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https://mathoverflow.net/questions/17152 | 26 | Sorry if this is a silly question. I was wondering, under what axioms of set theory is it true that if $\alpha$,$\beta$ are cardinals, and $2^\alpha=2^\beta$, then $\alpha=\beta$? Do people use these conditions to prove interesting results?
This question is prompted from a recent perusing of Johnson's "Topics in the ... | https://mathoverflow.net/users/1446 | When $2^\alpha = 2^\beta$ implies $\alpha=\beta$ ($\alpha,\beta$ cardinals) | François gives the correct affirmative answer. For the negative side, the usual method of proving that the negation of the Continuum Hypothesis is consistent with ZFC is to use the method of forcing to add Aleph2 many Cohen reals, so that 2ω = ω2 in the forcing extension V[G]. In this model V[G], it is also true that 2... | 21 | https://mathoverflow.net/users/1946 | 17154 | 11,478 |
https://mathoverflow.net/questions/17138 | 21 | I browsed Dirichlets Werke today and was kind of surprised by two remarks that he made on p. 354 (Über die Bestimmung ...) and p. 372 (Sur l'usage ...). In the second paper, he claims (my translation)
*I have applied these principles to a demonstration of the remarkable formula given by Legendre for expressing in an... | https://mathoverflow.net/users/3503 | Dirichlet and the prime number theorem | Dirichlet's remark from the first paper is extracted and translated on page 98 of The Development of Prime Number Theory by Narkiewicz. So this has not passed completely unnoticed. Narkiewicz remarks that Dirichlet believed that his analytic methods would enable him to prove Legendre's conjecture, and that Dirichlet ne... | 7 | https://mathoverflow.net/users/3304 | 17163 | 11,483 |
https://mathoverflow.net/questions/17115 | 28 | I am pretty sure that the following statement is true. I would appreciate any references (or a proof if you know one).
Let $f(z)$ be a polynomial in one variable with complex coefficients. Then there is the following dichotomy. Either we can write $f(z)=g(z^k)$ for some other polynomial $g$ and some integer $k>1$, or... | https://mathoverflow.net/users/4384 | Restriction of a complex polynomial to the unit circle | You're right. Quine proved in "[On the self-intersections of the image of the unit circle under a polynomial mapping](http://www.jstor.org/stable/2039005)" that if the degree is $n$ and $f(z)\neq g(z^k)$ with $k>1$, then the number of points with at least 2 distinct preimage points is at most $(n-1)^2$. An example show... | 20 | https://mathoverflow.net/users/1119 | 17164 | 11,484 |
https://mathoverflow.net/questions/17166 | 2 | It is well known that the 1-dimensional [heat equation](http://en.wikipedia.org/wiki/Heat_equation) $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a t}} \ \exp\left(-\frac{x^2}{4at}\right)$$.
**My question**
I am looki... | https://mathoverflow.net/users/1047 | Undergraduate Derivation of Fundamental Solution to Heat Equation | I think the term fundamental solution (at least sometimes) conventionally includes the integral around your $K$. I will assume this. If I recall correctly then the following argument is from "Partial Differential Equations" by Strauss.
A particularly simple solution follows from the self-similarity principle, i.e.
... | 9 | https://mathoverflow.net/users/3623 | 17167 | 11,486 |
https://mathoverflow.net/questions/17180 | 2 | Motivation
----------
The problem I am facing can be considered a variant of the standard set packing problem. However; instead of being given a list of sets, I am given a function $\nu : 2^N \rightarrow \{0,1\}$ and want to find a partitioning $P$ of $N$ that maximizes $g(P) = \sum\_{S \in P} \nu(S)$. This can be sh... | https://mathoverflow.net/users/4401 | Algorithmic aspects of maximizing a convex function over a convex set | For one, the KKT conditions still apply. But in general even simple problems can be tough. For instance, maximizing a p-norm over a fairly nice type of convex polytope ("parallelotope") is NP hard -- see Bodlaender et al, "Computational Complexity of Norm Maximization." The general wisdom is: minimizing convex function... | 4 | https://mathoverflow.net/users/1557 | 17183 | 11,494 |
https://mathoverflow.net/questions/12737 | 2 | Let S and S' be closed [Edit: orientable] surfaces, then it is well known that for S' to cover S it is necessary and sufficient that chi(S)|chi(S'). (Here 'chi' denotes the Euler characteristic).
However, if S and S' are punctured surfaces then the above condition is necessary but no longer sufficient.
>
> Is th... | https://mathoverflow.net/users/683 | Necessary and sufficient criteria for a surface to cover a surface | In a [recent paper](http://arxiv.org/abs/1003.0411) by Calegari, Sun, and Wang, the authors cite
W.S. Massey, Finite covering spaces of 2-manifolds with boundary. Duke Math. J. 41 (1974),
no. 4, 875-887.
which proves that Dmitri's conditions (1) and (2) are sufficient if the S and S' are compact, orientable with n... | 2 | https://mathoverflow.net/users/683 | 17194 | 11,502 |
https://mathoverflow.net/questions/17197 | 35 | This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on [this SBS post](http://sbseminar.wordpress.com/2008/09/10/a-curious-speculation/) and [this post I wrote](http://qchu.wordpress.com/2009/06/07/the-catalan-numbers-regular-languages-and-orthogonal-polynomials/) inspired by ... | https://mathoverflow.net/users/290 | How does this relationship between the Catalan numbers and SU(2) generalize? | The Catalan numbers enumerate (amongst everything else!) the bases of the Temperley-Lieb algebras. These algebras $TL\_n(q)$ are exactly $\operatorname{End}\_{U\_q \mathfrak{su}\_2}(V^{\otimes n})$ where $V$ is the standard representation.
If $q$ is a $2k+4$-th root of unity, the semisimplified representation theory ... | 28 | https://mathoverflow.net/users/3 | 17200 | 11,505 |
https://mathoverflow.net/questions/17208 | 2 | I believe that what I'm about to describe has a name—I'm almost certain that I've seen this in model theory and term-rewriting systems, possibly having something to do with ‘signature’ or ‘carrier’?—so please feel free to tell me if I'm using terms in bad, or non-standard, ways.
Consider a set $C$ of real constants. ... | https://mathoverflow.net/users/2383 | Decidable real arithmetic | There are at least two approaches to what you describe. In symbolic computation people ask questions of the sort you are asking, except that more generally one wants to know how to compute normal or canonical forms of terms, not just decide equality.
Another line of attack comes from computability theory. There is co... | 3 | https://mathoverflow.net/users/1176 | 17214 | 11,510 |
https://mathoverflow.net/questions/17195 | 2 | Let $A\_k$ be a sequence of real, rank $r$, $n$ x $m$ matrices such that $A\_k$ converges to a rank $r$ matrix $A$. Let $v\_k, u\_k$ be sequences of vectors such that $u\_k\rightarrow u$ and $A\_k v\_k=u\_k$. I would like to know if it is possible to show that there exist a vector $v$ such that $Av=u$. Appreciate any h... | https://mathoverflow.net/users/1172 | Sequence of constant rank matrices | I think it is best to settle this problem geometrically, that is if you think of matrices as linear maps from $\mathbb R^m$ to $\mathbb R^n$. The images of these maps are $r$-dimensional linear subspaces of $\mathbb R^n$. Let $X\_k$ denote the image of $A\_k$, then $u\_k\in X\_k$, and you want to prove that the limit v... | 5 | https://mathoverflow.net/users/4354 | 17220 | 11,515 |
https://mathoverflow.net/questions/17221 | 2 | If I have a group $G$ with two two subgroups $H$ and $I$ such that $G/H \approx G/I$, what can I say about the relationship between $H$ and $I$? Are they equal; isomorphic?
Sorry if this question is below the level of the site.
| https://mathoverflow.net/users/4409 | Can a Quotient of a Group by Two Different Subgroups be Isomorphic? | There exists groups $G$ with normal proper non-trivial subgroups $H$ such that $G \cong G/H$. Even finitely presented ones, as the one given by [Higman, Graham. A finitely related group with an isomorphic proper factor group. J. London Math. Soc. 26, (1951). 59--61. [MR0038347](http://www.ams.org/mathscinet-getitem?mr=... | 19 | https://mathoverflow.net/users/1409 | 17222 | 11,516 |
https://mathoverflow.net/questions/17223 | 4 | According to "The Geometry of Four-Manifolds" by Donaldson and Kronheimer, indefinite unimodular forms are classified by their rank, signature and type. This is the Hasse-Minkowski classification of indefinite forms, they say.
However, this seems to be a bit of a folklore theorem, as I cannot find a single citation f... | https://mathoverflow.net/users/1703 | What is a reference for the Hasse-Minkowski classification of indefinite forms? | For a survey of the topic check out Milnor and Husemoller, Symmetric Bilinear Forms, Springer, 1973, II.3. They cite Borevich-Shafarevich (Number Theory), O'Meara (Introduction to Quadratic Forms), and Serre (Cours d'Arithmétique).
| 5 | https://mathoverflow.net/users/1822 | 17224 | 11,517 |
https://mathoverflow.net/questions/17212 | 8 | Is there a description of finite groups whose all quotients have trivial center? Is it true that only direct products of non-abelian simple groups have this property?
| https://mathoverflow.net/users/4408 | Finite groups with centerless quotients | The answer to your second question is negative. Take a finite simple group $G$ and its nontrivial irreducible representation $V$ over finite field. Then the semi-direct product of $G$ and $V$ has a unique nontrivial quotient, the group $G$ itself.
| 10 | https://mathoverflow.net/users/4158 | 17225 | 11,518 |
https://mathoverflow.net/questions/17209 | 154 | I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice to include consequences of the generalized Riemann hypothesis (but specify which one is assumed).
| https://mathoverflow.net/users/828 | Consequences of the Riemann hypothesis | I gave a talk on this topic a few months ago, so I assembled a list then which could be appreciated by a general mathematical audience. I'll reproduce it here. (Edit: I have added a few more examples to the end of the list, starting at item m, which are meaningful to number theorists but not necessarily to a general au... | 255 | https://mathoverflow.net/users/3272 | 17232 | 11,521 |
https://mathoverflow.net/questions/17202 | 114 | I am interested in the function $$f(N,k)=\sum\_{i=0}^{k} {N \choose i}$$ for fixed $N$ and $0 \leq k \leq N $. Obviously it equals 1 for $k = 0$ and $2^{N}$ for $k = N$, but are there any other notable properties? Any literature references?
In particular, does it have a closed form or notable algorithm for computing ... | https://mathoverflow.net/users/4405 | Sum of 'the first k' binomial coefficients for fixed $N$ | I'm going to give two families of bounds,
one for when $k = N/2 + \alpha \sqrt{N}$ and one for when $k$ is fixed.
The sequence of binomial coefficients ${N \choose 0}, {N \choose 1}, \ldots, {N \choose N}$ is symmetric. So you have
$\sum\_{i=0}^{(N-1)/2} {N \choose i} = {2^N \over 2} = 2^{N-1}$
when $N$ is odd. ... | 78 | https://mathoverflow.net/users/143 | 17236 | 11,524 |
https://mathoverflow.net/questions/17230 | 10 | Let $\rho : S\_n \rightarrow \text{GL}(n, \mathbb{C})$ be the homomorphism mapping a permutation $g$ to its permutation matrix. Let $\chi(g) = \text{Trace}(\rho(g))$.
What is the value of $\langle \chi, \chi \rangle = \displaystyle \frac{1}{n!} \sum\_{g \in S\_n} \chi(g)^2$ ? Computing this expression for small $n$ y... | https://mathoverflow.net/users/4197 | Permutation representation inner product | You are computing the inner product of $\chi$ with itself.
Since $\chi=\mathrm{triv}+\mathrm{std}$ as a $S\_n$-module, with $\mathrm{triv}$ being the trivial reprsentation, and $\mathrm{std}$ its orthogonal complement, which is an irreducible $S\_n$-module, and since the inner product is, well, an inner product and ... | 23 | https://mathoverflow.net/users/1409 | 17237 | 11,525 |
https://mathoverflow.net/questions/17231 | 2 | D is a central division algebra over F. We know that we can always find a maximal subfield K inside D such that K/F is separable. I want to know can we always make it Galois?
| https://mathoverflow.net/users/2008 | Maximal subfield inside a central division algebra | I believe you are just asking if every central division algebra over F is a crossed product. This is not true, and the first example was given in:
Amitsur, S. A. "On central division algebras."
Israel J. Math. 12 (1972), 408-420.
[MR 318216](http://www.ams.org/mathscinet-getitem?mr=318216)
[DOI: 10.1007/BF02764632](h... | 6 | https://mathoverflow.net/users/3710 | 17240 | 11,527 |
https://mathoverflow.net/questions/17250 | 20 | Let $ \lambda\_1 \ge \lambda\_2 \ge \dots \ge \lambda\_{2n} $
be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any work or references that would consider the middle eigenvalues $\lambda\_n$ and $\lambda\_{n+1}$. In particular, the bounds on $R(G) = \m... | https://mathoverflow.net/users/4400 | The middle eigenvalues of an undirected graph | One thing that you might find useful is the [Cauchy interlacing theorem](https://en.wikipedia.org/wiki/Min-max_theorem#Cauchy_interlacing_theorem).
In response to the comment, presumably Tomaz's interest is in some particular sorts of graphs. It may be the case that such a graph has some well-understood subgraphs. Th... | 7 | https://mathoverflow.net/users/22 | 17254 | 11,536 |
https://mathoverflow.net/questions/17246 | 5 | Suppose a finite group $G$ acts on a standard probability space $(X, \mu)$ by measure-preserving actions (i.e. $\mu(g(A)) = \mu(A)$ for all $g \in G$ and $A \subset X$ measurable). In addition, suppose that for $g \in G$ and $g$ not the identity then $\mu(\{x:g(x) = x\})=0$. I am wondering if it is always possible to f... | https://mathoverflow.net/users/792 | Fundamental domains of measure preserving actions | I believe the answer is yes. You can first show, that once you have a set $A$ of non-zero measure, such that $g(A) = A$ for any $g \in G$, there is some $B \subset A$ of non-zero measure, such that $g(B) \cap h(B) = \emptyset$ for $g \ne h$. If this is shown, then you can prove your statement by Zorn-like argument, sho... | 4 | https://mathoverflow.net/users/896 | 17259 | 11,538 |
https://mathoverflow.net/questions/17233 | 3 | Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. Suppose the induced morphism of root data $f^\star: \Psi(G,T) \rightarrow \Psi(G,T)$ is a Frobenius morphism multiplying roo... | https://mathoverflow.net/users/717 | If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is then the isogeny itself a Frobenius? | The answer is "yes", but not in a good way: the descent it arises from is the split form, so this does not encode an interesting $\mathbf{F}\_ q$-structure. More precisely, $f$ arises from the $q$-Frobenius of a split descent down to $\mathbf{F}\_ q$ (and so even to the prime field $\mathbf{F}\_ p$). In particular, thi... | 5 | https://mathoverflow.net/users/3927 | 17263 | 11,542 |
https://mathoverflow.net/questions/17257 | 11 | What's the current state of Yang–Mills mass gap question, is there any place that does this problem? Especially I want to know if there is any progress (out of that mentioned in the introduction article by Witten and Jaffe). Is it too hard for a mathematician? Thanks!
| https://mathoverflow.net/users/2391 | What's the current state of Yang–Mills mass gap question? | There's some guidelines about open problems in the FAQ that you might want to read, but there was a good article by Faddeev last November that you should know about:
[Mass in Quantum Yang-Mills Theory, by Faddeev](http://arxiv.org/abs/0911.1013)
| 9 | https://mathoverflow.net/users/3623 | 17267 | 11,543 |
https://mathoverflow.net/questions/16966 | 3 | Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbb{C}$.
The polynomial current Lie algebra $\mathfrak{g}[t] = \mathfrak{g} \otimes \mathbb{C} [t]$
has the bracket
$$[xt^r, yt^s] = [x,y] t^{r+s}$$
for $x,y \in \mathfrak{g}$. It is graded with deg$(t) = 1$.
If we set $h=0$ in Drinfeld's fi... | https://mathoverflow.net/users/3316 | Why are relations of degree 3 or less enough in a presentation of the polynomial current Lie algebra g[t]? | For a nilpotent Lie algebra L, generators could be described in terms of $H\_1(L) = L/[L,L]$, and relations in terms of $H\_2(L)$. While this is not applicable directly to $\mathfrak g \otimes \mathbb C[t]$, it is close enough: it could be decomposed, for example, as the semidirect sum $\mathfrak g \oplus (\mathfrak g ... | 6 | https://mathoverflow.net/users/1223 | 17271 | 11,545 |
https://mathoverflow.net/questions/17205 | 13 | I am supervising an undergraduate for a project in which he's going to talk about the relationship between Galois representations and modular forms. We decided we'd figure out a few examples of weight 1 modular forms and Galois representations and see them matching up. But I realised when working through some examples ... | https://mathoverflow.net/users/1384 | Computing (on a computer) higher ramification groups and/or conductors of representations. | You can also compute some higher ramification groups in Sage. At the moment it gives lower numbering, not upper numbering, but here it is anyway:
`sage: Qx.<x> = PolynomialRing(QQ)`
`sage: g=x^8 + 20*x^6 + 146*x^4 + 460*x^2 + 1681`
`sage: L.<a> = NumberField(g)`
`sage: G = L.galois_group()`
`sage: G.ramificat... | 28 | https://mathoverflow.net/users/2481 | 17272 | 11,546 |
https://mathoverflow.net/questions/17226 | 22 | Recently, the Department of Mathematics at
our University issued a recommendation encouraging its members
to publish their research in non-specialized, mainstream mathematical journals. For
numerical analysts this will make an additional obstacle for their promotions. But even
for discrete mathematicians this recommend... | https://mathoverflow.net/users/4400 | Is discrete mathematics mainstream? | There are two different questions here, one objective and one subjective. I will try to give my view, for what it's worth. Bear with me.
First, you are asking what is the publication history of discrete mathematics? (Even if I suspect you know this much better than I do.) Well, originally there was no such thing as D... | 35 | https://mathoverflow.net/users/4040 | 17273 | 11,547 |
https://mathoverflow.net/questions/17255 | 11 | ### Informal Statement
In the $n\times n \times n$ grid, we can places rooks (those from chess) such that no two rooks can attack each other. One way to achieve this is to place a rook in position $(i,j,k)$ if and only if $i+j+k=0\mod n$. In general, there are "many" ways to do this.
Each such "attack-free" rook po... | https://mathoverflow.net/users/4416 | Counting colored rook configurations in the cube - when is it even? | This can be phrased as a problem concerning [Latin squares](http://en.wikipedia.org/wiki/Latin_square). Eg. a "rook set" is equivalent to a Latin square. For example:
```
123 100 010 001
231 <-> 001 100 010
312 010 001 100
```
A colouring of the Latin square is a partition of its entries (correspondi... | 13 | https://mathoverflow.net/users/2264 | 17274 | 11,548 |
https://mathoverflow.net/questions/13928 | 8 | Let $p$ be a prime number, $K$ a finite extension of $\mathbb{Q}\_p$, $\mathfrak{o}$ its ring of integers, $\mathfrak{p}$ the unique maximal ideal of $\mathfrak{o}$, $k=\mathfrak{o}/\mathfrak{p}$ the residue field, and $q=\operatorname{Card} k$.
Recall that a polynomial $\varphi=T^n+c\_{n-1}T^{n-1}+\cdots+c\_1T+c\_0$... | https://mathoverflow.net/users/2821 | When is the extension defined by an Eisenstein polynomial galoisian or abelian or cyclic ? | In the case where the ground field $K$ is $\mathbb{Q}\_p$, some old work of Lbekkouri has recently been published [here](https://doi.org/10.1007/s00013-009-0026-3 "On the construction of normal wildly ramified extensions over Q_p, (p \neq 2). Arch. Math. 93, 331 (2009)"). In particular, for that case, i.e. for finite t... | 3 | https://mathoverflow.net/users/3143 | 17287 | 11,559 |
https://mathoverflow.net/questions/17298 | 6 | Background
----------
The beth function is defined recursively by: $\beth\_0 = \aleph\_0$, $\beth\_{\alpha + 1} = 2^{\beth\_\alpha}$, and $\beth\_\lambda = \bigcup\_{\alpha < \lambda} \beth\_\alpha$. Since the beth function is strictly increasing and continuous, it is guaranteed to have arbitrarily large fixed points... | https://mathoverflow.net/users/4087 | Does the beth function have fixed points of arbitrarily large cofinality? | Yes. The definable class $C$ of fixed points $\beth\_\kappa = \kappa$ is closed and unbounded and therefore contains ordinals of all possible cofinalities. Specifically, let $\kappa\_\alpha$ denote the $\alpha$-th element of $C$. Note that $\kappa\_\delta = \sup\_{\alpha<\delta} \kappa\_\alpha$ for every limit ordinal ... | 4 | https://mathoverflow.net/users/2000 | 17300 | 11,569 |
https://mathoverflow.net/questions/17295 | 14 | Given a group of order $n$ where $n$ is either a specific number, or a number of a particular form, e.g. square-free, when does $n$ completely determine a particular group property among all groups of that order? Vipul's group theory wiki has several stubs on this topic, and in the language of his wiki, I will call thi... | https://mathoverflow.net/users/2616 | Results about the order of a group forcing a particular property. | The numbers n such that every group of order n is cyclic, abelian, nilpotent, supersolvable, or solvable are known. Most are described in an easy to read survey:
Pakianathan, Jonathan; Shankar, Krishnan. "Nilpotent numbers."
Amer. Math. Monthly 107 (2000), no. 7, 631-634.
[MR 1786236](http://www.ams.org/mathscinet-ge... | 17 | https://mathoverflow.net/users/3710 | 17304 | 11,573 |
https://mathoverflow.net/questions/17305 | 3 | Let $f \in \mathbf{F}\_q[T]$ be irreducible. I know that the ray class field for $\mathrm{Cl}((f)) \cong (\mathbf{F}\_q[T]/(f))^\times$ can be constructed by adjoining torsion points of a Carlitz module. Is there an easy explicit minimal polynomial for a generator of this extension?
| https://mathoverflow.net/users/nan | ray class field of rational function field | The minimal polynomial is $\phi\_f(X)/X$, where $\phi\_g$ (the Carlitz module) is defined by being $\mathbb{F}\_q$-linear in $g$, satisfy
$\phi\_{T^{n+1}} = \phi\_T(\phi\_{T^n})$ and $\phi\_T =X^q+TX$.
It even has the bonus of being an Eisenstein polynomial at $f$.
| 7 | https://mathoverflow.net/users/2290 | 17308 | 11,575 |
https://mathoverflow.net/questions/17268 | -2 | As we can see,there are some conditions given in a proposition.
If there are 2 propositions having approximate conclusions.Usually，we can name one propositions gives stronger conditions than the other.
My question is that whether there is a good system to weigh the conditions given in a proposition.e.g. a condition "un... | https://mathoverflow.net/users/4419 | how to weigh the conditions given in a proposition | I think that one may have there three kinds of valuations for "strength" of conditions in theorem.
* practical ones: for example some conditions may be easier to check, or even define whilst other may be difficult to check or even define in practical causes. Then You may use fro example "**computational complexity**"... | 1 | https://mathoverflow.net/users/3811 | 17311 | 11,578 |
https://mathoverflow.net/questions/17286 | 10 | This question is on "computing" the Grothendieck group of the projective $n$-space with $m$ origins ($m\geq 1$). For any (noetherian) scheme $X$, let $K\_0(X)$ be the Grothendieck group of coherent sheaves on $X$.
Firstly, let me sketch that $K\_0(\mathbf{P}^n) \cong K\_0(\mathbf{P}^{n-1})\oplus K\_0(\mathbf{A}^n)$.
... | https://mathoverflow.net/users/4333 | $K_0$ of a non-separated scheme | This is exactly the sort of example where it is relevant which version of K-theory you are employing. (In particular, which theorems you can employ here depends on this!) The issue here is a tad subtle.
To illustrate, let's restrict attention to the case of affine $n$-space $X$ with a doubled origin ($n\geq 2$). (An ... | 7 | https://mathoverflow.net/users/3049 | 17316 | 11,581 |
https://mathoverflow.net/questions/17323 | 2 | A paper I'm reading implicitly assumes the statement: Let $K\_0$ be the completion of $\mathbb {Q}\_ p^{un}$. Then any finite extension of $K\_0$ is complete with residue field $\bar {\mathbb {F}} \_p$. So here are a few questions:
1. Why is this true? Is it true in general that if you complete, and then take algebraic... | https://mathoverflow.net/users/3238 | Fiddling with p-adics | A finite extension of a complete field is complete. This is standard and proved in lots of places e.g. one of the first two chapters of Cassels-Froehlich, or Bosch-Guentzer-Remmert, or lots of other places. The residue field will only go up a finite amount too, so it must have stayed the same!
Your Q1 appears to hav... | 5 | https://mathoverflow.net/users/1384 | 17326 | 11,589 |
https://mathoverflow.net/questions/17331 | 1 | I have been asked to provide an "approximation at infinity" of an expression that at the end simplifies to $-\frac{b e^{-a t}-a e^{-b t}}{a-b}$, in a course about extreme value theory.
In the course, we saw "approximations" such as $2 t^{-\alpha }-t^{-2 \alpha }$ being approximated to $2 t^{-\alpha }$ whatever the vagu... | https://mathoverflow.net/users/3899 | Extreme value theory | Do you have information on the relative sizes of $a$ and $b$? If we have, say $a>b$, then we can say that as $t \to \infty$, $-\frac{b e^{-a t}-a e^{-b t}}{a-b}=-e^{-b t}(\frac{b e^{-(a-b) t}-a }{a-b}) \sim \frac{a e^{-b t}}{a-b} =\frac{e^{-b t}}{1-\frac{b}{a}}$. It is rather simple, but I don't think there's much more... | 0 | https://mathoverflow.net/users/4436 | 17336 | 11,597 |
https://mathoverflow.net/questions/17341 | 3 | Let $M$ be a smooth manifold. We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed $M$ into a Euclidean space of sufficiently large dimension, and we thus get a Riemannian metric on $M$ by restricting the Euclidean metric o... | https://mathoverflow.net/users/4437 | Are all Riemannian metrics induced by Euclidean metrics? [Nash Embedding Theorem] | Yes, see e,g, <http://en.wikipedia.org/wiki/Nash_embedding_theorem>
Briefly, every $C^1$--metric on a $C^1$-manifold is induced by a $C^1$-embedding $M^n\to\mathbf{R}^{2n+1}$; every $C^\infty$ metric on a $C^{\infty}$ manifold is induced by $C^\infty$-embedding $M^n\to\mathbf{R}^{n^2+5n+3}$.
| 12 | https://mathoverflow.net/users/2349 | 17343 | 11,603 |
https://mathoverflow.net/questions/17345 | 1 | we all know that if we consider the sheaf of germs of a holomorphic functions defined on a domain in C^n,we have too many beautiful theorems characterizing the geometry of the domain by consider the Cech-cohomology of the sheaf.Then i think that plurisubharmonic functions is in some sense a weaker function than holomor... | https://mathoverflow.net/users/4437 | the Cech-cohomology of the sheaf of germs of plurisubharmonic functions defined on a domain in C^n | As Petya has pointed out, plurisubharmonic functions on an open set do not form a group, so when one sheafifies, one gets a sheaf of sets, not groups; it has H^0, but no higher cohomology.
| 4 | https://mathoverflow.net/users/2349 | 17350 | 11,608 |
https://mathoverflow.net/questions/17344 | 20 | In a paper I developed some theory; some of the applications require extensive computations that are not part of the paper. I wrote a Mathematica notebook. I want to publish a PDF and .nb version somewhere to refer to from the paper. arXiv.org seems a good choice, but they won't accept the .nb file. I do not want to pu... | https://mathoverflow.net/users/4438 | Where to publish computer computations | You should upload the notebook along with the sources of the actual article to the arXiv, of course.
The official (but rather hard to find) advice on this from the arXiv is to place your code in a directory called /aux/. (This is [problematic](http://en.wikipedia.org/wiki/LPT#Naming) for windows users.)
You can se... | 31 | https://mathoverflow.net/users/3 | 17351 | 11,609 |
https://mathoverflow.net/questions/17325 | 32 | You don't need a metric to define the differential of a function,
and the cotangent bundle carries a canonical one-form.
But you do need a metric to define the gradient, and the
tangent bundle does not have a canonical vector field.
These are not difficult truths, but still... why the preference
toward "co"?
| https://mathoverflow.net/users/1186 | Why is cotangent more canonical than tangent? | If you want to differentiate functions *from* a manifold to (say) the real line R, then you want to use the cotangent bundle on the manifold.
If instead you want to to differentiate functions *to* the manifold from the real line (i.e. parameterised curves), then you want to use the tangent bundle on the manifold.
S... | 69 | https://mathoverflow.net/users/766 | 17355 | 11,610 |
https://mathoverflow.net/questions/17358 | 6 | Is finding a cycle of fixed even length in a bipartite graph any easier than finding a cycle of fixed even length in a general graph? This question is related to the question on [Finding a cycle of fixed length](https://mathoverflow.net/questions/16393/finding-a-cycle-of-fixed-length)
**Edit:** There is natural refin... | https://mathoverflow.net/users/4400 | Finding a cycle of fixed length in a bipartite graph | Finding a cycle of length 2k in an arbitrary graph is the same thing as finding a cycle of length 4k in the bipartite graph formed by subdividing every edge. So in general even cycles of fixed length are no easier to find in bipartite graphs than in arbitrary graphs. But it's possible that the lengths that are 2 mod 4 ... | 11 | https://mathoverflow.net/users/440 | 17359 | 11,612 |
https://mathoverflow.net/questions/17360 | 6 | A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason other than historical that we don't think of the Serre SS this way initially?
Any thoughts or references would be great!
| https://mathoverflow.net/users/3901 | Serre spectral sequence with spectra | Since the ordinary Serre spectral sequence is about a fibration of spaces, I don't think you can just talk about a "fibration of spectra" and expect that to be a generalization, since the suspension spectrum functor doesn't preserve fibration sequences. However, there is a version of the Serre spectral sequence involvi... | 10 | https://mathoverflow.net/users/49 | 17361 | 11,613 |
https://mathoverflow.net/questions/17376 | 0 | How can one show that rational functions satisfy a Lipschitz condition *in the chordal metric* on the Riemann sphere?
| https://mathoverflow.net/users/nan | How to prove that rational functions satisfy a Lipschitz condition in the *chordal metric*? | By compactness, you only need to prove the Lipschitz property locally. First prove that Möbius transforms are Lipschitz (easy – they are compositions of translations, multiplications by constants, and inversions). Then, by composing with suitable Möbius transforms, you only need to show that a rational function which m... | 2 | https://mathoverflow.net/users/802 | 17379 | 11,624 |
https://mathoverflow.net/questions/17371 | 14 | In the undergraduate toplogy course we were given examples of spaces that are $T\_i$ but not $T\_{i+1}$ for $i=0,\ldots,4$. However, no example of a space which is $T\_3$ but not $T\_{3.5}$ was given. Later I was told by a colleague that such examples are rare and difficult to construct.
I know there is an example of... | https://mathoverflow.net/users/2578 | Regular spaces that are not completely regular | These examples seem to be very difficult to construct. The problem is that any local compactness or uniformity will automatically boost your space to a Tychonoff space, and Tychonoff spaces are closed under passing to subspaces or products. Consequently, there's doesn't seem to be a "machine" for producing these kinds ... | 15 | https://mathoverflow.net/users/3049 | 17382 | 11,626 |
https://mathoverflow.net/questions/17396 | 24 | To be more precise (but less snappy): is there an example of a group $G$ with finitely many infinite-index subgroups $H\_1,\dots, H\_n$ and elements $k\_1,\dots, k\_n$ such that $G$ is the union of the left cosets $k\_1 H\_1 , ..., k\_n H\_n\ ?$ And what if we relax the requirement that these all be *left* cosets, and ... | https://mathoverflow.net/users/93 | Can a group be a finite union of (left) cosets of infinite-index subgroups? | No. This follows from a beautiful theorem of B.H. Neumann:
Let $G$ be a group. If $\{x\_iH\_i\}\_{i=1}^n$ is a covering of $G$ by cosets of proper subgroups, then $n \geq \min\_{i} [G:H\_i]$.
Explicitly, this is Lemma 4.1 in
[http://alpha.math.uga.edu/~pete/Neumann54.pdf](http://alpha.math.uga.edu/%7Epete/Neumann... | 38 | https://mathoverflow.net/users/1149 | 17398 | 11,636 |
https://mathoverflow.net/questions/17395 | 1 | As the title shows,we know that there is some points the series not approaching to the function.
Now,take the convergence theorem into consideration.As there is some the first break-points，the series is still convergent.And,the Gibbs phenomenon always takes place on the first break-points.
Why does Gibbs phenomenon... | https://mathoverflow.net/users/4419 | What does Gibbs phenomenon shows the nature of Fourier Series | A Fourier series truncated to order $n$ is the best approximation to the given function *in the $L^2$ sense* using trigonometric polynomials of order $n$. As such, small rapid deviations don't matter much. Since there is a limit to how big the derivatives of a trigonometric polynomial of fixed order can be (without the... | 11 | https://mathoverflow.net/users/802 | 17399 | 11,637 |
https://mathoverflow.net/questions/17388 | 10 | Suppose A and B are compact connected sets in the XY plane and XZ plane respectively in R^3. Suppose further that the the range of x-values taken by A and B are the same (i.e, projections of A and B onto the x-axis are the same closed interval). Is there always a connected set in R^3 whose projections onto XY and XZ pl... | https://mathoverflow.net/users/996 | existence of a connected set with given connected projections. | The answer is yes.
Let $\alpha,\beta:[0,1]\to[0,1]\times \mathbb R$ be two paths;
$\alpha(t)=\left(\alpha\_1(t),\alpha\_2(t)\right)$ and $\beta(t)=\left(\beta\_1(t),\beta\_2(t)\right)$.
Assume that $\alpha\_1(0)=\beta\_1(0)=0$, $\alpha\_1(1)=\beta\_1(1)=1$.
>
> **Claim.** The points $a=\left(0,\alpha\_2(0),\beta\... | 6 | https://mathoverflow.net/users/1441 | 17405 | 11,643 |
https://mathoverflow.net/questions/17411 | 8 | Where's the notion of interpretation (model) originally introduced?
I find it used in Skolem's paper "Logico-combinatorial investigations in the satisfiability or provability of mathematical propositions" (1920).
However, I do not find any explicit reference to semantical ideas in Frege's Begriffsschrift. Where did the... | https://mathoverflow.net/users/3872 | Where's the notion of interpretation (model) originally introduced? | **Updated answer:**
As best as I have been able to figure out, the pre-Tarskian notions of "semantics" in mathematical logic grew out of the "algebra of logic" introduced by Boole ("An investigation into the laws of thought," 1854, and some earlier papers) and elaborated by Charles Peirce, Schröder, and others.
It'... | 3 | https://mathoverflow.net/users/93 | 17419 | 11,651 |
https://mathoverflow.net/questions/17422 | 7 | Let $Q$ be a quadratic form in $n$ variables with integer coefficients. Let us say that $Q$ has the "special property" mod $p$, if the relation $Q(x\_1,...,x\_n)=0$ (mod $p$) implies that $(x\_1,...,x\_n)=(0,...,0)$ (mod $p$). (There must be a name for this property, but I don't know it, which is why I'm calling it "th... | https://mathoverflow.net/users/4467 | Quadratic forms that evaluate to zero mod p only when their input is zero. | The Chevalley–Warning theorem (see wikipedia) implies that any quadratic form in at least three variables has a non-trivial solution modulo p.
| 11 | https://mathoverflow.net/users/nan | 17426 | 11,654 |
https://mathoverflow.net/questions/17425 | 17 | Suppose I have a small category $ \mathcal{C} $ which is fibered over some category $\mathcal{I}$ in the categorical sense. That is, there is a functor $\pi : \mathcal{C} \rightarrow \mathcal{I}$ which is a fibration of categories. (One way to say this, I guess, is that $\mathcal{C}$ has a factorization system consisti... | https://mathoverflow.net/users/4466 | Homotopy Limits over Fibered Categories | I can't think of a reference for this. But here is what I would do:
Given any functor $\pi\colon C\to I$ (not necessarily fibered), there's a "homotopy right Kan extension" functor
$$\lim{}^\pi \colon Func(C,sS) \to Func(I,sS),$$
and a weak equivalence $\lim\_C = \lim\_I \lim{}^\pi$. There's a formula to compute $\l... | 13 | https://mathoverflow.net/users/437 | 17434 | 11,661 |
https://mathoverflow.net/questions/17424 | 1 | I am looking for reference talking about how torsion theory play roles in algebraic geometry. I will be really happy to see some concrete examples. Say, talking about torsion theory in $Coh(P^{1})$.
Thanks in advance
| https://mathoverflow.net/users/1851 | Looking for reference talking about torsion theory on coherent sheaves on projective space | Depending upon how strict you are with your definition of torsion theory a good source of examples is the theory of semi-orthogonal decompositions. A really nice example of this is the appearance of such decompositions which parallel operations in the minimal model program. A good introduction to this is [Kawamata's s... | 3 | https://mathoverflow.net/users/310 | 17438 | 11,663 |
https://mathoverflow.net/questions/17437 | 5 | Let $a\_1, ... a\_n$ be real numbers. Consider the operation which replaces these numbers with $|a\_1 - a\_2|, |a\_2 - a\_3|, ... |a\_n - a\_1|$, and iterate. Under the assumption that $a\_i \in \mathbb{Z}$, the iteration is guaranteed to terminate with all of the numbers set to zero if and only if $n$ is a power of tw... | https://mathoverflow.net/users/290 | Does this problem have a name? [Ducci Sequences] | This is known as [Ducci's problem](http://en.wikipedia.org/wiki/Ducci_sequence). "Ducci map" or "Ducci sequence" as key words should let you search on most of the articles studying properties of the above map.
| 11 | https://mathoverflow.net/users/2384 | 17439 | 11,664 |
https://mathoverflow.net/questions/17409 | 10 | There are some 'standard' applications of the adjoint functor
theorem (AFT) and the special adjoint functor theorem (SAFT), for
example, the existence of a free $\tau$-algebra (where
$\tau=$(operations,identities)) on a small set by the AFT,
Stone-Cech compactification by the SAFT, and, if I am not mistaken, the pro... | https://mathoverflow.net/users/2734 | What are the 'standard' applications of the duals of the adjoint functor theorems? | One example is the construction of [geometric morphisms](http://ncatlab.org/nlab/show/geometric+morphism). Any colimit-preserving functor between Grothendieck toposes has a right adjoint, so if it also preserves finite limits, then it is part of a geometric morphism. Of course, in many cases in practice, the right adjo... | 7 | https://mathoverflow.net/users/49 | 17445 | 11,669 |
https://mathoverflow.net/questions/17440 | 3 | We consider $n\times n$ complex matrices. Let $i\_+(A), i\_-(A), i\_0(A)$ be the number of eigenvalues of $A$ with positive real part, negative real part and pure imaginary. It is well known if two Hermitian matrices $A$ and $B$ are $\*-$congruent, then
$$(i\_+(A), i\_-(A), i\_0(A))=(i\_+(B), i\_-(B), i\_0(B)).\qquad{... | https://mathoverflow.net/users/3818 | A question on star-congruence. | An answer to the second question: Yes, a square complex matrix is always $\*$-congruent to its transpose, according to a more general result proved by Horn and Sergeichuk in "[Congruences of a square matrix and its transpose](http://arxiv.org/abs/0709.2489)". They prove the result for all fields with involution in char... | 4 | https://mathoverflow.net/users/1119 | 17446 | 11,670 |
https://mathoverflow.net/questions/17452 | 9 | I was wondering about the paper by Bernstein, Gel'fand, and Gel'fand on [Schubert Cells](http://iopscience.iop.org/0036-0279/28/3/R01;jsessionid=E4D2896AE55597FA4F4F63297F63FBC6.c3). This paper is fairly old(and often cited) so I figured someone must have represented this material. In particular, I was wondering if thi... | https://mathoverflow.net/users/348 | Expository treatment of Schubert Cells Paper | The Lecture Notes in Mathematics number 1689, "Schubert Varieties and Degeneracy Loci" by Fulton and Pragacz seems to be exactly what you're looking for. I think chapter 6 is particularly relevant.
| 5 | https://mathoverflow.net/users/788 | 17455 | 11,675 |
https://mathoverflow.net/questions/17458 | 3 | I can calculate a likelihood ratio statistic to measure how well my data fits either of two mutually exclusive models:
$\Lambda({\text{data}}) = \frac{ f( \theta\_0 | {\text{data}} )}{f( \theta\_1 | {\text{data}} )} $
Calculating this statistic is straightforward to me, but I'd like some measure of statistical conf... | https://mathoverflow.net/users/4476 | Confidence intervals for likelihood ratios | Yes, in principle, the distribution function of the likelihood ratios statistic can be computed, since it is a function of the random variables "data" and the model parameters theta\_0 and theta\_1. Suppose, that cumulative distribution function is calculated, the p-value can be computed giving a confidence estimate.
O... | 2 | https://mathoverflow.net/users/1059 | 17459 | 11,676 |
https://mathoverflow.net/questions/17258 | 3 | The most appealing statement of the Bessis-Moussa-Villani conjecture is as follows:
>
> **Conjecture:** For all Hermitian positive semidefinite $n\times n$ matrices $A$ and $B$,
> and all positive integer $m$, the polynomial function
> $$t \in \mathbb{R}\mapsto g(t) \equiv tr[(A + t B)^m] =
> \sum\limits\_{
> k=0... | https://mathoverflow.net/users/3818 | What's known about the 3rd coefficient in the BMV conjecture? | Look at <http://arxiv.org/abs/0802.1153>.
Here the fourth coefficient is shown to be non-negative which implies by
Hillar's decent theorem
<http://arxiv.org/abs/math/0507166>
also the third coefficient to be non-negative.
| 2 | https://mathoverflow.net/users/4479 | 17462 | 11,678 |
https://mathoverflow.net/questions/17472 | 3 | Consider $\mathbb{R}^n$ as measurable space with the Borel algebra. If $\mathbb{R}^n$ and $\mathbb{R}^m$ are isomorphic (in the category of measurable spaces, i.e. there are measurable maps in both directions, which are inverse to each other), can we conclude $n=m$? Note that this statement is stronger than the invaria... | https://mathoverflow.net/users/2841 | Dimension of the measurable space $\mathbb{R}^n$ | All uncountable standard Borel spaces are Borel isomorphic (see A. S. Kechris, Classical Descriptive Set Theory, page 90, Theorem 15.6).
| 16 | https://mathoverflow.net/users/3096 | 17475 | 11,684 |
https://mathoverflow.net/questions/2147 | 128 | What are really helpful math resources out there on the web?
Please don't only post a link but a short description of what it does and why it is helpful.
Please only one resource per answer and let the votes decide which are the best!
| https://mathoverflow.net/users/1047 | Most helpful math resources on the web | I occasionally find [mathoverflow.net](https://mathoverflow.net/) rather helpful.
In particular, there's a good list of answers to your specific question [here](https://mathoverflow.net/questions/2147/most-helpful-math-resources-on-the-web).
| 168 | https://mathoverflow.net/users/35575 | 17476 | 11,685 |
https://mathoverflow.net/questions/17466 | 7 | Is there a notion (for schemes or just locally ringed spaces) of cohomology with compact support? I guess there is for algebraic schemes over $\mathbf{C}$, but what about schemes in general? Does anybody have a good reference?
| https://mathoverflow.net/users/4481 | Cohomology with compact support for coherent sheaves on a scheme | Yes there is. Take $S$ a scheme and take $f \colon X \to S$ a compactifiable morphism of schemes. By definition this means that there exists a proper $S$-scheme Y which contains $X$ as an open subscheme. Then, given a compactification and a sheaf on $X$, you may define the cohomology with proper support of this sheaf a... | 8 | https://mathoverflow.net/users/4398 | 17494 | 11,698 |
https://mathoverflow.net/questions/17486 | 14 |
>
> Is it possible to construct smooth embedded of 2-discs $\Sigma\_1$ and $\Sigma\_2$ in $\mathbb R^3$ with the same boundary curve such that there is no pair of points $p\_1\in \Sigma\_1$ and $p\_2\in \Sigma\_2$ with parallel tangent planes?
>
>
>
**Comments:**
* The question is inspired by [this](https://ma... | https://mathoverflow.net/users/1441 | Two discs with no parallel tangent planes | Suppose two such disks Σ1 and Σ2 exist, and pull back TΣ2 to Σ1 by some homeomorphism. Viewed as a subbundle of TR3|Σ1, this plane bundle intersects TΣ1 in a line bundle L over Σ1 since no two tangent planes are parallel. Furthermore, L|∂Σ1 is exactly the bundle of lines tangent to ∂Σ1.
Since a line bundle over a dis... | 19 | https://mathoverflow.net/users/428 | 17497 | 11,700 |
https://mathoverflow.net/questions/17495 | 17 | Let $\mathbb{H}$ denote the quaternions. Let $G$ be a group, and define a *representation of $G$ on $\mathbb{H}^n$* in the natural way; that is, its a map $\rho:G\rightarrow Hom\_{\mathbb{H}-}(\mathbb{H}^n,\mathbb{H}^n)$ such that $\rho(gg')=\rho(g)\rho(g')$ (where $Hom\_{\mathbb{H}-}$ denotes maps as left $\mathbb{H}$... | https://mathoverflow.net/users/750 | What rings/groups have interesting quaternionic representations? | Recall that if G is a group, k a field, and V\_k an irreducible representation of G over k then End\_G(V\_k) is a division algebra D over k. For example, if $V\_{\mathbb{R}}$ is a real representation then the endomorphism ring is $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$. Such representations are typically called *re... | 12 | https://mathoverflow.net/users/22 | 17500 | 11,702 |
https://mathoverflow.net/questions/17511 | 6 | I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.
| https://mathoverflow.net/users/4488 | Best way to teach concept of real numbers using a hands-on activity? | What do you mean exactly? Show them that not all numbers are rational? That not all numbers are algebraic? I think that you could explain historically how the Pythagoreans thought all numbers were rational. Then challenge them to find a number which is not rational. You might be surprised with the creativity you get. M... | 11 | https://mathoverflow.net/users/1106 | 17513 | 11,712 |
https://mathoverflow.net/questions/17501 | 46 | I was asked the following question by a colleague and was embarrassed not to know the answer.
Let $f(x), g(x) \in \mathbb{Z}[x]$ with no root in common. Let $I = (f(x),g(x))\cap \mathbb{Z}$, that is, the elements of $\mathbb{Z}$ which are linear combinations of $f(x), g(x)$ with coefficients in $\mathbb{Z}[x]$. Then ... | https://mathoverflow.net/users/2290 | The resultant and the ideal generated by two polynomials in $\mathbb{Z}[x]$ | This quantity $D$ is known as the "congruence number" or "reduced resultant" of the polynomials f and g. I first saw this in a preprint by Wiese and Taixes i Ventosa, <http://arxiv.org/abs/0909.2724>. They ascribe the concept to a paper which I don't have a copy of:
M. Pohst. *A note on index divisors*. In *Computat... | 22 | https://mathoverflow.net/users/2481 | 17514 | 11,713 |
https://mathoverflow.net/questions/17516 | 15 | (Warning: a student asking)
Let $E$ be an elliptic curve over $\mathbf Q$. Let $P(a,b)$ be a (nontrivial) torsion point on $E$. Is there an easy description of the ring of algebraic integers of $\mathbf Q(a,b)$? I'm curious about the answer for general elliptic curves, but I'm not sure whether such an answer is possibl... | https://mathoverflow.net/users/3777 | The ring of algebraic integers of the number field generated by torsion points on an elliptic curve | [Comment: what follows is not really an answer, but rather a focusing of the question.]
In general, there is not such a nice description even of the number field $\mathbb{Q}(a,b)$ -- typically it will be some non-normal number field whose normal closure has Galois group $\operatorname{GL}\_2(\mathbb{Z}/n\mathbb{Z})$,... | 5 | https://mathoverflow.net/users/1149 | 17517 | 11,715 |
https://mathoverflow.net/questions/17523 | 5 | In "continuous" mathematics there are several important notions such as covering space, fibre bundle, Morse theory, simplicial complex, differential equation, real numbers, real projective plane, etc. that have a "discrete" analog: covering graph, graph bundle, discrete Morse theory, abstract simplicial complex, differ... | https://mathoverflow.net/users/4400 | Are there any important mathematical concepts without discrete analog? | A lot of ideas from topology and analysis don't have obvious discrete analogues to me. At least, the obvious discrete analogues are vacuous.
* Compactness.
* Boundedness.
* Limits.
* The interior of a set.
I think a better question is which ideas have surprisingly interesting discrete analogues, like cohomology or ... | 4 | https://mathoverflow.net/users/2954 | 17537 | 11,726 |
https://mathoverflow.net/questions/17560 | 139 | I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number.
1. If $n^x$ is an integer for every $n \in \mathbb{N}$ then $x$ must be an integer. This is a fun little puzzle.
2. If $2^x$, $3^x$ and $5^x$ are in... | https://mathoverflow.net/users/25 | If $2^x $and $3^x$ are integers, must $x$ be as well? | Still open, to the best of my knowledge. The $2^x,3^x,5^x$ result follows from the [Six Exponentials Theorem](http://en.wikipedia.org/wiki/Six_exponentials_theorem), q.v., and the $2^x,3^x$ would follow from the Four Exponentials Conjecture, q.v.
| 39 | https://mathoverflow.net/users/3684 | 17564 | 11,743 |
https://mathoverflow.net/questions/17545 | 53 | Because the theory of sheaves is a functorial theory, it has been adopted in algebraic geometry (both using the functor of points approach and the locally ringed space approach) as the "main theory" used to describe geometric data. All sheaf data in the LRS approach can be described by bundles using the éspace étalé co... | https://mathoverflow.net/users/1353 | Sheaves and bundles in differential geometry | If $X$ is a manifold, and $E$ is a smooth vector bundle over $X$ (e.g. its tangent bundle),
then $E$ is again a manifold. Thus working with bundles means that one doesn't have to leave
the category of objects (manifolds) under study; one just considers manifold with certain extra structure (the bundle structure). This ... | 112 | https://mathoverflow.net/users/2874 | 17570 | 11,746 |
https://mathoverflow.net/questions/17468 | 23 | Grothendieck, before he disappeared, was working on a manuscript called "Les Derivateurs", which detailed the theory of derivators. Prof. Cisinski has done work with them as he mentioned in this [post](https://mathoverflow.net/questions/17425/homotopy-limits-over-fibered-categories/17463#17463). However, most of his wo... | https://mathoverflow.net/users/1353 | Derivators (in English) | For a few references in English, there are the papers of Heller, the main one being:
A. Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (383) (1988)
There is also a paper I wrote with A. Neeman, in which there is a little introduction to derivators in the second half of:
Additivity for derivator K-theory, Adv. ... | 20 | https://mathoverflow.net/users/1017 | 17589 | 11,760 |
https://mathoverflow.net/questions/17483 | 5 | Is there a definition of what is a 'free monoid' which does not pre-suppose that the natural numbers has already been defined? The definitions that I have been able to track down all use the natural numbers (since sequences/words need them).
In other words, I am trying to define the *theory* of free monoids (as a sig... | https://mathoverflow.net/users/3993 | Defining 'free monoid' without Nat? | As I pointed out in the comments, the theory of free monoids is somewhat ill defined. It is still unclear what your logic is and what you really want, but you have two basic options which were proposed by Pete Clark and sigfpe. Here are a few additional remarks that may help you sort things out.
*Universal Property à... | 12 | https://mathoverflow.net/users/2000 | 17603 | 11,768 |
https://mathoverflow.net/questions/17614 | 105 | This question is of course inspired by the question [How to solve f(f(x))=cosx](https://mathoverflow.net/questions/17605/how-to-solve-ffx-cosx)
and Joel David Hamkins' answer, which somehow gives a formal trick for solving equations of the form $f(f(x))=g(x)$ on a *bounded* interval. [EDIT: actually he can do rather be... | https://mathoverflow.net/users/1384 | solving $f(f(x))=g(x)$ | Q1: No. Let $g(0)=1, g(1)=0$ and $g(x)=x$ for all $x\in\mathbb R\setminus\{0,1\}$.
Assuming $f\circ f=g$, let $a=f(0)$, then $f(a)=1$ and $f(1)=g(a)=a$ since $a\notin\{0,1\}$.
Then $g(1)=f(f(1))=f(a)=1$, a contradiction.
Q2: No. Let $g(x)=-x$ or, in fact, any decreasing function $\mathbb R\to\mathbb R$. Then $f$ must... | 121 | https://mathoverflow.net/users/4354 | 17621 | 11,778 |
https://mathoverflow.net/questions/17615 | 12 | As the title says.
In particular, I am interested in the story for a general reductive group $G$, say defined over $\mathbb{Q}$. I can imagine that mod-$\ell$ (algebraic) automorphic representation should correspond to conjugacy classes of homomorphisms from the motivic Galois group into $\widehat{G}(\overline{\mathb... | https://mathoverflow.net/users/1464 | Is there a canonical notion of "mod-l automorphic representation"? | This is in my mind a central open problem.
Here is an explicit example which I believe is still wide open. Serre's conjecture (the Khare-Wintenberger theorem) says that if I have a continuous odd irreducible $2$-dimensional mod $p$ representation of the absolute Galois group of the rationals then it should come from ... | 18 | https://mathoverflow.net/users/1384 | 17625 | 11,782 |
https://mathoverflow.net/questions/17626 | 0 | Let $G$ be a (complex algebraic) group, $H$ a subgroup, and ${\cal O}(G)$ and ${\cal O}(H)$ the coordinate algebras of complex regular functions of $G$ and $H$ respectively. Can ${\cal O}(H)$ ever embedded as a subalgebra of ${\cal O}(G)$? For example, I'm looking at $U(k-1)$ as a subalgebra of $SU(k)$ (embedded in the... | https://mathoverflow.net/users/4409 | Subgroup Groups and Coordinate Algebra Subalgebras | It may be possible in rare circumstances, but the natural thing is that $\mathcal{O}(H)$ is a quotient of $\mathcal{O}(G)$, not a subalgebra. The quotient map is dual to the inclusion $H\to G$.
I guess in the case that $G$ is a direct product $H \times K$, you get
$$ \mathcal{O}(G) \simeq \mathcal{O}(H) \otimes \math... | 1 | https://mathoverflow.net/users/703 | 17627 | 11,783 |
https://mathoverflow.net/questions/17610 | 10 | Let $V$ be a finite dimensional highest weight representation of a (semi)-simple Lie algebra. For each $n\ge 0$ take $a\_n$ to be the dimension of the space of invariant tensors in $\otimes^n V$.
In certain cases there is a formula for $a\_n$. For example, for $V$ the two dimensional representation of $sl(2)$ we get ... | https://mathoverflow.net/users/3992 | Can you find linear recurrence relation for dimensions of invariant tensors? | Finding the recurrence (and proving it is correct) can be done by the standard techniques for extracting the diagonal of a rational power series.
Let $\beta\_1$, $\beta\_2$, ..., $\beta\_N$ be the weights of $V$. Let $\rho$ be half the sum of the positive roots and $\Delta = \sum (-1)^{\ell(w)} e^{w(\rho)}$ be the W... | 8 | https://mathoverflow.net/users/297 | 17628 | 11,784 |
https://mathoverflow.net/questions/17638 | 16 | Let $p \equiv 1 \bmod 4$ be a prime number. Define the polynomial
$$ f(x) = \sum\_{a=1}^{p-1} \Big(\frac{a}{p}\Big) x^a. $$
Then $f(x) = x(1-x)^2(1+x)g(x)$ for some polynomial
$g \in {\mathbb Z}[x]$ (this follows from elementary properties
of quadratic residues).
For $p = 5$, we have $g = 1$; for $p = 13$, we find
... | https://mathoverflow.net/users/3503 | Irreducibility of polynomials related to quadratic residues | These are known as Fekete polynomials:
<http://en.wikipedia.org/wiki/Fekete_polynomial> .
I don't know of any results on their Galois groups.
| 9 | https://mathoverflow.net/users/4213 | 17642 | 11,789 |
https://mathoverflow.net/questions/17126 | 3 | Let P and Q be simple polytopes such that P = Q ∩ H and let H be a halfspace with normal vector n. Let projn(e) denote the length of the projection of edge e onto vector n.
Consider the set E of edges of Q that cross through H, ie, edges with one endpoint in H and one outside H.
For any e ∈ E, does there always ex... | https://mathoverflow.net/users/2122 | Edge-maximizing projective transformation on polytopes | Let $e$ be the desired edge to be maximized. let point $a$ and $b$ be the endpoint of the edge. Let $a$ lie in $H$, $b$ outside of $H$. Let $e$ pass through the boundry of $H$, $B$ at $c$. Let $Q$ and $H$ lie in a space of dimension $n$ let this space be in a space of dimension $n+1$. Take point $a'$
which lies in this... | 1 | https://mathoverflow.net/users/1098 | 17649 | 11,796 |
https://mathoverflow.net/questions/7664 | 3 | I am interested in the theory of reductive groups which is useful in the theory of automorphic forms. But the trouble boring me so long time is that I don't know the appropriate material for beginners or outsiders who wants to pave in this field and learn more about automorphic forms.
So my question is that what kind ... | https://mathoverflow.net/users/1930 | How do we study the theory of reductive groups? | Sit at a table with the books of Borel, Humphreys, and Springer. Bounce around between them: if a proof in one makes no sense, it may be clearer in the other. For example, Springer's book develops everything needed about root systems from scratch, and has lots of nice exercises relate to that stuff. On the other hand, ... | 22 | https://mathoverflow.net/users/3927 | 17651 | 11,798 |
https://mathoverflow.net/questions/17634 | 13 | Usually (eg, intro. in M. Rost's 'Cycle modules with coefficients'), for a variety, $X$, over a field one can define the Chow group of p-cycles, $CH\_p (X)$, as
$$CH\_p (X) = coker\; \left[\bigoplus\_{x\in X\_{p+1}} k(x)^\times \rightarrow \bigoplus\_{x\in X\_p} \mathbb{Z}\; \right]$$.
What about for an arithmetic s... | https://mathoverflow.net/users/4235 | Definition of Chow groups over Spec Z | One can define Chow groups over any Noetherian scheme $X$. Let $Z\_iX$ be the free abelian group on the $i$-dimensional
subvarieties (closed integral subschemes) of $X$. For any $i+1$-dimensional
subscheme $W$ of $X$, and a rational function $f$ on $W$, we can
define an element of $Z\_iX$ as follows:
$$ [div(f,W)] = \s... | 12 | https://mathoverflow.net/users/2083 | 17652 | 11,799 |
https://mathoverflow.net/questions/17653 | 15 | This question is related to this one: [Continued fractions using all natural integers](https://mathoverflow.net/questions/6222/continued-fractions-using-all-natural-integers). Suppose we have the set of natural numbers $N$ with order and we perform permutation on it. So we obtain the same elements with different order.... | https://mathoverflow.net/users/3811 | infinite permutations | The first thing to notice is that infinite permutations may have infinite support, that is, they may move infinitely many elements. Therefore, we cannot expect to express them as finite compositions of permutations having only finite support.
But if we allow (well-defined) infinite compositions, then the answer is t... | 10 | https://mathoverflow.net/users/1946 | 17655 | 11,801 |
https://mathoverflow.net/questions/17658 | 1 | Every closed immersion is a finite morphism. Can you give an example of quasi-projective varieties $X\subset Y$ such that inclusion $X\hookrightarrow Y$ is not finite? Same with Y projective?
Thanks!
**Edit:** Sorry this question is very simple, I made a mistake asking the question. For a corrected version, check o... | https://mathoverflow.net/users/3637 | Example of inclusion which is not a finite morphism | An open immersion is never finite unless it is also a closed immersion (for finite morphisms are proper). So you just need to take a non-empty open subset $X$ which is not a connected component in $Y$.
| 13 | https://mathoverflow.net/users/3485 | 17660 | 11,805 |
https://mathoverflow.net/questions/17594 | 9 | Let $G$ be a reductive group over an (say, algebraically closed) field $k$. Springer (in his book on algebraic groups) calls for a chosen maximal torus $T$ in $G$ a family $(u\_\alpha) \_{\alpha \in \Phi(G,T)}$ of immersions $u \_\alpha:\mathbf{G}\_a \rightarrow G$ such that
(i) $t u\_\alpha(c) t^{-1} = u\_\alpha( \a... | https://mathoverflow.net/users/717 | Realizations and pinnings (épinglages) of reductive groups | OK, here's the deal.
I. First, the setup for the benefit of those who don't have books lying at their side. Let $(G,T)$ be a split connected reductive group over a field $k$, and choose $a \in \Phi(G,T)$ (e.g., a simple positive root relative to a choice of positive system of roots). Let $G\_a$ be the $k$-subgroup g... | 17 | https://mathoverflow.net/users/3927 | 17673 | 11,815 |
https://mathoverflow.net/questions/17590 | 2 | Let $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational function of degree $d\geq2$. How can one prove, using the normalized spherical measure, that
$$\int\_{\mathbb{P}^1(\mathbb{C})}|(\phi^n)'(z)|\ d\mu (z) \sim d^n$$ as $n\to\infty$?
| https://mathoverflow.net/users/nan | Help determining the asymptotic behavior of an integral involving rational functions. | If I understand your problem correctly, i.e. that $\mu$ is the [usual metric on the Riemann sphere](http://en.wikipedia.org/wiki/Riemann_sphere#Metric), then you're asking if essentially all orbits are expansive, at least with respect to that metric.
This will almost always be the case. For example, it will be so whe... | 1 | https://mathoverflow.net/users/3993 | 17676 | 11,818 |
https://mathoverflow.net/questions/17526 | 7 | Hello all, if $a\_1,a\_2, \ldots a\_t$ are $t$ integers $\geq 2$, the set
$G(a\_1,a\_2, \ldots a\_t)=\lbrace N \geq 1 |$ In any sequence of $N$ consecutive
integers there is at least one not divisible by any of $a\_1,a\_2, \ldots a\_t\rbrace$
is nonempty (it contains $a\_1a\_2 \ldots a\_t$) so it has a minimal element
... | https://mathoverflow.net/users/2389 | Smallest integer not divisible by integers in a finite set | Given an integer $n$, the Jacobsthal function $g(n)$ is the least integer, so that among any $g(n)$ consecutive integers $a,a+1,\dots,a+g(n)-1$ there is at least one that is coprime to $n$. Let $\nu(n)$ count the distinct prime factors of $n$. You can define $$C(r)=\max\_{\nu(n)=r} g(n)$$ and as Jonas Meyer points out ... | 6 | https://mathoverflow.net/users/2384 | 17682 | 11,820 |
https://mathoverflow.net/questions/17617 | 52 | I'm merely a grad student right now, but I don't think an exploration of the sporadic groups is standard fare for graduate algebra, so I'd like to ask the experts on MO. I did a little reading on them and would like some intuition about some things.
For example, the order of the monster group is over $8\times 10^{53}... | https://mathoverflow.net/users/1876 | Why are the sporadic simple groups HUGE? | The question seems to be made of several smaller questions, so I'm afraid my answer may not seem entirely coherent.
I have to agree with the other posters who say that the sporadic simple groups are not really so large. For example, we humans can write down the full decimal expansions of their orders, where a priori ... | 77 | https://mathoverflow.net/users/121 | 17696 | 11,829 |
https://mathoverflow.net/questions/17692 | 9 | Is there any way to define the orientation of an orientable smooth manifold using sheaves (when our smooth manifold is viewed as a locally ringed space) without our definition being overly contrived?
| https://mathoverflow.net/users/1353 | Orientation of a smooth manifold using sheaves | In the study of (finite-dimensional?) paracompact and locally compact (?) spaces there is Verdier's topological duality theorem, expressed in terms of a dualizing complex (which is built up from a sheafification process using duals of compactly-supported cohomologies of open subspaces, or something like that). It is pu... | 9 | https://mathoverflow.net/users/3927 | 17700 | 11,830 |
https://mathoverflow.net/questions/17678 | 2 | Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties $X\subset Y$, $Z$ and a finite morphism $f:Y\to Z$ such that restriction $f:X\to f(X)$ is not finite? Same with Y -- proje... | https://mathoverflow.net/users/3637 | Example of restriction of a finite morphism which is not finite | Almost the same counterexample works. Take any non-closed (so non-finite) open immersion $U\hookrightarrow Z$. Then the trivial double cover $Z\sqcup Z\to Z$ is finite, but the restriction to $U\sqcup Z\to Z$ is not (but is still surjective).
| 2 | https://mathoverflow.net/users/1 | 17701 | 11,831 |
https://mathoverflow.net/questions/17709 | 4 | This is something I am stuck on (it might well be trivial- in which case this is an embarassing question):
Let V be a dimension r vector space over Fp, the field with p prime elements (I also care about this when V is over Zn with n composite). Let t be a **given** automorphism of V of order q (prime different from ... | https://mathoverflow.net/users/2051 | Order of "one minus automorphism" | Perhaps the answer is just that the order is what it is, and that sometimes it's q. Do you have a compelling reason to believe that there is any more structure to your question than that?
Here's an example of the situation. $V$ could be the finite field $F$ with $p^r$ elements, and $q$ could be a prime divisor of $p^... | 6 | https://mathoverflow.net/users/1384 | 17724 | 11,842 |
https://mathoverflow.net/questions/17722 | 4 | For any $\alpha \in \mathbb{R}$ and a parameter $Q$, we can write $\alpha = a/q + \theta$, for integers $a, q$ with $q \leq Q$, and real $\theta$ with $|\theta|\leq (qQ)^{-1}$, a simple application of the Dirichlet approximation theorem. I'm looking for a similar statement for number fields.
**Setup**: $K$ is a fixed... | https://mathoverflow.net/users/4426 | Dirichlet Approximation over a Number Field | Schmidt, Wolfgang M.
Diophantine approximation.
Lecture Notes in Mathematics, 785. Springer, Berlin, 1980.
| 3 | https://mathoverflow.net/users/2290 | 17725 | 11,843 |
https://mathoverflow.net/questions/17745 | 9 | Next semester I will teach an elementary statistic course for the first time (which I am actually quite excited about). A brief description can be found [here](https://www.math.ku.edu/cgi/hb/index.cgi?action=coursedetail&number=365&title=Elementary%20Statistics). I am told to expect very little math background from the... | https://mathoverflow.net/users/2083 | How to teach introductory statistic course to students with little math background? | A bit of background: a few years ago, I designed such a course, after noticing that many of our social science majors were ending up taking a precalculus course (spending much time learning trig), which was mostly useless for their later study. I created a case-study approach to probability and statistics for students ... | 16 | https://mathoverflow.net/users/3545 | 17754 | 11,860 |
https://mathoverflow.net/questions/17753 | 2 | Greetings,
I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar transitional-curriculum courses would be interesting as well.
The context is that we are creating a 2nd year analysi... | https://mathoverflow.net/users/1465 | Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis | If you want a reference that will not bore them, supplement the main text with "Metric Spaces: Iteration and Application" by Victor Bryant. The book is *short* and it shows in several contexts how the concept of a fixed-point property, via iteration, can be used to solve worthwhile problems. It is very nicely written a... | 5 | https://mathoverflow.net/users/3272 | 17768 | 11,867 |
https://mathoverflow.net/questions/17774 | 29 | I have been looking at Serre's conjecture and noticed that there are two conventions in the literature for a p-adic representation $\rho:\mbox{Gal}(\bar{\mathbb Q}/\mathbb Q)\to \mbox{GL}(n,V).$ In some references (eg Serre's book on $\ell$-adic representations), $V$ is a vector space over a finite extension of $\mathb... | https://mathoverflow.net/users/3000 | Does the image of a p-adic Galois representation always lie in a finite extension? | A proof of the result you're after is contained at the beginning of section two of a recent paper of Skinner [here](http://www.math.uiuc.edu/documenta/vol-14/10.html). Skinner mentions that references for this fact seem to be rare.
| 17 | https://mathoverflow.net/users/1979 | 17777 | 11,873 |
https://mathoverflow.net/questions/17782 | 15 | I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised.
In [this question](https://mathoverflow.net/questions/17101/how-do-you-explicitly-compute-the-p-torsion-points-on-a-general-elliptic-curve-in), in Charles Rezk's answer, I erroneously claim that his co... | https://mathoverflow.net/users/1384 | components of E[p], E universal in char p. | Speaking of "connected components" is a delicate thing since you really mean in a relative sense, and more specifically the etale quotient $H$ can have its open and closed non-identity part with very nontrivial $\pi\_1$-action (so more subtle than on geometric fibers over the base). But even if the $\pi\_1$-action is t... | 10 | https://mathoverflow.net/users/3927 | 17785 | 11,877 |
https://mathoverflow.net/questions/17662 | 11 | Prof. Conrad mentioned in a recent [answer](https://mathoverflow.net/questions/7664/how-do-we-study-the-theory-of-reductive-groups/17651#17651) that most of the (introductory?) books on reductive groups do not make use of scheme theory. Do any books using scheme theory actually exist? Further, are there any books that ... | https://mathoverflow.net/users/1353 | Books on reductive groups using scheme theory | Personally, I find the "classical" books (Borel, Humphreys, Springer) unpleasant to read because they work in the wrong category, namely, that of reduced algebraic group schemes rather than all algebraic group schemes. In that category, the isomorphism theorems in group theory fail, so you never know what is true. For ... | 32 | https://mathoverflow.net/users/930 | 17793 | 11,882 |
https://mathoverflow.net/questions/17758 | 13 |
>
> What is an example (as simple as possible, please!) of a closed hyperbolic three-manifold with a right-angled polyhedron as fundamental domain?
>
>
>
If we allow cusps then the Whitehead link or the Borromean rings are good answers (fundamental domains have not too many sides and the gluings can be understo... | https://mathoverflow.net/users/1650 | Closed hyperbolic manifold with right-angled fundamental domain | A. Yu. Vesnin has some articles on these Lobell manifolds. The first one describes how to construct arbitrarily many non-isometric closed hyperbolic manifolds from one right-angled polyhedron.
1. "Three-dimensional hyperbolic manifolds with a common fundamental polyhedron" Math. Notes 49 (1991), no. 5-6, 575--577
2. ... | 10 | https://mathoverflow.net/users/4325 | 17795 | 11,883 |
https://mathoverflow.net/questions/17792 | 9 | Let $G$ be a group and let $H$ be a subgroup. If $H$ is normal in $G$, then $G/H$ has a group structure. But in general, can there be a groupoid structure on $G/H$(left cosets or right cosets) that generalizes the normal case?
| https://mathoverflow.net/users/4538 | Groupoid structure on G/H? | One answer to your question is that there is always the notion of an "action groupoid", although this does not reproduce the *group* structure on $G/H$ when $H$ is normal.
Let $G$ be a group acting on a set $X$. (There are generalizations when both $X,G$ are groupoids.) Then the **action groupoid** $X//G$ is the grou... | 11 | https://mathoverflow.net/users/78 | 17799 | 11,887 |
https://mathoverflow.net/questions/17817 | 7 | It is typical to find a corollary that following theorems, but is it right to use the word corollary for a statement following a conjecture, where the statement is true only if the unproven conjecture is true?
| https://mathoverflow.net/users/4545 | Can a corollary follow a conjecture? | I think it's generally bad form to have a corollary dependent on an earlier conjecture. I recommend one of the following:
**Theorem**: Assuming Conjecture A, properties X, Y and Z are true.
or
**Theorem**: Conjecture A implies X, Y and Z.
Most importantly, it should be crystal clear that the result is dependen... | 12 | https://mathoverflow.net/users/2264 | 17818 | 11,899 |
https://mathoverflow.net/questions/17811 | 5 | I am confused regarding supermanifolds. Suppose I consider R^(1,2) (1 "bosonic", 2 "fermionic"), This map (x,a,b) -> (x+ab, a,b) (a,b are fermionic) is supposed to be a morphism of this supermanifold. But I thought a morphism should be a continuous map from R->R together with a sheaf map of the sheaf of supercommutativ... | https://mathoverflow.net/users/3709 | Morphisms of supermanifolds | The ring of functions on your supermanifold is $C^\infty(\mathbb{R}) \otimes \mathbb{C}[a,b]$, where $a$ and $b$ are odd. The even part is then $C^\infty(\mathbb{R}) \oplus C^\infty(\mathbb{R})ab$, where $(ab)^2 = 0$, so there is an even nilpotent direction. You might want to view it as a thickening in a perpendicular ... | 4 | https://mathoverflow.net/users/121 | 17823 | 11,901 |
https://mathoverflow.net/questions/17826 | 30 | I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also want answerers to pick up points for good answers, so...
Let $X,Y$ be smooth manifolds. A smooth map $f: Y \to X$ is a ... | https://mathoverflow.net/users/78 | Why should I prefer bundles to (surjective) submersions? | One would be that a fibre bundle $F \to E \to B$ has a homotopy long exact sequence
$$ \cdots \to \pi\_{n+1} B \to \pi\_n F \to \pi\_n E \to \pi\_n B \to \pi\_{n-1} F \to \cdots $$
This isn't true for a submersion, for one, the fibre in a submersion does not have a consistent homotopy-type as you vary the point in ... | 38 | https://mathoverflow.net/users/1465 | 17829 | 11,905 |
https://mathoverflow.net/questions/17836 | 5 | There are $n$ students in a class, and they must be divided into, say, $k$ groups. Each student ranks the other students in order of preference of working together. Is there a way to generally optimize student happiness (where happiness is based on working with preferred teammates). We could assume for simplicity that ... | https://mathoverflow.net/users/4241 | How to optimize student happiness in group work? | This is a generalization of the [stable roommate problem](http://en.wikipedia.org/wiki/Stable_roommates_problem) (which is the same thing where $k = n/2$, ie, groups of 2). In general, there exist groups in which under any pair of groups contain members who would both like to switch teams.
From wikipedia:
>
> F... | 9 | https://mathoverflow.net/users/1610 | 17841 | 11,914 |
https://mathoverflow.net/questions/17803 | 12 | This question is motivated by the recent paper [An invitation to higher gauge theory](http://golem.ph.utexas.edu/category/2010/03/an_invitation_to_higher_gauge.html) by Baez and Huerta, and the 2007 paper [Parallel Transport and Functors](http://arxiv.org/abs/0705.0452) by Schreiber and Waldorf.
Let $M$ be a smooth, ... | https://mathoverflow.net/users/78 | What is the infinite-dimensional-manifold structure on the space of smooth paths mod thin homotopy? | Okay, you asked for it!
>
> **Question:** What is the manifold structure on $P^1(M)$?
>
>
>
>
> **Answer:** There isn't one.
>
>
>
---
**Update:** The biggest failing is actually that the obvious model space is **not** a vector space. The space of paths mod thin homotopy in $\mathbb{R}^n$ does n... | 10 | https://mathoverflow.net/users/45 | 17843 | 11,916 |
https://mathoverflow.net/questions/17844 | -1 | I want to do something about ”games of incomplete information“，like "Computer poker program".I know,Albert university(in canada) have do a lot of things to that field,they write a program called: "Polaris"(deal with computer poker) .computer poker is more difficult than computer cheese,because the former don't have eno... | https://mathoverflow.net/users/4548 | Monte Carlo method and possible applications to computer poker? | Monte Carlo methods are appropriate for analyzing some systems involving chance, not incomplete information. Monte Carlo methods tell you nothing about how to model a poker strategy.
For general games of incomplete information, you should look up [game theory](http://en.wikipedia.org/wiki/Game_theory) (and not combin... | 7 | https://mathoverflow.net/users/2954 | 17851 | 11,921 |
https://mathoverflow.net/questions/17771 | 35 | Fix a prime $p$. What is the smallest integer $n$ so that there is a simplicial complex on $n$ vertices with $p$-torsion in its homology?
For example, when $p=2$, there is a complex with 6 vertices (the minimal triangulation of the real projective plane) with 2-torsion in its homology. I'm pretty sure that it's the s... | https://mathoverflow.net/users/4194 | Small simplicial complexes with torsion in their homology? | **UPDATE** This version is substantially improved from the one posted at 8 AM.
I now think I can achieve $\mathbb{Z}/p$ using $O( \log p)$ vertices. I'm not trying to optimize constants at this time.
Let $B$ be a simplicial complex on the vertices $a$, $b$, $c$, $a'$, $b'$, $c'$ and $z\_1$, $z\_2$, ..., $z\_{k-3}$,... | 16 | https://mathoverflow.net/users/297 | 17852 | 11,922 |
https://mathoverflow.net/questions/17809 | 10 | There was a rather [cute question last week](https://mathoverflow.net/questions/17269/let-g-be-a-graph-such-that-for-all-u-v-v-g-u-no-equal-to-v-n-u-n-v) about graphs where every pair of distinct vertices has an odd number of mutual neighbours.
The question was to show that such a graph must have an odd number of ver... | https://mathoverflow.net/users/1492 | Graphs where every two vertices have odd number of mutual neighbours | Take a Steiner triple system on $v$ points. Let $X$ be the graph with the $v(v-1)/6$ triples
as its vertices, two triples adjacent if the have exactly one point in common. We need
$v\equiv1,3$ modulo 6. Then two adjacent triples have exactly $(v+3)/2$ common neighbours, and two disjoint triples have exactly 9 common ne... | 4 | https://mathoverflow.net/users/1266 | 17853 | 11,923 |
https://mathoverflow.net/questions/17854 | 1 | I am talking about a relation that is what Wikipedia describes as [left-unique and right-unique](http://en.wikipedia.org/wiki/Relation_%28mathematics%29#Special_types_of_binary_relations). I never heard these terms before, but I have heard of the alternatives (injective and functional). The question is, *which terminol... | https://mathoverflow.net/users/840 | Do you need to say what left-unique and right-unique means? | Injective and functional are completely standard in this case. This is what you should use. The term "functional" is not overloaded, when you are using it to say that something is a function. Being functional means exactly that the relation is a function.
A relation that is injective and functional is precisely an in... | 6 | https://mathoverflow.net/users/1946 | 17855 | 11,924 |
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