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https://mathoverflow.net/questions/33436 | 2 | Let $M$ be a compact manifold, and let $M\_1,\ldots, M\_k$ (k>2) be embedded submanifolds. Suppose that $p\in\cap\_{i=1}^k M\_k$ and that for any subset $S$ of $\{1,\ldots, k\}$ and any $j\notin S$ that $\cap\_{i\in S}M\_i$ intersects $M\_j$ transversally at $p$.
I believe that in this case the fact that $\cap\_{i=1... | https://mathoverflow.net/users/5399 | Normal intersections of submanifolds | The matter being local, we can restrict to a nbd $U$ of $p$ and think that $M\_i$ is the zero set of some local submersion $g\_i:U\to\mathbb{R}^{n\_i}$. If I'm not wrong your transversality assumption then translates into the surjectivity of the differential at $p$ of the map $g:=(g\_1,\dots,g\_k):U\to\mathbb{R}^m$ (he... | 3 | https://mathoverflow.net/users/6101 | 33446 | 21,677 |
https://mathoverflow.net/questions/32843 | 4 | Hello, this is my first post here. I hope that it is not too vague; I will be as precise as I can, but I have more of a meta-problem so please forgive me if this is inappropriate. My question is essentially of the form "what mathematical tools are most appropriate for investigating another problem?" Of course, if it tu... | https://mathoverflow.net/users/7814 | Under what conditions will a unitary matrix fix a subspace which does not diagonalize the generating Hamiltonian? | OK, I think I have a comprehensive answer. Since this is physics-inspired, I'll use bra-ket notation.
First, in my comment above, you can see that if $U\_{t\_0}$ fixes a subspace $V\_n \subset V$, then $U\_{t\_0}$ is block diagonal:
>
> If we write $U$ in the basis $\{|v\_1\rangle,...,|v\_N\rangle\}$ and see how... | 1 | https://mathoverflow.net/users/5789 | 33449 | 21,680 |
https://mathoverflow.net/questions/33443 | 2 | On p.8 of <http://www.msri.org/publications/books/Book39/files/marker.pdf>, the author writes $\Gamma(\bar{d})$, when $\Gamma$ is, first of all, a set of formulas (not a single one), and it is a formula which has variables, not constants. This doesn't make sense. And what does he mean by $T+\Gamma(\bar{d})$? This would... | https://mathoverflow.net/users/1355 | Confusion about model theory notes | The notation $\Gamma(\bar d)$ means that for every formula $\psi(\bar v)\in\Gamma(\bar v)$ we substitute $\bar d$ for $\bar v$. The set $T+\Gamma(\bar d)$ is just the union $T\cup\Gamma(\bar d)$.
The interchanging of variables $\bar v$ and constants $\bar d$ in the proof comes from the fact that the constants $\bar d... | 3 | https://mathoverflow.net/users/400 | 33452 | 21,683 |
https://mathoverflow.net/questions/33432 | 2 | Extending my [earlier question about linear transformations](https://mathoverflow.net/questions/33303/linear-transformation-takes-a-polygon-to-another-one), what's the easiest way to test if there exists a projective linear transformation that takes one polygon to another (in $\mathbb{R}\mathbb{P}^2$)?
| https://mathoverflow.net/users/2503 | Projective transformation between polygons. | The answer Tim Gowers and Will Jagy gave you in response to your earlier question can be extended straightforwardly. A homogeneous linear transformation on R^3 has 3x3 - 1 = 4x2 degrees of freedom, so there is a unique projective-linear transformation between any given pair of non-degenerate quadrilaterals. Thus do wit... | 2 | https://mathoverflow.net/users/2036 | 33459 | 21,688 |
https://mathoverflow.net/questions/33478 | 63 | The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its intermediate coefficients?
For a linear operator $f : V \to V$, we have the beautiful formula
$$\chi(f) = det(f - t) ... | https://mathoverflow.net/users/2036 | Geometric interpretation of characteristic polynomial | A rather simple response is to differentiate the characteristic polynomial and use your interpretation of the determinant.
$$det(I-tf) = {t^n}det(\frac{1}{t}I-f) = (-t)^ndet(f-\frac{1}{t}I)= {(-t)^n}\chi(f)(1/t)$$
So if we let $\chi(f)(t) = \Sigma\_{i=0}^n a\_it^i$, then ${(-t)^n}\chi(f)(1/t) = (-1)^n\Sigma\_{i=0}... | 75 | https://mathoverflow.net/users/1465 | 33482 | 21,701 |
https://mathoverflow.net/questions/33470 | 12 | In general, the tensor product of two local rings is not local. For example, $\mathbb{C} \otimes\_{\mathbb{R}} \mathbb{C}\ $ is not a local ring.
Let $\mathbb{F}\_{p}$ denote the finite field with $p$ elements. Let $A,B$ be two complete local noetherian $\mathbb{Z}\_p$-algebras with residue field $\mathbb{F}\_p$. Le... | https://mathoverflow.net/users/1816 | Is tensor product of local algebras local? | Let $A=\mathbb F\_p[[t]],B=\mathbb F\_p[[u]]$. Then, $1\otimes1-t\otimes u$ is neither in $\mathfrak m\_A\otimes B+A\otimes\mathfrak m\_B$ nor a unit, so it is contained in some other maximal ideal of $A\otimes B$. (Proof that $1\otimes1-t\otimes u$ is not a unit: An element of $\mathbb F\_p[[t]][[u]]$ coming from $A\o... | 7 | https://mathoverflow.net/users/2035 | 33488 | 21,706 |
https://mathoverflow.net/questions/32322 | 7 | Consider the Koch curve $G \subseteq \mathbb{R}^2$. Clearly $G$ is the invariant set (IS) of the iterated function system (IFS) $\lbrace \phi\_1, \phi\_2, \phi\_3, \phi\_4 \rbrace$. Where (not wanting to jump between $\mathbb{R}^2$ and $\mathbb{C}$ but doing so for ease):
$\phi\_1(x) = \frac{1}{3} x$,
$\phi\_2(x) = ... | https://mathoverflow.net/users/3121 | Minimum number of contractions needed to obtain a particular invariant set | An interesting question. Of course there is some ambiguity in the formulation "when can we tell".
Certainly in explicit examples, one may be able to apply ad-hoc methods. For example, things are easier for the Sierpinski gasket and carpet, since these have identifiable features in terms of their complementary region... | 1 | https://mathoverflow.net/users/3651 | 33492 | 21,709 |
https://mathoverflow.net/questions/33489 | 12 | Let $X$ be a locally ringed space (or a scheme) and $M,N$ two $\mathcal{O}\_X$-modules such that $M \otimes N \cong \mathcal{O}\_X$. Does it follow that $M$ is invertible in the usual sense, namely that $M$ is locally free of rank $1$?
It is true if $M$ is locally of finite type (which is, of course, also necessary).... | https://mathoverflow.net/users/2841 | Justification of the term "invertible sheaf" | Any $M'\to M$ that induces an epi after tensoring with $N$ must also induce an epi after tensoring with $N\otimes M$ and therefore must be an epi. And locally such a finitely generated $M'$ must exist.
| 6 | https://mathoverflow.net/users/6666 | 33497 | 21,713 |
https://mathoverflow.net/questions/32444 | 2 | Ie, is there a way to probe it for regions of depth that involves a function, the domain of which is the Mandelbrot set itself, or a part of that set?
| https://mathoverflow.net/users/7739 | Is there a way to find regions of depth in the Mandelbrot set other than simply poking around? | It is a little bit difficult to answer the question as posed, because there is a question as to what you mean by "depth".
One of the previous answers mentions Misiurewicz points - parameters where the critical orbit is pre-periodic. Examples are the "branch points" and "tips" in the Mandelbrot set. However, these are... | 8 | https://mathoverflow.net/users/3651 | 33499 | 21,714 |
https://mathoverflow.net/questions/32442 | 4 | Using Matlab, how to generate a net of 3^10 points that are evenly located (or distributed) on the 8-dimensional unit sphere?
Thanks for any helpful answers!
| https://mathoverflow.net/users/7738 | How to generate a net on a 8-dimensional sphere | If it's really important for the points to be evenly distributed, and you don't mind doing a lot of calculation to get them that way, you can start with a randomly distributed set and then iterate over the entire set repeatedly, allowing each point in turn to make whatever small adjustment improves your chosen definiti... | 4 | https://mathoverflow.net/users/7936 | 33501 | 21,716 |
https://mathoverflow.net/questions/33477 | 6 | I just took a look at the nlab entry: [Nikolai Durov](http://ncatlab.org/nlab/show/Nikolai+Durov). It seems that Skoda never mentioned that what Durov introduced was a special case of generalized scheme theory. I did not read his [dissertation](http://arxiv.org/abs/0704.2030) carefully or completely. I wonder whether h... | https://mathoverflow.net/users/1851 | Did Durov's work give an example of noncommutative schemes? | No, there is no need for the Rosenberg [noncommutative scheme](http://ncatlab.org/nlab/show/noncommutative+scheme) that the categories be abelian in general; whatever being said in his 1998 paper he does not mean so. He defines a relative scheme over a base category given by exactness properties of direct and inverse i... | 6 | https://mathoverflow.net/users/35833 | 33514 | 21,724 |
https://mathoverflow.net/questions/33513 | 4 | Let $A$ be a local ring with nilpotent maximal ideal $\mathfrak{m}$ (i.e., some power of $\mathfrak{m}$ vanishes), and $M$ an $A$-module (not necessarily finitely generated). Let $\bar{S}\subset M/\mathfrak{m}M$ be a set of generators and $S$ a set of representatives of $\bar{S}$ in $M$. Then is it true that $S$ is a s... | https://mathoverflow.net/users/5292 | Non-finite version of Nakayama's lemma? | Dear Kwan,
Let $N$ be the submodule of $M$ generated by $S$. Then by assumption
$M = N +\mathfrak m M.$ Iterating this, we find that
$$M = N + \mathfrak m (N + \mathfrak m M) = N + \mathfrak m^2 M = \cdots
= N + \mathfrak m^n M$$
for any $n > 0.$ If we take $n$ large enough then $\mathfrak m^n = 0$ (by hypothesis).
... | 19 | https://mathoverflow.net/users/2874 | 33517 | 21,726 |
https://mathoverflow.net/questions/31279 | 10 | I've decided to start a wiki to do collaborative mathematics. However I don't have access or control over a server. So I need a wiki farm. I've tried out pbworks and wikidot, but their latex support is not as straightfoward as say wordpress.
Do you have a suggestion of which wiki farm to use?
| https://mathoverflow.net/users/nan | Suggestions for wiki farm with good latex support | If you're looking for a wiki that can handle LaTeX-style equations, then you should take a look at instiki. Not only does it display mathematics properly, it can also export pages to LaTeX.
<http://www.instiki.org/show/HomePage>
| 4 | https://mathoverflow.net/users/45 | 33521 | 21,730 |
https://mathoverflow.net/questions/30493 | 2 | Let $P$ be a (finitely generated) pro-$p$ group, and let $E$ be an infinite elementary abelian normal subgroup. Does $E$ necessarily contain a non-trivial finite normal subgroup of $P$? We can think of $E$ as consisting of sequences of elements of $C\_p$, with open subgroups $O\_X$, where $X$ is a finite subset of the ... | https://mathoverflow.net/users/4053 | Elementary abelian normal subgroups of a pro-p group | After talking to Charles Leedham-Green, I now have an example that answers the question (I think). See <http://mathoverflow.net:80/questions/33533/name-this-pro-p-group>. More interesting examples would still be nice though, particularly if they do not have $C\_p \wr C\_p$ as an image.
| 1 | https://mathoverflow.net/users/4053 | 33536 | 21,740 |
https://mathoverflow.net/questions/33540 | 4 | Does a module (over a commutative ring) always possess a minimal generating set? When the module is not finitely generated, the typical Zorn's lemma type argument doesn't seem to work. More precisely, if $(S\_{\alpha})\_{\alpha}$ is a chain of generating subsets of a module $M$, $M=\cap \_ {\alpha}(S \_ {\alpha})\supse... | https://mathoverflow.net/users/5292 | Existence of a minimal generating set of a module | $\mathbb{Q}$, as a $\mathbb{Z}$-module, has no minimal generating set.
By the way, in the paper "A characterization of left perfect rings" by Yiqiang Zhou, it is proven that a ring $R$ is left perfect if and only if every generating set of some $R$-module contains a minimal generating set.
| 9 | https://mathoverflow.net/users/2841 | 33542 | 21,742 |
https://mathoverflow.net/questions/33468 | 10 | I seem to remember reading somewhere that ZF+AD proves that $\omega\_1$ and $\omega\_2$ are measurable cardinals.
Is that right?
If so, can someone [point me to or give here] a [sketch or proof] of these results?
| https://mathoverflow.net/users/nan | Measurable cardinals under Axiom of Determinacy | An alternative proof for $\omega\_1$ goes via the set $D$ of Turing degrees (also known as degrees of unsolvability). $D$ is upward directed by the ordering $\leq\_T$ of Turing reducibility, so the cones $C\_d=\{p\in D:d\leq\_Tp\}$ generate a filter on $D$. The axiom of determinacy implies that this filter is an ultraf... | 15 | https://mathoverflow.net/users/6794 | 33546 | 21,743 |
https://mathoverflow.net/questions/33541 | 2 | Suppose we are working in an "arrows-only" definition of a category such as given in
Mac Lane's "Categories of the Working Mathematician" (1998) p.279 or on
[nlab](http://ncatlab.org/nlab/show/single-sorted+definition+of+a+category). How can we formulate the definition of a product in such a category? I do not see how ... | https://mathoverflow.net/users/7779 | What would be an "arrows-only" defintion of a product in a category? | Using the notation of nlab, the following is a fibered product: if $x, y$ are arrows with $t(x) = t(y)$, then their fiber product is the pair of arrows $u, v$ with $t(u) = s(x)$, $t(v) = s(y)$, and $s(u) = s(v)$ such that for any pair of arrows $a, b$ with $s(a) = s(b)$, $t(a) = t(u) \, (= s(x))$ and $t(b) = t(v) \, (=... | 3 | https://mathoverflow.net/users/6545 | 33547 | 21,744 |
https://mathoverflow.net/questions/33539 | 12 | Suppose that $G$ is a finitely presented group and $H$ is a finitely generated normal subgroup such that $G/H$ is infinite cyclic. Is it true that $H$ is finitely presented?
| https://mathoverflow.net/users/7307 | Finitely generated subgroups with infinite cyclic quotient | No. [Ollivier & Wise's version](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=ollivier&s5=wise&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq) of the Rips Construction gives, for any ... | 15 | https://mathoverflow.net/users/1463 | 33549 | 21,746 |
https://mathoverflow.net/questions/31007 | 22 | What are the units in $R[X,X^{-1}]$, where $R$ is a commutative ring with $1$? I know that the question for polynomial rings is a standard textbook exercise. However, I couldn't find a reference for Laurent polynomials, since most people only seem to consider coefficients in an integral domain.
| https://mathoverflow.net/users/7422 | What are the units in the ring of Laurent polynomials? | You can find a more general result in the [paper [1]](http://dx.doi.org/10.1007/s10440-008-9370-8), which determines the units and nilpotents in arbitrary group rings $\rm R[G]$ where $\rm G$ is a unique-product group - which includes ordered groups. As the author remarks, his note was prompted by an earlier [paper [2]... | 7 | https://mathoverflow.net/users/6716 | 33550 | 21,747 |
https://mathoverflow.net/questions/33504 | 5 | I got to thinking about this problem while sifting through the [math puzzles for dinner](https://mathoverflow.net/questions/29323?sort=votes&page=1#sort-top) thread. There's a fun puzzle by rgrig which asks the guests to prove that when they came to dinner two of them shook hands the same amount of times. The solution ... | https://mathoverflow.net/users/2233 | Degree sequences of multigraphs with bounded multiplicity | The answer to Question 1 is no. Here's a construction. First we construct a simple graph $G\_n$ on $n$ vertices for each $n\geqslant3$ which has every degree from $1$ to $n-1$ inclusive occurring, with the degree $n-1$ occurring only once. You do this recursively: to construct $G\_n$, just take the complement of $G\_{n... | 3 | https://mathoverflow.net/users/6771 | 33562 | 21,755 |
https://mathoverflow.net/questions/33545 | 3 | I have a question about surgery.
Let $G= \mathbb{Z}\_m \times \mathbb{Z}$ and $M$ be a oriented 3-manifold with G-action. i.e. There exists a map $f\colon M/G \to BG$, where $BG$ is classifying space.($BG=K(G,1)=K(\mathbb{Z}\_m \times \mathbb{Z},1)).$ Therefore, we can regard $(M/G,f)\in \Omega^{SO}\_3(K(G,1))$, wher... | https://mathoverflow.net/users/7776 | Equivariant Surgery problem | Yes. You need to extend the map to $BG$ over the surgery cobordism, which is possible.
First let me add that all your $G$-actions seem to be free and that the boundary of $V$ would consist of $r$ copies of $M$.
Now for the surgeries:
first take connected sum of V/G with 2 copies of $S^1\times S^2$'s (surgeries on ... | 2 | https://mathoverflow.net/users/1090 | 33565 | 21,757 |
https://mathoverflow.net/questions/33563 | 14 | I recently read the model-theoretic proof of the Nullstellensatz using quantifier elimination (see www.msri.org/publications/books/Book39/files/marker.pdf). I'm convinced that the Nullstellensatz is true, i.e. by showing that $\exists y\_1 \cdots \exists y\_n (\bigwedge\_{i} (f\_i(y) = 0)$ is equivalent to a quantifier... | https://mathoverflow.net/users/1355 | Intuition for Model Theoretic Proof of the Nullstellensatz | Your example is not an illuminating one, because you used two equations in two variables. You would therefore expect there, generically, to be a solution for most values of $(a,b,\ldots, l)$. I'll get back to your example, but let's start with the more informative case of two equations in one variable. Then I'll use th... | 19 | https://mathoverflow.net/users/297 | 33570 | 21,760 |
https://mathoverflow.net/questions/33571 | 3 | I'm studying category theory for the first time in a very succint book for computer scientists (I'm not actually a computer scientist, I'm a physicist, but my interest in cat theory is related to purely functional programming languages). But, as the book is very succint, may be it lacks some information so I have a que... | https://mathoverflow.net/users/757 | Finite categories and partial orders | Almost, this has nothing to do with finiteness: any category where the
homsets have at most one element each is a preorder. Define
$a\le b$ if there is an arrow from $a$ to $b$. Then $\le$ is reflexive
and transitive, by the category axioms. But it may fail to be
antisymmetric: one may have $a\le b\le a\ne b$. But a pr... | 3 | https://mathoverflow.net/users/4213 | 33573 | 21,762 |
https://mathoverflow.net/questions/33522 | 36 | The following statement is well-known:
Let $A$ be a commutative Noetherian ring and $M$ a finitely generated $A$-module. Then $M$ is flat if and only if $M\_{\mathfrak{p}}$ is a free $A\_{\mathfrak{p}}$-module for all $\mathfrak{p}$.
My question is: do we need the assumption that $A$ is Noetherian? I have a proof (... | https://mathoverflow.net/users/5292 | Flatness and local freeness | By request, my earlier comments are being upgraded to an answer, as follows. For finitely generated modules over any local ring $A$, flat implies free (i.e., Theorem 7.10 of Matsumura's CRT book is correct: that's what proofs are for). So the answer to the question asked is "no". The CRT book uses the "equational crite... | 35 | https://mathoverflow.net/users/3927 | 33574 | 21,763 |
https://mathoverflow.net/questions/33486 | 12 | In an earlier thread I had asked whether or not one can find a smooth 4-dimensional submanifold of $S^4$ whose fundamental group has an unsolvable word problem. The answer is yes, and the reference is in that thread.
see: [Word problem for fundamental group of submanifolds of the 4-sphere](https://mathoverflow.net/q... | https://mathoverflow.net/users/1465 | 4-manifolds in the 4-sphere such that it, *and* its complement have unsolvable word problem | The answer to the question in the beginning should be YES and it follows from the answer to your previous question. We just need to use the fact that if $G$ has unsolvable word problem then $G\*F$ too, where $F$ is a group and $G\*F$ is the free product.
To construct the example take the solution to the previous ques... | 6 | https://mathoverflow.net/users/943 | 33577 | 21,766 |
https://mathoverflow.net/questions/33510 | 11 | I am writing an article on Fermat's work in number theory and feel uncomfortable everytime I have to write "Fermat's Little Theorem": it's clumsy and belittles the fundamental character of Fermat's result. "Fermat's Theorem" is too ambiguous, and I don't really like acronyms such as Flt or Flit. Has anyone ever seen a ... | https://mathoverflow.net/users/3503 | Christening Fermat's Little Theorem | I think you shouldn't change the name. It's universally known as Fermat's Little Theorem, and especially if you're writing a survey or historical article, you're not in a place to try to revolutionize established mathematical nomenclature. There are many instances of unfortunate terminology in mathematics, but in my op... | 8 | https://mathoverflow.net/users/4183 | 33578 | 21,767 |
https://mathoverflow.net/questions/31249 | 15 | I define a 1-isomorphism between two groups as a bijection that restricts to an isomorphism on every cyclic subgroup on either side. There are plenty of examples of 1-isomorphisms that are not isomorphisms. For instance, the exponential map from the additive group of strictly upper triangular matrices to the multiplica... | https://mathoverflow.net/users/3040 | Order information enough to guarantee 1-isomorphism? | Here is a counterexample of order $32$.
$G$ and $H$ will each have $3$ elements of order $2$ and $28$ elements of order $4$. In both cases all three elements of order $2$ will have square roots. That insures that $F\_G=F\_H$. But in $G$ one of them will have $4$ square roots while the others each have $12$, and in $... | 16 | https://mathoverflow.net/users/6666 | 33583 | 21,769 |
https://mathoverflow.net/questions/33543 | 9 | Let $M$ be a filtered module over a filtered algebra $A$, and suppose $gr(M)$ is flat over $gr(A)$, where $gr$ means the associated graded module and algebra, respectively.
What can one say in general about the flatness of $M$ over $A$, or with relevant assumptions (for instance in the above, we should assume both fi... | https://mathoverflow.net/users/1040 | Associated graded and flatness | Let me suppose, as in your examples, that we have a base field $k$.
It is well known that to check that a right $A$-module $M$ is flat it is enough to show that whenever $I\leq\_\ell A$ is a left ideal, the map $M\otimes\_AI\to M\otimes\_A A$ induced by the inclusion $I\to A$ is injective. This condition can be rewri... | 8 | https://mathoverflow.net/users/1409 | 33584 | 21,770 |
https://mathoverflow.net/questions/32911 | 10 | An important invariant of a knot in $S^3$ is its *Alexander polynomial*, related also to *Reidemeister torsion*. Is there something like that for knotted surfaces in $S^4$? If not, what are the difficulties?
| https://mathoverflow.net/users/5196 | Alexander polynomial or Reidemeister torsion for knotted surfaces? | Yes for $S^2$, and more generally, depending on what you are after.
Given any torsion module $M$ over the PID $Q[t,t^{-1}]$, the order of $M$ is well defined in
$Q[t, t^{-1}]$ up to units, in the usual way. Moreover $M\otimes Q(t)=0$. These two facts are at the heart of why the Alexander polynomial is related to Rei... | 7 | https://mathoverflow.net/users/3874 | 33587 | 21,772 |
https://mathoverflow.net/questions/32511 | 9 | Edited question:
Are there any other non-trivial \*-homomorphisms between matrix algebras apart from the unitary homomorphisms?
Original question:
Does there exist a surjective (but not bijective) \*-homomorphism between matrix algebras over the complex numbers? If so, are there any nice examples?
(I had not r... | https://mathoverflow.net/users/6985 | *-homomorphisms between matrix algebras | The algebra $M\_n(\mathbb{C})$ of $n \times n$ complex matrices is Morita equivalent to $\mathbb{C}$. Which implies: Every unital representation of $M\_n(\mathbb{C})$ is isomorphic to a direct sum of copies of the defining representation. Thus your homomorphism $\rho:M\_n \to M\_k$ exists precisely when $k$ is a multip... | 16 | https://mathoverflow.net/users/1450 | 33592 | 21,774 |
https://mathoverflow.net/questions/33591 | 0 | If there is an arrow $h: (A\times B) \to (A\times B)$, then, necessarily $h = \left\lt f,g\right\gt$ for some $f: (A\times B) \to A$ and $g (A\times B) \to B$?
The book I'm studying defines a product $A\times B$ to be an object provided with two projecting arrows $\pi\_A : A\times B \to A$ and $\pi\_B: A\times B \to ... | https://mathoverflow.net/users/757 | Products of objects in categories | The reason you're having trouble proving the uniqueness is that you need to use the projections. There is not a unique map from $C$ to $A \times B$. But there is a unique map *given* specified maps from $C$ to $A$ and $B$ (i.e. such that the composition of the map from $C$ to the product with the projection gives the m... | 3 | https://mathoverflow.net/users/1355 | 33593 | 21,775 |
https://mathoverflow.net/questions/33581 | 11 | After reading Cohen and Voronov's notes on string topology, one can find the following construction: Suppose we have a topological space $X$ with continuous action of $S^1$. This means we have a map $\rho: S^1 \times X \to X$. If we choose a fundamental class $[S^1]$ of the circle, we can form the operator $\Delta: H\_... | https://mathoverflow.net/users/798 | What does this naive attempt at $S^1$-equivariant homology describe? | By $G$-equivariant homology of $X$ you mean the homology of the Borel construction or homotopy orbit space $EG\times\_GX$. (I only mention this because the same phrase can refer to other kinds of homology for $G$-spaces, satisfying a weaker kind of homotopy axiom. This "Borel homology" gives you an isomorphism for ever... | 14 | https://mathoverflow.net/users/6666 | 33595 | 21,777 |
https://mathoverflow.net/questions/33602 | 17 | In my answer to [this question](https://math.stackexchange.com/questions/640/why-are-differentiable-complex-functions-infinitely-differentiable/820#820) on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One wa... | https://mathoverflow.net/users/290 | What is a reasonable finitary analogue of the statement that harmonic functions are smooth? | There are quite a few properties shared by discrete and continuous harmonic functions. They also generalize in various forms to graphs. I don't know if the analogy is so close that there is a unique concept of smoothness for the discrete case, but see the last item for one point of contact.
1. Poisson formula: $f(p)$... | 15 | https://mathoverflow.net/users/6579 | 33610 | 21,787 |
https://mathoverflow.net/questions/33597 | 16 | I'm trying to refresh myself on quantum algorithms and have been skimming Childs and van Dam's [2008 RMP paper](http://arxiv.org/abs/0812.0380v1) among other things. From my preliminary surfing it looks like the known quantum algorithms still essentially fall into (in that their nontrivial quantum aspect is governed by... | https://mathoverflow.net/users/1847 | Are there any known quantum algorithms that clearly fall outside a few narrow classes? | Does the [Farhi-Goldstone-Gutman game tree evaluation algorithm](http://arxiv.org/abs/quant-ph/0702144) and the extensions of it fall into one of these classes? You might put it in quantum simulation/annealing because of the technique used, but I think that would be a mistake. You might also put it in quantum search be... | 15 | https://mathoverflow.net/users/2294 | 33612 | 21,789 |
https://mathoverflow.net/questions/33469 | 3 | The following integral came up in one of my applications:
$\int\_{-1}^1P\_n(x)T\_j(x)T\_k(x)\mathrm{d}x$
where $P\_n(x)$ is a Legendre polynomial, $T\_k(x)$ is a Chebyshev polynomial, and $j$, $k$, and $n$ are nonnegative integers.
I want to ask if there might be a closed-form representation for this integral. I ... | https://mathoverflow.net/users/7934 | Closed form for an orthogonal polynomial integral? | It turns out the identities I needed for resolving
$\int\_{-1}^1P\_n(x)T\_j(x)\mathrm{d}x$
into a closed form was well-hidden in Abramowitz and Stegun and Gradshteyn and Ryzhik.
As I had mentioned in the edit to my original question, the integral is 0 if $j<n$ by virtue of the orthogonality property of the Legend... | 5 | https://mathoverflow.net/users/7934 | 33613 | 21,790 |
https://mathoverflow.net/questions/33614 | 23 | Let $D$ be an elliptic operator on $\mathbb{R}^n$ with real analytic coefficients. Must its solutions also be real analytic? If not, are there any helpful supplementary assumptions? Standard Sobolev methods seem useless here, and I can't find any mention of this question in my PDE books.
I began thinking about this b... | https://mathoverflow.net/users/4362 | Does elliptic regularity guarantee analytic solutions? | While probably not the fastest approach I think that Hörmander: The analysis of linear partial differential equations, IX:thm 9.5.1 seems to give a (positive) answer to your question. It is overkill in the sense that it gives you a microlocal statement telling you that for $Pu=f$, $u$ is analytic in the same directions... | 7 | https://mathoverflow.net/users/4008 | 33615 | 21,791 |
https://mathoverflow.net/questions/28241 | 8 | Can the lattice stick number of a knot be bounded
by the stick number of the knot?
The stick number
$S(K)$ of a knot $K$ is the fewest number of segments
needed to realize it by a simple 3D polygon.
The lattice stick number $S\_L(K)$ is the fewest segments in a realization in the cubic
lattice, with all segments par... | https://mathoverflow.net/users/6094 | Lattice Stick Number vs. Stick Number of Knot | I wouldn't be surprised by something like a quadratic bound, or possibly something reasonable in terms of another complexity measure for the knot, but I see no hope for making $m$ constant. Consider the following construction: given $m$, choose some large number like $N=(10m)^6$ of points uniformly at random in the uni... | 8 | https://mathoverflow.net/users/7936 | 33616 | 21,792 |
https://mathoverflow.net/questions/33622 | 7 | The Poincare-Hopf theorem tell us that the sum of the indices of a vector field at isolated zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold. But how to construct a vector fiedls with isolated zeros?
| https://mathoverflow.net/users/3896 | How to construct a vector fields with isolated zeros? | Your question isn't very well defined. A manifold on its own is not an object where constructions come by easily. But there is a generic way to construct vector fields with isolated zeros. *Any* vector field can be approximated by one with isolated zeros. This is a consequence of Sard's theorem. So start off with the z... | 4 | https://mathoverflow.net/users/1465 | 33627 | 21,798 |
https://mathoverflow.net/questions/33476 | 6 | Sometimes, given an object A in an Abelian category, the Yoneda product on Ext(A, A) is graded-commutative, for example in cases where it coincides with the cup-product in singular cohomology. Are there any nice theorems about when the Yoneda product is graded-commutative in general? Thanks in advance.
| https://mathoverflow.net/users/7935 | When is the Yoneda product graded commutative? | I move this to a more proper answer to discuss some subtle points of the
question. The Eckman-Hilton argument (or more concrete calculations) shows, as
Chris points out, that $\mathrm{Ext}(A,A)$ is commutative when $A$ is the unit
for a monoidal category. The subtleties appear when we consider for instance the
ring $R=... | 6 | https://mathoverflow.net/users/4008 | 33633 | 21,801 |
https://mathoverflow.net/questions/33625 | 11 | [Qiaochu's question](https://mathoverflow.net/questions/33602/what-is-a-reasonable-finitary-analogue-of-the-statement-that-harmonic-functions-a) on a discrete analogue of harmonic function theory reminded me of some thoughts I had a long time ago about the relationship between cubical cohomology and de Rham cohomology.... | https://mathoverflow.net/users/2036 | Cubical cohomology and de Rham cohomology | M. Carmen Minguez in [this article](http://archive.numdam.org/article/CTGDC_1988__29_1_59_0.pdf) constructs a homomorphism between de Rham and Cubical Singular Cohomology without showing that is an isomorphism. This is done in the context of Synthetic Differential Geometry.
In general Synthetic Differential geometers... | 7 | https://mathoverflow.net/users/733 | 33635 | 21,803 |
https://mathoverflow.net/questions/33638 | 5 | I refer to "Sheaves in Geometry and Logic", by S. MacLane.
Let **C** be a category. Dealing with a *subobject* of an object $D \in \text{Ob}\_{\mathbf C}$, one defines an equivalence relation between morphisms towards *D*:
>
> Two monomorphisms $f:A\to D$, $g:B\to D$ with a common codomain *D* are called *equival... | https://mathoverflow.net/users/7952 | Is the subobject functor really a presheaf? | For a general category the subobjects do indeed not have to form a set.
In the context of MacLane/Moerdijk you only look at toposes and there one has a natural isomorphism $Sub\_{\mathbf{C}}(D) \cong Hom\_{\mathbf{C}}(D,\Omega)$, where $\Omega$ is the subobject classifier.
So it follows from the axioms of a topos... | 10 | https://mathoverflow.net/users/733 | 33643 | 21,809 |
https://mathoverflow.net/questions/32999 | 7 | This problem arises while studying the complexity of algorithms and I am quite unfamiliar with the subject.
Consider the set F of injective functions from {1..N} to {1..M}
we can define an association scheme on F x F by
(f,f') and (g,g') are in the same class if there is a permutation $\pi\in S\_M$ and a permutatio... | https://mathoverflow.net/users/6673 | Association scheme on injective functions | I don't think this scheme has a particular name, and am not aware of any study of it. Its Bose-Mesner algebra is the commutant of a multiplicity-free representation of the wreath product of $S\_m$ by $S\_n$. To get the eigenspaces you need to find the decomposition of the representation into irreducibles.
The most us... | 5 | https://mathoverflow.net/users/1266 | 33651 | 21,815 |
https://mathoverflow.net/questions/33676 | 3 | Does the existence of exotic smooth structure in $\mathbb{R}^4$ imply the existence of an atlas which has a $C^0$ mapping to the Cartesian atlas, but not a $C^k$ mapping (for some finite $k$)? Does the nonexistence of exotic smooth structure in $\mathbb{R}^n$, $n\neq 4$ imply that all atlases therein have smooth mappin... | https://mathoverflow.net/users/7956 | exotic smooth structure clarification | Regarding your 1st question, perhaps you meant to ask something else? Any atlas can be composed with a non-smooth homeomorphism to produce an atlas that isn't smooth in the standard sense. For example, $\mathbb R \to \mathbb R$ defined by $t \longmapsto t^{1/3}$ is an atlas on $\mathbb R$ but it's not $C^1$. This answe... | 7 | https://mathoverflow.net/users/1465 | 33677 | 21,827 |
https://mathoverflow.net/questions/25484 | 7 | Given two modules $M$ and $N$ there is a nice scheme parametrizing extensions
$0 \rightarrow M \rightarrow E \rightarrow N \rightarrow 0$
namely $\operatorname{Ext}^1(N,M)$ or, leaving out the trivial extension, the projective space $P(\operatorname{Ext}^1(N,M))$.
There are (at least) two natural generalizations
... | https://mathoverflow.net/users/5714 | Moduli of extensions of modules | in the situation you have in mind (sheaves on an algebraic variety),
such spaces are not too difficult to construct as Artin stacks. If you omit the condition that the i-th filtration quotient is isomorphic to a given one, then such a universal Artin stack is e.g. constructed in Bridgeland's introduction to Hall-algebr... | 4 | https://mathoverflow.net/users/7437 | 33678 | 21,828 |
https://mathoverflow.net/questions/33533 | 3 | Let $K$ be the set of open-closed subsets of $\mathbb{Z}\_p$. Let $M$ be the set of functions from $K$ to $C\_p$ that are additive under disjoint unions. Then $M$ can be regarded as an elementary abelian pro-$p$ group: multiplication is pointwise in $C\_p$, and a base of open subgroups is given by $\{ U\_n \}$, where $... | https://mathoverflow.net/users/4053 | Name this pro-$p$ group | I would call this group the pro-$p$ completion of $C\_p \wr \mathbb{Z}$. Alternatively, it is the group given by the pro-$p$ presentation <$ a, b| a^p, [a, a^b]>$, along similiar lines.
It looks a lot like $C\_p\wr \mathbb{Z}\_p$, except we have the product of as many copies of $C\_p$ as open subgroups of $\mathbb{Z... | 1 | https://mathoverflow.net/users/4100 | 33682 | 21,831 |
https://mathoverflow.net/questions/33672 | -2 | I'm trying to solve the following least squares problem:
$\underset{x}{\text{min}} ||Ax - \tilde{b}||\_2$
where $Ax = b$ and $\tilde{b} = b + w$
### Question:
How do I determine which probability distribution fits $w$ best?
Also, $A \in \mathbb{R}^{n\times n}$ is a large, and sparse Toeplitz matrix, $\tilde{b... | https://mathoverflow.net/users/1899 | Determine noise distribution | There's no way to answer in general what distribution data has. It has whatever distribution it has, but what you really want to know is whether a distribution which someone has identified adequately fits your data. So you have to propose a specific distribution first, then test for goodness of fit. Searching on "goodn... | 2 | https://mathoverflow.net/users/136 | 33694 | 21,838 |
https://mathoverflow.net/questions/33629 | 13 | Here are two well known facts, which put together leave me confused.
First, it's well known that intuitionistic logic is the logic of constructive mathematics. From every intutionistic proof, you can extract an algorithm which will compute the witness to that theorem (i.e., the famous "existence principle" of intuti... | https://mathoverflow.net/users/1610 | What happens when we print the digits of a real number? | [**Edit:** Following Carl Mummert's comments, I have improved the part about transformation of Cauchy sequences to digit expansions. Thanks to Carl for interesting observations.]
Your question is best divided into two subquestions:
1. Does every real number have a digit expansion?
2. What is the nature of the "proc... | 10 | https://mathoverflow.net/users/1176 | 33702 | 21,844 |
https://mathoverflow.net/questions/33687 | 7 | I'm writing some software to automatically compute things like Alexander modules, Alexander ideals, Milnor signatures and Farber-Levine pairings for 1 and 2-knot complements. So among other things I'd like to have a quick way of comparing ideals in the Laurent polynomial ring $\mathbb Z[t^{\pm 1}]$.
So my question:
... | https://mathoverflow.net/users/1465 | Ideals in the ring of single-variable Laurent polynomials with integer coefficients | It is fairly straightforward to adapt standard Gröbner basis techniques to such algebras, e.g. see the [paper [1]](https://doi.org/10.1007/s002000050108). See also the [paper [0]](https://doi.org/10.1007/11870814_12) which applies such algorithms to the problem at hand.
[[0](https://doi.org/10.1007/11870814_12)] Jesu... | 5 | https://mathoverflow.net/users/6716 | 33707 | 21,846 |
https://mathoverflow.net/questions/33713 | 5 | Let $T: X \to X$ be an Anosov diffeomorphism. Suppose $f: X \to \mathbb{R}$ is Holder continuous (say with exponent $\alpha$). The question arises as to when $f$ can be written as $g \circ T - g $ for some $\alpha$-Hölder $g: X \to \mathbb{R}$. It is easily checked that a necessary condition is that the periodic data v... | https://mathoverflow.net/users/344 | Existence conditions for twisted cohomological equations? | Part I:
The answer is yes under additional conditions:
1. Periodic data conditions are satisfied. That is, for any periodic point $p$
$$
\sum\_{x\in O(p)}f(x)=0.
$$
2. Exponent $\alpha$ is sufficiently close to 1.
3. Transformation $A$ is dominated by $T$. That is, the map $(x,v)\mapsto(Tx, Av)$ is partially hyperbol... | 7 | https://mathoverflow.net/users/2029 | 33725 | 21,855 |
https://mathoverflow.net/questions/33706 | 4 | This question is related to a previous question of mine:
[Determinacy interchanging the roles of both players](https://mathoverflow.net/questions/32966/determinacy-interchanging-the-roles-of-both-players)
Given any set A of sequences of natural numbers, every strategy (no matter for which player) is either winning ... | https://mathoverflow.net/users/6466 | Subsets of sequences of natural numbers vs. strategies under ZFC | 4 is not possible, the others that you list are.
By observing that a strategy is winning for player X in the game $G(A)$ if and only if it is losing for X in $G(A^c)$ we can reduce the total number of games necessary to construct by noting some equivalences. For example, if there is a game of the form 5, then by taki... | 7 | https://mathoverflow.net/users/2436 | 33728 | 21,858 |
https://mathoverflow.net/questions/33717 | 5 | What is a good reference for a geometrical viewpoint on the calculus of variations for physics, using differential forms etc. to derive Yang-Mills equations and other topics of the standard model?
Thanks for ideas.
| https://mathoverflow.net/users/7626 | literature on geometrical viewpoint on calculus of variations for physics | I think that the book
* David Bleecker: *Gauge Theory and Variational Principles*, Addison-Wesley, 1981
contains exactly what you are looking for.
| 4 | https://mathoverflow.net/users/3473 | 33729 | 21,859 |
https://mathoverflow.net/questions/33724 | 7 | Background
----------
When constructing the exterior algebra of a (finite-dimensional, complex) vector space $V$, there are two equivalent pictures. The first is the quotient picture. First you define the tensor algebra $T(V)$, define $\mathcal{J}$ to be the 2-sided ideal generated by elements of the form $x\otimes y... | https://mathoverflow.net/users/703 | Idempotency of the q-antisymmetrizer | This element has many expressions. It is characterised up to a scalar multiple by the property that $\sigma\_iA\_n=-q^{-1}A\_n$ for $i=1,2,\ldots ,n-1$. It also satisfies
$A\_n\sigma\_i=-q^{-1}A\_n$ for $i=1,2,\ldots ,n-1$. In particular it is central.
Using this property you can calculate $A\_n^2$. Note that
$\sigm... | 7 | https://mathoverflow.net/users/3992 | 33734 | 21,863 |
https://mathoverflow.net/questions/33568 | 0 | There is only one differentiable structure permitted in R^2, meaning, I think, that all atlases in R^2 are diffeomorphic to the Cartesian atlas. But, doesn't the polar coordinate system represent an atlas that is not truly diffeomorphic to the Cartesian atlas, due to the coordinate singularity it has at its origin?
| https://mathoverflow.net/users/7956 | differentiable structure and coordinates in R^2 | I think Matt Noonan's comment technically answered my question. The polar coordinate chart is not a valid atlas on $\mathbb{R}^2$ because the angular coordinate is discontinuous across the non-negative x-axis.
| -1 | https://mathoverflow.net/users/7956 | 33735 | 21,864 |
https://mathoverflow.net/questions/33567 | 8 | I'm trying to find the minimal (monic) polynomial $M(x)$ (over the rationals) for an algebraic number. I know the degree of the polynomial (call it $d$) and I have $d+1$ data points of the form $(x\_i, |M(x\_i)|)$. The $x\_i$ are all rational numbers, so the $|\cdot|$ is just regular absolute value.
If it wasn't for ... | https://mathoverflow.net/users/3400 | Monic polynomial from absolute value information | Here's a solution using lattice reduction:
1) Find degree $d$ polynomials $p\_i(x)$ such that $p\_i(x\_j) = |M(x\_i)| \delta\_{i j}$.
2) Let $c\_i$ be the coefficient of $x^d$ in $p\_i(x)$, and $c$ the $d+1$ long column vector whose coordinates are $c\_i$.
3) Find a matrix $U \in SL\_{d+1}(\mathbb{Z})$ such that ... | 2 | https://mathoverflow.net/users/2784 | 33737 | 21,865 |
https://mathoverflow.net/questions/33720 | 1 | Dear all;
Let $\Sigma$={a,b,c,d}, and $\delta$ be a function that returns a string $S$ of infinite length over $\Sigma$, where each character $s \in S$ has been chosen uniformly at random. My questions are the following:
1. What would be an intuitive notation
for $\delta$?
2. Is $\delta$, as I've
defined it above ... | https://mathoverflow.net/users/7995 | Permutation with repetitions of a vocabulary | For your first question, I would simply let $X$ be a random infinite string from the alphabet $\Sigma$. I don't see anything wrong with calling it $\delta$, either. I would treat it as a random variable, though. I don't really see the problem with treating it as a nondeterministic function either, but I'm not a compute... | 1 | https://mathoverflow.net/users/35336 | 33738 | 21,866 |
https://mathoverflow.net/questions/33736 | 2 | $(A,\mathfrak{m})$ a Noetherian local ring, $M\neq 0$ a finitely generated $A$-module. As I understand, $\mbox{Ext }^{j}(A/\mathfrak{m}, M) = 0$ for $j<\mbox{depth }(M)$ and for $j>\mbox{inj. dim }(M)$, while $\mbox{Ext }^{j}(A/\mathfrak{m}, M) \neq 0$ for $j=\mbox{depth }(M)$ and $j=\mbox{inj. dim }(M)$. And I cannot ... | https://mathoverflow.net/users/5292 | Homological dimensions of module | Yes. See Fossum, Foxby, Griffith, and Reiten, ["Minimal injective resolutions with applications to dualizing modules and Gorenstein modules"](http://archive.numdam.org/article/PMIHES_1975__45__193_0.pdf) (Theorem 1.1) and also Roberts, "[Two applications of dualizing complexes over local rings](http://archive.numdam.or... | 4 | https://mathoverflow.net/users/460 | 33743 | 21,869 |
https://mathoverflow.net/questions/33774 | 26 | Let $k$ be a field (I'm mainly interested in the case where $k$ is a number field, however results for other fields would be interesting), and $X$ a smooth projective variety over $k$.
By a zero cycle on $X$ over $k$ I mean a formal sum of finitely many (geometric) points on $X$, which is fixed under the action of t... | https://mathoverflow.net/users/5101 | Existence of zero cycles of degree one vs existence of rational points | There has been a lot of work on this problem, although nothing like a general answer is known. By way of abbreviation, the **index** of a nonsingular projective variety is the least positive degree of a $k$-rational zero cycle, so you are asking about the relationship between index one and having a $k$-rational point.
... | 29 | https://mathoverflow.net/users/1149 | 33778 | 21,884 |
https://mathoverflow.net/questions/33710 | 3 | Let $K$ be a non-Archimedean local field, i.e., complete with respect to a non-trivial, non-archimedean discrete absolute value, with finite residue field $k$ of characteristic $p\neq 0$. Also let $K\_s$ be a fixed separable closure of $K$, and $K\_{un}$ (resp. $K\_t$) the maximal unramified (resp. tamely ramified) ext... | https://mathoverflow.net/users/4351 | What are the open normal subgroups of the inertia group of a local field? | As all the responses indicate, the answer to my question is "yes." The most direct route seems to be the one suggested by KConrad. Explicitly, if $F/K\_{un}$ is Galois of degree $e$ (inside $K\_s$), then the ring of integers of $F$ is a DVR, and if $\Pi$ is a uniformizer for $O\_F$, then because $F/K\_{un}$ is totally ... | 3 | https://mathoverflow.net/users/4351 | 33782 | 21,887 |
https://mathoverflow.net/questions/33784 | 1 | As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }\_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what can come in as $X$. But when I consider $0\rightarrow \mathbb{Z}\rightarrow X\rightarrow \mathbb{Z}/(3)\rightarrow 0 $,$... | https://mathoverflow.net/users/5292 | Extension problem | $\mathrm{Ext} = \mathrm{Ext}^1$ does not classify the middle terms up to isomorphism. It classifies the short exact sequences up to equivalence, where the equivalence relation is generated by commutative diagrams having equalities on each end. It's possible for two inequivalent short exact sequences to have isomorphic ... | 6 | https://mathoverflow.net/users/460 | 33789 | 21,890 |
https://mathoverflow.net/questions/33145 | 3 | Suppose we have a [Markov Random Field](http://en.wikipedia.org/wiki/Markov_random_field) P(X1,...,Xn) on graph G. Suppose we know P(Xi,Xj) for every edge (i,j). Can we recover P(X1,...,Xn)?
If G is a tree, then there's a formula for joint (product of edge marginals divided by product of node marginals). Is there a n... | https://mathoverflow.net/users/7655 | Recovering joint distribution from marginals | No, a counter-example can be constructed as follows:
Let G be the complete graph on 3 vertices, where each vertex is a binary random variable. Let the joint distribution for each pair of vertices be independent Bernoulli with probability 1/2.
There are multiple joint distributions which satisfy such edge marginals... | 2 | https://mathoverflow.net/users/8019 | 33794 | 21,892 |
https://mathoverflow.net/questions/33798 | 7 | Let $E\_k$ be the normalized Eisenstein series of weight k and let p be an odd prime. Then
$$
E\_{p^m(p-1)} = 1 \mod p^{m+1},
$$
and so the p-adic limit $\lim E\_{p^m(p-1)} = 1$ is a p-adic modular form of weight 0. (It is even overconvergent.)
>
> **Question:** Suppose f is a p-adic modular form whose q-expan... | https://mathoverflow.net/users/2 | Can a the q-expansion of a p-adic modular form be a non-constant polynomial? | It is. I want to argue the following way: if the polynomial is non-constant then after scaling it has integral coefficients and so the reduction of the p-adic form mod p^n will be a *classical* form whose q-expansion is a non-constant polynomial. But I think Katz proved in his Antwerp paper that the only modular forms ... | 13 | https://mathoverflow.net/users/1384 | 33800 | 21,895 |
https://mathoverflow.net/questions/33805 | 5 | The fact that the atlas using $\phi: x \mapsto x^{1/3}$ on $\mathbb{R}$ is diffeomorphic to the trivial atlas using $\psi: x \mapsto x$ on $\mathbb{R}$ highlights my ignorance of diffeomorphisms and atlases. Apologies in advance for clustering several questions, but I'm not sure how to disentangle them.
First of all,... | https://mathoverflow.net/users/7956 | Why is the x->x^1/3 atlas on R diffeomorphic with the x->x atlas on R? |
>
> The fact that the atlas using $x \mapsto x^{1/3}$ on $\mathbb{R}$ is diffeomorphic to the trivial atlas using $x \mapsto x$ on $\mathbb{R}$ highlights my ignorance of diffeomorphisms and atlases.
>
>
>
Other differential topologists should weigh in on this to confirm or deny, but as a differential topologist... | 21 | https://mathoverflow.net/users/45 | 33818 | 21,904 |
https://mathoverflow.net/questions/33816 | 1 | I have a Banach space $B$ and a continuous linear transformation $F:B \rightarrow B\times B$. One of the induced transformations $F(1):B \rightarrow B$ and $F(2):B \rightarrow B$ into the factors of the product has closed range. Must F have closed range? I have the max norm on the product, i.e., $\|F(x) \| = max\{\|F(1... | https://mathoverflow.net/users/8027 | Closed range for a continuous linear transformation | No. Consider $F(x)=(f(x),0)$ where $f$ does not have closed range.
| 2 | https://mathoverflow.net/users/2554 | 33824 | 21,908 |
https://mathoverflow.net/questions/33788 | 11 | This question is inspired by [Emerton's question](https://mathoverflow.net/questions/28776/does-the-ideal-class-of-the-different-of-a-number-field-have-a-canonical-square) whether the ideal class of the different has a canonical square root.
Consider the diagram (of elements; the groups these lie in are the ideal cl... | https://mathoverflow.net/users/3503 | Does the ideal class of the different have a functorial square root? | Taking $L$ to be the Hilbert class field of $K$, such a construction
would imply that the Steinitz class of $L/K$ is always trivial.
Yet this is false - take $K = \mathbb{Q}(\sqrt{-15})$ for example.
(EDIT): If $L = K(\sqrt{\alpha})$ is a tamely ramified extension of $K$, then
the Steinitz class is represented by an... | 12 | https://mathoverflow.net/users/nan | 33827 | 21,911 |
https://mathoverflow.net/questions/29347 | 3 | Given a finite set of statements known to be true, I need to derive all the "non-redundant" statements in disjunctive form using only literals that can be derived from this set of statements, i e all statements on the form (a ∨ b ∨ c ∨ ...) where a, b ... are literals. By non-redundant I mean that I do not wish to incl... | https://mathoverflow.net/users/7049 | Deriving the complete set of "non-redundant" true statements in disjunctive form in propositional logic | I believe your problem is: Given a boolean function $\phi$, find the set of its prime clauses (aka prime implicates). This is equivalent to finding all the prime implicants of $\lnot\phi$.
You will find in the paper [A Knowledge Compilation Map](http://scholar.google.co.uk/scholar?cluster=4020517321592725796) many re... | 2 | https://mathoverflow.net/users/840 | 33835 | 21,914 |
https://mathoverflow.net/questions/33817 | 32 | It is an open problem to prove that $\pi$ and $e$ are algebraically independent over $\mathbb{Q}$.
* What are some of the important results leading toward proving this?
* What are the most promising theories and approaches for this problem?
| https://mathoverflow.net/users/4361 | Work on independence of pi and e | People in model theory are currently studying the complex numbers with exponentiation.
Z'ilber has an axiomatisation of an exponential field (field with exponential function) that looks like the complex numbers with exp. but satisfies Schanuel's conjecture.
He proved that there is exactly one such field of the size of ... | 35 | https://mathoverflow.net/users/7743 | 33837 | 21,916 |
https://mathoverflow.net/questions/33833 | 8 | I've been reading the book of Hilton, Mislin, and Roitberg on Localization of Nilpotent Groups and Spaces. In Section II.2 they define a principal refinement at stage $n$ of a Postnikov system $$\cdots \to X\_n\overset{p\_n}{\to} X\_{n-1}\to \cdots$$
to be a factorization of $p\_n$ into a finite sequence of fibrations ... | https://mathoverflow.net/users/6646 | What is a principal refinement of a Postnikov system? | The key idea is that not all fibrations $E \to B$ with fibre an Eilenberg-MacLane space $K(\pi,n)$ can be constructed by pulling the principal path fibration $K(\pi,n) \to PK(\pi,n+1) \to K(\pi,n+1)$ along a classifying map $B \to K(\pi,n+1)$. If you can construct the fibration in this way then the classifying map is t... | 10 | https://mathoverflow.net/users/4910 | 33845 | 21,919 |
https://mathoverflow.net/questions/33842 | 23 | The Suzuki and Ree groups are usually treated at the level of points. For example, if $F$ is a perfect field of characteristic $3$, then the Chevalley group $G\_2(F)$ has an unusual automorphism of order $2$, which switches long root subgroups with short root subgroups. The fixed points of this automorphism, form a sub... | https://mathoverflow.net/users/3545 | Suzuki and Ree groups, from the algebraic group standpoint | It is not really a question of inner forms. What happens is that the
*algebraic group* $G\_2$ has an extra endomorphism $\varphi$ whose square
is the Frobenius map (over the appropriate finite field). Just as for any
algebraic group over a finite field $F$ its rational points over $F$ are the
fixed points of the Froben... | 17 | https://mathoverflow.net/users/4008 | 33847 | 21,921 |
https://mathoverflow.net/questions/32986 | 27 | This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO.
Consider the standard assumptions for ruler and compass constructions: We have an infinitely large sheet of paper, which we associate with the complex plan... | https://mathoverflow.net/users/7641 | How fast are a ruler and compass? | *Edit on July 31: Now the upper bound is tight (up to replacing n by O(n)). The improvement over the older version is in the argument after Lemma 1. Now we consider the Weil absolute logarithmic height of an algebraic number instead of the length.*
Here is a proof that D(n) < 22cn for some positive constant c>0 for s... | 17 | https://mathoverflow.net/users/7982 | 33857 | 21,928 |
https://mathoverflow.net/questions/33854 | 6 | How does one solve the diophantine equation $x^2 + y^2 = z^2 + w^2 $? Can solutions be parameterized in three variables analogously to the Pythagorean triples case?
| https://mathoverflow.net/users/1358 | integer solutions to quadratic forms | Here is the standard geometric argument: after extracting common factors, you are asking for rational points on the quadric $Q\colon x^2-w^2=z^2-y^2$ in $\mathbb{P}^3$, which is isomorphic to $\mathbb{P}^1\times\mathbb{P}^1$ (as Matt Young's comment explains). Projection from a point on $Q$ gives a birational map $p:Q\... | 15 | https://mathoverflow.net/users/5480 | 33864 | 21,932 |
https://mathoverflow.net/questions/33861 | 10 | Looking at the chart of cardinals in Kanamori's book, one realizes that all large cardinals are implied by stronger ones and imply weaker ones. For instance measurable implies Jonsson which implies zero sharp which implies weakly compact which implies Mahlo which implies inaccessible. So it seems as if all these large ... | https://mathoverflow.net/users/3859 | Large cardinals | By the well-known Levy-Solovay theorem, large cardinal properties are preserved under "small" forcing. Therefore CH is an assumption above ZFC which is not settled by large cardinal axioms.
| 10 | https://mathoverflow.net/users/3532 | 33870 | 21,935 |
https://mathoverflow.net/questions/32801 | 8 | A secret sharing scheme such as [Shamir's secret sharing](http://en.wikipedia.org/wiki/Shamir%27s_Secret_Sharing) allow to perform addition and multiplication for secret values so far as there is at least 3 participants. Addition of two secret values is done locally at each party by adding the corresponding local share... | https://mathoverflow.net/users/7685 | Is there a two-party multiplicative and additive secret sharing scheme ? | Unconditionnaly secure 2-party computation does not exist (unfortunately). This is derived from the impossibility of Oblivious Transfer. Also note that unconditionnaly secure OT is also impossible if the 2 parties are quantum.
| 1 | https://mathoverflow.net/users/6673 | 33884 | 21,942 |
https://mathoverflow.net/questions/33883 | 5 | Suppose $E$ is a vector bundle with structure group $G$ and let $P = F(E)$ be the frame bundle. Let $\mathfrak{g}\_P$ denote the associated bundle to the adjoint representation of $G$ on its Lie algebra (i.e. $\mathfrak{g}\_P = P \times\_G \mathfrak{g}$ where $\mathfrak{g}$ is the Lie algebra of $G$). Given a section o... | https://mathoverflow.net/users/8041 | How do you exponentiate a section of the adjoint bundle to get a gauge transformation? | I'm not sure what it is that you tried, but the exponential map should work. First of all, let $U\_i$ be a trivialising cover for $P$ and its associated bundles. A section through $\mathfrak{g}\_P$ is given by functions $\omega\_i: U\_i \to \mathfrak{g}$ which, on overlaps, transform according to
$$\omega\_i(p) = \op... | 7 | https://mathoverflow.net/users/394 | 33887 | 21,944 |
https://mathoverflow.net/questions/33877 | 10 | Let A be an algebra (or dg algebra). Where can I find a proof of HH\_\*(A) = HH\_\*(Mod\_A) and HH^\*(A) = HH^\*(Mod\_A)? (And does this hold for any A?) Here Mod\_A is, e.g., the category of left A-modules.
One reason why this is interesting/important/useful is because many categories which arise "in nature" are of ... | https://mathoverflow.net/users/83 | Hochschild (co)homology of A and of Mod_A | Basically this follows from the fact that the derived category of bimodules over two algebras is equivalent to the (suitably defined) functor category between the derived category of modules of each algebra. Say, Toen's paper on derived Morita equivalence. Then, the identity functor is given by the algebra itself inter... | 8 | https://mathoverflow.net/users/947 | 33889 | 21,945 |
https://mathoverflow.net/questions/33893 | 4 | There is a result on the dimension bound for ${M\_{g,n}}/S\_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is
$H\_{i}({M\_{g,n}}/S\_n)=0$, for $i\ge 6g-7+2n$ except $(g,n)=(0,3),(0,2),(0,1),(0,0),(1,1)$. This result (see Costello: Gromov-Witten potential associated to a TCFT) can b... | https://mathoverflow.net/users/2391 | Homology dimension of the mapping class group of a surface with boundary | There is a fibration sequence
$$\mathbb{S}(\Sigma\_g) \to \mathcal{M}\_{g}^1 \to \mathcal{M}\_g$$
where $\mathcal{M}\_g$ is the moduli space of Riemann surfaces, $\mathcal{M}\_{g}^1$ is the moduli space of Riemann surfaces with a single boundary, and $\mathbb{S}(\Sigma\_g)$ is the sphere bundle associated to the tangen... | 2 | https://mathoverflow.net/users/318 | 33899 | 21,950 |
https://mathoverflow.net/questions/33896 | 25 | I was always bothered by the definition of the cross product given in e.g. a calculus course because it's never made clear how one would go about defining the cross product in a coordinate-free manner. I now know, not one, but *two* ways of doing this, and I can't quite see how they're related:
* The cross product is... | https://mathoverflow.net/users/290 | How are these two ways of thinking about the cross product related? | To expand on Victork Protsak's comment, if $V$ is an $n$-dimensional real vector space with inner-product, the inner-product gives an isomorphism $V\to V^\*$ and hence $V\otimes V \to \mathrm{End}(V)$. Under this isomorphism, $\Lambda^2(V)$ is identified with skew-adjoint endomorphisms of $V$, which is precisely the Li... | 22 | https://mathoverflow.net/users/380 | 33903 | 21,953 |
https://mathoverflow.net/questions/33888 | 6 | For finite sets $A$ and $B$, it is clear that $A \subseteq B$ and $|A| \geq |B|$ implies $A = B$. While an obvious fact, it can sometimes be a nice shortcut in proofs.
Analogously, if $V$ and $W$ are finite-dimensional vector spaces such that $V \subseteq W$ and $dim\ V \geq dim\ W$ then $V = W$. This is an especiall... | https://mathoverflow.net/users/2036 | Proving equality of varieties by dimension counting | I guess this answers at least part of the question, so I will post it as an answer. As in my comment above: the correct generalization is that if $X \subset Y$ is a closed subvariety, if $X$ and $Y$ are both irreducible, and if $\dim X = \dim Y$, then $X = Y$. At this level of generality this is an application of the n... | 8 | https://mathoverflow.net/users/6545 | 33908 | 21,957 |
https://mathoverflow.net/questions/15204 | 13 | $\bf Definition.$ We define the space bounded communication in the following way. A and B are
supernatural beings capable of computing anything but
they only have a limited amount of memory and that is shared. The
minimum size of this common memory that they can use to evaluate
a given function $f$ for which both of th... | https://mathoverflow.net/users/955 | Space Bounded Communication Complexity of Identity | Here's an argument that I believe shows that $\log n - \omega(1)$ is impossible. (This argument came out of a discussion I had with Steve Fenner.)
Let Alice's input be $x\in\{0,1\}^n$ and let Bob's input be $y\in\{0,1\}^n$. Assume the shared memory stores states in $\{0,1\}^m$, and its initial state is $0^m$. We are ... | 8 | https://mathoverflow.net/users/7641 | 33912 | 21,959 |
https://mathoverflow.net/questions/33910 | 5 | There is classical description of cohomology ring of projective bundle. Is there an analog in quantum cohomology?
| https://mathoverflow.net/users/8051 | Quantum cohomology of projective bundles | There is a quantum Leray-Hirsch Theorem, do to Maulik and Pandharipande [here](http://arxiv.org/abs/math/0412503). It relates the Gromov-Witten theory of the projective line bundle to that of the base.
| 4 | https://mathoverflow.net/users/622 | 33926 | 21,968 |
https://mathoverflow.net/questions/33879 | 7 | Take multi-sorted first-order logic with equality, complex scalars, 1xn vectors, nx1 vectors, nxn matrices, addition and multiplication for each pair of sorts they make sense for, and hermitian transpose (which is conjugation on scalars). Is it decidable what sentences are [true for all n]? (there are 4 sorts, what sen... | https://mathoverflow.net/users/nan | Decidability of matrix algebra | The second problem (where real scalar variables and the comparison relation are also allowed) is equivalent to the first problem. Here is a standard argument showing this:
* A complex scalar variable z can be restricted to real values by requiring $z=\bar{z}$.
* The comparison x≤y can be replaced by $\exists z.\ x+z\... | 2 | https://mathoverflow.net/users/7982 | 33927 | 21,969 |
https://mathoverflow.net/questions/33923 | 4 | Let $(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}\_t)$ be a given
probability space with usual conditions, on which $W$ is a standard
Brownian motion. For $x \ge 0$, consider
$$X(t) = x + \int\_0^t \sigma (X(s)) dW(s)$$
Assume $\sigma \in C^{0,1/2}\_{loc}$, $\sigma(0) = 0$, $\sigma>0$ on $(0,\infty)$.
By [Karatzas an... | https://mathoverflow.net/users/5656 | Distribution of running maximum of a local martingale | No. It is true that $\mathbb{P}(X^\*\_T>\beta)=O(\beta^{-1})$, but you don't have a`little-o' bound. In fact it fails, and $\beta\,\mathbb{P}(X^\*\_T>\beta)$ converges to a strictly positive value, precisely when $X$ fails to be a martingale.
If $S$ is the first time at which $X$ hits $\beta>x$ then continuity gives ... | 4 | https://mathoverflow.net/users/1004 | 33938 | 21,974 |
https://mathoverflow.net/questions/33924 | 9 | Background: I am trying to work out some Ext calculations for finite flat group schemes over a ring where p is nilpotent. I know how to do these calculations for finite group schemes over a finite field $k$ using the anti-equivalence with Dieudonne modules. I also know how to do this for finite flat group schemes over ... | https://mathoverflow.net/users/8055 | How does one classify finite flat group schemes over a ring where p is nilpotent? | Since the case of interest is $W\_2(k)$ with perfect $k$ of characteristic $p > 2$, the answer is given by Ioan Berbec's 2009 paper "Group schemes over artinian rings and applications. In that paper (esp. section 3) he defines an essentially surjective additive functor to a certain semi-linear algebra category and prov... | 10 | https://mathoverflow.net/users/3927 | 33941 | 21,977 |
https://mathoverflow.net/questions/23943 | 59 | NEW CONJECTURE: There is no general upper bound.
Wadim Zudilin suggested that I make this a separate question. This follows
[representability of consecutive integers by a binary quadratic form](https://mathoverflow.net/questions/23690/)
where most of the people who gave answers are worn out after arguing over indefin... | https://mathoverflow.net/users/3324 | Can a positive binary quadratic form represent 14 consecutive numbers? | I just wanted to remark that if $p$ is a prime such that
$\ell$ splits in $F = \mathbb{Q}(\sqrt{-p})$ for all $\ell \le N$,
then one may prove the existence of $N$ consecutive integers which
are norms of integers in $\mathcal{O}\_F$, providing one is willing
to *assume* a standard hypothesis about prime numbers, namely... | 22 | https://mathoverflow.net/users/nan | 33961 | 21,992 |
https://mathoverflow.net/questions/33962 | 5 | Is the countability of the set of irrational algebraic numbers somehow reflected in a characteristic property of their decimal expansions?
| https://mathoverflow.net/users/8064 | Question on the decimal expansion of algebraic numbers | The answer is not known, and it is a conjecture of Borel that the answer is no. See [Words and Transcendence](http://arxiv.org/abs/0908.4034) by M. Waldschmidt for references, in particular your previous claim that every digit in irrational algebraic numbers occurs infinitely often is an open problem. We don't know if ... | 5 | https://mathoverflow.net/users/2384 | 33967 | 21,995 |
https://mathoverflow.net/questions/33945 | 17 | Let $\mathcal{O}(\mathbb{C})$ be the ring of entire functions, that is, those functions $f : \mathbb{C} \to \mathbb{C}$ which are holomorphic for all $z \in \mathbb{C}.$ For each $z\_0 \in \mathbb{C}$.
Are there any other maximal ideals in $\mathcal{O}(\mathbb{C})$ besides these obvious ones?
If anyone can give a c... | https://mathoverflow.net/users/4872 | What is the spectrum of the ring of entire functions? | Here's a more analytic description of exactly what knowing $V(f)$ tells you. Let us say $f$ ~ $g$ if their vanishing sets are the same, and moreover there exist positive constants $c,C$ such that $c\cdot ord\_g(z)< ord\_f(z)< C\cdot ord\_g(z)$ as $z$ ranges over the vanishing set. Then knowing $V(f)$ is the same is as ... | 12 | https://mathoverflow.net/users/5513 | 33969 | 21,997 |
https://mathoverflow.net/questions/33963 | 3 | Let $C\_1, \dots, C\_n$ be a family of disjoint simple curves in a surface $\Sigma$. If $C$ is any simple curve in $\Sigma$, it turns out that we can map $C$ to a curve $C'$ (via a homeomorphism of $\Sigma$) such that $C'$ only intersects each $C\_i$ at most twice. I believe I have a proof of this result that works for... | https://mathoverflow.net/users/2233 | Removing intersections of curves in surfaces | There are two questions here.
1) The fact about Dehn twists and isotopies is really a consequence of the fact that the [mapping class group](http://en.wikipedia.org/wiki/Mapping_class_group) of an orientable surface is generated by Dehn twists. For non-orientable surfaces, this is not true -- you also need the so-cal... | 3 | https://mathoverflow.net/users/317 | 33970 | 21,998 |
https://mathoverflow.net/questions/33942 | 12 | I believe this question is due to Erdős and Graham, and I think it is still open: does the base 3 expansion of $2^n$ avoid the digit 2 for infinitely many $n$?
If we concatenate the digits of $2^i$, $i \geq 0$, we produce the number $0.110100100010000...$. This number is not simply normal in base 2, so it is not norm... | https://mathoverflow.net/users/35336 | Do the base 3 digits of $2^n$ avoid the digit 2 infinitely often -- what is the status of this problem? | As of a few months ago, the status of the problem was: still unsolved. See the slides [Jeff Lagarias](http://www.math.lsa.umich.edu/~lagarias/) put up from a talk he gave in September 2009: <http://www.math.lsa.umich.edu/~lagarias/TALK-SLIDES/ternary-fields-2009sep.pdf>
An older reference is
<http://citeseerx.ist.ps... | 6 | https://mathoverflow.net/users/3684 | 33980 | 22,003 |
https://mathoverflow.net/questions/33964 | 4 | Let $\Delta \subset \mathbb{R}^n$ be an $n$-dimensional integral polytope, let $f$ be a Laurent polynomial in $n$-variables with coefficients in an extension of the integers and Newton polytope $\Delta$. We can view $f$ as both a Laurent polynomial over $\mathbb{C}$ and $\mathbb{F} \_ q$ for some prime power $q$. Suppo... | https://mathoverflow.net/users/8044 | Relationship between topological cohomology and $\ell$-adic cohomology | The way to study the topology of the situation was introduced by Khovanski in
"Newton polyhedra, and toroidal varieties" Funkcional. Anal. i Priložen. 11
(1977), no. 4, 56--64, 96. His result (if I have interpreted it correctly) is
that $X$ may be compactified as a hypersurface in a projective toric variety to
a smooth... | 6 | https://mathoverflow.net/users/4008 | 33981 | 22,004 |
https://mathoverflow.net/questions/25030 | 10 | Suppose that we have a 2d-regular graph whose edges are colored such that the edges of each color form a cycle of length 2d. (So if the graph has 2n vertices, then there are n colors.) Is it true that there always is a perfect matching containing one edge of each color?
Remarks. For d=2 there is a simple proof by Zol... | https://mathoverflow.net/users/955 | Can we select a rainbow matching if each degree is 6 and each colorclass is a C_6? | This is a little embarrassing, but it turned out that not even a (non-rainbow) matching is guaranteed to exist. The problem was solved on [this](http://www.renyi.hu/~emlektab/) workshop by a number of people, presented by Tamas Terpai. They raised the same question for bipartite graphs, for which a matching must always... | 2 | https://mathoverflow.net/users/955 | 33992 | 22,008 |
https://mathoverflow.net/questions/32889 | 24 | [K] refers to Kontsevich's paper "Deformation quantization of Poisson manifolds, I".
Background
----------
Let $X$ be a smooth affine variety (over $\mathbb{C}$ or maybe a field of characteristic zero) or resp. a smooth (compact?) real manifold. Let $A = \Gamma(X; \mathcal{O}\_X)$ or resp. $C^\infty(X)$.
Denote t... | https://mathoverflow.net/users/83 | A few questions about Kontsevich formality | To (1): Daniel is right, there is a map of homotopy Gerstenhaber algebras between the two algebras. However the full story is quite complicated and to show that the hochschild cochains form a homotopy Gerstenhaber algebra is hard, it's known as the Deligne conjecture. I don't know the details of the proof.
Recall tha... | 6 | https://mathoverflow.net/users/109 | 34002 | 22,015 |
https://mathoverflow.net/questions/33990 | 3 | can you give me a good paper (in the sense of a simple introduction) about Sasaki-Einstein manifolds?
Thank you and best regards
Florian M.
| https://mathoverflow.net/users/7015 | Paper about Sasaki-Einstein manifolds | Well, now there is a great *textbook* for Sasaki-Einstein geometry, by Boyer-Galicki: Sasakian Geometry.
[Here is a link to the book at Oxford University Press](https://global.oup.com/academic/product/sasakian-geometry-9780198564959), DOI: [10.1093/acprof:oso/9780198564959.001.0001](https://doi.org/10.1093/acprof:oso/9... | 4 | https://mathoverflow.net/users/6871 | 34006 | 22,019 |
https://mathoverflow.net/questions/34016 | 2 | Let $k$ be a field. What are the $k$-rational points of the affine $k$-scheme $\mathrm{Spec}(k[[t]])$, where $k[[t]]$ is the power series ring over $k$ (equivalently, what are the $k$-algebra morphisms $k[[t]] \rightarrow k$?)
I'm only sure about one point, namely the map $t \mapsto 0$. Do I have to assume some sort ... | https://mathoverflow.net/users/8070 | What are the k-rational points of k[[t]]? | $k[[t]]$ is a local ring with maximal ideal $(t)$ and the kernel of every $k$-homomorphism $k[[t]] \to k$ is a maximal ideal, thus *the* maximal ideal. Thus it factors as $k[[t]] \to k[[t]]/(t) = k \to k$ and $t \mapsto 0$ is the unique $k$-rational point.
| 5 | https://mathoverflow.net/users/2841 | 34018 | 22,026 |
https://mathoverflow.net/questions/33995 | 17 | In the first pages of SGA4 I read
>
> [...] *Cependant le seul univers connu est l'ensemble des symboles du type* {Ø,{Ø},{Ø,{Ø}}, ... } *etc. (tous les éléments de cet univers sont des ensembles finis et cet univers est dénombrable). En particulier, on ne connaît pas d'univers qui contienne un élément de cardinal i... | https://mathoverflow.net/users/7952 | Is {Ø,{Ø},{Ø,{Ø}}, ... } the only known universe? | The universe that Grothendieck intends to suggest by his notation is known in set theory as HF, the class of hereditarily finite sets, the sets that are finite and have all elements finite and elements-of-elements, and so on (the transitive closure should be finite). The set HF is the same as $V\_\omega$ in the Levy hi... | 21 | https://mathoverflow.net/users/1946 | 34024 | 22,031 |
https://mathoverflow.net/questions/34007 | 81 | Nowadays, the usual way to extend a measure on an algebra of sets to a measure on a $\sigma$-algebra, the Caratheodory approach, is by using the outer measure $m^\* $ and then taking the family of all sets $A$ satisfying $m^\* (S)=m^\* (S\cap A)+m^\* (S\cap A^c)$ for every set $S$ to be the family of measurable sets. I... | https://mathoverflow.net/users/35357 | Demystifying the Caratheodory Approach to Measurability | Here is an argument that may give some intuition:
Assume that $m^{\*}$ is an outer measure on $X$, and let us assume furthermore that this outer measure is finite:
$m^\* (X) < \infty$
Define an "inner measure" $m\_\*$ on $X$ by
$m\_\* (E) = m^\* (X) - m^\* (E^c) $
If $m^\*$ was, say, induced from a countably ... | 52 | https://mathoverflow.net/users/7392 | 34029 | 22,035 |
https://mathoverflow.net/questions/34010 | 43 | In an answer to the popular question on common false beliefs in mathematics
[Examples of common false beliefs in mathematics](https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/23568#23568)
I mentioned that many people conflate the two different kinds of formal Laurent series f... | https://mathoverflow.net/users/1149 | Explicit elements of $K((x))((y)) \setminus K((x,y))$ | Suppose $R$ is a domain with field of fractions $F$. Let $f\in F[[y]]$ and suppose that $f\in Frac(R[[y]])$. Then $f=h/g$ with $g,h\in R[[y]]]$ and we may assume that $g=b\_0+b\_1y+\dots$ with $b\_0\ne0$. Therefore
$$
b\_0g^{-1}=(1+(b\_1/b\_0)y+\dots)^{-1}\in R[[y/b\_0]]
$$
and so $f\in b\_0^{-1}R[[y/b\_0]]$.
So wh... | 15 | https://mathoverflow.net/users/5480 | 34033 | 22,036 |
https://mathoverflow.net/questions/34030 | 3 | Let $f \colon X \to S$ be a proper morphism form a seperated algebraic space $X$ to an affine noetherian scheme $S$.
Given a coherent sheaf $F$ on $X$, we know from Knutson's book, that the finiteness theorem holds: The $R^qf\_\*F$ are finite $\mathcal O\_S$-modules. For complete $S$, we also have the theorem of form... | https://mathoverflow.net/users/5273 | Semicontinuity and cohomological flatness for algebraic spaces | This is definitely true and there ought to (but may not) exist a reference. One
way to prove it is just to check that the usual proof for schemes extends. If we
follow Hartshorne for instance the crucial part is Proposition III:12.2 which
can be proven using an étale affine cover instead of an open affine one.
| 7 | https://mathoverflow.net/users/4008 | 34035 | 22,038 |
https://mathoverflow.net/questions/34041 | 1 | Hi--
Where can I find a proof of this theorem: For each $r \in \mathbb{Z}\_{+}$,
there exists a complex entire function $f(z)$ such that $f(r) \neq 0$ but
$f(r+1)=f(r+2)=\cdots =0$,
i.e. $f(z) \in I\_{r+1}$ but $f(z) \neq I\_{r}$, where
$I\_{r}= \{ f \in R \ | f(r)=f(r+1)= \cdots =0\}$
where $R$ is the ring of comp... | https://mathoverflow.net/users/1483 | Weierstrass Theorem | Why don't you start with a function with zeros at the integers,
for instance $\sin\pi z$, and then somehow eliminate the zero at $r$?
| 5 | https://mathoverflow.net/users/4213 | 34043 | 22,043 |
https://mathoverflow.net/questions/34044 | 6 | I have seen that if $G$ is a finite group and $H$ is a proper subgroup of $G$ with finite index then $ G \neq \bigcup\limits\_{g \in G} gHg^{-1}$. Does this remain true for the infinite case also?
| https://mathoverflow.net/users/1483 | Group cannot be the union of conjugates | Not in general. Every matrix in $\text{GL}\_2(\mathbf C)$ is conjugate to an invertible upper triangular matrix (use eigenvectors), and the invertible upper triangular matrices are a proper subgroup.
| 23 | https://mathoverflow.net/users/3272 | 34046 | 22,045 |
https://mathoverflow.net/questions/33936 | 16 | Recall the notion of *Lie algebroid* ([n Lab](http://ncatlab.org/nlab/show/Lie+algebroid), [Wikipedia](http://en.wikipedia.org/wiki/Lie_algebroid)). One motivation for studying Lie algebroids is that they are infinitesimal versions of Lie groupoids, and Lie groupoids present stacks. In particular, Lie groupoids are the... | https://mathoverflow.net/users/78 | What is the 2-category whose 0-objects are Lie algebroids? | Here is one answer, but also sort of a no-go observation.
We are interested in creating a bicategory whose objects are Lie algebroids. The 1-morphisms and 2-morphisms are something yet to be determined. We have a couple simple requirements that we are going to demand:
1. We want a functor from the bicategory of Lie... | 7 | https://mathoverflow.net/users/184 | 34048 | 22,047 |
https://mathoverflow.net/questions/34055 | 7 | Can anyone suggest me an ingenious proof of the transcendence of $\pi$. I have seen Lindemann's proof but it appears intricate.
| https://mathoverflow.net/users/1483 | Transcendence of PI | There is a very nice book, "Irrational Numbers" by Ivan Niven. Available in paperback from the M.A.A. Evidently he gives a proof in the M.A.A.'s American Mathematical Monthly, volume 46 (1939) pages 469-471. His comment in the notes for chapter 9 of the book has "Proofs of the transcendence of $e$ and $\pi$ are not so ... | 11 | https://mathoverflow.net/users/3324 | 34056 | 22,050 |
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