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https://mathoverflow.net/questions/48499 | 2 | I saw the following theorem in the wiki page:
<http://en.wikipedia.org/wiki/H%C3%B6lder_condition>
---
if $f$ satisfies the $\alpha$-Hölder condition
$| f(x) - f(y) | \leq C \, |x - y|^{\alpha}$ for some $\alpha>1/2$,
then
$||f||\_{A} = \sum\_i |c\_{i}|\leq C c\_{\alpha}$
where $c\_{\alpha}$ only depends on... | https://mathoverflow.net/users/11368 | absolute convergence of the Fourier coeff for Hölder continuous function | The proof is outlined in Stein-Shakarchi's book Fourier Analysis, Chapter 3, Exercise 16.
| 3 | https://mathoverflow.net/users/10583 | 48507 | 30,565 |
https://mathoverflow.net/questions/48505 | 9 | I am not quite sure about the terminology, but let's call a $n$-dimensional manifold of *finite type* if it has a finite open cover $U\_1,\ldots,U\_k$ such that all intersections are either empty or diffeomorphic to ${\mathbb R}^n$. Every compact manifold is of finite type (endow it with a Riemannian metric and use geo... | https://mathoverflow.net/users/10675 | Finite type vs. finite dimensional cohomology? | It does matter what kind of cohomology you use:
First think of the connected sum of an infinite number of real projective spaces end to end. De Rham cohomology cannot tell that these were not spheres, but mod 2 cohomology shows that the thing is not of finite type.
Now take it further. Take a (necessarily non simp... | 12 | https://mathoverflow.net/users/6666 | 48512 | 30,567 |
https://mathoverflow.net/questions/48498 | 10 | Fix $n$ and let $H\_1$ and $H\_2$ be two hypersurfaces in ${\bf A}^n$ (not necessarily smooth or irreducible, but we'll assume reduced). If the complements $U\_i = {\bf A}^n \setminus H\_i$ are isomorphic as schemes, does this imply that the $H\_i$ are isomorphic?
From asking other people, the answer is yes via Euler... | https://mathoverflow.net/users/321 | Are complements of non-isomorphic affine hypersurfaces necessarily non-isomorphic? | Your question appears as the 'Complement problem' in Hanspeter Kraft's article [Challenging problems on affine n-space](http://archive.numdam.org/ARCHIVE/SB/SB_1994-1995__37_/SB_1994-1995__37__295_0/SB_1994-1995__37__295_0.pdf) and it seems from that article that the problem is still open for $H\_1, H\_2$ irreducible. ... | 14 | https://mathoverflow.net/users/3996 | 48513 | 30,568 |
https://mathoverflow.net/questions/48526 | 7 | What is known about the additivity of Lebesgue measure under the Axiom of Determinacy?
That is, for what cardinals $\kappa$ do we have
with $|I| = \kappa$, for all functions $f : I \to 2^\mathbb{R}$, $\; \lambda(\displaystyle\bigcup\_{i\in I}\ f(i)) = \operatorname{sup}(\{\lambda(\displaystyle\bigcup\_{i\in J}\ f... | https://mathoverflow.net/users/nan | How additive is Lebesgue measure in ZF+AD ? | Ricky:
I think I see how to answer the problem under a stronger assumption. Rather than $\mathsf{AD}$, work in $$ {\sf AD}^+ + V=L({\mathcal P}({\mathbb R})). $$ This is a bit unsatisfying, since it is very possible the question can be answered assuming only $\mathsf{AD}$. In any case, $\mathsf{AD}^+$ is potentially ... | 5 | https://mathoverflow.net/users/6085 | 48530 | 30,579 |
https://mathoverflow.net/questions/48539 | 15 | Let $M$ be a finitely generated module over the commutative noetherian ring $R$. Let ${\cal C}(M)$ be the set of all primes $P$ in $R$ such that $R/P$ appears as a quotient in every composition series for $M$. Clearly ${\cal C}(M)$ includes all the primes associated to $M$. But it can contain other primes as well. For ... | https://mathoverflow.net/users/10503 | Primes that must occur in every composition series for a given module | I think this is a subtle question. The best result I am aware of is the following paper "Filtrations of Modules, the Chow Group, and the Grothendieck Group", by Jean Chan. She proved the following: Let $\mathcal F$ be any composition series of $M$. Let $c\_i(\mathcal F)$ be the formal sum of primes of height $i$. Then ... | 10 | https://mathoverflow.net/users/2083 | 48541 | 30,586 |
https://mathoverflow.net/questions/48399 | 2 | Let $G$ be a discrete group.
>
> Do you know characterizations of amenable groups which use the space $\ell\_1(G)$ and convolution?
>
I only know Johnson's theorem:
>
> A group is amenable if and only if the Banach algebra $\ell\_1(G)$ is amenable.
>
>
>
Different characterizations are welcome.
| https://mathoverflow.net/users/5210 | Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution | I like the following characterization, due to Kaimanovich and Vershik (conjectured by Furstenberg):
>
> A (countable) group is amenable if and only if there is an everywhere positive $\ell^{1}$-function $\mu$ of norm $1$ on $G$ such that $\|g\mu^{\ast n} - \mu^{\ast n}\|\_{1} \to 0$ for all $g \in G$.
>
>
>
As... | 6 | https://mathoverflow.net/users/11081 | 48548 | 30,592 |
https://mathoverflow.net/questions/27879 | 2 | I have an immersion of a 2-simplicial complex S in $\mathbb{R}^3$, and then a piecewise linear motion of that immersion over an interval of time [0,1].
>
> Is there an existing name for the map $f:S\times[0,1]\to\mathbb{R}^3\times[0,1]$?
>
>
>
Update: here is a more detailed definition of f. Let $i\_t$ be the ... | https://mathoverflow.net/users/6757 | Name for the motion of an immersion? | The map $f$ is usually called a *level-preserving homotopy* (or *level-preserving regular homotopy* if each $i\_t$ is an immersion). At least this seems to be the accepted terminology in Geometric Topology.
| 3 | https://mathoverflow.net/users/8103 | 48551 | 30,595 |
https://mathoverflow.net/questions/48558 | 20 | This is a follow-up to [this question](https://mathoverflow.net/questions/48453) (in fact, this is what originally motivated me to ask that one.)
Let's say that a sequence $\{s\_i\}$ of positive reals *covers* a set $X\subset\mathbb R$ if there is a collection if intervals $\{I\_i\}$ such that $X\subset\bigcup I\_i$ ... | https://mathoverflow.net/users/4354 | A collection of intervals that can cover any measure zero set | No. If you can cover every set of measure $0$ by your sequence of intervals, you can certainly scale (shrink all intervals some number of times) and still have covering (just cover the expanded set by the original sequence) . If $\sum s\_j<+\infty$, then $\sum H(s\_j)<+\infty$ for some measuring function $H$ with $H(x)... | 19 | https://mathoverflow.net/users/1131 | 48560 | 30,598 |
https://mathoverflow.net/questions/48564 | 2 | I'm trying to read the paper of Jost and Karcher on the existence of harmonic coordinates on a ball whose size only depend on the injectivity radius and a two sided bound on the curvature.
Unfortunately, my german skills are quite low and make the reading really slow. Does there exist another place to find this proof... | https://mathoverflow.net/users/8887 | Harmonic coordinates on Riemannian manifolds | Perhaps [Jost's account of it](http://www.ams.org/mathscinet-getitem?mr=756629) from his lectures on harmonic mapppings between Riemannian manifolds contains a detailed description.
There's also some related results due to [Hebey and Herzlich](http://www.ams.org/mathscinet-getitem?mr=1620864)
| 5 | https://mathoverflow.net/users/3948 | 48566 | 30,600 |
https://mathoverflow.net/questions/48571 | 14 | Hi,
Let $R$ be equipped with the usual Borel structure. Let $F$ be a Borel subset and $E$ be a closed subset of $R$. Then $F+E=(f+e: f\in F, e \in E \)$ is Borel? If yes, is it true for any locally compact topological group? Thanks in advance.
| https://mathoverflow.net/users/11387 | Borel set plus a closed set = Borel | No. Erdös and Stone showed that the sum of two subsets $E$, $F\subset\mathbb R$ may not be Borel even if one of them is compact and the other is $G\_\delta$ (see [*"On the Sum of Two Borel Sets"*](http://www.jstor.org/pss/2037209), Proc. Am. Math. Soc., Vol. 25, (1970), pp. 304-306).
Their argument works for every c... | 20 | https://mathoverflow.net/users/5371 | 48579 | 30,606 |
https://mathoverflow.net/questions/48589 | 5 | Does anybody know of any finitely generated semigroups that are not residually finite and whose group of units (if there is an identity) is trivial? Basically, I'm looking for finitely generated semigroups that are not residually finite, but I don't just want to punt and look at groups (like Thompson's groups or someth... | https://mathoverflow.net/users/8434 | examples of finitely generated semigroups that are not residually finite | Take $S=\langle a, b, c \mid ab^na=0 \hbox{ for all prime } n, ab^ma=c \hbox{ for all composite } m\rangle$. If there exists a homomorphism of that semigroup onto a finite semigroup $F$, and $\bar x$ is the image of $x$ ($x=a,b,c$), then $\bar b^n=\bar b^{n+m}$ for some $n,m$. Therefore for every prime $p>n$, and every... | 7 | https://mathoverflow.net/users/nan | 48592 | 30,612 |
https://mathoverflow.net/questions/47981 | 9 | Is it possible to define the Ricci flow with surgery in dimension 2 and use it to classify the surfaces?
I know this is overkill, there are simpler ways to classify surfaces, but I would like to understand the Ricci flow with surgery in dimension 3 and perhaps that this is simpler in dimension 2.
| https://mathoverflow.net/users/10217 | Ricci flow with surgery in dimension 2 | (1) The normalized Ricci flow (NRF) on compact surfaces always exists for all time and does not have singularities. Moreover, NRF fixes the conformal class of the metric.
(2) Let $r$ be the integral of the curvature, which is constant under any flow. Hamilton and Osgood-Phillips-Sarnak (independently, I think) showed... | 18 | https://mathoverflow.net/users/1179 | 48595 | 30,614 |
https://mathoverflow.net/questions/48597 | 3 | Given a smooth manifold U, we have a map $\wedge^2\Gamma(U,TU)\to \Gamma(U,TU)$ given by $X\wedge Y\mapsto [X,Y]$, where $TU$ denotes the tangent bundle. Is it possible to describe the map $\Gamma(U,T^\*U)\to \Gamma(U,\wedge^2 T^\*U)$ corresponding to this map.
| https://mathoverflow.net/users/11395 | Dual of The Lie Bracket | To expand on Leonid's comment, if $\omega$ is a 1-form and $X,Y$ are vector fields, then
$$ d\omega(X \wedge Y) = X \omega(Y) - Y \omega(X) - \omega([X,Y]). $$
If the first two terms were not there, then one could say, as in Victor's answer, that the exterior derivative is (minus) the transpose of the Lie bracket of ve... | 13 | https://mathoverflow.net/users/394 | 48603 | 30,619 |
https://mathoverflow.net/questions/48598 | 5 | Hi,
If I have a divergence free vector field defined on a smooth manifold, and I apply some diffeomorphism, what can I say about what happens to the vector field? The example I am using is of an open or closed (pick one) disk embedded in 3 dimensions and we contract the disk to a line. Perhaps the question could be a... | https://mathoverflow.net/users/10007 | transformation properties of divergence (of a vector field) | The divergence of a vector field depends on a volume form, which is a nowhere-zero $n$-form on an oriented $n$-manifold. Given a volume form, most diffeomorphisms change it, but there is a large and rich group of diffeomorphisms that preserve it. For instance,
Calabi showed (by an elementary construction) that there is... | 25 | https://mathoverflow.net/users/9062 | 48606 | 30,621 |
https://mathoverflow.net/questions/48607 | 5 | Example : Consider the (open, not compactified) moduli space of stable maps $ \mathcal M\_g(1,d)$ of maps of smooth curves of genus $ g$ to $\mathbb P^1$. To each map
we associate its branch divisor, which is an element of $ Sym^r(\mathbb P^1)$. Then we should have a morphism from $ \mathcal M\_g(1,d)$ to $ Sym^r(\math... | https://mathoverflow.net/users/4275 | Proving that a map is a morphism | It might help to identify $Sym^r(\mathbb P^1)$ with the Hilbert scheme of degree $r$ effective divisors on $\mathbb P^1$. The Hilbert scheme represents a functor, as does the space $\mathcal M\_g(1,d)$, and so you can (try to) construct a map from one to the other by thinking in terms of the functors they represent.
... | 10 | https://mathoverflow.net/users/2874 | 48608 | 30,622 |
https://mathoverflow.net/questions/48609 | 1 | I found the proof by Kronecker that the expression
$$X = p^e + p^{e-1} + ... + p^2 +p + 1$$
is irreducible. Four people translated the German, two in Detroit and two in Kiel, Germany. See [Kronecker's proof of irreducibility](http://oddperfectnumber.org/docs/KroneckerProofOfIrreducibility.pdf).
Kronecker also p... | https://mathoverflow.net/users/11384 | Proof that the factors of sigma(p^e) have two forms | It's well-known that if $a$ is an integer then a prime factor of the number
$\Phi\_n(a)$ is either a factor of $n$ or congruent to $1$ modulo $n$.
Here $\Phi\_n$ is the $n$-th cyclotomic polynomial. The reason is that if
$p$ divdes $\Phi\_n(a)$ but not $n$ then $a$ has order exactly $n$
in the multiplicative group $(\m... | 5 | https://mathoverflow.net/users/4213 | 48612 | 30,625 |
https://mathoverflow.net/questions/23440 | 8 | It's a standard fact that, for finite CW complexes, the relative (edit: oriented) bordism group $\Omega\_n(X,A)$ coincides with the homology $[H\_\ast(X,A;\Omega\_\ast(pt))]\_n\simeq H\_n(X,A)$ for $n<5$ (edit: this should be $n<4$). One can prove this using the Atiyah-Hirzebruch spectral sequence, and all papers I've ... | https://mathoverflow.net/users/2051 | Reference request for relative bordism coinciding with homology in low dimensions | Yuli Rudyak gave a perfect answer to this question in May, but it was by e-mail. Because his might be of interest to others, I reproduce it in full.
>
> Look Theorem IV.7.37 of my book "On Thom spectra, orientability, and cobordism", Corrected reprint, Springer, 2008.
There is proved explicitly that the map $E\... | 4 | https://mathoverflow.net/users/2051 | 48635 | 30,638 |
https://mathoverflow.net/questions/48638 | 21 | I was seeking a binary operator on natural
numbers that is intermediate between
the sum and the product, and explored this natural
candidate:
$$x \star y = \lceil (x y + x + y)/2 \rceil \;.$$
Then I wondered which numbers are prime with respect to $\star$,
i.e., only have one factoring.
For example,
$11 = 1 \star ... | https://mathoverflow.net/users/6094 | Why are this operator's primes the Sophie Germain primes? | Q1. $p$ is $\star$-prime iff equation $xy+x+y=2p$ has no solution and $xy+x+y=2p-1$ has exactly one solution, i.e. $(x+1)(y+1)=2p+1$ has no solution (which is iff $2p+1$ is prime) and $(x+1)(y+1)=2p$ has only one solution $\{x,y\}=\{1,p-1\}$. This last holds iff $p$ is prime.
Q2. Why partitions?! The number of $\sta... | 20 | https://mathoverflow.net/users/4312 | 48640 | 30,640 |
https://mathoverflow.net/questions/48643 | 8 | Given two rings $R\_1$ and $R\_2$ (with or without identity: it's not specified). If $R\_1[x]$ is isomorphic to $R\_2[y]$ (No such requirement that the isomorphism sends the constant terms to constant terms), can we deduce that $R\_1 \cong R\_2$?
I feel there might be a counterexample but it's quite hard to find one.... | https://mathoverflow.net/users/10333 | Can we deduce that two rings $R_1$ and $R_2$ are isomorphic if their polynomial ring are isomorphic? | This was recently asked and answered at <https://math.stackexchange.com/questions/13504/does-rx-cong-sx-imply-r-cong-s>
As answered there, it is possible to find two non-isomorphic commutative rings whose polynomial rings in one variable are isomorphic.
An example is given in <http://www.ams.org/journals/proc/1972-034-... | 9 | https://mathoverflow.net/users/4614 | 48645 | 30,643 |
https://mathoverflow.net/questions/48521 | 7 | (I am assuming choice.)
Suppose that ${\mathbb P}=(P,{\lt})$ is a partially ordered set (a poset), and that $\kappa\le\lambda$ are ordinals. The notation $$ {\mathbb P}\to(\kappa)^1\_\lambda $$ means that whenever $f:P\to\lambda$, we can find some $i\in\lambda$ and some subset $H$ of $P$ such that $(H,{\lt})$ is orde... | https://mathoverflow.net/users/6085 | Partitions of partial orders | Assume that $P\rightarrow(\kappa)^1\_\lambda$, yet for some $\alpha<\kappa^+$, $P\not\rightarrow(\alpha)^1\_\lambda$. Accordingly, there is a decomposition $P=\bigcup\_{\xi<\lambda}P\_\xi$ such that no $(P\_\xi,<)$ does contain a chain of type $\alpha$. For some $\xi<\lambda$ we must have $P\_\xi\rightarrow (\kappa)^1\... | 7 | https://mathoverflow.net/users/6647 | 48654 | 30,650 |
https://mathoverflow.net/questions/48642 | 25 | Does anyone know how to parametrize the boundary of the Mandelbrot set? I am not a fractal-geometer or a dynamical systems person. I just have some idle curiosity about this question.
The Mandelbrot set is customarily defined as the set $M$ of all points $c\in\mathbb{C}$ such that the iterates of the function $z\maps... | https://mathoverflow.net/users/11264 | Parametrization of the boundary of the Mandelbrot set | Lasse's answer expanded: Let $\psi$ be the map of the exterior of the unit disk onto the exterior of the Mandelbrot set, with Laurent series
$$
\psi(w) = w + \sum\_{n=0}^\infty b\_n w^{-n} =
w - \frac{1}{2} + \frac{1}{8} w^{-1} - \frac{1}{4} w^{-2} +
\frac{15}{128} w^{-3} + 0 w^{-4} -\frac{47}{1024} w^{-5} + \dots
$$
T... | 24 | https://mathoverflow.net/users/454 | 48657 | 30,652 |
https://mathoverflow.net/questions/48664 | 6 | This question is about the beaviour of 4-genus of knots with respect to connected sum.
Let us indicate with $T(k)$ a Torus knot of type $(2,k)$, $k$ is an odd integer.
Fix an orientation for every $T(k)$ so that $T(-k)$ represents the same knot with reversed orientation.
$T(k)\sharp T(-k)$ is a slice knot. More gen... | https://mathoverflow.net/users/5001 | When is a connected sum of torus knots a slice knot? | I believe the answer to your last two questions is *yes*, and it follows from Litherland's (1979) computation of the Tristram-Levine invariants of torus knots. See Kearton's survey here:
<http://www.maths.ed.ac.uk/~aar/slides/durham.pdf>
i.e. a connect sum of torus knots is slice if and only if the prime summands ... | 10 | https://mathoverflow.net/users/1465 | 48667 | 30,659 |
https://mathoverflow.net/questions/48679 | 2 | I've been going through Fermats proof that a rational square is never congruent. And I've stumbled upon something I can't see why is. Fermat says: ''If a square is made up of a square and the double of another square, its side is also made up of a square and the double of another square'' Im having difficulties underst... | https://mathoverflow.net/users/11407 | Fermats proof that a rational square is never congruent | In other words, Fermat is saying that if $x^2=y^2+2z^2$, then $x=c^2+2d^2$ for some $c$, $d$. I take it you know how to show that the solutions of $x^2+y^2=z^2$ are given by $x=2kmn$, $y=(m^2-n^2)k$, $z=(m^2+n^2)k$. Maybe if you subject $x^2=y^2+2z^2$ to the same kind of analysis, you get Fermat's claim.
| 5 | https://mathoverflow.net/users/3684 | 48680 | 30,666 |
https://mathoverflow.net/questions/47850 | 3 | By polar decomposition, every continuous linear function $f \colon H \to K$ between Hilbert spaces can be written uniquely as $f = \widehat{f} \circ |f|$ for a positive operator $|f| \colon H \to H$ and a partial isometry $\widehat{f} \colon H \to K$ with $\ker(\widehat{f})=\ker(|f|)$. The binary operation $(f,g) \maps... | https://mathoverflow.net/users/10368 | Associativity of polar decomposition | Ah, (quite) some fiddling with Mathematica gave a counterexample.
In the notation of anon's answer, take
$$
A' = \begin{pmatrix} 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & 0 \end{pmatrix}, \qquad
B = \begin{pmatrix} 0 & 1 \\\\ 1 & 0 \end{pmatrix}, \qquad
C = \begin{pmatrix} 0 & 1/\sqrt{2} \\\\ 1 & 0 \\\\ 0 &
1/\sqrt{2} \end{p... | 1 | https://mathoverflow.net/users/10368 | 48681 | 30,667 |
https://mathoverflow.net/questions/48678 | 4 | Let $H, K$ be incomparable subgroups of $G$. The following is **false**:
$
N\_G(H \cap K) = H \cap K \quad \Rightarrow \quad N\_G(H)=H \text{ and }
N\_G(K)=K
$
Here is a counter-example:
$
G = A\_6, \quad H = (C\_3 \times C\_3) : C\_2, \quad \quad K = S\_4.
$
(see [link text](http://www.math.hawaii.edu/~william... | https://mathoverflow.net/users/9124 | A basic question about selfnormalizing subgroups | I think a counterexample is $G=S\_3\times S\_3\times S\_3$.
Say $U$ is the "diagonal" $S\_3$ in $G$; so if $G\lt S\_9$ is generated by $(123)$, $(12)$, $(456)$, $(45)$ ,$(789)$ and $(78)$, then $U$ is generated by $(123)(456)(789)$ and $(12)(45)(78)$, and it is self-normalizing.
However $H=\langle U,(123)\rangle$ ... | 6 | https://mathoverflow.net/users/3132 | 48682 | 30,668 |
https://mathoverflow.net/questions/48690 | 25 | Any Grothendieck topos E is the "classifying topos" of some geometric theory, in the sense that geometric morphisms F→E can be identified with "models of that theory" internal to the topos F. For the topos of sheaves on a site C, the corresonding theory may tautologically be taken to be "the theory of cover-preserving ... | https://mathoverflow.net/users/49 | What does an etale topos classify? | It classifies what the Grothendieck school calls "strict local rings". The points of such a topos are strict Henselian rings (Henselian rings with separably closed residue field). See Monique Hakim's thesis (*Topos annelés et schémas relatifs* $\operatorname{III.2-4}$) for a proof and a more precise definition of what ... | 26 | https://mathoverflow.net/users/1353 | 48691 | 30,672 |
https://mathoverflow.net/questions/48694 | 3 | How can I prove that $\text{Tor}\_1(R/I,R/J) = (I \cap J)/IJ$, where $R$ is a ring and $I, J$ ideals.
Moreover, if we suppose $R=I+J$, how do I prove that $\text{Tor}\_1(R/I,R/J)=0$?
Ps: No, this is not a homework question.
| https://mathoverflow.net/users/11414 | How to calculate Tor(R/I, R/J) ?? | Hints: 1) First prove that $I\otimes(R/J)=I/IJ$ . 2) If $I+J=R$, write $1=i+j$ and use the fact that $x=1x$.
| 4 | https://mathoverflow.net/users/10503 | 48696 | 30,674 |
https://mathoverflow.net/questions/48677 | 8 | Consider a finite rank complex bundle $E$ over $S^1$ with connection $\nabla$. Let $Q\_0, Q\_1 \in C^\infty(S^1, E)$ be pseudo-differential operators. $Q\_0$ is defined by the symbol $\sigma\_0(x, \xi) = i \xi$, and $Q\_1$ by $\sigma\_1(x,\xi) = 2H(\xi)-1$ where $H$ is the Heaviside step function. Both of these have di... | https://mathoverflow.net/users/11411 | Spectral theory of pseudo-differential operators | You should have a look at p. 267, 23.35.2 of Dieudonné's Treatise on Analysis, part 7, as well as Lawson-Michelson, Spin Geometry, p. 196, Thm. 5.8, and check if your symbols are elliptic (they seem to be) and notice that your base manifold is compact. This might help.
| 3 | https://mathoverflow.net/users/11176 | 48698 | 30,676 |
https://mathoverflow.net/questions/48702 | 10 | It is known that the sequence $d\_1 \geq d\_2 \geq \ldots \geq d\_n$ of nonnegative integers is the degree sequence of a graph if and only if the sum of the $d\_i$ is even and we have
\[
\sum\_{i = 1}^k d\_i \leq k(k - 1) + \sum\_{i = k + 1}^n \min(d\_i, k)
\]
for all $k \in \{1, \ldots, n\}$. (This is the Erdős–Gallai... | https://mathoverflow.net/users/4658 | Classification of degree (bi-)sequences of bipartite graphs? | So you are looking for the [Gale-Ryser theorem](http://mathworld.wolfram.com/Gale-RyserTheorem.html). There is a version of Havel-Hakimi for bipartite graphs as well. It says that the pair $(P,Q)$ is bigraphic if and only if the pair $(P',Q')$ is bigraphic, where $(P', Q')$ is obtained from $(P, Q)$ by deleting the lar... | 10 | https://mathoverflow.net/users/2384 | 48713 | 30,685 |
https://mathoverflow.net/questions/48707 | 3 | Let $K(x,y): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by $K(x,y) = e^{-< x,y>^2}$ where $<\cdot,\cdot>$ denote the canonical inner product. Define integral operator $T:C(\mathbb{R}^n) \to C(\mathbb{R}^n)$ by $(Tf)(x) = \int\_{\mathbb{R}^n}K(x,y)f(y) dy$ whenever this integration makes sense. (Here, $C(\ma... | https://mathoverflow.net/users/11416 | Integral kernel of form $e^{-<x,y>^2}$ | The answer to your second question is **No**. Take
$$f\_n(x)=\sqrt{1+x^2}-\sqrt{n^{-2}+x^2}.$$
You have $0\le f\_n\le f\_{n+1}$ and the limit $f$ is $\sqrt{1+x^2}-|x|$, which is not analytic at $x=0$.
| 2 | https://mathoverflow.net/users/8799 | 48723 | 30,690 |
https://mathoverflow.net/questions/48729 | 2 | Let $\alpha:\mathbb Z\longrightarrow \mathbb Z$ be a quadratic polynomial taking only integral values on the integers and consider the sequence of square-matrices with coefficients
$x^{\alpha(i+j)}$ for $i,j\in \{0,\dots,n\}$. The determinant of such a matrix involves a (not nessecarily positive) power of $x$ and a po... | https://mathoverflow.net/users/4556 | A determinant involving only cyclotomic factors | You can always factor out $x^{\alpha(i)}$ from each row and $x^{\alpha(j)}$ from each column, then multiply by $x^{n\alpha(0)}$ and you reduce to $\det(x^{\beta ij})=\prod (x^{\beta i}-x^{\beta j})$ by Vandermonde, where $\beta$ is some integer.
| 3 | https://mathoverflow.net/users/2384 | 48730 | 30,693 |
https://mathoverflow.net/questions/48718 | 20 | There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite rings, you get a compact ring; for example, the $p$-adic integers $\mathbb{Z}\_p$ are obtained as a limit of
$$
\cdots \tw... | https://mathoverflow.net/users/8410 | Is every compact topological ring a profinite ring? | Every compact topological ring has "enough" open ideals and is thus profinite. See for example Sect. 5.1 in
Luis Ribes, Pavel Zalesski, *Profinite Groups*, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge
| 12 | https://mathoverflow.net/users/2841 | 48735 | 30,694 |
https://mathoverflow.net/questions/48683 | 10 | The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra
$$
k[X\_1,...,X\_n]/I\_\Delta
$$
where $I\_\Delta$ is the ideal generated by the $X\_{i\_1}...X\_{i\_r}$ with ${i\_1,...,i\_r}\notin \Delta$.
Somebody told me that this construction helps to study va... | https://mathoverflow.net/users/2625 | Geometric motivation for the Stanley-Reisner correspondence | First when it comes to comparison with the simplicial complex it should be
realised that the Stanley-Reisner ring corresponds to the cone over the complex.
There is a non-homogeneous version of it where one replaces the linear subspace
through the origin by the affine space parallel to the linear space but passing
thro... | 6 | https://mathoverflow.net/users/4008 | 48738 | 30,696 |
https://mathoverflow.net/questions/48740 | 5 | Let $(X,d)$ be a metric space.
Let $(x\_n)\_{n\in\mathbb N}$ be a geodesic ray: $d(x\_n,x\_m)=\vert n-m\vert$.
>
> Is it true that, for all $y\in X$, the sequence $(d(x\_{n+1},y)-d(x\_n,y))$ converges to $1$ as $n$ goes to infinity ?
>
>
>
I am particularly interested in the case of $\delta$-hyperbolic spac... | https://mathoverflow.net/users/915 | Geodesic rays and horofunctions | **Yes**. Let $a\_n:=d(y,x\_n)-|n|$. By the triangle inequality, we see that $(a\_n)\_{n\ge0}$ is non-increasing and bounded. Therefore it has a finite limit $a$.
Now if $n\ge0$,
$d(y,x\_{n+1})-d(y,x\_n)=a\_{n+1}-a\_n+1\rightarrow a-a+1=1$.
| 9 | https://mathoverflow.net/users/8799 | 48743 | 30,699 |
https://mathoverflow.net/questions/48744 | 0 | Let $X$ be a set, $G$ the group of bijection on $X$. Then it is well-known that if $|X|\geq 3$, $Z(G)$ is trivial. However, I cannot see a way of extending this proof to $X$ being an infinite set (other than when a bijection only moves finitely many points). Indeed, I have been somewhat stumped trying to prove that $Z(... | https://mathoverflow.net/users/6503 | Center of a Symmetric Group on an Infinite Set | I think this is easy. We shall prove that $f=\mathrm{id}$ for every $f\in Z\left(G\right)$. In fact, let $x\in X$ be such that $f\left(x\right)\neq x$ (if no such $x$ exists, we are done anyway). Now let $y\in X$ be such that $y\neq x$ and $y\neq f\left(x\right)$ (such an $y$ exists due to $\left|X\right|\geq 3$). Then... | 5 | https://mathoverflow.net/users/2530 | 48746 | 30,701 |
https://mathoverflow.net/questions/48737 | 9 | suppose $M$ is a manifold , $G$ is a lie group (may be finite) ,then let $G$ act on $M$ freely , $N=M/G$ is then a manifold ,so my question is what relations may be between the homology and cohomology of $M$ and $N$ ?
In the surface case ,if $G$ is a finite group ,then we can get no differences between the cohomolgy a... | https://mathoverflow.net/users/4437 | homology and cohomology of a quotient manifold | To elaborate a little on Kostya's answer, for the case where $G$ acts freely you may want to look up the Cartan-Leray spectral sequence (eg Chapter 7 of the book by [Brown](http://books.google.co.uk/books?id=PMqb2DppvCsC&printsec=frontcover&dq=cohomology+of+groups&source=bl&ots=Jtg5xSGYAK&sig=Oijriey_a0jEe_gw7uMp0ByoG_... | 3 | https://mathoverflow.net/users/8103 | 48766 | 30,711 |
https://mathoverflow.net/questions/48693 | 16 | What can we say about the connected sum of a manifold $M^n$ with an Exotic sphere? Is is possible some of them are still diffemorphic to $M^n$. Is it possible to classifying all the exotic smooth structures for a given $M^n$?
| https://mathoverflow.net/users/4760 | Smooth structures on the connected sum of a manifold with an Exotic sphere | Surgery theory provides a framework for classifying closed higher-dimensional manifolds, but unfortunately, a definitive classification is known only for a very few homotopy types. Here is how surgery attempts to classify smooth structures on a given manifold $M$.
A basic object is a smooth structure set $S(M)$, whi... | 12 | https://mathoverflow.net/users/1573 | 48768 | 30,713 |
https://mathoverflow.net/questions/48773 | 0 | Let $f: X\rightarrow Y$ be a morphism of varieties. Let $0\rightarrow F\rightarrow E\rightarrow G\rightarrow 0$ be a short exact sequence of locally free sheave of finite
rank. If direct images of above sheaves are locally free, then is it true that it
induces a short exact sequence $0\rightarrow f\_\*F\rightarrow f\_\... | https://mathoverflow.net/users/11433 | direct image functor | Let $X,Y$ be defined over the field $k$ and take $f$ to be the structure map $f:X\to {\rm Spec}\, k$. Then let $E\to G$ be a surjective morphism of sheaves that is not surjective on global sections, e.g., $$E=\mathcal O\_{\mathbb P^1}(-1)\oplus \mathcal O\_{\mathbb P^1}(-1)\to G=\mathcal O\_{\mathbb P^1}.$$
Then $f\_\*... | 3 | https://mathoverflow.net/users/10076 | 48785 | 30,725 |
https://mathoverflow.net/questions/47807 | 2 | Given $A\in Mat\_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat\_{n\times n}(R)$ such that $XAY=diag(a\_1, \ldots , a\_n)$ for some $a\_i$?
The interesting answer for me is if $A=(x\_{i,j})$ and
$R=\mathbb Z [x\_{i,j}]$
(free associative non-commuta... | https://mathoverflow.net/users/11072 | on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring | I've understood that if $R=Mat\_n(K)$ then for every $A\in Mat\_m(R)$ exist $B\in Mat\_m(R)$, s.t. $AB=\lambda Id$. $A=(a\_{ij,kl})$ is a $mn\times mn$-matrix over $K$ and $B=A^{V}$ is a $m\times m$-matrix over $R$.
| -3 | https://mathoverflow.net/users/11072 | 48811 | 30,743 |
https://mathoverflow.net/questions/48809 | 7 | Hey everyone,
I would like to know if anybody could help me find references for the following.
Take a suitably well defined entire function $f(x)$ and it's derivative $\tilde{f}(x)$ to which the roots $x\_n$ and $\tilde{x}\_n$ are associated. The function $f(x)$ may have infinitely many roots, though naturally one ... | https://mathoverflow.net/users/11439 | Relationships between the roots of an entire function and the roots of its derivative | I don't know if there are any general results about these, but
when $f$ is a polynomial, these must be in essence results about
symmetric functions. If $f(z)=z^n+a\_{n-1} z^{n-1}+\cdots+a\_1z+a\_0$
then $Z(1)=-a\_1/a\_0$, $Z(2)=Z(1)^2-2a\_2/a\_0$ etc. In this case
your results surely specialize to polynomial identities... | 4 | https://mathoverflow.net/users/4213 | 48815 | 30,746 |
https://mathoverflow.net/questions/48819 | 3 | Starting with a principal $GL\_n$ bundle, we could form the vector bundle associated to a representation $GL\_n\to GL(V)$. Could someone please explain how we get a connection on this vector bundle.
| https://mathoverflow.net/users/11395 | Various definitions of Connections on bundles-2 | Let $P \to M$ is a $G$-principal bundle with a connection $\omega$ on it and $R:G \to GL (V)$ is a representation, inducing a Lie algebra map $r: \mathfrak{g} \to \mathfrak{gl}(V)$. Assume that $P$ is trivial and pick a section $s$ of $P$. The section identifies sections of $P \times\_G V$ with functions $M \to V$ and ... | 4 | https://mathoverflow.net/users/9928 | 48821 | 30,749 |
https://mathoverflow.net/questions/48728 | 1 | I've been learning about sheaves and have, I think, understood its characterisation
as a space that glues together locally, and its presentation as a space over a base to which it is locally homeomorphic.
However I've come across the following description of this equivalence (Tom Leinster, Sheaves do not belong to al... | https://mathoverflow.net/users/6408 | Extracting the Sheaf and espace étalé condition from an abstractly given equivalence between these two spaces | The general definition of sheaf, is much more general than a local homeomorphism over a space- however, it is sheaves of this later kind that came about first. In the general framework, you have a small category $C$ and a $\textbf{Grothendieck topology}$ on $C$, which declares what families of arrows $\left(c\_i \to c\... | 1 | https://mathoverflow.net/users/4528 | 48828 | 30,754 |
https://mathoverflow.net/questions/48488 | 9 | I have been looking at the numerical behavior of a particular quantity (of no direct importance here, though if you must know the gory details start with figure 17 [here](http://arxiv.org/abs/1009.2127)) associated to the geodesic flow on a surface of constant negative curvature and genus $g$. The behavior is quantitat... | https://mathoverflow.net/users/1847 | How does the mixing time of a geodesic flow on a surface vary with the genus? | Here is another thought that struck me on the way home, that I should have realized earlier. Suppose that $S$ is a closed hyperbolic surface, of genus $g$. Then the area of $S$ is $-2\pi\chi(S) = 2\pi(2g - 2)$. Since the area of a disk in the hyperbolic plane is exponential in its radius, it follows that the diameter o... | 4 | https://mathoverflow.net/users/1650 | 48840 | 30,763 |
https://mathoverflow.net/questions/48843 | 6 | For a given symmetric and positive semidefinite $n \times n$ matrix $A$, we want to solve the problem $$\max\_{||x|| = 1, \ x\geq 0} x^T A x.$$
Here, $x\geq 0$ indicates that $x$ must be component-wise non-negative. Without the $x\geq 0$ constraint, the solution $x$ must be the eigenvector corresponding to the largest ... | https://mathoverflow.net/users/11443 | Non-negative quadratic maximization | First, note that the condition that $A$ be positive semidefinite (PSD) doesn't buy you anything. Replacing $A$ by $A+kI$ changes the objective value of any feasible solution by $k$, so if we could solve the given problem when the matrix in the objective is PSD, we could just choose some $k$ large enough to make $A+kI$ ... | 11 | https://mathoverflow.net/users/5963 | 48847 | 30,768 |
https://mathoverflow.net/questions/48716 | 7 | Here is something I've wondered about from time to time: The continental divide in North America is commonly described as the geographic line curve seperating points where a drop of water would drain to the Atlantic from those where it would drain to the Pacific. My question is how to characterize such a curve mathemat... | https://mathoverflow.net/users/8008 | Characterize a continental divide | As Thierry and Gerry mentioned, Brian Hayes wrote an article "Dividing the Continent"
in *American Scientist* ([Volume 88, Number 6, page 481](http://www.americanscientist.org/issues/pub/dividing-the-continent)),
reprinted in his book, *Group Theory in the Bedroom
and Other Mathematical Diversions*. His focus is algo... | 2 | https://mathoverflow.net/users/6094 | 48858 | 30,774 |
https://mathoverflow.net/questions/48849 | 44 | Allow me to take advantage of your collective scholarliness...
The symmetric group $\mathbb S\_n$ can be presented, as we all know, as the group freely generated by letters $\sigma\_1,\dots,\sigma\_{n-1}$ subject to relations
$$
\begin{aligned}
&\sigma\_i\sigma\_j=\sigma\_j\sigma\_i, && 1\leq i,j<n, |i-j|>1;\\\\
&... | https://mathoverflow.net/users/1409 | Transpositions of order three | Following up what was mentioned in the comments for $n$ up to $5$. In "Factor groups of the braid group" Coxeter showed that the quotient of the Braid group by the normal closure of the subgroup generated by $\{\sigma\_i^k \ | \ 1\le i\le n-1\}$ is finite if and only if $$\frac{1}{n}+\frac{1}{k}>\frac{1}{2}$$
In your c... | 50 | https://mathoverflow.net/users/2384 | 48862 | 30,777 |
https://mathoverflow.net/questions/48774 | -3 | I have an NxN matrix of linear constraints that is not of full rank. In other words, some of the constraints are linear combinations of other constraints. The "standard" linear algebra tools (determinants, matrix inversion, etc.) seem to only be useful for testing for linear dependence. I'd like to not only discover th... | https://mathoverflow.net/users/4833 | Eliminating redundant linear constraints? | At sort-of request of the author, I'm making my comment into an answer:
The algorithm you're looking for is [Gaussian elimination](http://en.wikipedia.org/wiki/Gaussian_elimination#General_algorithm_to_compute_ranks_and_bases).
| 2 | https://mathoverflow.net/users/4658 | 48867 | 30,779 |
https://mathoverflow.net/questions/48761 | 3 | This question is motivated by my previous [post](https://math.stackexchange.com/questions/13344/proof-for-an-integral-involving-sinc-function) in SE (math.stackexchange.com).
Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire (i.e. analytic everywhere), or continuous,
function, $f(x)$ on $\mathbb{R}... | https://mathoverflow.net/users/10583 | Continuous or analytic functions with this property of sinc function | Seems like the Poisson formula is well-forgotten nowdays. Assume that $f$ is a real valued even Schwartz function and pass to its Fourier transform $g$ (with $2\pi$ in the exponent). Then the conditions are $$g(0)=\int\_{\mathbb R} g^2=\sum\_{k\in\mathbb Z} g(k)=\int\_{[-1/2, 1/2]}G^2$$ where $G(x)=\sum\_{k\in\mathbb Z... | 8 | https://mathoverflow.net/users/1131 | 48871 | 30,783 |
https://mathoverflow.net/questions/48538 | 2 | hello,
I am having a hard time following this isotopy put forth by Milnor in On the Total Curvature of Knots
>
> For each $c$ and $p$ in
> $\mathbb{R}^{n-1}$ such that $\|c-p\|
> > < r$, there is an isotopy, $f\_u^{c\;
> > p} (\gamma), 0 \le u \le 1$, of
> $\mathbb{R}^{n-1}$ onto itself which
> transforms $c$ ... | https://mathoverflow.net/users/11324 | isotopy doesn't make sense (Milnor) | [I'm entering my comment above as an answer here, so as to stop this question being bumped to the front page in the future.]
There is simply a typo: the definition of $f\_u^{cp}(\gamma)$ should be
$\gamma + u$(etc. ...).
| 4 | https://mathoverflow.net/users/2874 | 48873 | 30,785 |
https://mathoverflow.net/questions/48759 | 15 | I remember a friend in graduate school throwing an early edition of Jurgen Neukirch's Algebraic Number Theory book against a wall (so hard that it split the binding) after he had worked for a number of days to reconcile something he realized was an error (or typo) in the book.
For the life of me, I cannot remember wh... | https://mathoverflow.net/users/6269 | Has anyone found an error in an early version of Neukirch? | In my edition of Neukirch, Chapter I.9, Exercise 2:
If $L|K$ is a Galois extension of algebraic number fields, and $\mathfrak{P}$ a prime ideal which is unramified over $K$ (i.e. $\mathfrak{p} = \mathfrak{P} \cap K$ is unramified in $L$), then there is one and only one automorphism $\phi\_{\mathfrak{P}} \in G(L|K)$ s... | 9 | https://mathoverflow.net/users/1050 | 48877 | 30,788 |
https://mathoverflow.net/questions/48839 | 13 | Let $(G, n)$ be a pair where $G$ is an abelian group and $n \in \mathbb{N}$. Recall that an **Eilenberg-MacLane space** is a connected CW complex $X$ such that $\pi\_r(X) = G$ if $r=n$ and $0$ otherwise. An Eilenberg-MacLane space is unique up to homotopy equivalence. A direct argument is given in Hatcher's book based ... | https://mathoverflow.net/users/344 | Is there a functorial proof that Eilenberg-MacLane spaces are unique up to homotopy equivalence? | I think you're referring to the argument on page 366 of Hatcher. But there is an earlier argument for $n=1$ in Chapter 1B, which works just as well in general (as Eric Wofsey points out above).
Hatcher deduces
>
> Theorem 1B.8: The homotopy type of a
> CW complex $K(G,1)$ is uniquely
> determined by $G$.
>
>
... | 14 | https://mathoverflow.net/users/250 | 48886 | 30,792 |
https://mathoverflow.net/questions/48883 | 2 | Given a morphism $\phi : X\rightarrow Y$ of smooth projective varieties and a non-constant
map $f : C\rightarrow Y$, where $C$ is amooth projective curve, how to construct a smooth
projective curve $C'$ with non-constant map $g : C'\rightarrow X$ such that $\phi \circ g = f\circ h$, where $h : C'\rightarrow C$.
| https://mathoverflow.net/users/8141 | smooth projective curve | First form the fibre product $Z$ of $X$ and $C$ over $Y$, to get maps $f':Z \to X$ and
$\phi':Z \to C$ such
that $\phi\circ f' = f \circ \phi'.$
Now if the map $\phi'$ is not surjective (equivalently, dominant, since all varieties in sight are projective) then we can't find $C'$. (This happens when the image of $f$ ... | 11 | https://mathoverflow.net/users/2874 | 48890 | 30,794 |
https://mathoverflow.net/questions/48854 | 12 | (Hi. This is my first question here.)
A well known result in complex analysis says that there is an $\varepsilon\gt 0$ such that if $f$ is holomorphic in (a neighborhood of) the closed disk ${\mathbb D}$ of radius 1, and $f'(0)=1$, then $f({\mathbb D})$ contains a disk of radius $\varepsilon$.
This is due to Landau... | https://mathoverflow.net/users/11449 | Landau's constant | You can find some answers in Section 10.1 of Reinhold Remmert's *Classical Topics in Complex Function Theory* (Springer GTM 172) -- or in the original german edition *Funktionentheorie 2*.
A simple proof is given that Landau's constant $L$ verifies $L>\frac{3}{2}\sqrt{2}-2\simeq 0.1213$.
A more involved proof of $L... | 3 | https://mathoverflow.net/users/730 | 48894 | 30,797 |
https://mathoverflow.net/questions/48888 | 3 | There are many facts about integer gcds which can be proved by appealing to unique prime factorization (up to sign). for example $\gcd(a^2,b^2)=\gcd(a,b)^2$. One way to get the machinery (if that is not too strong a word) about primes and unique factorization is to start with the fact that for all integers $a,b$ there ... | https://mathoverflow.net/users/8008 | What is the divisibility theory for Bezout Domains? | I find your question interesting but a little vague. Let me give you a couple of references: perhaps you can use them to sharpen your question. (Or perhaps they're not what you're looking for at all: we'll see...)
First, there are certainly plenty of treatments of factorization from the perspective of Bezout domains ... | 8 | https://mathoverflow.net/users/1149 | 48897 | 30,800 |
https://mathoverflow.net/questions/48880 | 8 | Suppose one day I came up with a proof that 0 = 1 in some formal system such as PA or ZFC that cannot prove its own consistency (unless it is inconsistent). Would it be possible to have a zero-knowledge proof of this? In other words, would it be possible for me to convince you with high probability that I had derived s... | https://mathoverflow.net/users/11318 | Zero-knowledge proof that 0 = 1 | In this setting, the adversary seeks to find a deduction $\phi\_0, \dots, \phi\_n$ of $P \wedge \neg P$ quickly. If ZFC, for example, is inconsistent, there exists such a deduction and hence there exists a (constant time) adversary, which simply publishes $\phi$.
In order to have a zero-knowledge proof problem, one ... | 4 | https://mathoverflow.net/users/9470 | 48916 | 30,807 |
https://mathoverflow.net/questions/48912 | 11 | Does anybody know a reference for basic properties of tensor products of (finite) algebraic extensions of fields?
Ideally, I would like a description of $L \otimes\_k K$ for arbitrary finite extensions $L, K$ of $k$ but I would settle for a reference for results such as
1) If $K / k$ is Galois with group $G$ then $... | https://mathoverflow.net/users/8324 | Reference for tensor products of fields | Dear anon, the most complete reference might be Bourbaki's Algèbre, Chapter V.
**For question 1**, I suggest Bourbaki's Algèbre, Chapter V, §10, 4. Descente galoisienne, Corollaire . There the Master proves the more general result that the canonical morphism
$$K\otimes\_{k} K\to K^G: x\otimes y\mapsto (x \sigma (y))... | 12 | https://mathoverflow.net/users/450 | 48924 | 30,813 |
https://mathoverflow.net/questions/48213 | 17 | Suppose that $G$ is a semi-simple Lie group that acts smoothly (i.e., $C^\infty$) on a smooth, finite dimensional manifold $M$. Does it follow that the action of $G$ is linearizable near any stationary point of the action? This was conjectured by Steve Smale and myself in 1965, and was proved for the case that $M$ and ... | https://mathoverflow.net/users/7311 | Is a smooth action of a semi-simple Lie group linearizable near a stationary point? | There are smooth counter-examples by Cairns and Ghys [Ens. Math. 43, 1997], for instance a smooth non-linearizable action of $SL(2,\mathbb{R})$ on $\mathbb{R}^3$ (fixing the origin) or of $SL(3,\mathbb{R})$ on $\mathbb{R}^8$.
By contrast, they show that any $C^k$ action of $SL(n,\mathbb{R})$ on $\mathbb{R}^n$ (same $n$... | 16 | https://mathoverflow.net/users/6451 | 48925 | 30,814 |
https://mathoverflow.net/questions/48929 | 4 | It is known that for triangle-free graphs, if they are $d$-regular, then $2d\leq n$, where $n$ is the number of vertices. In words, the degree is less than or equal to half the number of vertices (complete bipartite for $2d = n$).
>
> My question is, for all $d, n$ with $2d\leq n$, can we always find a
> $d$-regula... | https://mathoverflow.net/users/11464 | Existence of triangle-free graphs for regular graphs of degree at most n/2 | **Yes**, it is always possible to find regular triangle-free graphs of any degree up to half the number of vertices (as long as the number of vertices is even). To see this, by Hall's Theorem the edges of $K\_{n,n}$ can be partitioned into $n$ disjoint perfect matchings. The union of $d$ of these perfect matchings is a... | 12 | https://mathoverflow.net/users/2233 | 48931 | 30,816 |
https://mathoverflow.net/questions/48910 | 43 | A friend of mine introduced me to the following question: Does there exist a smooth function $f: \mathbb{R} \to \mathbb{R}$, ($f \in C^\infty$), such that $f$ maps rationals to rationals and irrationals to irrationals and is nonlinear?
I posed this question earlier in math.stackexchange.com ([link to the question](ht... | https://mathoverflow.net/users/4165 | Smooth functions for which $f(x)$ is rational if and only if $x$ is rational | There are such functions. Moreover any diffeomorphism $f\_0:\mathbb R\to\mathbb R$ can be approximated by such $f$. For the sake of simplicity I assume that $f\_0'\ge 2$ everywhere.
Enumerate the rationals: $\mathbb Q=\{r\_1,r\_2,\dots\}$, and construct a sequence $f\_0,f\_1,f\_2,\dots$ of self-diffeomorphisms of $\m... | 35 | https://mathoverflow.net/users/4354 | 48932 | 30,817 |
https://mathoverflow.net/questions/48928 | 33 |
>
> Question: Given a finite group $G$, how do I find the smallest $n$ for which $G$ embeds in $S\_n$?
>
>
>
Equivalently, what is the smallest set $X$ on which $G$ acts faithfully by permutations?
This looks like a basic question, but I seem not to be able to find answers or even this question in the literatur... | https://mathoverflow.net/users/3132 | Smallest n for which G embeds in $S_n$? | Maybe this part of the answer helps. It was sufficient for many tasks, but fails at some reasonable problems.
A permutation action is a multi-set of conjugacy classes of subgroups of a group. The degree of the action is the total of the indices of representatives from each class (with multiplicity). The kernel of the... | 11 | https://mathoverflow.net/users/3710 | 48937 | 30,819 |
https://mathoverflow.net/questions/48936 | 5 | Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M\_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ with entries $A,B,C,D\in M\_2(R)$. Asssume further, that they all commute.
So my question is: Can one then compute the determin... | https://mathoverflow.net/users/3969 | Iterated calculation of determinants | The answer is **Yes**. This is done in Exercise 120 of my web page [link text](http://www.umpa.ens-lyon.fr/~serre/DPF/exobis.pdf). You can replace $4$ and $2$ by numbers $n$ and $m$ with $m$ dividing $n$.
**Later**. The required complement. Let me take the situation of the question, but with $A,B,C,D$ being $m\times ... | 11 | https://mathoverflow.net/users/8799 | 48940 | 30,822 |
https://mathoverflow.net/questions/48908 | 37 | There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the [Invariant Subspace Problem](http://garden.irmacs.sfu.ca/?q=op/invariant_subspace_problem), he says (paraphrasing) "...this question is still open. It is also an open question whether or not this question is interesting... | https://mathoverflow.net/users/9124 | Is the Invariant Subspace Problem interesting? | 1. The invariant subspace problem for Banach spaces was solved in the negative for Banach spaces by Per Enflo and counterexamples for many classical spaces were constructed by Charles Read. The problem is open for reflexive Banach spaces. On the other hand, S. Argyros and R. Haydon recently constructed a Banach space $... | 42 | https://mathoverflow.net/users/2554 | 48941 | 30,823 |
https://mathoverflow.net/questions/48934 | 6 | Let $X$ be a geometrically integral smooth projective variety over a number field $k$. Then if $X$ is everywhere locally soluble, we have $Pic(X) = H^0(k,Pic (\overline{X}))$, where $\overline{X}=X \times\_k \overline{k}$. This can be proved using the Hochschild-Serre spectral sequence and the fundamental exact sequenc... | https://mathoverflow.net/users/5101 | An example where $Pic(X) = H^0(k,Pic(\overline{X}))$? | No, it is possible for $X$ not to have points everywhere locally and still have every $k$-rational divisor class be represented by a $k$-rational divisor (this is the "Picard equality" you want).
For instance, the Picard equality holds if $X$ has points everywhere locally except at a single place. Many examples of ge... | 6 | https://mathoverflow.net/users/1149 | 48945 | 30,825 |
https://mathoverflow.net/questions/48951 | 1 | It is asserted in *A Course in Metric Geometry* by Burago, Burago, Ivanov that
>
> there can be no more than continuum of mutually nonisometric compact spaces
>
>
>
How is this proven?
Its clear that there must be at least a continuum of mutually nonisometric compact spaces, i.e. $([0,\alpha], d\_{\mathbb{R... | https://mathoverflow.net/users/1540 | How to show the cardinality of nonisometric compact metric spaces is the continuum | I think "compact" can be even weakened here to "separable and complete" (and regarding your first guess, total boundedness is essentially used to prove that compact implies separable). Here's a sketch: any such space is determined, up to isometry, by the restriction of the metric to a countable dense subset. Thus the n... | 6 | https://mathoverflow.net/users/1044 | 48955 | 30,833 |
https://mathoverflow.net/questions/48532 | 15 | Embed the hyperoctahedral group $H\_n$ into the symmetric group $S\_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S\_{2n}$ by partitions of $2n$ and label representations of $H\_n$ by pairs of partitions whose sizes add up to $n$ in the standard... | https://mathoverflow.net/users/321 | Branching rule from symmetric group $S_{2n}$ to hyperoctahedral group $H_n$ | Okay I found it in the article by Koike and Terada, "Littlewood's formulas and their application to representations of classical Weyl groups". Here is the theorem. Define $d^\nu\_{\lambda, \mu}$ as the coefficients in the product of plethysms:
$\displaystyle (s\_\lambda \circ h\_2)(s\_\mu \circ e\_2) = \sum\_\nu d^\n... | 9 | https://mathoverflow.net/users/321 | 48972 | 30,843 |
https://mathoverflow.net/questions/48970 | 33 | An apparently elementary question that bugs me for quite some time:
>
> **(1)** Why are the integers with the cofinite topology not path-connected?
>
>
>
Recall that the open sets in the cofinite topology on a set are the subsets whose complement is finite or the entire space.
Obviously, the integers are con... | https://mathoverflow.net/users/11081 | Why are the integers with the cofinite topology not path-connected? | I happen to have been thinking about this question recently. The proof I like uses the fact that a nested sequence of *open* intervals has non-empty intersection provided neither end point is eventually constant. Now one inductively constructs a sequence of such intervals as follows. Each interval is a component of the... | 40 | https://mathoverflow.net/users/1459 | 48977 | 30,847 |
https://mathoverflow.net/questions/48980 | 1 | Let $X$ be a quasi-projective variety. Suppose that we (perhaps partially, if either enough is known) compactify to $\bar{X}$ with $\bar{X}\setminus X=D$ is a divisor. Say that we know the canonical bundle $K\_X$. Then $K\_{\bar{X}}=K\_X+nD$ for some $n$.
1. Is $n$ always negative? The examples I'm thinking of are fo... | https://mathoverflow.net/users/622 | Canonical bundle of compactifications | perhaps I am misunderstanding what you ask, but the answer is **no** to both questions. Take an arbitrary projective variety $\overline X$, actually for simplicity let $\overline X$ be normal. Then it has a canonical divisor, say $K\_{\overline X}$ and choose an actual representative, $K=\sum\_i a\_i K\_i$ where the $a... | 4 | https://mathoverflow.net/users/10076 | 48983 | 30,851 |
https://mathoverflow.net/questions/48966 | 9 | Let $M$ be a contact manifold, and let $F$ be an oriented 1-dimensional foliation that is transverse to the contact structure.
Is there a contact form $\alpha$ whose associated Reeb vector field generates the foliation $F$?
| https://mathoverflow.net/users/5690 | In a contact manifold, is every tranverse 1-foliation given by some Reeb vector field? | Answer is negative. A possible solution is to construct a transversal vector field without closed trajectories. By Taubes theorem (see his paper "The Seiberg–Witten equations and the Weinstein conjecture") it is impossible. One could easily prove that such a field exists for a standard $3$-torus with contact structure ... | 8 | https://mathoverflow.net/users/2823 | 48987 | 30,855 |
https://mathoverflow.net/questions/48988 | 1 | What are the most obvious generalizations of Berlekamp Massey algorithm [1] to matrix sequences?
>
> [1] [Massey, J. L.](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.80.9932), "Shift-register synthesis and BCH decoding", IEEE Trans. Information Theory IT-15 (1): 122–127
>
>
>
| https://mathoverflow.net/users/4415 | Matrix version of Berlekamp Massey algorithm | You might find [this](http://repository.lib.ncsu.edu/ir/handle/1840.16/3825) dissertation interesting. The author has collected many references of the Matrix versions of the Berlekamp-Massey algorithm (citing Rissansen, Dickinson, Thome and other papers) and its applications.
| 3 | https://mathoverflow.net/users/2384 | 48992 | 30,858 |
https://mathoverflow.net/questions/48752 | 3 | Has anyone read Atiyah and Bott's famous paper "The moment map and equivariant cohomology"?
I have some trouble with the equaations appearing between equations (4.18) and (4.19) in page 13 of the original paper. The paper claims
$$D\lambda a = D(a-i(X)a\theta) = da-i(X)da \theta + i(X)au,$$
whereas I think that it sh... | https://mathoverflow.net/users/4437 | could you help me understand an equation in a paper by Atiyah & Bott? | I think that the map $\lambda$ should be defined as follows:
$$\lambda a = a - \theta i(X)a.$$
In this way,
$$D\lambda a = D( a - \theta i(X)a) = da + u i(X)a + \theta di(X)a$$
and using that $di(X) a + i(X) da = 0$ and that $\lambda(i(X) a) = i(X) a$, one finds
$$D\lambda a = \lambda( da + i(X)a u ),$$
which is (4.1... | 4 | https://mathoverflow.net/users/394 | 49005 | 30,866 |
https://mathoverflow.net/questions/49015 | 17 | This question might be astoundingly naive, because my understanding of modular forms is so meek. It occurred to me that the reason I was never able to penetrate into the field of modular forms, automorphic forms, the Langland's program and so forth was because my appeal is to things that have the feel of SGA1, and thos... | https://mathoverflow.net/users/5756 | Did Grothendieck write about modular forms? | No.
That is perhaps a little too categorical, but a mathscinet search with Grothendieck as author and "modular form" or "forme modulaire" as "anywhere" gives no result. I don't remember him mentionning modular forms in "Recoltes et Semailles" either.
More to the point, it is a commonplace in the field of modular an... | 36 | https://mathoverflow.net/users/9317 | 49023 | 30,876 |
https://mathoverflow.net/questions/49034 | 0 | My question is the following:
Can a (probabilistic, deterministic, ndtm) oracle turing machine $A$ calling an oracle residing in a superior (more difficult) complexity class $B$, have less power then the oracle called ($A^B < B$).
My believe is that this new class should be at least as powerful as $B$?
For exam... | https://mathoverflow.net/users/11502 | weaker oracle machine ? | If you think of Turing machines that have an additional oracle tape and the complexity
class $C$ under consideration is below linear time, then for any nontrivial oracle $X$ we have $X\not\in C^X$, just because you don't have enough time to actually ask the oracle.
In all other cases (i.e., if you allow for linear ... | 5 | https://mathoverflow.net/users/7743 | 49036 | 30,880 |
https://mathoverflow.net/questions/48968 | 6 | This is a simple terminology question: I want to know if the involution $z \mapsto z^{-1}$ on Laurent polynomials (over some ring, I happen to be working over $\mathbb{Z}$ but that's not important) has a special name.
My motivation is perhaps a little unusual for this site. I'm doing some computations that involve ma... | https://mathoverflow.net/users/45 | Is there a name for the involution on Laurent polynomials? | I would call it the antipode. If your base ring is commutative, then the Laurent polynomials are the coordinate ring of the multiplicative group, and the antipode gives you the inversion on the group scheme.
| 7 | https://mathoverflow.net/users/121 | 49042 | 30,884 |
https://mathoverflow.net/questions/49008 | 5 | This question is twofold.
1) What is the best reference on root systems?
2) Do complex root systems exist?
| https://mathoverflow.net/users/6209 | Complex root systems | To supplement what Pete says, I'd emphasize that "root systems" have been defined in a variety of ways for a variety of purposes related to Lie theory or to some type of "reflection" group. (For instance, Vinay Deodhar proposed a general notion in the setting of Coxeter groups, which I used in my 1990 book.) The tradit... | 19 | https://mathoverflow.net/users/4231 | 49050 | 30,891 |
https://mathoverflow.net/questions/49052 | 1 | Let S be a sphere of unit radius in three dimensional Euclidean space, R^3. Given a
positive real number e, does there always exist a convex polyhedron P in R^3 such that:
(1) S is a subset of P (2) The boundary of P is homeomorphic to the boundary of S (3) The
volume of P does not exceed the volume of S by more than e... | https://mathoverflow.net/users/4423 | A question about convex polyhedra | For small $\epsilon$ let $Q$ be the convex symmetric hull of a finite $\epsilon$ net for the boundary of $S$ and let $P=(1+\epsilon) Q$.
| 4 | https://mathoverflow.net/users/2554 | 49053 | 30,892 |
https://mathoverflow.net/questions/49071 | 19 | I submitted a paper to a journal. It was accepted and published.
I had also posted a version of the paper on arXiv. The arXiv version of the paper and the journal's version are not identical, because the journal's version uses their house LaTeX style. However, the two versions are otherwise the same, and share the sa... | https://mathoverflow.net/users/437 | Should a published paper with a published correction be replaced on arXiv? | In any case I think you should update the arXiv paper, that's where most people will read your paper anyways. Besides, writing a separate errata paper seems a bit unnecessary in this case. In the arXiv version you could probably add a footnote about the change of numbering.
On the other hand, when people reference a... | 7 | https://mathoverflow.net/users/3996 | 49074 | 30,908 |
https://mathoverflow.net/questions/48367 | 6 | Let $X$ be a compact Kahler manifold with $c\_1(X) = 0$. Any Kahler metric $\omega$ on $X$ gives a Laplacian $\Delta\_\omega$ and the $(1,1)$-form $\omega$ is harmonic with respect to this Laplacian.
1. Take $\omega$ to be Ricci-flat. Is any other Ricci-flat metric $\alpha$ harmonic with respect to $\omega$?
2. If th... | https://mathoverflow.net/users/4054 | Harmonic forms on Ricci-flat Kahler manifolds | First let me give some cheap examples of CY manifolds that satisfy condition of 1) while they are not tori quotients.
Example. Let $X$ be any CY manifold with $h^{1,1}(X)=1$, for example $X$ can be a quintic in $\mathbb CP^4$. Then the conclusion of 1) holds, because all Kahler Ricci flat metric on $X$ are proportion... | 5 | https://mathoverflow.net/users/943 | 49091 | 30,920 |
https://mathoverflow.net/questions/48708 | 4 | (I'm just adding the completeness condition to $V$ from [this 2 month old question of mine](https://mathoverflow.net/questions/42731/are-coordinate-functions-on-topological-vector-spaces-always-continuous), because I realized it's relevant to whether Bill Johnson's answer to [this 4 month old question of mine](https://... | https://mathoverflow.net/users/nan | Are coordinate functionals on complete vector spaces always continuous? | Consider $\ell\_1:=\ell\_1(\Bbb{N}\cup \{0\})$ as the dual of $c$, the space of convergent sequences indexed by $\Bbb{N}$, where the action is given by $e\_0^\*(x)=\lim\_n x(n)$ and $e\_n^\*(x)=x(n)$ for $n\ge 1$. Put the bw$^\*$ topology on $\ell\_1$ under this pairing, which is the largest locally convex topology tha... | 2 | https://mathoverflow.net/users/2554 | 49095 | 30,923 |
https://mathoverflow.net/questions/49102 | 3 | I'm thinking about making an attempt at counting the number of Latin squares of order 12. Currently I'm in (what I call) the "sanity check" stage, where we check that the ranges that need searching through aren't too massive.
The standard method of counting Latin squares involves building up from Latin rectangles. We... | https://mathoverflow.net/users/2264 | What is the number of k-regular subgraphs of $K_{12,12}$? | The "nonisomorphic" part is not relevant, if you are only looking for an estimate (the isomorphsm group of almost all graphs is trivial), so to first order you are looking for the number of $k$-regular bi-partite graphs on $n$ (in your case $n=12$) vertices. At that point, the key words are "configuration model", inven... | 2 | https://mathoverflow.net/users/11142 | 49104 | 30,930 |
https://mathoverflow.net/questions/48989 | 23 | Let $p=37$. Since $p$ divides the numerator of $B\_{32}$, by Ribet's proof of the converse of Herbrand's theorem, we know that the class group of ${\bf Q}(\mu\_p)$ has size divisible by $p$. More specifically, the $p$-part of the class group decomposes under the action of $\Delta = {\rm Gal}({\bf Q}(\mu\_p)/{\bf Q})$, ... | https://mathoverflow.net/users/86179 | Does the everywhere unramified extension of Q(mu_37) of degree 37 grow into a Z_37-extension? | $\newcommand{\L}{\mathcal{L}}$
$\newcommand{\Q}{\mathbf{Q}}$
$\newcommand{\Z}{\mathbf{Z}}$
$\newcommand{\F}{\mathbf{F}}$
The answer is yes. It suffices (as Brian mentions) to show that the corresponding space
of extensions is one dimensional.
Let $p$ be an odd prime, let $k$ be an integer, and let $\omega$ be the m... | 22 | https://mathoverflow.net/users/nan | 49112 | 30,936 |
https://mathoverflow.net/questions/49027 | 5 | Can there exist two functions $f,g: \mathbb R \to \mathbb R$ so that $f$ is continuous, $g$ is discontinuous, and their graphs $\Gamma\_f, \Gamma\_g \subseteq \mathbb R^2$ are related by an isometry? (I think you can assume the isometry is a rotation.)
The graph of a continuous function must be path connected, so a n... | https://mathoverflow.net/users/11499 | Can the graph of a continuous function be a rotation of the graph of a discontinuous function? | I think the question has an easy answer, and it's essentially a 1st course in analysis type question that reduces pretty quickly to an intermediate value theorem application. Here's a more general statement. Let $f : \mathbb R \to \mathbb R$ be continuous and $h : \mathbb R^2 \to \mathbb R^2$ a homeomorphism. Then if $... | 6 | https://mathoverflow.net/users/1465 | 49114 | 30,937 |
https://mathoverflow.net/questions/48865 | 14 |
>
> Suppose $f:X\to Y$ is a morphism of *smooth connected* schemes (over some base). Say $Z\subseteq Y$ is a closed subscheme with complement $U$ so that $f$ pulls back (restricts) to isomorphisms on $Z$ and $U$. Does it follow that $f$ is an isomorphism?
>
>
>
If we drop the condition that $X$ and $Y$ have to b... | https://mathoverflow.net/users/1 | If a map restricts to an isomorphism on a closed subscheme and its open complement, must it be an isomorphism? | This is essentially the argument in [Bhargav](https://mathoverflow.net/users/986/bhargav)'s comment. [Matt Satriano](https://mathoverflow.net/users/28/matt-satriano) showed me the separatedness argument.
We first apply the valuative criterion for separatedness to show that $f:X\to Y$ is separated. In fact, this shows... | 5 | https://mathoverflow.net/users/1 | 49117 | 30,940 |
https://mathoverflow.net/questions/49131 | 16 | Do higher genus Teichmuller modular forms have, or are they expected to have, implications for number theory that generalize the sorts of results that flow from the study of classical modular forms?
| https://mathoverflow.net/users/10909 | Teichmuller modular forms and number theory | @David Hansen: Teichmuller modular forms are basically the natural analogue of Siegel modular forms when one considers sections of line bundles on $M\_g$ instead of $A\_g$. Search for papers by Ichikawa.
I don't know very much about Teichmuller modular forms but my advisor knows a bit about them and from what I can t... | 21 | https://mathoverflow.net/users/1310 | 49135 | 30,953 |
https://mathoverflow.net/questions/49136 | 2 | Let $C$ be the set of all vectors of dimension $n$ such that each of its entries are one of $-1$, $0$ and $1$ and also that the every $v \in C$ has at least $\frac{n}{100}$ $1$'s and at least $\frac{n}{100}$ $-1$'s. For a matrix $A$ (its dimensions are $\Lambda n \times n$ for some sufficiently large constant $\Lambda$... | https://mathoverflow.net/users/10858 | Norms over some subspaces | The answer is negative. You can achieve at most $O(\sqrt{n\log n})$.
Since more than half of the $\pm 1$ vectors are in $C$,
$$\|A\|\_{C,\infty} \leq 2 Aver(\|Av\|\_\infty),$$
where the Average is taken over all $\pm 1$ vectors.
To give an upper bound on $Aver(\|Av\|\_\infty)$, note that each of the coordinate... | 4 | https://mathoverflow.net/users/6921 | 49140 | 30,955 |
https://mathoverflow.net/questions/49144 | 3 | Let $\Gamma\subset\mathbb{P}^2$ be a singular plane curve and let us consider $f:C\to\Gamma$ the normalization map. Given $L\in\mathrm{Pic}(\Gamma)$, is it always true that $h^0(\Gamma,L)=h^0(C, f^\*L)$? If not, can you exhibit a counterexample?
| https://mathoverflow.net/users/33841 | Pull-back of line bundles under the normalization map. | Alternately, for any such $\Gamma$ and $C$, we always have a short exact sequence $$0 \to O\_{\Gamma} \to f\_\* O\_C \to F \to 0,$$
where $F$ is a finite length $\mathcal{O}\_{\Gamma}$-module supported where $\Gamma$ is non-smooth.
Twisting by a very positive (ie, high multiple of an ample) Cartier divisor $L$ and t... | 6 | https://mathoverflow.net/users/3521 | 49145 | 30,958 |
https://mathoverflow.net/questions/49134 | 7 | Let $i : U \to X$ be a quasi-compact open immersion of schemes. Under which conditions is the natural map
$i\_\* M \otimes i\_\* N \to i\_\* (M \otimes N)$
for all $M,N \in \text{Qcoh}(U)$ an isomorphism? We may assume that $X=\text{Spec}(A)$ is affine. If $U$ is affine, then we may reduce to the case $M=N=\mathcal... | https://mathoverflow.net/users/2841 | When does the direct image functor commute with tensor products? | Here's a counterexample: over a field $k$, let $X=\mathbb{A}^4=\mathbb{A}^2\times \mathbb{A}^2$, and $U$ the complement of the origin. Put $Y=\mathbb{A}^2\times\{0\}$, $Z=\{0\}\times\mathbb{A}^2$, $Z'=Z\cap U$, $Y'=Y\cap U$.
Take $M=\mathcal{O}\_{Y'}$ and $N=\mathcal{O}\_{Z'}$. Then $M\otimes N$ is zero, while $ i\_... | 10 | https://mathoverflow.net/users/7666 | 49149 | 30,961 |
https://mathoverflow.net/questions/49156 | 5 | Does the Zariski topology on a ring (not commutative in common) form a compact or paracompact space and why?
| https://mathoverflow.net/users/11530 | Zariski topology and compact \paracompact space? | It is compact (or quasi-compact?).
Let $R$ be a ring. The Zariski topology has closed sets given by $V(I) = \{P \mid I \subset P\}$. So let $I\_\alpha$ be a family of ideals such that $\bigcap V(I\_\alpha) = \emptyset = V(R)$.
This means that $\sum I\_\alpha = R$. However, as $1 \in R$, we can write $1 \in \sum I\_... | 0 | https://mathoverflow.net/users/1703 | 49162 | 30,968 |
https://mathoverflow.net/questions/49111 | 11 | Kaledin and Verbitsky have shown that symplectic varities have a remarkably nice deformation theory *as symplectic varieties.*
Let $X$ be a symplectic variety (a smooth quasi-projective variety over $\mathbb{C}$ equipped with a nondegenerate closed algebraic 2-form $\Omega$); then there is a universal formal deformat... | https://mathoverflow.net/users/66 | Is the generic deformation of a symplectic variety affine? | Being "affine" in this case does not make much sense,
because the hyperkaehler deformation is a complex manifold, without
a fixed algebraic structure. Simpson produced an example of a
hyperkaehler deformation of a space of flat bundles
admitting several algebraic structures, both
inducing the same Stein complex stru... | 7 | https://mathoverflow.net/users/3377 | 49165 | 30,970 |
https://mathoverflow.net/questions/48764 | 10 | Let $\mathfrak{g}$ be a semi-simple finite dimensional Lie algebra over $\mathbb C$.
Is it true that
$$
\dim \mathfrak h \ge \mathop{rank} \mathfrak g
$$
for **any** maximal abelian subalgebra $\mathfrak h \subset \mathfrak g$ ? Here by maximal I mean that $\mathfrak h$ is not properly contained in any other abelian s... | https://mathoverflow.net/users/605 | Minimal dimension of maximal abelian subalgebras | I think the answer to my question is no. A search at Mathscinet revealed the following:
>
> Gerstenhaber conjectured in [Ann. of
> Math. (2) 73 (1961), 324--348;
> MR0132079 (24 #A1926)] that a maximal
> commutative subalgebra of the
> associative ring of $n\times n$
> matrices $M\_n(k)$ always has dimension
>... | 8 | https://mathoverflow.net/users/605 | 49167 | 30,972 |
https://mathoverflow.net/questions/49173 | 18 | Recall the notion of [locally presentable category (nLab)](http://ncatlab.org/nlab/show/locally+presentable+category): $\DeclareMathOperator{\Hom}{Hom}$
**Definition:** Fix a regular cardinal $\kappa$; a set is *$\kappa$-small* if its cardinality is strictly less than $\kappa$. A *$\kappa$-directed category* is a pos... | https://mathoverflow.net/users/78 | What's an example of a locally presentable category "in nature" that's not $\aleph_0$-locally presentable? | The category of Banach spaces and contractions (over the reals or any other complete normed field, I think) is an example of an $\aleph\_{1}$-presentable category which is not $\aleph\_{0}$-presentable. The point is that the ground field is a strong generator and its represented functor $Ban(k,-)$ commutes with $\aleph... | 27 | https://mathoverflow.net/users/11081 | 49178 | 30,976 |
https://mathoverflow.net/questions/49187 | 0 | Is there a name for such class of modules $M$ such that $M\rightarrow N\rightarrow 0$ splits for every $N$?
| https://mathoverflow.net/users/5292 | Projectively splitting module | These are the [semisimple modules](https://en.wikipedia.org/wiki/Semisimple_module). ("Semisimple" is usually defined by the condition that every injection into M splits, but that's clearly equivalent to your condition.)
| 6 | https://mathoverflow.net/users/10503 | 49188 | 30,981 |
https://mathoverflow.net/questions/49175 | 15 | Here is a word that I think should be adopted by the category theorists. (If there is another synonymous word already in existence, please let me know.)
**Definition:** A category $C$ is *saft* if every cocontinuous functor $C \to D$ has a right adjoint.
The word "saft" is an abbreviation for "special adjoint funct... | https://mathoverflow.net/users/78 | Is every saft category cocomplete? | Theo, your question is in the neighborhood of what is called "total cocompleteness" or "totality". A category $C$ is *total* if the Yoneda embedding $y: C \to Set^{C^{op}}$ has a left adjoint. There is a similar notion of totality in the enriched case.
Total categories have this "saft" property you are discussing: e... | 15 | https://mathoverflow.net/users/2926 | 49193 | 30,983 |
https://mathoverflow.net/questions/49192 | 15 | If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely speculative.] Is there a shortish list of theorems such that any construction verifying the properties on the list would ... | https://mathoverflow.net/users/460 | What formal properties should resolution of singularities have? | Here are some properties of resolution of singularities that I use often:
1. The map is an isomorphism (or some version of *very nice*) over the smooth locus is important. *EDIT:* This usually shows up because I want to make statements about properties of the singular locus itself. One common application of this is t... | 21 | https://mathoverflow.net/users/3521 | 49195 | 30,984 |
https://mathoverflow.net/questions/49170 | 7 | It's known that all abelian groups are regularly realizable over $\mathbb{Q}(x)$, but it occurred to me that I don't even have an example of a cyclic regular extension of $\mathbb{Q}(x)$ handy.
So: what is an example of a regular realization of $C\_5$ over $\mathbb{Q}(x)$?
| https://mathoverflow.net/users/5756 | What is an example of a regular realization of $C_5$ over $\mathbb{Q}(x)$? | Emma Lehmer's quintic
\begin{align\*}
y^5 +& x^2y^4 - (2x^3 + 6x^2 + 10x + 10)y^3 +\\
&(x^4 + 5x^3 + 11x^2 + 15x + 5)y^2 + (x^3 + 4x^2 +10x + 10)y + 1
\end{align\*}
has Galois group $C\_5$ over ${\mathbf Q}(x)$ and the splitting field over ${\mathbf Q}(x)$ is a regular extension. Her paper is "Connection between Gauss... | 11 | https://mathoverflow.net/users/3272 | 49198 | 30,985 |
https://mathoverflow.net/questions/49209 | -3 | This may have a simple answer, but I'm not getting anywhere.
If $U$ is an open set in $\mathbb{R}^n$ (usual topology), and $p:[0,1] \to U$ is a continuous path, from $x=p(0)$ to $y=p(1)$, with $x,y \in \mathbb{R}^n$, then can we find an $\epsilon \in (0,1)$ such that $\cup\_{t \in [0,1]} B(p(t), \epsilon) \subseteq U... | https://mathoverflow.net/users/7895 | epsilon tube around continuous path in open set in R^n | $\epsilon\_t = \operatorname{sup}(\{r\in (0,1) : \operatorname{B}(p(t),r)\subseteq U\}) = \operatorname{inf}(\{d(p(t),u) : u\in U\}) = d(p(t),U)$
$t\mapsto \epsilon\_t \; = \; t\mapsto d(p(t),U) = (x\mapsto d(x,U))\circ (t\mapsto p(t))$
$x\mapsto d(x,U)$ and $(t\mapsto p(t))$ are both continuous, so $(x\mapsto d(x,... | 1 | https://mathoverflow.net/users/nan | 49211 | 30,991 |
https://mathoverflow.net/questions/46716 | 14 | Is the following true?
For any $c\in (0,1)$ there exists $f(c)>0$ such that for any subset $A\subset \{1,2,\dots,n\}$ of cardinality $|A|\geq cn$, the set
$$B=\left\{ k \in \{1,2,\dots,n!\} \colon \text{ there is } a \in A \text{ that divides } k\right\}$$
of numbers having at least one divisor in $A$ satisfies $$... | https://mathoverflow.net/users/4312 | numbers divisible by at least one of many numbers | The asymptotic version is certainly false. Let $\varepsilon (x,y)$ be the density of numbers having a divisor in the interval $[x,y]$, then Besicovitch proved in "On the density of certain sequences of integers" that $$\liminf\_{x\to \infty} \varepsilon (x,2x)=0.$$
Later, Erdos improved this to $$\varepsilon(x,2x)\sim(... | 3 | https://mathoverflow.net/users/2384 | 49223 | 30,995 |
https://mathoverflow.net/questions/49226 | 8 | The spectrum of a graph is the (multi)set of eigenvalues of its adjacency matrix (or Laplacian, depending on what you're interested in). In general, two non-isomorphic graphs might have the same spectrum.
Prompted in part by [this discussion on reverse engineering a graph from its spectrum](https://cstheory.stackexc... | https://mathoverflow.net/users/972 | Classes of graphs for which isospectrum implies isomorphism? | Maximum degree 2 would be such a class (which includes regular of degree $2$ as a subclass). Transitive graphs (by which I mean that the relation of being connected by an edge is transitive) are another example (there is a less obscure description of that class of graphs but I wanted it to sound mysterious for a few mo... | 5 | https://mathoverflow.net/users/8008 | 49231 | 31,002 |
https://mathoverflow.net/questions/49210 | 2 | Let $A$ be a small category, and let $X:=Psh(A)$ denote the category of presheaves on $A$. It is a theorem that for any such category $X$, there exists a small set $M$ of monomorphisms admitting the small object argument such that $LLP(RLP(M))$ is exactly the class of all monomorphisms of $X$. Recall that a separated s... | https://mathoverflow.net/users/1353 | Abstract classes of anodyne maps (relative to an interval) in a presheaf category are stable under smash products with monomorphisms? | This condition is equivalent to the following condition:
The class $\mathrm{An}=\mathrm{An}(\Lambda\_I(S))$ of anodynes generated by a small set $S$ of monomorphisms with respect to the cylinder $I$ is closed under finite Cartesian products with objects of $X=\mathrm{Psh}(A)$. That is, for any anodyne $f:K\hookrighta... | 0 | https://mathoverflow.net/users/1353 | 49241 | 31,009 |
https://mathoverflow.net/questions/49215 | 4 | My question might be an easy or could be a bit complicate and classic. Actually I am trying to understand why the discriminant of a binary quadratic form is a "the fundamental invariant" under $GL(2,\mathbb{Z})$-action i.e any other invariant is a polynomial of the discriminant. Also I am interested to know about gener... | https://mathoverflow.net/users/8419 | Invariant theory | Finding the number of generators for the invariants of binary forms is a classic and very hard problem in invariant theory. For forms of small degree one can find a description in Hilbert's book on invariant theory (ISBN 978-0521449038). In the 19th century it was solved for forms of degree at most 8, and it has recent... | 3 | https://mathoverflow.net/users/51 | 49267 | 31,022 |
https://mathoverflow.net/questions/48733 | 8 | Hello everyone
In connection with calculating the Fourier coefficients of some quasi-modular forms which I have been looking at lately, I have come across the following type of sum
$$ S\_{a,b}(N) := \sum\_{t=1}^{N-1} \ \ \sum\_{(n,m) \in I\_{N-t,t}} \frac{1}{m^a n^b} $$
where $$ I\_{k,l} = \{ \ (m,n) \ \big| \ \... | https://mathoverflow.net/users/11423 | Curious divisor-like sums | This function seems related to Percy MacMahon's generalized sum-of-divisors functions. A generating function can be given as follows:
Let $A\_k(q) = \sum\_n a\_{n,k}q^n$ where $a\_{n,k}$ is given by the sum $\sum s\_1 \cdots s\_k$ taken over all ways of writing $k = m\_1s\_1 + \cdots m\_ks\_k$ with $0 < m\_1 < \cdots... | 0 | https://mathoverflow.net/users/1703 | 49270 | 31,024 |
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