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https://mathoverflow.net/questions/45688 | 9 | Hi,
It is well known that the fundamental group of a topological group is abelian, and that every group is the fundamental group of some topological space.
My question is: Does every abelian group arise as the fundamental group of some topological group or are there other restrictions?
| https://mathoverflow.net/users/10217 | Abelian groups as fundamental groups of topological groups | Yes, if $G$ is an abelian group, its classifying space $BG$ is an *abelian* topological group whose $\pi \_1$ is $G$. You can find details in John Baez's wonderful post, hearteningly called "Classifying Spaces Made Easy"
<http://math.ucr.edu/home/baez/calgary/BG.html>
or in the answer by Chris Schommer-Priess to t... | 16 | https://mathoverflow.net/users/450 | 45690 | 28,941 |
https://mathoverflow.net/questions/45687 | 3 | Hi,
Is it true that for every locally compact separable metric space $E$ there exists a sequence $(K\_n)\_{n\in\mathbb{N}}$ of compact subsets of $E$ such that $K\_n\subset\stackrel{\circ}{K\_{n+1}}$ and $\cup K\_n = E$ ?
I’m almost sure this is false but I can’t find a counterexample.
Thank you.
| https://mathoverflow.net/users/10217 | Locally compact separable metric spaces | I think the following argument works under your hypothesis:
Consider $\mathcal{B}=\{B\_n\}$ a countable basis of the topology of $E$ such that $\overline{B\_n}$ is compact for any $n$ (this exists since $E$ is a separable metric space, thus, it has a countable basis and then a basis like this is constructed using loc... | 4 | https://mathoverflow.net/users/5753 | 45691 | 28,942 |
https://mathoverflow.net/questions/45683 | 1 | Hi, I am new to queueing theory. I am interested in a question that I feel should be fairly basic, yet I haven’t really found a clear solution to it. Hopefully somebody here can help me.
We have a single server system, with an infinite queue, and with slotted time. At the beginning of every slot, a number of jobs arr... | https://mathoverflow.net/users/5873 | Stability of discrete queue (new twist) | You are asking the queue $Q\to (Q+Y)^+$ to be ergodic, where $Y$ is your $X-C$, and the stationary distribution of this queue to be integrable. Ergodicity requires that $E(Y)<0$, i.e. $\mu < p$. Integrability holds as soon as $Y$ (or, equivalently, $X$) is square integrable.
Amongst many other places, you might want... | 2 | https://mathoverflow.net/users/4661 | 45701 | 28,947 |
https://mathoverflow.net/questions/45700 | 9 | The de-Rham cohomology ring of U(n) is the exterior algebra generated by the odd-dimensional classes x\_1, x\_3, ..., x\_(2n-1). Moreover, on a Lie group every cohomology class is represented by a unique invariant form (both left and right). I ask two questions:
1) if we represent U(n) with matrices U = [z\_ij], what i... | https://mathoverflow.net/users/10758 | Cohomology of the unitary group | Here's an answer to the first question.
Let $\theta = U^{-1} dU$ be the left-invariant Maurer-Cartan one-form: it is a matrix of one-forms. Then
$$\omega\_{2n+1} = \mathrm{Tr} \theta^{2n+1}$$
is the desired bi-invariant form. Here $\theta^{2n+1}$ stands both for the wedge and matrix product of $\theta$ with itself $2... | 11 | https://mathoverflow.net/users/394 | 45702 | 28,948 |
https://mathoverflow.net/questions/45094 | 3 | Let $u^i\_j$, $i,j = 1, . . . N$, and det$\_q^{-1}$ be the standard generators of the quantum group $U\_q(N,C)$, and define the matrices $U$ and $U^{\ast}$ by setting $U\_{ij} := u^i\_j$ and $U^{\ast}\_{ij}:=(u^j\_i)^{\ast}$. It is "well known" that $U^{\ast}U=UU^{\ast}=1$. How does one prove this?
Moreover, how does... | https://mathoverflow.net/users/2612 | Prove that $U^*U=UU^*=1$ for $U_q(N,C)$ | I assume that $U^\ast\_{ij}=S(u\_j^i)$, where $S$ is the antipode. (See for example the book by Klimyk & Schmüdgen, Section 9.2.4.) If so, then $UU^\ast=1=U^\ast U$ follows from the antipode axiom
$\mu\circ(\mathrm{Id}\otimes S)\circ \Delta =\varepsilon =\mu\circ( S\otimes\mathrm{Id})\circ \Delta$
after applying bo... | 2 | https://mathoverflow.net/users/10756 | 45707 | 28,951 |
https://mathoverflow.net/questions/45697 | 5 | I would like to classify the integers $m \geq 2$ for which the four quadratic polynomials
$3k^2$, $3k^2+2k$, $3k^2+3k+1$, and $3k^2+5k+2$ together represent all integers modulo $m$. That is, every integer modulo $m$ should be in the range of at least one of these polynomials (where all operations are carried out modulo... | https://mathoverflow.net/users/10757 | Four polynomials representing all integers modulo m | For a prime $p>2$, fix a nonsquare $c$. If you find $y$ such that $y/3$ is a non-square (i.e. $y/3=cx^2, x\ne0$) and $y/3 - 1/9 = cz^2, z\ne 0$, then $y$ is not represented by the first two polynomials and I can't be bothered completing the square to write the conditions for the other two. Bottom line is, you find such... | 7 | https://mathoverflow.net/users/2290 | 45709 | 28,952 |
https://mathoverflow.net/questions/45708 | 3 | Apparently there is a notion of for example a $G$-connection on a discrete set. I've understood that this is a standard tool in for example lattice gauge theory. I'm looking for references to learn more about this (i.e. discrete 1-forms, connections, etc.) in the discrete setting.
More specifically, suppose I have a... | https://mathoverflow.net/users/9545 | Discrete G-connections | I'd begin by looking at Oeckl's "Discrete gauge theory":
[http://www.amazon.com/Discrete-Gauge-Theory-Lattices-Tqft/dp/1860945791](http://rads.stackoverflow.com/amzn/click/1860945791)
For a finite group $G$, the notion of a $G$-connection is easy to define; it is usually done when you have not just a graph but a 2-c... | 6 | https://mathoverflow.net/users/10745 | 45712 | 28,954 |
https://mathoverflow.net/questions/45670 | 6 | In group cohomology, for $H$ a finite-index subgroup of $G$ and $M$ a $G$-module, there is a transfer (or corestriction) map $Cor : H^\* (H;M) \to H^\*(G;M)$.
In homotopy theory, there is a transfer map for finite covering spaces $\bar{X} \to X$, and it exists for all coefficient systems on $X$. It is given on homolo... | https://mathoverflow.net/users/318 | Transfer homomorphisms with coefficients | I believe the answer is yes. The idea should be to generalize the construction of the group-theoretic transfer. I'll describe how I think this ought to go. I'm pretty sure what I'm describing is a known construction, but I haven't been able to dig up a reference which describes it.
Let $f:X\to Y$ be a map of spaces (... | 7 | https://mathoverflow.net/users/437 | 45724 | 28,961 |
https://mathoverflow.net/questions/41334 | 6 | This is a fairly minor, technical question, but I'll toss it out in case someone has a good idea on it.
Suppose $(X,<\_X)$ and $(Y,<\_Y)$ are well-founded orderings (not necessarily linearly ordered, though I don't think it matters). Consider the ordering ${<}$ on $X\times Y$ given by $(x',y') < (x,y)$ if $x'\leq x$ ... | https://mathoverflow.net/users/8991 | Symmetric Proof that Product is Well-Founded | I've found a symmetric proof.
First, every well-founded relation admits an ordinal rank function, an assignment of points to ordinals that respects the relation. For example, in your case $\alpha\_x=\sup\{\ \alpha\_u+1\mid u\mathrel{\lt\_X} x\ \}$ is the canonical rank function for $X$ and $\beta\_y=\sup\{\ \beta\_w... | 3 | https://mathoverflow.net/users/1946 | 45725 | 28,962 |
https://mathoverflow.net/questions/45680 | 3 | How can I prove that the Cartier dual of αp is again αp (using the Yoneda lemma)? It should be something like $\alpha\_p(R) \to (\alpha\_p(R) \to \mu\_p(R)),x \mapsto (y \mapsto exp\_{p−1}(x+y)$, where $exp\_{p−1}$ is the truncated exponential sequence. My problem is that this isn't a homomorphism.
| https://mathoverflow.net/users/nan | Cartier dual of \alpha_p | It is probably a bad idea to try to compute the Cartier dual but better to let
Cartier do that for you... If $G$ is a flat commutative finite group scheme with
affine algebra, the commutative and cocommutative $A$ which is the flat over the
base ring $R$. Then the Cartier dual is the spectrum of the dual Hopf algebra
$... | 7 | https://mathoverflow.net/users/4008 | 45727 | 28,964 |
https://mathoverflow.net/questions/45738 | 1 | Given two smooth projective surfaces $X$ and $Y$ over some algebraically closed field.
Given a torsion free coherent sheaf $M$ on $X$. One has the projections $\pi\_X$ and $\pi\_Y$ from the product $X\times Y$. Then we have $\pi\_X^{\\*}M$ on $X\times Y$.
Question: Is $\pi\_X^{\\*}M$ flat over $Y$?
One has to show ... | https://mathoverflow.net/users/3233 | Are pullbacks from a factor of a product scheme flat over the other factor? | Certainly it is flat, and you don't even have to assume that $M$ is torsion free. Indeed, the question is local, so you can assume that $X$ and $Y$ are affine, so the question is: given $k$-algebras $A$ and $B$ and an $A$-module $M$ check that $M\otimes\_A(A\otimes\_k B)$ is flat over $B$. But this is easy --- $M\otime... | 3 | https://mathoverflow.net/users/4428 | 45741 | 28,968 |
https://mathoverflow.net/questions/45613 | 3 | Can Anyone prove the following conjecture?
Consider $k$ rational function vectors $V\_1(x\_1,\cdots,x\_n),\cdots,V\_k(x\_1,\cdots,x\_n)$. They are called \textbf{linearly dependent} if there exists rational functions $\alpha\_1(x\_1,\cdots,x\_n),\cdots,\alpha\_k(x\_1,\cdots,x\_n)$ which are not identically zero such ... | https://mathoverflow.net/users/10735 | Rank-1 decomposition conjecture for matrix with linear function elements | The answer is **NO**. Take $l=m=3$ and the generic skew-symmetric matrix
$$A:=\begin{pmatrix} 0 & x & y \\\\ -x & 0 & z \\\\ -y & -z & 0 \end{pmatrix}.$$
The rank is $k=2$, yet it cannot be written $A=A^1+A^2$ where both $A^j$ would be linear in the indeterminates $x,y,z$ and rank-one.
>
> Sketch of the proof. Such... | 2 | https://mathoverflow.net/users/8799 | 45743 | 28,969 |
https://mathoverflow.net/questions/45432 | 2 | I am looking for a comprehensible material covering Plancherel formula for $SL(n,\mathbb{R})$ and $SL(n,\mathbb{C})$. Of course, I wouldn't mind reading an explanation for general semisimple Lie groups.
(I am reading Varadarajan's Introduction to harmonic analysis on semisimple Lie groups and I'm a bit lost there.)
... | https://mathoverflow.net/users/6818 | Plancherel formula for special linear group | Knapp's book *Representation theory of semisimple groups : an overview based on examples* has a chapter on various versions of the Plancherel theorem. It is also done (very carefully, with all the details, I suppose) in volume 2 of Wallach's *Real Reductive Groups*.
| 3 | https://mathoverflow.net/users/9962 | 45744 | 28,970 |
https://mathoverflow.net/questions/45735 | 6 | To a (projective smooth) algebraic surface $S$ over an algebraically closed field and a divisor $D$ of $S$, we can associate $n= \dim |D|$ and $g$, the genus of a generic member of $|D|$. I would like to fix $g$ and vary $S,D$ so as to make $n$ as large as possible. Is $n$ unbounded and, if not, what is the optimal bou... | https://mathoverflow.net/users/2290 | Maximal dimension of linear system of curves of fixed genus on a surface | $n$ is unbounded, as it is shown by the following example.
EDIT: my example did not work, as pointed out by quim in the comments.
The example he suggests however works: take $S$ the blowup of $P^2$ at a point $x$ and $D$ the strict transform of a curve of degree $d$ with a singular point of multiplicity $d-1$ at $x$.... | 11 | https://mathoverflow.net/users/10610 | 45749 | 28,973 |
https://mathoverflow.net/questions/20771 | 12 | **Background:**
Let $G$ be a profinite group. If $M$ is a discrete $G$-module, then $M=\varinjlim\_U M^U$, where the direct limit is taken with respect to inclusions over all open normal subgroups of $G$, and one naturally has $H^n(G,M)\simeq\varinjlim H^n(G/U,M^U)$, where the cohomology groups on the right can be re... | https://mathoverflow.net/users/4351 | In what sense (if any) does the cohomology of profinite groups commute with projective limits? | Hi Keenan,
You're right that the projective limit of discrete $G$-modules is not necessarily discrete. To take the cohomology of such "topological $G$-modules" you can use continuous cochain cohomology and this continuous cochain cohomology commutes with inverse limits under certain conditions. See section 7 of chapt... | 3 | https://mathoverflow.net/users/10766 | 45751 | 28,975 |
https://mathoverflow.net/questions/45748 | 0 | Here are some direct questions at the interface of algebraic and differential geometry:
(1) Is there an easy characterisation of those affine algebraic varieties which are Kahler?
(2) Is there an easy characterisation of those affine algebraic varieties which are symmetric spaces?
(3) Is there an easy characteris... | https://mathoverflow.net/users/1867 | What are the Compact Symmetric Kahler Algebraic Varieties? | If we ignore the trivial case of the affine line, then irreducible symmetric spaces come in pairs compact - non-compact. The compact ones are naturally projective varieties, while the non-compact ones are affine varieties. Thus question (4) is problematic, unless you mean "locally symmetric" or a more general notion of... | 1 | https://mathoverflow.net/users/9927 | 45753 | 28,977 |
https://mathoverflow.net/questions/45730 | 1 | Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there exists a matrix in $G$ that maps $u$ to $v$. Are there any non-trivial examples of such a $G$?
| https://mathoverflow.net/users/3960 | Transitive Semigroups of $2\times 2$ matrices | Here is a complete answer:
*Every semigroup* $S$ *of invertible* $2\times 2$\*-matrices which is transitive on\* $\mathbb R^2$ *is either conjugate to* $SO\_2(\mathbb R) \times \mathbb R^+$ *or* $SO\_2(\mathbb R) \times \mathbb R$ *or it is a product of* $SL\_2(\mathbb R)$ *and a multiplicative subgroup of* $\mathbb ... | 3 | https://mathoverflow.net/users/9927 | 45757 | 28,978 |
https://mathoverflow.net/questions/45731 | 8 | The concept of a subobject classifier is of course standard and ubiquitous. But is there any nontrivial example of an unrestricted slice classifier?
Specifically, what I mean by this is, is there any non-preorder category with pullbacks with a morphism m into an object X such that ALL other morphisms can be taken as ... | https://mathoverflow.net/users/3902 | Is it possible for a nontrivial category to have a slice classifier? | It looks like such categories may be rather easy to construct. The following example should give the general idea: take the category of sets $V\_\alpha$ of cardinality less than or equal to $\alpha$, for some infinite cardinal $\alpha$. The morphism classifier will be the set $C$ of cardinals up to and including $\alph... | 7 | https://mathoverflow.net/users/2926 | 45768 | 28,982 |
https://mathoverflow.net/questions/45715 | 10 | Let $I\subseteq{\mathbb C}[X\_1,\dotsc,X\_n]$ be an ideal, and
let $V\subseteq{\mathbb C}^n$ be the corresponding algebraic set
($V$ consists of those $x$ at which all $f\in I$ vanish).
Is it true that then there exists an integer $N$ such that:
if a function $f\in{\mathbb C}[X\_1,\dotsc,X\_n]$
and all its par... | https://mathoverflow.net/users/9878 | Strong Nullstellensatz | The answer is yes. Let $I = q\_1 \cap q\_2 \cap \cdots \cap q\_k$ be the primary decomposition of $I$. Let $p\_i$ be the radical $\sqrt{q\_i}$ and let $N\_i$ be large enough that $q\_i \supseteq p\_i^{N\_i}$. I claim that we can take $N= \max(N\_i)$. We'll abbreviate $\mathbb{C}[x\_1, \ldots, x\_n]$ to $A$.
Let $f$ b... | 10 | https://mathoverflow.net/users/297 | 45777 | 28,985 |
https://mathoverflow.net/questions/45782 | 6 | I'm looking for the name of a certain n-category definition. (Someone explained it to me a couple of years ago. I remember the definition, but not the name. Without the name it's difficult to search for a citation. I want the citation in order to explain something we're *not* doing in a paper.)
For background, consid... | https://mathoverflow.net/users/284 | What's the name of this flavor of n-category? | Ronnie Brown has a related idea, contained in this article:
>
> Moore hyperrectangles on a space form a strict cubical omega-category
>
> [arXiv](http://arxiv.org/abs/0909.2212)
>
>
>
discussed briefly [here at the nLab](http://ncatlab.org/nlab/show/Moore+path+category).
If you are instead thinking of a g... | 7 | https://mathoverflow.net/users/4177 | 45783 | 28,987 |
https://mathoverflow.net/questions/45784 | 76 | Suppose $f\_n$ is a sequence of real valued functions on $[0,1]$ which converges pointwise to zero.
1. Is there an uncountable subset $A$ of $[0,1]$ so that $f\_n$ converges uniformly on $A$?
2. Is there a subset $A$ of $[0,1]$ of cardinality the continuum so that $f\_n$ converges uniformly on $A$?
Background: Ego... | https://mathoverflow.net/users/2554 | Does pointwise convergence imply uniform convergence on a large subset? | I did some Googling and came up with something that looks relevant, [Theorem 10](https://books.google.com/books?id=WwmvxtDlz9UC&lpg=PA124&ots=lcdSy9gacd&dq=point%2520set%2520theorem%2520morgan&pg=PA88#v=onepage&q&f=false) quoted below from Morgan's *Point set theory*. It cites works of Sierpiński from the late 1930s, b... | 21 | https://mathoverflow.net/users/1119 | 45786 | 28,989 |
https://mathoverflow.net/questions/45770 | 4 | I encountered this problem in my research and it is turning out to be a surprisingly difficult one(for me, at least).
Suppose we have a univariate nonlinear function $f(x)$ where $x \in [L,U]$. Our goal is to approximate this nonlinear function with $n$ piecewise-continuous linear functions $g\_{i}(x)$ within the gi... | https://mathoverflow.net/users/7851 | Optimal knot placement for fitting piecewise-continuous linear functions to a nonlinear function | In the limit of fine subdivision, the local goodness of fit depends on only two things: the absolute value of the local second derivative, and the local density of knots. For a segment of constant second derivative $a$ over an interval of length $c$, the integrated squared error over the interval comes out to $a^2c^5/1... | 5 | https://mathoverflow.net/users/7936 | 45793 | 28,994 |
https://mathoverflow.net/questions/45776 | 1 | Given a toric ideal, say $J$, in a polynomial ring $k[x\_1,...,x\_n]$ we can find a finite
generating set for $J$. Is it possible, perhaps under additional assumptions on the structure of $J$, to give a finite minimal generating set for $J$ such that every subset of generators also generates a toric ideal.
If not, ... | https://mathoverflow.net/users/10775 | Do subsets of generators of a toric ideal generate a toric ideal? | I believe this is a counterexample. Toric ideals are prime by definition (assuming that I am remembering correctly). Then I think that the ideal of the twisted cubic $C\subseteq \mathbb P^3$ ought to do it.
$J=\langle xz-y^2, xw-yz,wy-z^2\rangle$.
Any two of the three generators will intersect in the union of $C$ a... | 2 | https://mathoverflow.net/users/4 | 45797 | 28,998 |
https://mathoverflow.net/questions/45787 | 10 | An element $F\in \mathbb{C}[[x,y]]$ defines a germ of plane curve.
We assume $F(0,0)=0$.
The multiplicity $mult$ of the germ is defined to be a minimal number $i$
such that $F\in m^i$ where $m=(x,y)$ is the maximal ideal in $\mathbb{C}[[x,y]]$.
Other standard invariants of the germ are Milnor number:
$$
\mu=\dim \mat... | https://mathoverflow.net/users/10578 | Minimum of Milnor number for the curve singularities of fixed multiplicity | As Roy remarked in his answer, the delta invariant of the germ of plane curve singularity $f(x,y)=0$ at $p$ is equal to
$\delta(f) = \sum \frac{m\_q(m\_q-1)}{2}$,
where the sum is extended over all the points $q$ which are "infinitely near" to $p$ and $m\_q$ denotes the multiplicity at $q$.
Then $\delta(f)$ is mi... | 7 | https://mathoverflow.net/users/7460 | 45803 | 29,001 |
https://mathoverflow.net/questions/45802 | 53 | I believe this is the right place to ask this, so I was wondering if anyone could give me advice on research at the undergraduate level.
I was recently accepted into the [McNair Scholars program](http://www.unh.edu/mcnair/). It is a preparatory program for students who want to go on to graduate school. I am expected... | https://mathoverflow.net/users/nan | Undergraduate math research | Since you are a student who's already interested in going on to graduate school and is specifically asking about finding a topic to study at your undergraduate level program at McNair, please *disregard the negative nattering nabobs* whose answers and comments suggest that undergraduates have no place or business in tr... | 45 | https://mathoverflow.net/users/8676 | 45810 | 29,008 |
https://mathoverflow.net/questions/45812 | 8 | I'm on my last semester of a math B.Sc. and about to start studying for a math M.Sc in the same institute.
It now seems like a good time to start thinking of a PhD.
I'm interested in both algebra and algorithms. So I read a little in some computational algebra books (comp. group theory to be precise) and it looks lik... | https://mathoverflow.net/users/nan | Computational algebra: where? | The first piece of advice I would give you is not to prematurely limit yourself to a narrow area, unless your M.Sc only takes a year. If it takes at least two years, then you will still learn plenty of exciting mathematics before you have to make a reasonably definitive choice. If you are thinking of doing a PhD in the... | 16 | https://mathoverflow.net/users/35416 | 45816 | 29,011 |
https://mathoverflow.net/questions/45844 | 23 | The standard proof of the Hahn-Banach theorem makes use of Zorn's lemma. I hear that, however, Hahn-Banach is strictly weaker than Choice. A quick search leads to many sources stating that Hahn-Banach can be proven using the ultrafilter theorem, but I cannot seem to find an actual proof. So...
1. What is the ultrafil... | https://mathoverflow.net/users/8452 | Hahn-Banach without Choice | The ultrafilter theorem is the statement that any filter on a set can be extended to an ultrafilter. It is perhaps more common to see it sated as the (Boolean) Prime ideal theorem: Every Boolean algebra admits a prime ideal.
The Hahn-Banach theorem is actually equivalent to the statement that every Boolean algebra a... | 22 | https://mathoverflow.net/users/6085 | 45846 | 29,031 |
https://mathoverflow.net/questions/45863 | 13 | A topologist came to me with this question, but everything I think should work doesn't.
How many triangulations are there of a polyhedron with n vertices?
By a "triangulation" of a polyhedron P we mean a decomposition of P into 3-simplices whose interiors are disjoint, whose vertices are vertices of P, and whose un... | https://mathoverflow.net/users/1060 | Triangulations of polyhedra | You should read Section 6.1 in the excellent monograph "Triangulations" by De Loera, Rambau and Santos (here is a [slightly dated version](http://www.math.ucdavis.edu/~deloera/BOOK/OLDVERSIONS/april2010.pdf)). It deals with triangulations of cyclic polytopes - exactly the subject of your question. Not only it answers y... | 21 | https://mathoverflow.net/users/4040 | 45865 | 29,045 |
https://mathoverflow.net/questions/45704 | 2 | Suppose we have a PEL type $(H,\phi ,\*;T,O,V)$ where H is a rational nonsplit quaternion algebra, $\phi$is an embedding of Q-algebra $\phi : H-->M(2,R)$, and \* is a positive anti involution of H; O is the maximal order of H , and V level structure . Associate these datum a shimura curve parametric fake elliptic curve... | https://mathoverflow.net/users/3945 | Shimura datum of family of fake elliptic curves | Actually, fake elliptic curves are discussed in Chapter 9 of Lang's Introduction to algebraic and abelian functions.
In order to describe the map to $GSP(4,Q)$, recall that there is the standard antiinvolution $x\to x'$ on the quaternion $Q$-algebra $H$ such that both $tr(x)=x+x'$ and $Norm(x)=xx'=x'x$ are rational ... | 11 | https://mathoverflow.net/users/9658 | 45872 | 29,049 |
https://mathoverflow.net/questions/26821 | 64 | Last year a paper on the arXiv (Akhmedov) claimed that Thompson's group $F$ is not amenable, while another paper, published in the journal "Infinite dimensional analysis, quantum probability, and related topics" (vol. 12, p173-191) by Shavgulidze claimed the exact opposite, that $F$ is amenable. Although the question o... | https://mathoverflow.net/users/6503 | Is Thompson's Group F amenable? | While I did not participate in most of the checking of Shavgulidze's
argument, I can offer the following partial account of the situation. I
am told the paper was correct except for a lemma (or sequence of them)
claiming that a sequence of auxiliary measures had certain properties.
These were Borel measures on the $n$-... | 54 | https://mathoverflow.net/users/10774 | 45891 | 29,064 |
https://mathoverflow.net/questions/45881 | 1 | I guess this is a well known fact/definition for many people. It is mentioned in many places that if $\Gamma$ is a lattice of a vector space(vector bundle/affine bundle) $V$, then there is a dual lattice $\check{\Gamma}$ in $V^\* $ and the torus $V/\Gamma$ has a dual torus $V^\*/\check{\Gamma}$. What does this mean? Wh... | https://mathoverflow.net/users/10799 | What's dual torus and mirror manifold? | The usual answer is that the dual lattice is
$\check{\Gamma}=\{f\in V^\* | f(\gamma)\in \mathbb{Z}\ \forall \gamma \in \Gamma\}$. It is defined for any lattice $\Gamma\subset V$ - no extra information needed.
| 4 | https://mathoverflow.net/users/10745 | 45895 | 29,065 |
https://mathoverflow.net/questions/44420 | 2 | I am reading some aspects of Mirror Symmetry and in mirror symmetry the $N=2$ SCFT on a Calabi Yau Manifold can be divided into two sectors each of which is a topological sigma model, A-Model and B-Model. After some research through some literature about the topological models, it seems that the topological models are ... | https://mathoverflow.net/users/9534 | Are there non-supersymmetric and/or non-Calabi-Yau topological sigma models? | I believe that A-model does not require a Calabi-Yau target space. In fact, A-model is well-defined on any almost complex manifold, which was Witten's original construction (Comm. Math. Phys. Volume 118, Number 3 (1988), 411-449). On the other hand, B-model can only be defined on a Calabi-Yau manifold, which follows fr... | 2 | https://mathoverflow.net/users/7035 | 45897 | 29,067 |
https://mathoverflow.net/questions/45880 | 1 | ### Trying to draw the Amoeba
With Mathematica, it's possible to graph $e^{-k x} + e^{-k y} = 1,e^{-k x} - e^{-k y} = 1$ and $e^{-k x} + e^{-k y} = -1$ to get the amoeba of **1 + x + y** when **k = 1**. Then by plotting when $k \to \infty$, these graphs converge to the tropical polynomial
<http://www.freeimagehost... | https://mathoverflow.net/users/1358 | How to Tropicalize a Polynomial in Two Variables? | For the first amoeba you mentioned, I think your equations should be $e^{-kx}\pm e^{−ky}=\pm e^{-k}$, not $e^{-kx}\pm e^{−ky}=\pm e^{0}$.
For the main question, I think you should be using an equation like $e^{-k}\pm e^{-kx}\pm e^{-ky} \pm e^{-k(x+y+1)} = 0$... so really you want a curve like $c+x+y+cxy$, where when ... | 3 | https://mathoverflow.net/users/2363 | 45899 | 29,069 |
https://mathoverflow.net/questions/45879 | 7 | Let $f\colon \mathbb R^1\to \mathbb R^3$ be a continuous and injective map. Is $\mathbb R^3\setminus f(\mathbb R^1)$ a path-connected space?
| https://mathoverflow.net/users/4298 | Space-discriminating injective curve | Yes. In fact, $\mathbb R^3$ could be any 3-manifold and $f(\mathbb R^1)$ any countable union of embedded segments.
**Lemma 1**. Let $U$ be an open ball in $\mathbb R^3$ and $f:I\to\mathbb R^3$ an embedding. Then $U\setminus f(I)$ is path-connected.
*Proof.* See [Hatcher](http://www.math.cornell.edu/~hatcher/AT/ATpa... | 14 | https://mathoverflow.net/users/4354 | 45902 | 29,071 |
https://mathoverflow.net/questions/45923 | 6 | While skimming the book *Concrete Mathematics*, (edit: first edition) I came across the following problem, which is listed there as a Research Problem: (Chapter 5, Exercise 96)
>
> Is ${2n \choose n}$ divisible by the square of a prime for all $n > 4$.
>
>
>
This problem looked to me much simpler than a divisi... | https://mathoverflow.net/users/8430 | Divisibility of a binomial coefficient by $p^2$ -- current status | This is/was known as the Erdős square-free conjecture, and seems to now be solved. See the bottom of [this page](http://en.wikipedia.org/wiki/Square-free_integer#Erd.C5.91s_Squarefree_Conjecture).
| 13 | https://mathoverflow.net/users/8103 | 45929 | 29,080 |
https://mathoverflow.net/questions/45928 | 27 | Inspired by a recent Math.SE question entitled [Where do we need the axiom of choice in Riemannian geometry?](https://math.stackexchange.com/questions/10102/where-do-we-need-the-axiom-of-choice-in-riemannian-geometry), I was thinking of the [Arzelà--Ascoli theorem](http://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascol... | https://mathoverflow.net/users/4832 | Does Arzelà-Ascoli require choice? | There is a canonical way of checking the literature for most questions of this kind. Since they come up with some frequency, I think having the reference here may be useful.
>
> First, look at "Consequences of
> the Axiom of Choice" by Paul Howard
> and Jean E. Rubin, Mathematical
> Surveys and Monographs, vol 5... | 45 | https://mathoverflow.net/users/6085 | 45931 | 29,082 |
https://mathoverflow.net/questions/45932 | 10 | André and Quillen both gave constructions of the relative cotangent complex for commutative rings, so pretty immediately that gives us that we understand the cotangent complex for affine schemes. Illusie generalized the cotangent complex construction from "rings over A" for a ring A to "rings over $\mathcal{O}\_X$" for... | https://mathoverflow.net/users/1353 | Is Illusie's generalization of the cotangent complex to arbitrary ringed toposes necessary in algebraic geometry? | As BCnrd points out, gluing cotangent complexes is a nontrivial thing. You might still ask whether it is really necessary for Illusie to work in the generality of a ringed topos. Would using a ringed space suffice? For standard deformation problems (deformation of a morphism or deformation of a scheme) working on the u... | 11 | https://mathoverflow.net/users/32 | 45941 | 29,087 |
https://mathoverflow.net/questions/45905 | 13 | If $K$ is a number field, is it always possible to find a finite extension $L/K$ such that $L$ is the composite of fields $L\_1,\ldots, L\_n$, with the property that at most one prime ramifies in $L\_i/\mathbb{Q}$? Equivalently, is the composite of all extensions ramified at $\leq 1$ place all of $\overline{\mathbb{Q}}... | https://mathoverflow.net/users/5513 | Expressing a number field as a composite of extensions ramified at one place | I believe the answer is `No', and Franz Lemmermeyer's example $K=Q(2^{1/3})$ and his strategy of the proof do the trick.
Suppose this particular $K$ is contained in the compositum $F$ of $L\_i$, with every $L\_i$ ramified at only one prime. Assume each $L\_i$ is Galois over $Q$ (otherwise replace it by its Galois cl... | 15 | https://mathoverflow.net/users/3132 | 45949 | 29,093 |
https://mathoverflow.net/questions/45950 | 39 | Occasionally I find myself in a situation where a naive, non-rigorous computation leads me to a divergent sum, like $\sum\_{n=1}^\infty n$. In times like these, a standard approach is to guess the right answer by assuming that secretly my non-rigorous manipulations were really manipulating the Riemann zeta function $\z... | https://mathoverflow.net/users/78 | Is there a "quantum" Riemann zeta function? | The paper by Cherednik [On q-analogues of Riemann's zeta function](https://doi.org/10.1007/s00029-001-8095-6 "Sel. math., New ser. 7, 447–491 (2001). zbMATH review at https://zbmath.org/?q=an:1001.11033") ([arXiv:math/9804099](https://doi.org/10.48550/arXiv.math/9804099)) gives precisely the definition you're after:
$$... | 42 | https://mathoverflow.net/users/2149 | 45958 | 29,098 |
https://mathoverflow.net/questions/45936 | 3 | Hi there,
Assuming X and Y are modal formulae and diamond X is satisfiable and diamond Y is satisfiable, how would one show that they X AND Y is satisfiable?
I don't think it requires much effort?
I think you need to choose one world and one model where X AND Y is true and that would mean it is satisfiable?
So ... | https://mathoverflow.net/users/10814 | Modal logic - satisfiability | Your question as originally written (which Henry correctly diagnosed as problematic in two ways) does not match the more reasonable aim reflected in your comments to Henry's answer. Specifically, your comments make it sound like you want to show that the satisfiability of both $\diamond X$ and $\diamond Y$ implies the ... | 6 | https://mathoverflow.net/users/4137 | 45959 | 29,099 |
https://mathoverflow.net/questions/45871 | 7 | There seem to be intimate connections between the different definitions of von Neumann module. The two that I'm aware of are Hilbert von Neumann modules and correspondences (in the sense of Connes). I was wondering if there is any significant interplay between the two - for instance if any of the ideas around Jones' su... | https://mathoverflow.net/users/10779 | Subfactor theory and Hilbert von Neumann Algebras | Answers: (i) Yes, if we replace states by weights (not every
von Neumann algebra admits a faithful state);
(ii) Yes (for all von Neumann algebras); (iii) All of them.
Suppose M is an arbitrary von Neumann algebra and p≥0 is a real number.
Then we define a right L\_p(M)-module as a right M-module equipped
with an inne... | 7 | https://mathoverflow.net/users/402 | 45964 | 29,102 |
https://mathoverflow.net/questions/45966 | 2 | Hello,
i was reading your article about non metrizability of \*R.
i was able to prove that the interval open topology is not metrizable by proving that the intersection of decreasing hyper-intervals contains an interval. But i do not get how you used this argument for any metric, since we do not know how our nested bal... | https://mathoverflow.net/users/10820 | Metrization of hyperreals | I am not sure whom you are addressing in your question, but
some of your remarks relate to issues brought up at [this
MO
question](https://mathoverflow.net/questions/10870/which-topological-spaces-admit-a-nonstandard-metric). If not, could you let us know to which post you were referring?
It is quite common to consid... | 9 | https://mathoverflow.net/users/1946 | 45972 | 29,109 |
https://mathoverflow.net/questions/45954 | 4 | As far as I understood, there are two way to assign weights to a mixed Hodge complex (a way that does not depend on shifts, and another one that does). There is a clever way to relate these to ways using Deligne's decalage. Does there exist a conceptual explanation of these matters (including decalage)?
| https://mathoverflow.net/users/2191 | Do Deligne's decalage and two filtrations for mixed Hodge complexes have a conceptual explanation? | Unfortunately, I don't have a good conceptual explanation for any of this.
So here's a somewhat more technical explanation of how *decalage* is used in Hodge theory.
The actual business of constructing mixed Hodge structures tends to be quite involved.
In Deligne's approach, the Hodge and weight filtrations $F$, $W$ ar... | 6 | https://mathoverflow.net/users/4144 | 45977 | 29,114 |
https://mathoverflow.net/questions/45953 | 18 | I'm curious about the following:
Is every real $n$-manifold isomorphic to a quotient of $\mathbb{R}^n$?
Thanks.
EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking about topological quotients. Specifically, is there a surjective map $\mathbb{R}^n\to M$ such that $M$ has the quotien... | https://mathoverflow.net/users/2857 | Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$? | Hahn–Mazurkiewicz Theorem: Suppose $X$ is a nonempty Hausdorff topological space.
Then the following are equivalent:
1. there is a surjection $[0,1]\to X$,
2. $X$ is compact, connected, locally connected and second-countable.
It follows that a Hausdorff space satisfying the conditions of (2) is a quotient
of $I = [... | 33 | https://mathoverflow.net/users/3634 | 45980 | 29,116 |
https://mathoverflow.net/questions/45756 | 10 | Considering the success of a [previous question](https://mathoverflow.net/questions/29118/need-help-proving-that-sum-limits-j0k-1-1j1k-j2k-2-binom2k1) involving Eulerian numbers, I thought I might throw this question into the mix. It comes from some localization computations in GW theory, but in this form is purely com... | https://mathoverflow.net/users/6240 | Expressions involving Eulerian numbers of the second kind: trying to show $\sum_{m=0}^{n} (-1)^m(m)m!(2n-m-2)!\left\langle\left\langle n\atop m\right\rangle\right\rangle\neq0$ for even $n$. | Following Mike Spivey's comment above I will consider $B(n)$ in (3). It turns out that your conjecture is true, because for odd $n$ the sum $B(n)$ is a certain weighted $L^2$ norm of the Eulerian polynomial of the second kind of order $\frac{n+1}{2},$ with the sign of $(-1)^{\frac{n-1}{2}}\, . $
The product of the tw... | 12 | https://mathoverflow.net/users/6101 | 45985 | 29,117 |
https://mathoverflow.net/questions/45534 | 9 | Recall that an $R$-algebra $R\to S$ is called formally smooth (resp. formally unramified resp. formally étale) if given any lifting problem of the form
$$\begin{matrix}
R&\to &T\\
\downarrow&{}^?\nearrow&\downarrow\\
S&\to&T/J\end{matrix}$$
where $J$ is a square-zero nilpotent ideal of $T$, there exists at least o... | https://mathoverflow.net/users/1353 | Formally smooth morphisms, the cotangent complex, André-Quillen cohomology, and representability of nilpotent extensions as trivial extensions over a cofibrant replacement | Your question is:
>
> Why, morally, do we need to look at the S-modules over the cofibrant replacement
> of S to capture the lifting data from the rest of the square-zero extensions
> that we would need to characterize formal smoothness (resp. formal étaleness)?
>
>
>
Here's how I think about it, though I ... | 7 | https://mathoverflow.net/users/437 | 45987 | 29,119 |
https://mathoverflow.net/questions/46011 | 16 | Let $K$ be a compact metric space, and $(E,d\_E)$ a complete separable metric space.
Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with
the metric $d(f,g)=\sup\_{k\in K}\ d\_E (f(k),g(k))$. Is the space $C$ separable?
The result is true when $E$ is the real line; this is Corollary 11.2.5 ... | https://mathoverflow.net/users/nan | Is the space of continuous functions from a compact metric space into a Polish space Polish? | Yes, it appears e.g. as Theorem 4.19 in Chapter I of Kechris' *Classical Descriptive Set Theory*. (The relevant page is visible in Google Books if it's not in your library.)
| 13 | https://mathoverflow.net/users/4137 | 46015 | 29,136 |
https://mathoverflow.net/questions/46024 | 2 | It is known that solving systems of linear equations is reducible to SVD in a straightforward way; if you want to solve $\mathbf{Ax}=\mathbf{b}$, then you can perform SVD on $\mathbf{A}$ and minimize $||\mathbf{UDVx}-\mathbf{b}||$.
However, is there a reverse reduction that is also very efficient? That is, if you ca... | https://mathoverflow.net/users/5534 | What is the relationship between singular value decomposition and solving linear systems? | I don't think so. Solving linear equations is an algebraic problem, where the scalar field is arbitrary: $\mathbb R$ or $\mathbb C$, but also $\mathbb Q$, $\mathbb F\_{p^n}$, $k(X)$, $\mathbb Q\_p$, $\mathbb Q(\alpha)$ ($\alpha$ algebraic). In many cases, there is no analogue of SVD at all. This is why Gauss eliminatio... | 6 | https://mathoverflow.net/users/8799 | 46026 | 29,143 |
https://mathoverflow.net/questions/46019 | 20 | Suppose $F: C\to D$ is an additive functor between abelian categories and that
$$0\to X\xrightarrow f Y\xrightarrow g Z\to 0$$
is and exact sequence in $C$. Does it follow that $F(X)\xrightarrow{F(f)} F(Y)\xrightarrow{F(g)} F(Z)$ is exact in $D$? In other words, is $\ker(F(g))=\mathrm{im}(F(f))$?
**Remark 1:** If... | https://mathoverflow.net/users/1 | Is there an additive functor between abelian categories which isn't exact in the middle? | Consider the abelian category of morphisms of vector spaces, i.e., the objects are linear maps $f:U\to V$, and the morphisms are commutative squares. Let the functor $Im$ assign to a morphism $f$ its image $Im(f)$. Consider the short exact sequence of morphisms $(0\to V)\to (U\to V)\to (U\to 0)$. The functor $Im$ trans... | 28 | https://mathoverflow.net/users/2106 | 46035 | 29,148 |
https://mathoverflow.net/questions/45716 | 7 | Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace $V$ of the even, smooth functions on the interval $[-1,1]$.
Can you construct a (bounded) operator $D:\mathcal{S}(\m... | https://mathoverflow.net/users/10400 | A Question concerning the Fourier Transform of $\mathbb{R}$ | You don't need to do things the rough way; there is enough freedom for the smooth approach.
Take any even $C\_0^\infty$ descent $\Phi$ from $[-1,1]$ and define $Pf=\Phi f$ and $Qf=\mathcal F^{-1}(\Phi\mathcal F f)$. Now take the standard $D=I-(I-PQ)^{-1}(1-P)$. This works in $L^2$ for the same reason as it does with ... | 5 | https://mathoverflow.net/users/1131 | 46045 | 29,156 |
https://mathoverflow.net/questions/46044 | 0 | Hi -- is there any analogue or adjustment of, say, Schwartz Bayesian (or other) information criterion that would be applicable to model selection in ridge regression with a given ridge parameter $\eta$, i.e. $\hat{\beta}\_\eta = (X'X+\eta I)^{-1}X'Y$?
Thanks!!
| https://mathoverflow.net/users/10837 | Information criteria for ridge regression | The ridge estimator corresponds to the posterior mean under a Normal linear regression model with a conjugate Normal-inverse-gamma prior on the regression coefficients: $\beta \mid \sigma^2, \lambda \sim \mbox{N}(0, \lambda^{-1}\sigma^2 \mbox{I})$ and $\sigma^2 \sim \mbox{IG}(a,b)$ for known hyperparameters $a$ and $b$... | 1 | https://mathoverflow.net/users/8719 | 46053 | 29,160 |
https://mathoverflow.net/questions/46046 | 4 | Let $k$ be a positive integer. Is it true that any finite group $H$ of cardinal $4k+2$ whose center contains an element $h$ of order $2$ is isomorphic to the direct product $H=(\mathbb{Z}/2\mathbb{Z})\times G$, where $G=H/\{1,h\}$?
An equivalent statement would be: Let $G$ be a finite group of odd cardinal. Is it tr... | https://mathoverflow.net/users/10675 | A question about finite groups. | There is a reasonably simple argument that any group $H$ of twice odd order has a normal subgroup $N$ of index 2. Given that, if you know also that $H$ has a central subgroup $Z$ of order 2, then it is straightforward to show that $H \cong N \times Z$.
The argument goes like this. By Cayley's Theorem, there is an iso... | 11 | https://mathoverflow.net/users/35840 | 46056 | 29,162 |
https://mathoverflow.net/questions/45831 | 33 | This is an excellent question asked by one of my students. I imagine the answer is "no", but it doesn't strike me as easy.
Recall the set up of the [stable marriage problem](http://en.wikipedia.org/wiki/Stable_marriage_problem). We have $n$ men and $n$ women. Each man has sorted the women into some order, according t... | https://mathoverflow.net/users/297 | Can assignment solve stable marriage? | David's question is sufficiently more precise than the question of Donald Knuth referred to in the answer below. I believe it can be answered in the negative for $n \geq 3$, as follows.
Let $\{m\_1,m\_2,\ldots,m\_n\}$ and $\{w\_1,w\_2,\ldots,w\_n\}$ be the vertices of the parts of our graph $K\_{n,n}$. Consider a cho... | 19 | https://mathoverflow.net/users/8733 | 46059 | 29,164 |
https://mathoverflow.net/questions/46068 | 104 | Let $n$ be a large natural number, and let $z\_1, \ldots, z\_{10}$ be (say) ten $n^{th}$ roots of unity: $z\_1^n = \ldots = z\_{10}^n = 1$. Suppose that the sum $S = z\_1+\ldots+z\_{10}$ is non-zero. How small can $|S|$ be?
$S$ is an algebraic integer in the cyclotomic field of order $n$, so the product of all its Ga... | https://mathoverflow.net/users/766 | How small can a sum of a few roots of unity be? | In this paper they talk about this problem for 5 instead of 10 roots.
<http://www.jstor.org/stable/2323469>
EDIT: In view of Todd Trimble's comment, here's a summary of what's in the paper.
Let $f(k,N)$ be the least absolute value of a nonzero sum of $k$ (not necessarily distinct) $N$-th roots of unity. Then
... | 61 | https://mathoverflow.net/users/10769 | 46069 | 29,169 |
https://mathoverflow.net/questions/43267 | 13 | I am reading the Witten's topological twisting for $N = 2$ Superconformal Field Theory(SCFT) <http://arxiv.org/abs/hep-th/9112056>
In this paper Witten constructed 2 TQFTs i.e. A-model and B-model from an $N=2$ SCFT with a Kahler target manifold. My queries are the following :
1. When we do the twist, we're actually ... | https://mathoverflow.net/users/9534 | Witten's topological twisting | First, a historical note: the twisting procedure for $N=2$ SCFTs is due to [Eguchi and Yang](http://inspirebeta.net/record/296815); although a twisting of sorts had already appeared in Witten's Topological Quantum Field Theory paper of 1988.
Let me give quick answers to your questions:
1. Twisting *per se* does not... | 12 | https://mathoverflow.net/users/394 | 46071 | 29,171 |
https://mathoverflow.net/questions/46028 | 3 | A category is called **well-powered** if the partial order of subobjects of any object is a set. Is there a German translation which is well-established in the literature? I don't want to use the English term in a German text.
| https://mathoverflow.net/users/2841 | German translation for "well-powered category" | Pareigis and Schubert both use "lokal klein" in their textbooks.
Of course there is a clash with the other meaning of "locally small" (i.e. that
Hom(a,b) is always a set), but this is usually satisfied by their definitions
of category.
| 3 | https://mathoverflow.net/users/10840 | 46072 | 29,172 |
https://mathoverflow.net/questions/46079 | 4 | So I just started learning about quasicategories... Alright, that's an understatement: I just listened to Julie Bergner talk about quasicategories, and then started reading Moritz Groth's short course about them. I did a quick google search for this question, but there's a high probability that I just wasn't using the ... | https://mathoverflow.net/users/6936 | Do quasi-categories have a `completion'? | Maybe this is not exactly what you are looking for, but it might be a place to start.
There is an analog of the Yoneda embedding for quasi-categories which takes the form $$j : \mathcal C \rightarrow Map(\mathcal C^{op}, \mathcal S)$$ where here $\mathcal S$ is the quasicategory of spaces (that is, the coherent nerve... | 5 | https://mathoverflow.net/users/4466 | 46085 | 29,181 |
https://mathoverflow.net/questions/46094 | 7 | Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Is the Hilbert space $L^2(H,\gamma)$ separable?
| https://mathoverflow.net/users/238 | If $H$ is a separable Hilbert space, is $L^2(H)$ separable? | By Example 7.14.13 in Volume 2 of Bogachev's *Measure Theory*, every Radon measure on $H$ is separable, so that $L^2(H,\gamma)$ is also separable. It is not necessary that $H$ is a Hilbert space, just that every compact subset of $H$ be metrizable.
| 14 | https://mathoverflow.net/users/nan | 46097 | 29,191 |
https://mathoverflow.net/questions/46022 | 11 | I just found an English translation of Serre's FAC at Richard Borcherds' Algebraic Geometry course web page. I really want to read it sometime. I am beginner in Algebraic Geometry, just started learning Scheme theory (Hartshorne Ch II). My question is :
Shall I read coherent sheaves and the cohomology from the transl... | https://mathoverflow.net/users/9534 | Serre's FAC versus Hartshorne as an introduction to sheaves in algebraic geometry | As always, the source you use may be related to what your goals are. To give some perspective, recall there are several ways to define sheaf cohomology, and Serre and Hartshorne feature different methods. Serre used Cech cohomology, and there the important long exact sequence property does not always hold. He was able ... | 18 | https://mathoverflow.net/users/9449 | 46099 | 29,192 |
https://mathoverflow.net/questions/46087 | 28 | The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are monomorphisms, and the fibrations are Kan fibrations.
Simplicial sets are combinatorial objects, so morally their model structure ... | https://mathoverflow.net/users/1709 | Model structure on Simplicial Sets without using topological spaces | Quillen's original proof (in *Homotopical Algebra*, LNM 43, Springer, 1967) is purely combinatorial (i.e. does not use topological spaces): he uses the theory of minimal Kan fibrations, the fact that the latter are fiber bundles, as well as the fact that the classifying space of a simplicial group is a Kan complex. Thi... | 30 | https://mathoverflow.net/users/1017 | 46112 | 29,200 |
https://mathoverflow.net/questions/46061 | 5 | I'm searching for a translation for the term "intertwiner" in German.
| https://mathoverflow.net/users/10400 | What is the German translation for intertwiner? | I (native speaker) learned the term "Vertauschungsoperator" in my undergraduate courses. Unfortunately I cannot cite any reference right now, except the fact that the lecturer of the courses, Prof H.S. Holdgruen, is very sensible in his use of the german language. Therefore I estimate the probability very high that thi... | 11 | https://mathoverflow.net/users/43085 | 46118 | 29,203 |
https://mathoverflow.net/questions/46104 | 1 | Consider an entire function $f : \mathbb{C} \rightarrow \mathbb{C}$! We search the function
$$ g: (a,b) \rightarrow \mathbb{C},$$ which solves the following equation locally: $g'(t)=f(g(t))$ and $g(0)=f(x\_0)$.
I can compute the inverse $G$ of $g$, if $f(x\_0) \neq 0$, i.e.
$$ G(y) = \int\limits\_{f(x\_0)}^y \frac{d ... | https://mathoverflow.net/users/10400 | A simple ordinary differential equation | It is hard to guess what you are looking for. Take the apparently simpler case where $f$ is a polynomial, say of degree $d$. If $d = 1$ you have an explicit solution in terms of the exponential function (because your $G$ is logarithmic). If $d = 2$ the solution can be written in terms of trigonometric functions. If $d ... | 2 | https://mathoverflow.net/users/7311 | 46119 | 29,204 |
https://mathoverflow.net/questions/46106 | 9 | Let $\mathfrak{M}\_g$ denote the moduli stack of Riemann surfaces of genus $g$, it is a smooth complex analytic stack, and is the analytic stack underlying $\mathsf{M}\_g$, the moduli stack of complex algebraic curves of genus $g$. Alternatively, it is the quotient stack $\mathcal{T}\_g // \Gamma\_g$ of the mapping cla... | https://mathoverflow.net/users/318 | Picard group of $\mathfrak{M}_g$ | This will be a bit sketchy, but hopefully you can fill in the necessary steps.
If you see a problem, let me know.
To avoid getting distracted with stacks, let me use a level $n\ge 3$ structure, and
write $M=M\_{g}[n]$. Let $j:M\hookrightarrow \bar M$ be the Satake (not Deligne-Mumford) compactification.
This is the n... | 9 | https://mathoverflow.net/users/4144 | 46128 | 29,209 |
https://mathoverflow.net/questions/46124 | 3 | Let $H$ be an infinite-dimensional, separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Let $H^\*$ denote the space of continuous linear functionals on $H$. By the Riesz representation theorem, $$H^\* = \{ \langle k, \cdot \rangle : ... | https://mathoverflow.net/users/238 | If $H$ is a separable Hilbert space, is its dual dense in $L^2(H)$? | (Rewritten to give an answer more useful to future visitors.)
First of all, as noted in comments, there is no (countably additive) Gaussian measure on $H$ with covariance operator the identity.
However, if we take $\gamma$ to be some other Gaussian measure, the answer is no, $H^\*$ is not dense in $L^2(H, \gamma)$... | 7 | https://mathoverflow.net/users/4832 | 46135 | 29,213 |
https://mathoverflow.net/questions/46133 | 12 | This question was asked on NMBRTHRY by Kurt Foster:
If $p$ is a prime number and $\mathbb{F}\_p$ the field of $p$ elements, the zeroes of the Artin-Schreier polynomial
$x^p - x - 1 \in \mathbb{F}\_p[x]$
obviously have multiplicative order dividing $1 + p + p^2 + \dots + p^{p-1} = (p^p - 1)/(p-1)$ (express the nor... | https://mathoverflow.net/users/2784 | Multiplicative order of zeros of the Artin-Schreier Polynomial | I've never seen it ascribed to Shafarevich, but it is an old question. As a question, it equivalent to determining the period mod p of the sequence of Bell numbers discussed, e.g. in:
Levine, Jack; Dalton, R. E.
Minimum periods, modulo p, of first-order Bell exponential integers.
Math. Comp. 16 1962 416–423.
But t... | 10 | https://mathoverflow.net/users/2290 | 46137 | 29,214 |
https://mathoverflow.net/questions/46136 | 3 | An object $M$ of an abelian category is called of finite type iff for every directed set of subobjects $M\_i$ of $M$ whose sum is $M$ there exists some $i$ with $M = M\_i$. Is the direct sum $M \oplus N$ of two objects $M,N$ of finite type again of finite type?
So let $P\_i \subseteq M \oplus N$ be a directed set of ... | https://mathoverflow.net/users/2841 | $M \oplus N$ is of finite type if $M,N$ are of finite type? | At least if we have a Grothendieck category everything seems OK: Suppose
$0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow0$ is exact with $M'$ and
$M''$ of finite type. Assume $\{M\_i\}$ is a directed collection of subobjects of
$M$ such that $\sum\_i M\_i=M$. We then have $M'=M'\bigcap\sum\_i M\_i=\sum\_i
M'\bi... | 3 | https://mathoverflow.net/users/4008 | 46148 | 29,217 |
https://mathoverflow.net/questions/46155 | 3 | Let $X$ be an n point finite set equipped with a metric $d$. An isometry is a map $\varphi$ $:X \mapsto X$ satisfying that $d(\varphi(x),\varphi(y))=d(x,y)$ for any $x,y \in X$. Identify $X$ with the set {$1,2,...,n $}. We can say every isometry is an element of $S\_n$(the permutation group of n elements) and the isome... | https://mathoverflow.net/users/7360 | A question about the isometry group of a finite metric space | As secretman says, the condition that there exist $i$ and $j$ in $X$ such that no isometry carries $i$ to $j$ is precisely to say that the action of the isometry group on $X$ is not *transitive*. If that's really your question, that seems to be all that can be said, and I doubt there any books on the subject.
One mig... | 4 | https://mathoverflow.net/users/1149 | 46169 | 29,226 |
https://mathoverflow.net/questions/46156 | 22 | Let $F\subset\mathbb{Q}^2$ a closed set. Does there exists some closed and connected set $G\subset\mathbb{R}^2$ such that $F=G\cap\mathbb{Q}^2$?
For example if $F=\{a,b\}$, you can take $G$ the reunion of two lines of different irrational slopes passing through $a$ and $b$. This is a connected set and the intersectio... | https://mathoverflow.net/users/10217 | Is every closed set of Q² the intersection of some connected closed set of R² with Q² | Enumerate all rational points outside your set. Then cover these points by open balls by induction as follows: the next ball is centered at the first rational point not covered so far, its radius is so small that is does not intersect $F$ and the previous balls and is chosen so that the boundary of the ball does not co... | 45 | https://mathoverflow.net/users/4354 | 46174 | 29,229 |
https://mathoverflow.net/questions/46116 | 18 | I have been reading "The Geometry of Schemes" by Eisenbud and Harris and have a question about Exercise III-43. There, one should show that there is a bijection between the sets
$\{(n+1)\mbox{-tuples of elements of }A\mbox{ that generate the unit ideal }\}$
and
$\{ \mbox{maps} \mbox{ Spec} A \to \mathbb{P}^n\_A$ such... | https://mathoverflow.net/users/10844 | A-valued points of projective space | Yes, you (and BCnrd) are absolutely correct and the quoted statement is wrong.
Over any scheme $S$, the $S$-points of $\mathbb P^n$ are the surjections $\mathcal O\_S^{\oplus n+1} \to F$ with invertible $\mathcal O\_S$-module $F$. More generally, the $S$-points of the grassmannian $Gr(m,k)$ are the surjections $\math... | 12 | https://mathoverflow.net/users/1784 | 46175 | 29,230 |
https://mathoverflow.net/questions/46176 | 10 | I'm wondering if finite unramified morphism between reduced schemes decomposes as closed immersions and etale morphisms. Suppose I have a morphism between reduced schemes which is finite, surjective and unramified, is it necessarily etale? I think this is certainly true if both source and target are curves, but I'm not... | https://mathoverflow.net/users/1657 | What are unramified morphisms like? | Finite, surjective, and unramified does not imply etale. E.g. suppose that $Y$ is a proper closed subscheme of $X$, and we consider the map $X \coprod Y \to X$ defined as the disjoint union of the identity on $X$, and the given closed immersion $Y \to X$
on $Y$.
Then this map is finite, unramified, and surjective, bu... | 15 | https://mathoverflow.net/users/2874 | 46179 | 29,232 |
https://mathoverflow.net/questions/46181 | 3 | The following is a corollary of the Briançon-Skoda theorem:
If $R$ is a regular Noetherian ring of Krull dimension $d$ and $f\_1,f\_2,...,f\_{d+1}\in R$. Then, $f\_1^df\_2^d...f\_{d+1}^d \in (f\_1^{d+1},f\_2^{d+1},...,f\_{d+1}^{d+1})R$
Now, given $R=k[[x\_1,...,x\_d]]$ where $k$ is a field, then $R$ is a regular l... | https://mathoverflow.net/users/10775 | About a corollary of the Briançon-Skoda theorem | It depends on what do you mean by "direct". There is no elementary proof, as far as I known, even for polynomials when $d=2$, see page 8 of this [note](http://www.math.lsa.umich.edu/%7Ehochster/balt.ps). When $d=1$, the statement is an easy exercise: one can write $f\_1= da, f\_2=db$ with $(a,b) = R$ since $R$ is an UF... | 3 | https://mathoverflow.net/users/2083 | 46190 | 29,238 |
https://mathoverflow.net/questions/46196 | 3 | I just recently learned the Ham Sandwich Theorem in my algebraic topology class. If we take the measure to be the counting measure and let $n=2$, then the theorem tells us that given a set of black and white points in the plane, we can draw a line that'll divide the plane so that there is an equal number of black and w... | https://mathoverflow.net/users/8162 | Algorithm for Ham Sandwich with Points | The paper by
Lo, Matoušek, and Steiger entitled *"Algorithms for Ham-Sandwich Cuts"* gives an $O(n)$ algorithm, where $n$ is the number of points. That's the best you can do, since you need to consider all such points.
| 3 | https://mathoverflow.net/users/491 | 46200 | 29,242 |
https://mathoverflow.net/questions/46183 | 0 | Suppose X is a pure dimensional projective complex scheme, reducible and non-reduced but without embedded components of lower dimension. Let $X=\cup X\_i$ be the decomposition such that $X\_i$ is set-theoretically irreducible and $\dim(X\_i\cap X\_j)<\dim(X\_i)$.
Let $F\_X$ be a torsion free sheaf of $\mathcal{O}\_X... | https://mathoverflow.net/users/2900 | How the multi-rank of a torsion free sheaf on a non-reduced scheme is defined? | I don't know about commonly accepted, but here is one definition of rank (not multi-rank) that appears in theory of [stable sheaves](http://www.numdam.org/item?id=PMIHES_1994__79__47_0). Fix an ample line bundle $L$ and consider the Hilbert polynomial $P\_{L}(F, t)$, defined by
$$
P\_{L}(F, n) = \chi(F \otimes L^{\... | 2 | https://mathoverflow.net/users/5337 | 46201 | 29,243 |
https://mathoverflow.net/questions/46149 | 10 | Following along a similar line to the question asked here: [Is there an explicit bound on the number of tetrahedra needed to triangulate a hyperbolic 3-manifold of volume V?](https://mathoverflow.net/questions/38082/is-there-an-explicit-bound-on-the-number-of-tetrahedra-needed-to-triangulate-a-hy)
Let $K$ be a (hyper... | https://mathoverflow.net/users/3121 | Lower bound on number of tetrahedra needed to triangulate a knot complement | **Revision**
Let $t(K)$ be the minimal number of tetrahedra needed to triangulate a knot complement. Let $c(K)$ be the minimal crossing number of a knot.
As Ryan Budney points out in the comments, $t(K)\leq C c(K)$ for some constant $C$. One may show that this lower bound is optimal, indirectly using a result of La... | 13 | https://mathoverflow.net/users/1345 | 46204 | 29,245 |
https://mathoverflow.net/questions/46143 | 10 | We approach the problem of finding a metric of constant curvature on a surface (i.e. a $C^\infty$ 2-manifold). Specifically, what we want to do is, given a surface $M$ and a metric $g\_0$, show that there exists a new metric $g$ of the form $g=e^{2u}g\_0$ for some $u\in C^\infty (M)$ such that $g$ has constant curvatur... | https://mathoverflow.net/users/10850 | Finding constant curvature metrics on surfaces for the case of positive Euler characteristic | If you like the case $K=-1$ better, one way to do this is to choose 3 points $\{x,y,z\} \subset S^2$, and use the uniformization theorem to find a complete conformally equivalent metric on $P=S^2-\{x,y,z\}$ with constant curvature $K=-1$. There is a unique such metric on $P$, which is conformally equivalent to $\mathbb... | 8 | https://mathoverflow.net/users/1345 | 46205 | 29,246 |
https://mathoverflow.net/questions/46138 | 16 | Every book which treats dual spaces of normend spaces states that $(c\_0)' = \ell^1$ and $(\ell^1)' = \ell^\infty$ and some also describe $(\ell^\infty)'$.
However, is anything known about higher order duals in general? Does taking the normed dual of a given normed space $X$ stabilize? In other words: Defining $X^{(1... | https://mathoverflow.net/users/9652 | Does "taking the dual space" stabilize? | As noted previously, we can restrict our attention to Banach spaces.
A Banach space $X$ is reflexive if the canonical embedding into its double-dual is onto.
Now let $X$ be any Banach space. We define $X^{(\alpha)}$ for all ordinals $\alpha$ as follows:
$X^{(0)}=X$, $X^{(\alpha+1)}=(X^{(\alpha)})^\prime$, and for... | 16 | https://mathoverflow.net/users/7743 | 46210 | 29,250 |
https://mathoverflow.net/questions/46207 | 7 | Let $X$ be a smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete linear systems associated to powers of the canonical class $nK\_{X}$. I would like to understand some examples of how intrica... | https://mathoverflow.net/users/5124 | Basepoints in the canonical system of algebraic surfaces | In the general case when $n=\dim X$ is arbitrary
Kollár proves an analogue for minimal varieties of general type, although the bound is much worse than in the surface case.
In the situation at hand it implies that $\vert ((n+3)!)K\_X\vert$ is basepoint-free (actually one can write $(n+2)(n+2)!$ instead of $(n+3)!$, but... | 8 | https://mathoverflow.net/users/10076 | 46213 | 29,251 |
https://mathoverflow.net/questions/41386 | 4 | Can anybody give a definition of the equalizer completion of a cartesian category?
Is the method to get more or less as the regular and exact completions in the way that are given in: <http://ncatlab.org/nlab/show/regular+and+exact+completions>?
How is in that case the behaviour of the forgetful functor FL-->FP (... | https://mathoverflow.net/users/3338 | Equalizer completion | There are general results about how to freely add limits or colimits to a category. They are formally dual, but people normally state the colimit variety because they involve the category of presheaves.
To freely add some class of colimits to a given category $C$, you form the closure of the representables in the pr... | 12 | https://mathoverflow.net/users/10862 | 46214 | 29,252 |
https://mathoverflow.net/questions/46185 | 2 | The nodes of a surface are special cases of more general singularities. For example, the Cayley cubic has four nodes.
The full set of singularities of a surface can be characterized by finding all points where the partial derivatives are all zero. However, not all singularities are nodes. Some are cusps or other kind... | https://mathoverflow.net/users/10859 | How can I compute the full set of nodes of a surface? | A node (as in Cayley's surface) is a double point with nondegenerate tangent cone.
To check whether a given point on a surface in A^3 is a node in this sense, change coordinates so that it is the origin and write the equation as 0=F\_2+F\_3+... with F\_i homogeneous of degree i. The point is a node iff F\_2 is irreduci... | 4 | https://mathoverflow.net/users/1939 | 46216 | 29,254 |
https://mathoverflow.net/questions/46217 | 3 | A colleague of mine recently asked me if this set family had a name (see definition of *this* below) . I didn't know the answer, so I thought I would consult the MO oracle.
Let $\mathcal{S}:=\{ S\_1, \dots, S\_k \}$ be a family of subsets of $[n]$. Consider the family $\mathcal{F}\_{\mathcal{S}}$ formed by taking all... | https://mathoverflow.net/users/2233 | Does this set family have a name? | I am not sure how standard this is, but It makes sense to call this family the [atoms](http://en.wikipedia.org/wiki/Atom_(order_theory)) of the corresponding lattice of sets obtained from $\mathcal S$. They can be illustrated as the different regions in a Venn diagram.
| 4 | https://mathoverflow.net/users/2384 | 46218 | 29,255 |
https://mathoverflow.net/questions/46224 | 2 | Is it possible to find an example of an $\mathbb{R}$-Cartier divisor $D$ on an irreducible variety $X$ that is non-trivial, nef, effective and numerically rigid?
By "numerically rigid" I mean that if $E$ is another $\mathbb{R}$-Cartier effective divisor such that $E$ is numerically equivalent to $D$ then $D=E$.
Fo... | https://mathoverflow.net/users/7276 | Numerically rigid nef divisor | Take a minimal surface $S$ of general type with $p\_g=1$, $q=0$ and zero torsion.
Then $S$ contains a unique effective canonical curve $K$, which is nef and numerically rigid.
In fact, since $q=0$ and there is no torsion, we have $\textrm{Pic}^0(S)=0$, the Neron - Severi group $\textrm{NS}(S)$ coincides with the Pi... | 3 | https://mathoverflow.net/users/7460 | 46225 | 29,257 |
https://mathoverflow.net/questions/46220 | 2 | Is there a classification of Sasaki or Sasaki-Einstein manifolds?
What are important examples?
| https://mathoverflow.net/users/7015 | Classification of Sasaki manifolds | The book [Sasakian Geometry](http://books.google.com/books?id=ERYZAQAAIAAJ) by Charles Boyer and the late Krzysztof Galicki contains a wealth of information on this topic. As far as I know there is no classification, but there are of course tons of examples.
| 4 | https://mathoverflow.net/users/394 | 46226 | 29,258 |
https://mathoverflow.net/questions/46202 | 3 | Today in my introductory algebraic geometry class we defined the so-called Rees algebra associated with an ideal $I$ of a ring $R$ (with strong conditions on $R$, if you like: I don't mind restricting to finitely generated reduced algebras $R$ over an algebraically closed field $k$). If we want to think of (maximal) Pr... | https://mathoverflow.net/users/3544 | Rees algebra for non-radical ideals | The blowup of $Spec R$ along $I$ and $I^m$ give isomorphic results. This is Hartshorne exercise II.7.11.a. In general, any birational projective morphism is realized as the blowing up of the target along some ideal sheaf, which in general will be quite complicated (e.g. not radical, but not a power of a radical ideal e... | 5 | https://mathoverflow.net/users/397 | 46233 | 29,263 |
https://mathoverflow.net/questions/41756 | 4 | I'm playing with MATLAB's svd function to compute the svd of
```
[ 1 4 7 10
2 5 8 11
3 6 9 12 ]
```
When I type [U1, ~, ~] = svd(X), I get
```
U1 =
-0.5045 0.7608 0.4082
-0.5745 0.0571 -0.8165
-0.6445 -0.6465 0.4082
```
But when I compute the svd of... | https://mathoverflow.net/users/5287 | Making MATLAB svd robust to transpose operation | You may be interested in the following article which addresses this issue:
R. Bro, E. Acar and T. G. Kolda. Resolving the sign ambiguity in the singular value decomposition. Journal of Chemometrics 22(2):135-140, February 2008. (doi:10.1002/cem.1122)
The MATLAB code can be downloaded here:
<http://www.mathworks.... | 7 | https://mathoverflow.net/users/10872 | 46257 | 29,274 |
https://mathoverflow.net/questions/46247 | 7 | It is well-known that the universal cover $\tilde G$ of a connected Lie group $G$ has a Lie group structure such that the covering projection $\tilde G\to G$ is a Lie group morphism. Of course $\tilde G$ might not be linear even though $G$ is, but this is not the point here.
My question is: assume that $G$ is a not n... | https://mathoverflow.net/users/10675 | Is there a good definition of the universal cover for non-connected Lie groups? | The group $Pin\_-(2)$ is an example of what you're looking for.
It can be described explicitly as a subgroup of the group of unit quaternions:
$Pin\_-(2)=$ { $a+bi| a^2+b^2=1$ } $\cup$ { $cj+dk| c^2+d^2=1$ } $\subset \mathbb H^\times$.
Its main interesting properties are:
- The conjugation action of $\pi\_0$... | 12 | https://mathoverflow.net/users/5690 | 46260 | 29,275 |
https://mathoverflow.net/questions/46258 | 14 | [Belyi's theorem](http://en.wikipedia.org/w/index.php?title=Belyi%27s_theorem) states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic function $\phi: X\to\mathbb{P}^1\mathbb{C} $ ramified at m... | https://mathoverflow.net/users/10400 | Generalizations of Belyi's theorem | The compactification is the usual one coming up in
the theory of modular forms, with the cusps being
orbits of $\Gamma$ on $\mathbb{Q}\cup\{\infty\}$.
As for the proof, I like
[this paper](http://uk.arxiv.org/abs/math/0108222) by Bernhard Koeck.
| 9 | https://mathoverflow.net/users/4213 | 46261 | 29,276 |
https://mathoverflow.net/questions/46180 | 10 | If $X\_1, \ldots , X\_k$ are i.i.d normal random variables with mean $0$ and variance $1$, then is there a way to sample $Y\_1, \ldots , Y\_m$ for $m=\omega(k)$ such that each of the $Y\_i$'s is a normal random variable with mean $0$ and variance $1$ and they are pairwise independent?
| https://mathoverflow.net/users/10858 | How to sample pairwise independent gaussians | Here is the answer I promised in my last comment.
Instead of considering ${\rm N}(0,1)$ variables, we may consider uniform$[0,1)$ variables.
Indeed, if $Z\_i$ are i.i.d. ${\rm N}(0,1)$ variables, then, with $\Phi(\cdot)$ denoting the ${\rm N}(0,1)$ distribution function, $U\_i := \Phi (Z\_i)$ are i.i.d. uniform$[0,1)... | 9 | https://mathoverflow.net/users/10227 | 46264 | 29,278 |
https://mathoverflow.net/questions/46102 | 8 | I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics.
The classical Schwarz Christoffel theorem does the job for euclidean polygons (see e.g. <http://en.wikipedia.org/wiki/Schwarz-Christoffel_mapping>).
Does anybody know of a s... | https://mathoverflow.net/users/10400 | Conformal Mappings for hyperbolic polygon | See Harmer and Martin's work on [Conformal Mappings from the Upper Half Plane to Fundamental Domains on the Hyperbolic Plane](http://www.math.auckland.ac.nz/Research/Reports/view.php?id=499).
Some of the ideas developed by [Christopher Bishop](http://www.math.sunysb.edu/~bishop/) in the context of computational geome... | 6 | https://mathoverflow.net/users/5372 | 46269 | 29,283 |
https://mathoverflow.net/questions/46262 | 1 | Some time ago, I asked about inite interpolation by
a nondecreasing polynomial here at [Finite interpolation by a nondecreasing polynomial](https://mathoverflow.net/questions/16673/finite-interpolation-by-a-nondecreasing-polynomial). This turned out to be an already solved problem; it also turned out that the degree of... | https://mathoverflow.net/users/2389 | Finite interpolation by nondecreasing indefinitely differentiable functions in a finite-dimensional space | This is to expand Qiaochu Yuan's comment. For $1\le i < n$ let $b\_i:=(x\_{i+1}+x\_i)/2$ be the mid-point of the $i$-th interval, and let $0 < \epsilon \leq \min\_{1\le i < n} (x\_{i+1}+x\_i)/2\, .$ Start with the linear space $V\_0$ of all continuous functions on $\mathbb{R}$ that are affine on each component interval... | 2 | https://mathoverflow.net/users/6101 | 46272 | 29,285 |
https://mathoverflow.net/questions/46249 | 9 | Thinking about the question [Four polynomials representing all integers modulo m](https://mathoverflow.net/questions/45697/four-polynomials-representing-all-integers-modulo-m/45709#45709) lead me to the following complementary question:
If $S$ is a set of positive integers, say that a positive integer $m$ is *covered... | https://mathoverflow.net/users/2784 | Density of congruence classes covered by a set | Denote by $P$ the set of prime powers not covered by $S$ (for each prime $p$, take only the smallest non-covered its power). If $\sum\_{x\in P} 1/x=+\infty$, then $\prod\_{x\in P} (1-1/x)$ is 0, so the product over some finite subset is arbitrarily small. But this product is a density of numbers without forbidden remai... | 3 | https://mathoverflow.net/users/4312 | 46277 | 29,287 |
https://mathoverflow.net/questions/46252 | 134 | Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after reflection and some research,
I find little support for my unpremeditated claim.
Just sticking to the topological boundary (a... | https://mathoverflow.net/users/6094 | Is the boundary $\partial S$ analogous to a derivative? | The surface area $|\partial S|$ of a (bounded, smooth) body $S$ is the derivative of the volume $|S\_r|$ of the $r$-neighbourhoods $S\_r$ of $S$ at $r=0$:
$$ |\partial S| = \frac{d}{dr} |S\_r| |\_{r=0}.$$
Thus, for instance, the boundary $\partial D\_r$ of the disk $D\_r$ of radius $r$ has circumference $\frac{d}{d... | 137 | https://mathoverflow.net/users/766 | 46285 | 29,288 |
https://mathoverflow.net/questions/46115 | 27 | Let $$G=\mathbb{Z}/p\_1^{e\_1}\times\cdots\times\mathbb{Z}/p\_n^{e\_n}$$ be any finite abelian group.
What are $G$'s subgroups? I can get many subgroups by grouping the factors and multiplying them by constants, for example: If $$G=\mathbb{Z}/3\times \mathbb{Z}/9\times \mathbb{Z}/4\times \mathbb{Z}/8,$$ then I can ta... | https://mathoverflow.net/users/nan | Subgroups of a finite abelian group | These three answers were originally comments. I am answering the part of the question which was deleted:
**Is there anything else (interesting) to say about the collection of subgroups of an [finite] abelian group.**
1. This paper: Ganjuškin, A. G. Enumeration of subgroups of a finite abelian group (theory). Compu... | 4 | https://mathoverflow.net/users/nan | 46300 | 29,296 |
https://mathoverflow.net/questions/45892 | 9 | I am not sure if this is appropriate for MO. If not, I shall be happy to take it to SE.
For a local ring $(R,m)$, given any proper ideal $I$, the (Krull) dimension (from here on dimension means Krull dimension) of the associated graded ring of $R$ with respect to $I$, $gr\_I(R)=\oplus\_{n\geq 0}\frac{I^n}{I^{n+1}}$ i... | https://mathoverflow.net/users/10775 | (Krull) dimension of any associated graded ring of a ring R equals the dimension of R | Though I heartily agree with Victor Protsak's comment, I will add some references. These might be useful for you, at least if you haven't seen them before. The references add a restriction, however, by assuming that $I$ is an ideal of finite co-length.
Then Corollary 12.5 of Eisenbud's Commutative Algebra uses the th... | 4 | https://mathoverflow.net/users/4 | 46307 | 29,301 |
https://mathoverflow.net/questions/45668 | 3 | We consider *finite* algebras for a given signature, in the sense of universal algebra (for example, they might be groups, rings, or lattices). An algebra $A$ is *irreducible* when $A \cong B \times C$ implies that $B$ or $C$ is the one-point algebra.
Is it the case that a $\Sigma$-algebra can be expressed as a carte... | https://mathoverflow.net/users/1176 | Is the decomposition of an algebra into irreducible components essentially unique? | Let $A, B$ be the algebras with two elements $0,1$ under addition mod $2$ and unary operation $x'=x$ in $A$ and $x'=1-x$ in $B$. Then $A \times B \cong B \times B$, though $A$ and $B$ are not isomorphic.
B. Jonsson
Construct a 12-element commutative semigroup which does not have the unique factorization property.... | 3 | https://mathoverflow.net/users/454 | 46310 | 29,304 |
https://mathoverflow.net/questions/46312 | 4 | Let $A$ be a closed set and let $B$ be a connected set such that $A \subset B$.
Does there always exist a closed connected subset $C$ of $B$ that contains $A$?
What if $B$ is path connected, is there always a path-connected $C$? A connected $C$?
| https://mathoverflow.net/users/10876 | Closed connected subset of a connected set | The answer to your first question is no. Let $A$ be a sequence on the $x$ axis converging to the origin in the real plane $\mathbb{R}^2$, which is the background topological space for these examples. Let $B$ be a cone over the sequence, connecting each point in the sequence to the point $(0,1)$, together with the origi... | 5 | https://mathoverflow.net/users/1946 | 46313 | 29,305 |
https://mathoverflow.net/questions/46305 | 11 | For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define cohomology classes $H^{p,q}\_{\bar\partial}(M)$.
In general is there any relation between $h^r(M)$ and $h^{p,q}(M)$?
| https://mathoverflow.net/users/5259 | Non-Kahler Complex manifolds | For any compact complex manifold there is a spectral sequence with $E\_1$ term $H^{p,q}(M)$ which converges to $H^{p+q}(M)$. If $M$ were Kahler, then this spectral sequence would degenerate at the $E\_2$ page, giving the familiar Hodge decomposition on cohomology.
In general, there is still a filtration on the cohomo... | 13 | https://mathoverflow.net/users/7762 | 46316 | 29,306 |
https://mathoverflow.net/questions/46321 | 6 | Suppose we have two closed-form expressions with $k$ unknowns which are hard to test for equality but easy to evaluate numerically over $\mathbb{R}^k$. One could then approach the problem of equality testing by checking equality numerically at several points. The interesting questions are then -- for which kinds of exp... | https://mathoverflow.net/users/7655 | Numeric equality testing? | Maple has a procedure **testeq** which is a "random polynomial-time equivalence tester". It works in this way. The 1986 paper [New results for random determination of equivalence of expressions](http://portal.acm.org/citation.cfm?id=32465&dl=ACM&coll=DL&CFID=111332393&CFTOKEN=61560774) by Gaston H. Gonnet might be a st... | 7 | https://mathoverflow.net/users/8008 | 46327 | 29,312 |
https://mathoverflow.net/questions/46325 | 8 | I'm interested in looking at the details of Gauss' method of determining the sign of the Gauss sum in his "Summatio Quarumdam Serierum Singularium", and I was wondering if anyone knew if there was an English or French translation available at all? I've tried searching on the internet and haven't found anything. Also, a... | https://mathoverflow.net/users/5431 | English or French translation of Gauss' "Summatio Quarumdam Serierum Singularium" | ["*The determination of Gauss sums*"](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-5/issue-2/The-determination-of-Gauss-sums/bams/1183548292.full) by Berndt and Evans (Bull. Amer. Math. Soc., Vol. 5, Number 2 (1981), 107-129.) contains an exposition of the original ... | 9 | https://mathoverflow.net/users/5371 | 46329 | 29,314 |
https://mathoverflow.net/questions/46253 | 1 | If I would ask for $\phi'(x) = f( \phi(x))$ and $\phi(0)=f(0)$, I would get that the inverse of $\phi$ is forced to be of the form:
$$\phi^{-1}(z) = \int\_{0}^z \frac{1}{f(x)} d x.$$
Now it is natural to ask whether there is something similiar for the problem $\phi''(x) =f( \phi(x),\phi'(x))$?
Or a little bit mo... | https://mathoverflow.net/users/10400 | Solving nonlinear ODE's | The answer is **NO** in general because your differential system is overdetermined: it has several equations and only one unknown function. Take the polynomials $\omega^2 X\_0-X\_2$ and $X\_2^2-X\_1^2-X\_0^2$ with $\omega^4\ne1$. The solutions of the first equation are $ae^{\omega s}+be^{-\omega s}$, but none of them s... | 5 | https://mathoverflow.net/users/8799 | 46334 | 29,319 |
https://mathoverflow.net/questions/46314 | 6 | My question is related to several notions of hyperbolicity, applied to Kähler manifolds (projective, in general). Kähler hyperbolicity was introduced in [this paper of Gromov's](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-33/issue-1/K%C3%A4hler-hyperbolicity-and-L_2-Hodge-theory/10.4310/j... | https://mathoverflow.net/users/10857 | Hyperbolicity for algebraic varieties and relation to curves on them | Take a very general hypersurface $j \colon X \hookrightarrow \mathbb{P}^n$ of sufficiently high degree, $3 \leq n \leq 4$. Then $X$ is Kobayashi hyperbolic (Kobayashi actually conjectured that this is true for all $n$ and Siu recently outlined a strategy for proving this, see Diverietti's comment below).
On the other... | 8 | https://mathoverflow.net/users/7460 | 46339 | 29,324 |
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