Is it possible to bound $f(x) - f(x')$ in terms of $f(x\_i) - f(x')$?
That is, a bound of the form
$$f(x) - f(x') \le... | https://mathoverflow.net/users/30157 | Convex functions: bounding the difference | $$f(x) - f(x') \le \sum\_i \left(f(x\_i) - f(x')\right)$$
iff the function $g(x) = f(x + x') - f(x')$ is subadditive.
| 2 | https://mathoverflow.net/users/13650 | 117066 | 66,138 |
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