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https://mathoverflow.net/questions/135536 | 5 | Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated that $0!-1!+2!-3!+\dots$ equals 0.59637..., even though he had no definition of what an expression like $0!-1!+2!-3!+\dot... | https://mathoverflow.net/users/3621 | Procedure-based (as opposed to definition-based) concepts | Having mentioned Euler and twentieth century physicists in the question, Pierre Cartier's '[Mathemagics? (A Tribute to L. Euler and R. Feynman)](http://www.kurims.kyoto-u.ac.jp/EMIS/journals/SLC/wpapers/s44cartier1.pdf)'
should provide good cases for you.
>
> My thesis is: there is another way of doing mathematics,... | 2 | https://mathoverflow.net/users/35400 | 135614 | 74,640 |
https://mathoverflow.net/questions/135617 | 2 | I would like to know if there is any difference between
(1) an algebraic Hecke character
(2) a Hecke character
(3) a Grössencharakter
All of the above in the setting of ellitpic curves with complex multiplication. Thanks!
| https://mathoverflow.net/users/36133 | Hecke Character vs Grossencharakter | Grossencharacters and Hecke characters are different names for the same thing.
| 4 | https://mathoverflow.net/users/36429 | 135619 | 74,641 |
https://mathoverflow.net/questions/135634 | 2 | First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question [on MSE](https://math.stackexchange.com/questions/434659/hilberts-finiteness-theorem-for-connected-semisimple-lie-groups-over-mathbbc), ... | https://mathoverflow.net/users/22998 | Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's "Classical Groups" | Your basic question about a reference does go back to Weyl's complete reducibility theorem (I'd have to check his book on classical groups, but it isn't just a result about classical Lie groups). For a standard modern proof in the wider context of semisimple (or more generally, reductive) algebraic groups over an algeb... | 4 | https://mathoverflow.net/users/4231 | 135636 | 74,651 |
https://mathoverflow.net/questions/135631 | 4 | I'm wondering whether anyone knows a reference or proof for finding the Fourier transform of $f(t):=(t+1)^{1/2}t\_+^{1/2}$? (Here $t\_+=\max (t,0)$.)
| https://mathoverflow.net/users/19433 | Fourier transform of tempered distribution | $$\tfrac{1}{2}\int\_{0}^{\infty}e^{i\omega t}dt=\tfrac{1}{2}\pi\delta(\omega)+\tfrac{1}{2}i\omega^{-1}$$
$$\int\_{0}^{\infty}te^{i\omega t}dt=-i\frac{d}{d\omega}\int\_{0}^{\infty}e^{i\omega t}dt=-i\pi\frac{d}{d\omega}\delta(\omega)-\omega^{-2}$$
$$\int\_{0}^{\infty}[\sqrt{t(1+t)}-t-\tfrac{1}{2}]e^{i\omega t}dt=\ome... | 8 | https://mathoverflow.net/users/11260 | 135647 | 74,655 |
https://mathoverflow.net/questions/135656 | 6 | I wish to find the coefficients of the Laurent expansion of the inverse of the Vandermonde determinant, that is, the Laurent expansion at 0 of
$$\prod\_{1\leq i<j \leq n}(x\_j-x\_i)^{-1}.$$
We can iteratively do a series expansion in $x\_1,x\_2,$ and so on,
(the full expansion depends on the order of expansion).
An... | https://mathoverflow.net/users/1056 | Laurent expansion of inverse of vandermonde determinant | In your question, the exponent of $x\_2$ is $a\_2+k$. I am assuming that is a typo, and you intended $c\_a(k)$ to be the coefficient of
$$x\_1^{a\_1+k} x\_2^{a\_2} x\_3^{a\_3} \cdots x\_{n-2}^{a\_{n-2}} x\_{n-1}^{a\_{n-1}} x\_n^{a\_n-k}.$$
If so, I can answer all of your questions.
It will be convenient to switch th... | 11 | https://mathoverflow.net/users/297 | 135664 | 74,659 |
https://mathoverflow.net/questions/134708 | 6 | I'm looking for a reference or proof of the following. Let $K/\mathbb{Q}$ be a finite Galois extension of degree $n$. Let $a\_1,\ldots,a\_n$ be Galois conjugate elements in the ring of integers of $K$ with $a\_1, \ldots, a\_k$ ($1 \le k < n$) lying on the unit circle (hence are not roots of unity). Show there are infin... | https://mathoverflow.net/users/2043 | Simultaneous Powers Far From 1 | There is no proof because the desired result is false!
Indeed, for any $\theta \gt 0$ there exists an algebraic integer
of degree $n$ with $k \lt n$ conjugates $a\_1,\ldots,a\_k$ on the unit circle
such that for each $m$ at least one of $a\_1^m,\ldots,a\_k^m$
is within $\theta$ of $1$.
Suppose $\theta \geq \pi/r$ fo... | 9 | https://mathoverflow.net/users/14830 | 135667 | 74,662 |
https://mathoverflow.net/questions/135669 | 19 | Recall that two $k$-algebras $A, B$ are Morita equivalent iff their categories of left modules are equivalent. However, this relation turns out to be rather fine and one introduces a coarser equivalence relation of *derived Morita equivalence* by using (bounded) derived categories of modules, along with their triangula... | https://mathoverflow.net/users/16981 | Homotopy-theoretic derived Morita equivalences | Derived Morita equivalence is the same as higher derived Morita equivalence. Clearly, $\Leftarrow$ is obvious, and $\Rightarrow$ follows from Theorem 2.6 in Dugger-Shipley's 'K-theory and derived equivalences' Duke Math. J. 124 (2004), no.3, 587--617. This is surprising at a first glance, it follows from the fact that ... | 24 | https://mathoverflow.net/users/12166 | 135683 | 74,666 |
https://mathoverflow.net/questions/135681 | 2 | Here is the problem:
I have 3 subsets ( called $S\_1$, $S\_2$ and $S\_3$) that each have $N$ elements (arbitrary elements).
So I have one element $X$. I want to know if I can get $X$ making the sum of one element of $S\_1$, one of $S\_2$ and one of $S\_3$.
Say for example $N = 4$ and $S\_1 = \{ 1, 3,-1, 5\}$, $S... | https://mathoverflow.net/users/36458 | Algorithm to find if an element X can be represented with the sum of one number of each subset in $O(n^2)$? | First sort $S\_2$ and $S\_3$.
Divide into $n$ subproblems: for each $a\in S\_1$, look for $b\in S\_2,c\in S\_3$ such that $b+c=X-a$. Let's do one such subproblem in $O(n)$ time.
Let $b\_j$ be the $j$-th element of $S\_2$ and $c\_k$ be the $k$-th element of $S\_3$. The idea is that for $j=1,2,\ldots,n$ you find the ... | 6 | https://mathoverflow.net/users/9025 | 135693 | 74,667 |
https://mathoverflow.net/questions/135686 | 1 | In the definition of pre-quantization of representation $f\to \hat{f}$, (here $\hat{f}$ is Hermitian operator)of $C^{\infty}(M)$ on $L^2(M,L,\mu)$ where $\mu$ is Hermitian form, suppose that there exists some $(L,\mu,\Delta)$, such that $\Delta$ has the curvature $\omega$. Let given $(L,\mu)$, then why the choices of $... | https://mathoverflow.net/users/nan | choices of connection in prequantization | This is a general fact in differential cohomology, which in degree two classifies Hermitian line bundles with connection.
Differential cohomology has an exact sequence
$$
0 \to \frac{H^{k-1}(M,\mathbb{R})}{H^{k-1}(M,\mathbb{Z})} \to \hat H^k(M,\mathbb{Z}) \to H^k(M,\mathbb{Z}) \times\_{H^k(M,\mathbb{R})} \Omega^k\_{... | 2 | https://mathoverflow.net/users/3473 | 135694 | 74,668 |
https://mathoverflow.net/questions/135668 | 3 | Let $Q(i)$ be the extension of the rational numbers $Q$ obtained by adjoining a root i of the polynomial $X^2 + 1$.
Consider the algebra B defined by the Hilbert symbol $(-2, -5)$ over $Q(i)$. So, by definition,
$$
B := Q(i)[[\alpha, \beta]]/(\alpha^2 = -2, \beta^2 = -5, \alpha\beta = -\beta\alpha)
$$
here $[[\alpha, ... | https://mathoverflow.net/users/4398 | Morphism of algebras | No. Consider the matrix corresponding to $i$. It is semisimple with eigenvalues $i,i,-i,-i$ . So it's centralizer is $8$-dimensional - clearly, it is $M\_2(\mathbb Q(i))$. Since this algebra is ramified, it does not inject into the split algebra.
| 2 | https://mathoverflow.net/users/18060 | 135709 | 74,676 |
https://mathoverflow.net/questions/135641 | 14 | Let $X$ be an $E\_\infty$-space (not necessarily grouplike). Let $x \in \pi\_0 X$ be an element; say that $x$ is **strictly commutative** if there is a map of $E\_\infty$-spaces $\mathbb{Z}\_{\geq 0} \to X$ that takes $1 \mapsto x$. (The terminology is abusive, as for an element to be strictly commutative is extra data... | https://mathoverflow.net/users/344 | Strictly commutative elements of $E_\infty$-spaces | In the "easier" grouplike case, as you say, this is related to spaces of units, and Jacob and Neil have mentioned things about $gl\_1$. This thing exhibits strange behaviour, and was an object of close study (along with some serious calculation) a number of years ago.
Here's an example of something that may seem coun... | 18 | https://mathoverflow.net/users/360 | 135712 | 74,677 |
https://mathoverflow.net/questions/135697 | 2 | This is a follow-up question after [this](https://mathoverflow.net/questions/130868/ramification-in-division-field-of-abelian-varieties)
The set-up is almost the same as before,
Let $k$ be a number field, $p$ be a rational prime. Let $A$ be an abelian variety over $k$ which has a good reduction at all primes $\mat... | https://mathoverflow.net/users/21090 | Ramification in Division field of Abelian Varieties II | F.Voloch's $p=2$ counterexample $y^2 = x(x^2-d)$ with $d \equiv 1 \bmod 4$
still works. Yes, the curve has bad reduction at $2$, but it has
*potential* good reduction, so it will work over some number field $k$.
An explicit $p=2$ example over ${\bf Q}$ is the curve $[1,1,1,0,0]$, a.k.a.
$X\_1(15): y^2+xy+y=x^3+x^2$, ... | 2 | https://mathoverflow.net/users/14830 | 135715 | 74,678 |
https://mathoverflow.net/questions/135717 | 13 | I am looking for examples (references) of pairs of non Quillen-equivalent model categories having the same homotopy categories.
The motivation is of course that I have two model categories and all the attempts to prove that they are Quillen equivalent have failed so far, but the homotopy categories are equivalent as... | https://mathoverflow.net/users/24563 | Examples of non Quillen-equivalent model categories having equivalent homotopy categories | The ones which usually come up to my mind as soon as I think of this:
* The categories of modules over $\mathbb Z/p^2$ and $\mathbb F\_p[\epsilon]/(\epsilon^2)$, $p$ a prime integer. These rings are quasi-Frobenius, so their module categories have a model structure where cofibrations are monomorphisms, fibrations are... | 17 | https://mathoverflow.net/users/12166 | 135724 | 74,682 |
https://mathoverflow.net/questions/88818 | 6 | In the paper by Weinstein and Xu: Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), there are two versions of Lie groupoid cohomology. The same differential $\delta$ can be in fact applied to globally smooth cochains or only to cochains which are globally continous and smooth in a n... | https://mathoverflow.net/users/6032 | Continuous and smooth Lie groupoid cohomology | Sorry for answering so late, I just was pointed to this question. Before answering your questions in the case of a Lie group (i.e., one object) let me point out that the difference between $\mathbb{R}$-coefficients and $\mathbb{S}^1$-coefficients it not negligible but is somehow the whole heart of the story.
1. The c... | 7 | https://mathoverflow.net/users/5937 | 135727 | 74,685 |
https://mathoverflow.net/questions/135654 | 8 | Is there a generalization of Brownian motion to general metric spaces (which should probably be length spaces)?
This should be a process satisfying
$$d(B\_t, B\_s) \sim \mathcal{N}(0, t-s)$$
and such that $d(B\_t, B\_s)$ and $d(B\_s, B\_r)$ are independent; however, this will be not enough in general (this not even d... | https://mathoverflow.net/users/16702 | Brownian motion on Metric spaces | I place my comment as answer, as it seems to at least partly satisfy the OP.
If you have a metric *measure* space, you can define a random walk by jumping uniformly in a ball of radius $\varepsilon$, then make $\varepsilon\to0$ while rescaling time to get a continuous stochastic process. But then you certainly need h... | 2 | https://mathoverflow.net/users/4961 | 135729 | 74,687 |
https://mathoverflow.net/questions/135723 | 2 | I would appreciate a reference to the following statement, which, I was having an impression, is known:
Let $L, M$ be field extensions of finite degree of a number field $K$, such that $L \cap M = K$. Then probabilities of any two chosen splitting types in $L$ and $M$ respectively, of a randomly chosen prime ideal i... | https://mathoverflow.net/users/10591 | Independence of Chebotarev densities | As stated, this is false: take $L$ and $M$ to be two distinct intermediate cubic subfields of a Galois $S\_3$-extension. In particular, they are isomorphic. If a prime of $K$ splits completely in one, then it also splits completely in the other.
Perhaps you wanted to assume that both $L$ and $M$ are Galois and disjoi... | 6 | https://mathoverflow.net/users/35416 | 135732 | 74,688 |
https://mathoverflow.net/questions/135734 | 1 | Let $\pi\colon P\to X$ be a locally trivial principal $G$-bundle over a Hausdorff paracompact space $X$, where $G$ is a topological group (we work in the category of topological spaces, as I do not think smooth structures are relevant to the question I am asking).
The group $\mathrm{Gau}(P)$ of gauge transformations ... | https://mathoverflow.net/users/36502 | Non-compact structure group and compactly supported gauge transformations | First you just look at what happens on one fiber: left (or right) translation of the structure group by a group element. No point of the fiber is fixed (if you act by a nonidentity element). This fiber is already not a compact set, and you are sliding its points around. If the set of nonfixed points in $P$ sits in a co... | 1 | https://mathoverflow.net/users/13268 | 135735 | 74,689 |
https://mathoverflow.net/questions/135696 | 2 | I have been thinking about the concept of specialization of algebras defined over the field of rational functions $k = \mathbb{C}(t)$ (I'm using $\mathbb{C}$ for my work, but the question can be asked over any field).
In my examples the algebras are quadratic, but I don't think that this is essential.
The main question... | https://mathoverflow.net/users/703 | Specialization of PBW-algebras over rational function field | I fail to figure out where exactly you see potential problems.
Diamond lemma indeed says that each ambiguity is resolvable, that is for each "common multiple" of some $W\_\sigma$ and $W\_\tau$, WLOG $aW\_\sigma=W\_\tau b$, the element $af\_\sigma-f\_\tau b$ can be reduced to zero using your rewriting rules. Reducing... | 1 | https://mathoverflow.net/users/1306 | 135736 | 74,690 |
https://mathoverflow.net/questions/135733 | 4 | Is there a database for sequences indexed by partitions similar to Sloane's OEIS? I mean, I am aware that in the OEIS there are some arrays indexed by partitions, but I feel as though most of such sequences that frequently appear in combinatorial literature are not there.
One example of a sequence I'd really like to... | https://mathoverflow.net/users/1306 | Databases for sequences indexed by partitions | Thanks for asking this question! It really is a perfect occation for me advertising (once again) the combinatorial statistic finder <http://www.FindStat.org>!
To search the database for partitions, see <http://www.FindStat.org/StatisticFinder/IntegerPartitions>
| 8 | https://mathoverflow.net/users/21291 | 135740 | 74,692 |
https://mathoverflow.net/questions/135749 | 7 | Is an algebraic space over a DVR, whose special fibre (and all its infinitesimal neighborhood) and generic fibre are schemes, actually a scheme?
| https://mathoverflow.net/users/3848 | Is an algebraic space over a DVR, whose special fibre and generic fibre are schemes, actually a scheme? | The answer is negative. Let me give a counter-example by modifying the example of a singular complex algebraic surface that is not a scheme given by Knutson in [Algebraic Spaces, p.21-22].
Let me work over $\mathbb{C}$. Consider the pencil in $\mathbb{P}^2$ generated by two smooth cubic curves $C$ and $C'$ intersecti... | 16 | https://mathoverflow.net/users/2868 | 135754 | 74,695 |
https://mathoverflow.net/questions/135653 | 5 | There seems to be something curious about the so-called Van-Vleck-Morette determinant, as I cannot find any source that properly defines it in terms of expressions previously defined in that source and that actually proves properties.
To my understanding, a possible definition is
$$ \triangle(p, q) = (\det(g\_{ij}(x(... | https://mathoverflow.net/users/16702 | Van Vleck-Morette Determinant | The Van-Vleck Morette determinant often arises in semiclassical analysis, for example in computations of asymptotic expansions of heat kernels or Feynman path integrals. In these applications it is not easy to trace its origin as a transformation Jacobian.
The space of initial data of the geodesic initial data proble... | 7 | https://mathoverflow.net/users/1059 | 135762 | 74,697 |
https://mathoverflow.net/questions/135755 | 0 | As a unified approach if we have an ( read any) *infinite* board game described as $\mathcal{G}$ using a particular axiom set ***A***..
**can a sentence be devised in *A* which automatically answers the basic questions for the game** ie
- Won in k moves given any state of the game
- Draw in l moves given any... | https://mathoverflow.net/users/34859 | Infinite board games: sentences about | I explain in my answer to Richard Stanley's question on [the decidability of infinite chess](https://mathoverflow.net/a/86755/1946) that this is precisely how Dan Brumleve, Philip Schlicht and I proved that the mate-in-$n$ problem of infinite chess is decidable. Namely, we introduced the first-order *structure of chess... | 8 | https://mathoverflow.net/users/1946 | 135766 | 74,699 |
https://mathoverflow.net/questions/135748 | 8 | Suppose that $G$ is a Lie group with a transitive action on a smooth manifold $M$. The regular theory of Lie groups tells us that $G$ and $M$ are diffeomorphic if the isotropy group is trivial.
The proof that I know goes, as usual, by proving that the canonical map $G/H \rightarrow M$ is bijective and its differentia... | https://mathoverflow.net/users/35946 | Does a free action always induce a diffeomorphism? | The answer is no, in general, even for both $G$ and $M$ connected.
A simple example is the following:
Take $G=Diff(S^1)$, the diffeomorphism group with the the $C^\infty$-topology, which is a regular Frechet Lie group. Let $M=Diff(S^1)$ (smooth diffeomorphisms), but now with the Sobolev $H^1$-topology, which is a topol... | 10 | https://mathoverflow.net/users/26935 | 135768 | 74,700 |
https://mathoverflow.net/questions/135765 | 29 | Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list [can be found here.](https://www-users.cse.umn.edu/%7Etlawson/hovey/) Some of the problems I know about have been worked on quite a bit since the time of writing. The list is very goo... | https://mathoverflow.net/users/4517 | Open problems in algebraic topology and homotopy theory | I have made a note of some problems in the area of [Nonabelian algebraic topology and homological algebra](https://web.archive.org/web/20151016170311/http://pages.bangor.ac.uk/%7Emas010/pdffiles/PROBAMTX.PDF) in 1990, and in Chapter 16 of the book in the same area and [advertised here](https://web.archive.org/web/20151... | 11 | https://mathoverflow.net/users/19949 | 135770 | 74,701 |
https://mathoverflow.net/questions/135767 | 3 | Let $g \geq 1$ be an integer, and $\mathrm{Sp}\_{2g}(\mathbb{Z})$ be the symplectic group of $2g \times 2g$ integral matrices. In this case, we are defining the symplectic group a the automorphism group of a $2g$-dimensional vector space which preserves an alternating form $\langle \cdot, \cdot \rangle$, with respect t... | https://mathoverflow.net/users/24757 | subgroups of $\mathrm{Sp}_{2g}(\mathbb{Z})$ generated by powers of transvections | There is a Theorem of Tits in Comptes Rendus of Academy of Paris "Systemes generateurs grupes de congruence" ...(early 80's) where he proves this (the finiteness of index of the group generated by powers of the root group generators) for any Chevalley group of $Q$ rank at least two. For the symplectic group, the root g... | 6 | https://mathoverflow.net/users/23291 | 135786 | 74,708 |
https://mathoverflow.net/questions/135788 | 13 | For many examples of word-hyperbolic groups which I have seen in the context of low-dimensional topology, the ideal boundary is either homeomorphic to a n-sphere for some n or a Cantor set. So, I was wondering if this is generally true or there are some examples of hyperbolic groups whose boundaries are neither a spher... | https://mathoverflow.net/users/27570 | Topology of boundaries of hyperbolic groups | There are plenty of other possibilities. Here are a few examples:
* The boundary of the fundamental group of an acylindrical hyperbolic 3-manifold with totally geodesic boundary is homeomorphic to a Sierpinski carpet. (This appears in the Kapovich and Kleiner paper mentioned below, but must be standard---the point is... | 14 | https://mathoverflow.net/users/1463 | 135794 | 74,711 |
https://mathoverflow.net/questions/135782 | 14 |
>
> [Quantum game theory](https://en.wikipedia.org/wiki/Quantum_game_theory) is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:
>
>
> 1. Superposed initial states,
> 2. Quantum entanglement of initial states,
> 3. Superposition of strategies ... | https://mathoverflow.net/users/36526 | Is quantum game theory reducible to classical game theory? | There are several versions of quantum game theory. First, there are Eisert-Wilkens type models in which players submit strategies that are quantum superpositions of classical strategies. In these models, a classical game {\bf G} is replaced by a quantum game {\bf G}, which is itself a classical game with a much larger ... | 9 | https://mathoverflow.net/users/10503 | 135796 | 74,713 |
https://mathoverflow.net/questions/135690 | 3 | Call an univariate polynomial $f(x) = \sum\_{i=0}^{n}a\_{i}x^{i} \in \Bbb{Z}[x]$ symmetric if $a\_{i} = a\_{n-i}$ and $a\_{0} = a\_{n} > 0$.
For a given $\sum\_{i=0}^{n}a\_{i}$ and $a\_{i} \geq 0$, how many symmetric polynomials $f(x) = \sum\_{i=0}^{n}a\_{i}x^{i} $ over $\Bbb{Z}[x]$ are there such that for two fixed... | https://mathoverflow.net/users/10035 | Counting polynomials with same coefficient sum and a given value at a point | The polynomials $f\_1=x^4+9x^3+9x+1$ and $f\_2=3x^4+14x^2+3$ both have coefficient sum $f\_1(1)=f\_2(1)=20$ and the common value $f\_1(2)=f\_2(2)=107.$ The key is that the difference $f2-f1=g=2x^4-9x^3+14x^2-9x+2$ is symmetric with $g(1)=g(2)=0.$ we could take $f\_2=f\_1+g$ for any symmetric degree $4$ polynomial with ... | 4 | https://mathoverflow.net/users/8008 | 135797 | 74,714 |
https://mathoverflow.net/questions/135783 | 8 | Assume that $ B \hookrightarrow A $ is an injective ring homomorphism.
>
>
> >
> > **Question:** If $\mathfrak{p}$ is an associated prime of $B$, is there an associated prime $ \mathfrak{q}$ of $A$ such that $ \mathfrak{q} \cap B = \mathfrak{p}$?
> >
> >
> >
>
>
>
Some comments:
* This is true if $ ... | https://mathoverflow.net/users/4002 | Lying over theorem for associated primes | Let me explain why the statement is true. Let $i:B\to A$ be an injection of noetherian rings and $\mathfrak{p}$ be an associated prime of $B$ : it is the annihilator of a nonzero element $f\in B$.
Let us first localize at $\mathfrak{p}$. More precisely, replacing $B\to A$ by $B\_{\mathfrak{p}}\to A\otimes\_B B\_{\mat... | 6 | https://mathoverflow.net/users/2868 | 135800 | 74,715 |
https://mathoverflow.net/questions/135806 | 7 | Given a finite index inclusion, $N\subset M$, of $II\_1$ factors we can construct two towers of finite dimensional algebras known as the $\textit{standard invariant}$. For low index, this has allowed for a complete calculation of subfactors. However, at index 6 this breaks downs because there is an uncountable family o... | https://mathoverflow.net/users/5732 | Are subfactor planar algebras hard to classify at index 6? | At the very least there's one Bisch-Haagerup subfactor standard invariant for every quotient group of $\mathbb{Z}/2 \ast \mathbb{Z}/3 \cong \mathrm{PSL}\_2(\mathbb{Z})$. That's already way too much to try to classify. Even if you restrict to finite depth, there's way too many finite quotients of the modular group.
On... | 12 | https://mathoverflow.net/users/22 | 135810 | 74,721 |
https://mathoverflow.net/questions/135793 | 2 | I have often seen the statement that Hyper-Kahler (HK) manifolds have torsion-free connections. In general relativity, however, one is usually taught that the connection is something that you can "choose". You might choose to use the Levi-Civita connection or not. Is it not possible for a HK manifold to choose a connec... | https://mathoverflow.net/users/36541 | Why can't hyper-kahler manifolds have a connection with torsion? | You may be understandably confused by a terminological inconsistency. People study so-called ***"[Hyper-Kähler manifolds with torsion](http://www.scielo.org.ar/scielo.php?script=sci_arttext&pid=S0041-69322008000200012)"*** (a.k.a. HKT manifolds), which by definition have
* 3 complex structures $I$, $J$, $K$ satisfyin... | 6 | https://mathoverflow.net/users/19276 | 135811 | 74,722 |
https://mathoverflow.net/questions/135820 | 23 | I want to know whether there exist a non-square number $n$ which is the quadratic residue of every prime.
I know it is very elementary, and I think those kind of number are not exist, but I don't know
how to prove.
| https://mathoverflow.net/users/32866 | Does there exist a non-square number which is the quadratic residue of every prime? | This is actually an elementary consequence of quadratic reciprocity,
generalizing the familiar proof à la Euclid that that are infinitely many
primes of the form $4k+3$
(i.e. primes of which $-1$ is not a quadratic residue).
We want to show that there exists $p$ such that $(n/p) = -1$.
[As stated Paul's question asks ... | 27 | https://mathoverflow.net/users/14830 | 135829 | 74,730 |
https://mathoverflow.net/questions/135545 | 1 | I found the following exercise at page 85 of the Strade-Farnsteiner's book "Modular Lie algebras and their representation": Let $D$ be a finite-dimensional division ring over a field $F$ of characteristic $p>0$. Consider $D$ as a restricted Lie algebra via the Lie product $[x,y]=xy-yx$ and $p$-mapping $x \mapsto x^p$. ... | https://mathoverflow.net/users/17582 | Semisimple elements in division algebras | The claim is true under the assumption that the ground field $F$ is perfect. This follows from Exercise 2 in the same page which is, in turn, an immediate consequence of Theorem 3.7 in the same section of the book of Strade and Farnsteiner.
On the other hand, if $F$ is not perfect then the previous result does not hold... | 3 | https://mathoverflow.net/users/14653 | 135873 | 74,745 |
https://mathoverflow.net/questions/131270 | 5 | Given an action of a group $G$ on a topological space $X$, the associated *homotopy quotient* is $$X\_G := (EG \times X)/G,$$ where $EG$ is the total space of a universal principal $G$-bundle $EG \to BG$ and the quotient is by the (free) diagonal action of $G$ on $EG \times X$.
The $G$-*equivariant cohomology* of $X$ i... | https://mathoverflow.net/users/5792 | Is a Lie group equivariantly formal under conjugation by a maximal torus? | The answer is yes, if $G$ is connected.
This follows from the following theorem that I believe is due to Borel and can be found in his Seminar on Transformation Groups.
Theorem: Let $M$ be a compact $T$-manifold (this can be weakened) with fixed point set $M^T$. The sum of rational Betti numbers of $M$ is greater ... | 5 | https://mathoverflow.net/users/36588 | 135877 | 74,746 |
https://mathoverflow.net/questions/135882 | 3 | So a $2$-group presented by a crossed module
$H\overset{t}{\to} G$,
where $t$ has nontrivial kernel and cokernel, is given just by the data of that kernel and cokernel, the action of the cokernel on the kernel, and the class in the group cohomology $H^3(\text{coker } t, \text{ker } t)$ classifying the extension abo... | https://mathoverflow.net/users/36591 | topological class of a flat 2-bundle with nontrivial extension class | Here is the general story (as in section 4.3 of *[Principal infinity-bundles -- General theory](http://arxiv.org/abs/1207.0248)*).
So consider
$$
A \stackrel{i}{\to} \hat G \stackrel{p}{\to} G
$$
a "central" extension of higher groups, hence a long homotopy fiber sequence of the form
$$
\array{
\mathbf{B}A &... | 2 | https://mathoverflow.net/users/381 | 135895 | 74,753 |
https://mathoverflow.net/questions/135830 | 9 | In Grothendieck's Brauer group papers, he uses deformation theory to bootstrap the theory of central simple algebras over a field to the theory of Azumaya algebras over rings (and schemes). I am surprised to be unable to find any parallel discussion of octonion algebras in the literature, and wonder if someone may know... | https://mathoverflow.net/users/35361 | Deformation theory of octonion algebras? | I can answer my own question, upon finding the 1959 paper "The arithmetics of octaves and the group ${\rm{G}}\_2$" by van der Blij and Springer in the library this morning (pp. 406-418 in volume 21 of Indag. Math.). They work over a field, but their style of calculation on pages 407-412 (ignoring everything about "radi... | 7 | https://mathoverflow.net/users/35361 | 135897 | 74,754 |
https://mathoverflow.net/questions/135898 | 4 | Background
==========
I've just learned a bit about linear codes. [Hamming codes](http://en.wikipedia.org/wiki/Hamming_code) have the property that up to one bit in a block can be corrupted, and we still communicate the message correctly. This is done by making all the non-zero vectors in $\newcommand\ZZ{\mathbb Z}(\... | https://mathoverflow.net/users/1 | Is there a code which corrects corruption of any two bits in a block? | Here is the Tompkins proof, from page 86 of From Error-Correcting Codes Through Sphere Packings to Simple Groups, Volume 21 By Thomas M. Thompson
<http://books.google.com/books?isbn=0883850370>
Assume such a code exists, w/o loss of generality it contains 0.
Each 90-tuple with exactly three 1s must be in a Hamming sp... | 4 | https://mathoverflow.net/users/36061 | 135900 | 74,755 |
https://mathoverflow.net/questions/135886 | 13 | Let $\mathcal{C}$ be a small category and $\Sigma$ a collection of morphisms in $\mathcal{C}$. Denote by $F\_\Sigma:\mathcal{C} \to \mathcal{C}[\Sigma^{-1}]$ the usual quotient functor from $\mathcal{C}$ to its [localization](http://ncatlab.org/nlab/show/localization) about $\Sigma$.
>
> Are there conditions on $\S... | https://mathoverflow.net/users/18263 | When does localization preserve homotopy type of classifying spaces? | You can find some sufficient conditions in terms of simplicial localization of Dwyer and Kan. In Prop. 3.7 of *Simplicial Localizations of Categories* they prove that it holds when $\Sigma$ is free and $\mathcal{C} = \mathcal{D} \* \Sigma$ where $\*$ denotes the coproduct of categories with a fixed set of objects (aka ... | 9 | https://mathoverflow.net/users/12547 | 135905 | 74,758 |
https://mathoverflow.net/questions/135911 | 26 | Liouville's theorem gives such a proof for antiderivatives of functions like $e^x/x$ or $e^{x^2}$, and differential Galois theory extends that to Bessel functions, say. But what tools exist for implicit functions like Lambert's W?
| https://mathoverflow.net/users/17164 | How to prove Lambert's W function is not elementary? | It seems that the non-linear differential equation satisfied by the Lambert $W$ function is simple enough for this question to have already been answered. This paper proves that the Lambert $W$ is non-elementary by appealing to a result of Rosenlicht (1969):
>
> Bronstein, M., Corless, R. M., Davenport, J. H. and J... | 30 | https://mathoverflow.net/users/2622 | 135922 | 74,766 |
https://mathoverflow.net/questions/135876 | 1 | Let $M$ be a pre-symplectic manifold.In recent years several Geometrists are working on $TM \bigoplus T^\*M$ which has fascinated the complex and Poisson geometry. In recent decade also Nigel Hitchin introduced [generalized complex manifold](http://en.wikipedia.org/wiki/Generalized_complex_structure) on this manifold a... | https://mathoverflow.net/users/nan | prequantization on $TM \bigoplus T^*M$ | Hassan asks me in the comment section of my other reply to post the following as a separate reply, too.
The issue of geometric quantization of Poisson manifolds came up in the discussion section, and its potential relation to higher and/or generalized complex geometry. Indeed, one can find the following nice story he... | 4 | https://mathoverflow.net/users/381 | 135928 | 74,767 |
https://mathoverflow.net/questions/135710 | 10 | Conjecture: Let $m$ and $n$ be fixed positive integers and let $f(k)$ be the probability that a Binomial($k(m+n)$, $p$) random variable is less than $kn$. Then for sufficiently small $p$, $f(k)$ is an increasing function of $k$.
Unless I've made a mistake, the conjecture holds if you replace the binomial with its nor... | https://mathoverflow.net/users/136 | Binomial distribution conjecture | Unless I've made a mistake: the conjecture is true for $p<kn/(kn+km+1)$ and possibly other values as well.
Requiring $k(m+n)$ to be an integer, $f(k) = \displaystyle\sum\_0^{\lfloor kn\rfloor}(km+kn)Cr p^r (1-p)^km+kn-r$ and a similar expression with the same limits gives a lower bound for $f(k +1/(m+n))$. Considerin... | 5 | https://mathoverflow.net/users/36639 | 135941 | 74,772 |
https://mathoverflow.net/questions/135939 | 2 | I am trying to calculate **analytic** solution (or *locus*) of *zeros* of a very large **multi-variable** function which is consisted of thousands of **nonlinear trigonometric** terms. All the variables are **real numbers**. The function **is not differential**.
(The equation is a *Singular Value* of the *constrains ... | https://mathoverflow.net/users/36632 | Finding zeros of a multi-variable nonlinear trigonometric function | In the particular case you give, you can make the substitution $t = tan(\theta)/2,$ which transforms your equation into a single polynomial in three variables, of which you want to find all *real* solutions (you can then back out the $\theta$). If your trig functions depend nicely on the argument, this reduction will a... | 1 | https://mathoverflow.net/users/11142 | 135942 | 74,773 |
https://mathoverflow.net/questions/135781 | 7 | Perhaps this question is too elementary, but if it's written down anywhere, I'd love to know about it. Suppose I have a power series $f\in R[[x,y]]$ for some commutative, unital ring. I've recently been trying to write down an explicit formula for the power series in three variables, $f(f(x,y),z)-f(x,f(y,z))$, written ... | https://mathoverflow.net/users/11546 | Explicit formula for associator of commutative power series | Okay, so I'm pretty sure I have a sort-of answer for this for $f=x+y+\sum\_{i,j>0}a\_{ij}x^iy^j$, though it's not a closed form at all.
With a bit of fiddling one can see that $$ f\circ f = \sum\_{i,j>0}a\_{ij}\left(x+y+\sum\_{l,k>o}a\_{lk}x^ly^k\right)^iz^j-\sum\_{i,j>0}a\_{ij}x^i\left(y+z+\sum\_{l,k>o}a\_{lk}y^lz^... | 3 | https://mathoverflow.net/users/11546 | 135943 | 74,774 |
https://mathoverflow.net/questions/135945 | 8 | Let AC denote the Axiom of Choice. Let PP denote the so-called "Partition Principle" which states that "If S is a non-empty set and T is a non-empty set of pairwise disjoint subsets
of S, then S can be mapped onto T". It is well known that "AC implies PP" is provable in ZF,
but the question of whether "PP implies AC" i... | https://mathoverflow.net/users/4423 | A question about the Axiom of Choice | ~~To my best knowledge, the Banaschewski-Moore paper from some twenty years ago is pretty much the last recorded progress on the topic.~~
The two main papers on the subject are the Banaschewski-Moore paper and a paper by Higasikawa, both are from more or less twenty years ago.
>
> 1. Bernhard Banaschewski, Gregor... | 12 | https://mathoverflow.net/users/7206 | 135947 | 74,775 |
https://mathoverflow.net/questions/135426 | 3 | I would like to have an Itô Diffusion
$$ X\_t = \int\_0^t b(s) \mathrm{d}s + \int\_0^t \sigma(s) \mathrm{d}B\_s.$$
where the (vector- and matrix-valued, respectively) functions $b$ and $\sigma$ have lower regularity than the usual requirements that they be Lipschitz continuous.
**My question is: Can such a process be... | https://mathoverflow.net/users/16702 | Ito Diffusions with low regularity? | You can have weak solutions with quite low regularity. In the compact case, continuity of drift and diffusion is sufficient for existence of a weak solution, and in the non-compact case you will need to impose some growth conditions on the coefficient. This is a theorem of Skorokhod(1965), and for its proofs see [a rec... | 3 | https://mathoverflow.net/users/2968 | 135951 | 74,776 |
https://mathoverflow.net/questions/135817 | 4 | Let $G$ be a simple algebraic group, $\mathcal{O}=\mathbb{C}[[t]], \mathcal{K}=\mathbb{C}((t))$ and let $\text{Gr}\_G=G(\mathcal{K})/G(\mathcal{O})$ be the affine Grassmanian. My main question:
* Why does $\text{Gr}\_G$ parametrize the data of a $G$-bundle on $\mathbb{P}^1$, and a trivialization away from a point $x... | https://mathoverflow.net/users/2623 | Describing the affine Grassmanian via $G$-bundles on $\mathbb{P}^1$ | I am not sure I understand your question correctly, it might be you asking something deeper.
But let me write standard words (which you probably know).
Let me consider the case of vector bundles, general case is similar.
Remark 1. If you have any manifold and two charts U\_1 U\_2 you can construct a bundle taking an ... | 4 | https://mathoverflow.net/users/10446 | 135952 | 74,777 |
https://mathoverflow.net/questions/135950 | 15 | While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem:
>
> What is the volume of the largest symmetric convex subset $S$ of $\mbox{Mat}\_3(\mathbb{R})$ such that every matrix in this subset has determinant at most $1$? In particular, ... | https://mathoverflow.net/users/2363 | Geometry of numbers for three by three matrices? | The volume of $S\_1$ is $\frac{16767 \pi^4}{560} \approx 2916.53$.
The idea is to realize that the set $S\_1$ can be equivalently described as the nuclear norm (sum of the singular values $\sigma\_i$) of $M$ being less than or equal to 3, i.e., $S\_1 = \{M : \sum\_i \sigma\_i(M) \leq 3 \}$.
This is an orthogonally... | 11 | https://mathoverflow.net/users/11841 | 135962 | 74,782 |
https://mathoverflow.net/questions/135908 | 6 | In May and Sigurdsson ["Parametrized homotopy theory"](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf) there is a general treatment of Hurewicz style model structures in Chapter 4, see definitions 4.2.1 and 4.2.2. I am trying to adapt these to a more general setting. There is an "observation" after Lemma 4.2.4 s... | https://mathoverflow.net/users/1176 | Why is the path fibration a strong Hurewicz fibration? | Andrej, thank you for your reading of May and Sigurdsson, which I wish we
had made more accessible. I don't have time to check anything, but my
recollection is that the verification of the observation is easy, whether
or not it is formal, and works equally well in homological analogues. I
think we inserted it only f... | 7 | https://mathoverflow.net/users/14447 | 135964 | 74,783 |
https://mathoverflow.net/questions/135894 | 4 | Let $G$ be a connected, simply-connected complex semisimple Lie group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$, and let $$\frak{g}=\frak{t}\oplus\bigoplus\_{\alpha\in\Delta}\frak{g}\_{\alpha}$$ be the corresponding decomposition into weight spaces. Given a non-zero root vector $e\_{\alpha}\in\fra... | https://mathoverflow.net/users/25358 | Orbits of Root Vectors | If $\alpha$ is a long root (e.g. if $\mathfrak g$ is simply-laced), then $e\_\alpha$ is in the $G$-orbit of the high weight vectors. Projectively, its orbit looks like $G/P\_\Theta$ where $P\_\theta$ is the parabolic using those simple roots that are perpendicular to the highest root.
But you asked for the root vecto... | 3 | https://mathoverflow.net/users/391 | 135975 | 74,787 |
https://mathoverflow.net/questions/135963 | 4 | Let $p$ be a fixed prime and $r$ a fixed prime power. Let $x\_{p'}$ denote the largest divisor of a positive integer $x$ such that $x\_{p'}$ is not divisible by $p$. (For example, $60\_{2'}=15$.) I would like to prove:
**Claim**: $\dfrac{(r^n-1)\_{p'}}{n}\rightarrow\infty$ as $n\rightarrow \infty$.
Sorry if this is... | https://mathoverflow.net/users/32259 | A limit concerning prime numbers | This is true, and due to Walter Feit (On large Zsigmondy Primes, PAMS 1988). (What he shows [Theorem B in the quoted paper] is the following:
Let $N$ be a positive integer. Then, for all but finitely many pairs of integers $<a, n>$ with $a>1$ and $n>2$ there exists a Zsigmondy prime with $|a^n-1|\_p > n N +1.$ A Zsig... | 3 | https://mathoverflow.net/users/11142 | 135979 | 74,788 |
https://mathoverflow.net/questions/135130 | 1 | Thomas Bauer shows in <http://arxiv.org/pdf/alg-geom/9712019v1.pdf> that for a complex abelian variety a nef line bundle is numerically equivalent to an effective divisor (this is shown in Lemma 1.1). It seems to me (by comments in other papers) that this is known over a general algebraically closed field, but I have y... | https://mathoverflow.net/users/14143 | Nef classes on abelian varieties in positive characteristic | Here is a sketch of a purely algebraic proof based on the theory developed in Chapter 3 of Mumford's "Abelian Varieties".
Let $L$ be a nef line bundle on the abelian variety $A$ of dimension $g$. If $K(L)$ is finite, then the index of $L$ is a well defined integer between $0$ and $g$ and we need to show that $g=0$. B... | 3 | https://mathoverflow.net/users/519 | 135980 | 74,789 |
https://mathoverflow.net/questions/135978 | 2 | Suppose $R$ is a local ring and let $I\subset R$ be some nontrivial ideal. Are there conditions that we can place on $I$ so that if $R/I$ is regular, then so is $R$?
I am aware of the result that states: if $R$ is already a regular local ring, then $R/I$ is regular iff $I$ is generated by a subset of a regular syste... | https://mathoverflow.net/users/36661 | If the quotient of a local ring is regular, does that imply that the original ring must be regular? | The quoted result relies on the following elementary characterization of local regular rings:
>
> Let $R$ be a local ring with maximal ideal $\mathfrak m$ and $x\in\mathfrak m-\mathfrak m^2$. Then $R$ is regular iff $R/(x)$ is regular and $x$ doesn't belong to any minimal prime.
>
>
>
| 2 | https://mathoverflow.net/users/23950 | 135986 | 74,791 |
https://mathoverflow.net/questions/135921 | 6 | Suppose $C$ is an integral nodal curve with one node. It is claimed in the arxiv version of a paper by Bogomolov, Hassett, Tschinkel that the dualizing sheaf and the sheaf of differentials are related by the formula $$\Omega\_C \simeq \omega\_C \otimes I\_p,$$ where $I\_p$ is the ideal sheaf of a node: ([see p.10](http... | https://mathoverflow.net/users/36622 | Kaehler differentials on a nodal curve | As said the OP in the comments, in $\widehat{O}\_{C,p}$, the canonical map $\Omega^1\to \omega$ has a kernel $T$, isomorphic to the vector space generated by $sdt$. As $O\_{C,p}\to \widehat{O}\_{C,p}$ is flat, this implies that in $O\_C$, the kernel of the canonical map $\Omega^1\_C\to\omega\_{C}$ is a vector space of ... | 1 | https://mathoverflow.net/users/nan | 135989 | 74,794 |
https://mathoverflow.net/questions/135955 | 6 | A simple fact: Given a vector field on a compact manifold with boundary, if the vector field points inward along the boundary, then it must vanish somewhere in the interior. (EDIT: As pointed out in the accepted answer and in a comment, the Euler characteristic must be nonzero for this to be true.)
My question: Is th... | https://mathoverflow.net/users/1179 | vanishing of vector field in infinite dimensions | The simple fact in question is false in any dimension greater than one.
Consider the strip $ \mathbb{R} \times [-\pi/2,\pi/2] \subset \mathbb{R}^2$. At a point $(x, y)$ take the vector $(-sin(y), cos(y))$. This does not depend on $x$ so descends to a vector field on the annulus $\mathbb{R}/\mathbb{Z} \times [-\pi/2, ... | 5 | https://mathoverflow.net/users/4707 | 135993 | 74,796 |
https://mathoverflow.net/questions/135995 | 8 | If $\kappa$ is an inaccessible cardinal then $V\_\kappa$ is a model of $\sf ZFC\_2$ ($\sf ZFC$ with a second-order replacement axiom).
If there are many inaccessible cardinals then there are many models of $\sf ZFC\_2$, but one can add all sort of $\varphi$ which describe $V\_\kappa$ completely. For example if $\varp... | https://mathoverflow.net/users/7206 | Indescribability of cardinals and categoricity of $V_\kappa$ | Categoricity should perhaps be conceived of not as a large
cardinal notion, but rather as an *anti*-large cardinal notion,
since most large cardinal concepts express some degree of
reflection, which is the opposite of categoricity.
For example, if $\kappa$ is $\Pi^n\_m$-categorical, then clearly it
is not $\Pi^n\_m$-... | 9 | https://mathoverflow.net/users/1946 | 136003 | 74,800 |
https://mathoverflow.net/questions/133458 | 20 | I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for mathematics and mathematical physics? Is there any property which connects this theory to several areas of math?
**More prec... | https://mathoverflow.net/users/nan | What is Kirillov's method good for? | This question is quite general, I'll write just my own point of view, and hope others add more to get a complete picture.
**0)** Let me [quote A. Kirillov himself](http://www.ams.org/journals/bull/1999-36-04/S0273-0979-99-00849-6/S0273-0979-99-00849-6.pdf):
>
> "In conclusion I want to express the hope that the o... | 20 | https://mathoverflow.net/users/10446 | 136006 | 74,801 |
https://mathoverflow.net/questions/136007 | 1 | Suppose $K$ is an extension of $\mathbb{Q}\_{2}$ (I could ask this for any $p$, but I'm especially interested in $p = 2$) which contains the Hilbert Class Field (i.e., maximal unramified extension) of $\mathbb{Q}\_{2}$ and also contains all roots of unity (including $2$-power ones!) Then does $K$ contain all roots of a... | https://mathoverflow.net/users/24757 | If $K=\langle$HCF of $\mathbb{Q}_{p}$, $\mathbb{Q}^\mathrm{cycl}\rangle$, does $K$ also contain all roots of elements of $\mathcal{O}_{K}^{\times}$? | Think of it this way : $K$ is an abelian extension of $\mathbf{Q}\_p$. Now, the the extension $\mathbf{Q}\_p(\root{p^m}\of u)$ need not even be galoisian over $\mathbf{Q}\_p$ for some appropriate $u\in\mathbf{Z}\_p^\times$, so it cannot be contained in $K$. (As it happens, $K$ is the *maximal* abelian extension of $\ma... | 5 | https://mathoverflow.net/users/2821 | 136011 | 74,802 |
https://mathoverflow.net/questions/132873 | 4 | I have a bunch of papers that claim that, from the equation for shape energy:
$$ F = \frac{1}{2}k\_c \int (c\_1+c\_2-c\_0)^2 dA + \Delta p \int dV + \lambda \int dA$$
one can use "methods of variational calculus" to derive the following:
$$\Delta p - 2\lambda H + k(2H+c\_0)(2H^2-2K-c\_0H)+2k\nabla^2H=0$$
But I'm having... | https://mathoverflow.net/users/20343 | Deriving Helfrich's shape equation for closed membranes | The follow-up paper by Helfrich you are searching for is
*Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders,*
Ou-Yang Zhong-can and Wolfgang Helfrich,
Physical Review A **39**, 5280-5288 (1989).
>
> A... | 1 | https://mathoverflow.net/users/11260 | 136013 | 74,803 |
https://mathoverflow.net/questions/136012 | 9 | It seems to be well-known that the six-transitive finite groups are the symmetric and alternating groups, and the only other four-transitive finite groups are the Mathieu groups (the statement can be found in Cameron's 1999 "Permutation Groups", p 110), but I can't seem to figure out where the result was first stated (... | https://mathoverflow.net/users/11142 | Where was it first stated that there are no 4-transitive finite groups other than symmetric, alternating and Mathieu groups? | In *Pacific Journal of Math* **4** (1954), pp 219-226, Marshall Hall, Jr. writes in the paper "On a theorem of Jordan" that:
>
> In 1872, Jordan showed that a finite quadruply transitive group in which only the identity fixes four letters must be one of the following: the symmetric group on four or five letters, th... | 10 | https://mathoverflow.net/users/3959 | 136026 | 74,807 |
https://mathoverflow.net/questions/136005 | 4 | Consider two square matrices $A, B \in \mathbb{R}^{n \times n}$ and let $\| \cdot\|\_1$ and $\|\cdot\|$ be, respectively, the trace norm (the sum of singular values) and the usual operator norm (the maximum of singular values).
Is there a known bound for the following quantity?
$$
\sup\{\alpha > 0: \; \alpha \, \text... | https://mathoverflow.net/users/36687 | An inequality involving operator and trace norms | Since I cant comment, I will leave this thought here. Since $||\cdot||\_1$ and $||\cdot||$ (as you defined them) are dual norms, it must be that tr$((A+B)^TX)\leq||A+B||\_1$ for any $X$ such that $||X||\leq 1$. Therefore, tr$(A^TB)\leq ||A+B||\_1 - ||B||^2<||A+B||\_1$ (since tr$(B^TB)\geq||B||^2$).
(edit: fixed typo ... | 4 | https://mathoverflow.net/users/36154 | 136027 | 74,808 |
https://mathoverflow.net/questions/136025 | 14 | As mentioned by @MichaelZieve in his comment re [Quadratic residue](https://mathoverflow.net/a/135823/9072), Chebotarev's density theorem was preceded by an allegedly much easier theorem of Frobenius (Mike Zieve is certainly not the only one to mention that the Frobenius theorem is much easier than Chebotarev) -- the d... | https://mathoverflow.net/users/11142 | Frobenius density theorem | You can find Frobenius's theorem (and proof) on p.134 of Janusz, Algebraic Number Theory, 1973, and many other places. It really is easier. It was known long before Chebotarev's theorem, and Chebotarev had to come up with new ideas to prove his theorem (which in turn helped Artin prove his reciprocity law).
| 15 | https://mathoverflow.net/users/36469 | 136028 | 74,809 |
https://mathoverflow.net/questions/136017 | 38 | Let $X$ be an algebraic variety over an alg. closed field with zero char. and let $f:X\to \mathbb{A}^n$ be a smooth surjective morphism, such that all fibers (at closed points) are isomorphic to $\mathbb{A}^m$. Does it follow that $X\cong \mathbb{A}^{n+m}$?
If not, is it true with some additional assumptions? I know... | https://mathoverflow.net/users/14379 | Is an affine fibration over an affine space necessarily trivial? | I feel like I already answered this question, but it might have been a variant with fibers isomorphic to tori. Let the base $B$ be $\mathbb{A}^2$ with coordinates $s$ and $t$. Begin with $B\times \mathbb{P}^3$, where homogeneous coordinates on $\mathbb{P}^3$ are $[x,y,z,w]$. Let $S$ be the Cartier divisor in $B\times \... | 27 | https://mathoverflow.net/users/13265 | 136043 | 74,817 |
https://mathoverflow.net/questions/135949 | 37 | Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a permutation. Is $u$ a polynomial of degree $2$?
Note that the property clearly holds when $u$ is a polynomial of degre... | https://mathoverflow.net/users/9890 | A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself | Yes, $u$ must be a polynomial of degree $2$.
I had to draw on a few unexpected ingredients to prove this;
perhaps there's a simpler proof.
[**EDIT** *Or maybe not: Peter Mueller's answer reports that
this was "an open problem on planar functions for many years",
and gives links to three independent papers c.1990 that i... | 41 | https://mathoverflow.net/users/14830 | 136046 | 74,818 |
https://mathoverflow.net/questions/136008 | 6 | In the definition of [pre-quantization](http://en.wikipedia.org/wiki/Geometric_quantization) on a [symplectic manifold](https://en.wikipedia.org/wiki/Symplectic_manifold) $(M,\omega)$, we represent a function $f\in C^{\infty}(M)$(with Lie algebra structure) to $\hat{f}$ in the Hilbert space $L^2(M,L,\mu)$ associated to... | https://mathoverflow.net/users/nan | Prequantization and Hilbert space | OK, so here are just a few thought on this large topic of quantization. First of all, the question of irreducibility can equally well be asked for deformation quantization (as mentioned by other answers) and thus does not refer to any particular feature/problem of geometric quantization. The physical idea behind this r... | 4 | https://mathoverflow.net/users/12482 | 136050 | 74,820 |
https://mathoverflow.net/questions/135925 | 7 | $\DeclareMathOperator{\ab}{Ab}\DeclareMathOperator{\qcoh}{QCoh}$
[This entry in the nlab](http://ncatlab.org/nlab/show/module#ModulesOverARingInTermsOfStabilizedSlices) shows that for $A$ a (commutative unital) ring, the category $\mathsf{Mod}\_A$ of $A$-modules is equivalent to the category $\ab(\mathsf{CRing}/A)$ of ... | https://mathoverflow.net/users/6856 | When does the categorical definition of a module work? | It is not true, but something similar is true. If $X$ is a scheme, the functor you describe is actually $\mathsf{Qcoh}(X) \to \mathsf{Ab}(\mathsf{QAlg}(X)/\mathcal{O}\_X)$, where $\mathsf{QAlg}(X)$ denotes the category of quasi-coherent algebras on $X$, and $\mathsf{QAlg}(X)/\mathcal{O}\_X$ is the slice category consis... | 9 | https://mathoverflow.net/users/2841 | 136062 | 74,826 |
https://mathoverflow.net/questions/65524 | 10 | It seems to me this come up very often when we talk about group action on (étale) cohomology groups.
For example, let $X$ be a scheme over $\mathrm{spec}\mathbb{Z}$, $\mathcal{F}$ an $\ell$-adic sheaf on $X$, and $V\_\ell= H^i(X\otimes \overline{\mathbb{Q}}, \mathcal{F})$. The Galois group of $\overline{\mathbb{Q}}$... | https://mathoverflow.net/users/5738 | What is transport of structure in cohomology setting? | I am not sure this is adding anything substantial to the comments, but maybe it helps
to spell things out. At least this used to confuse me in the past. I found a remark in a recent preprint of
Deligne-Flicker helpful: *Transport of structures (Bourbaki Ens Ch. IV) is the principle that any isomorphism $Y\_1 \to Y\_2$ ... | 9 | https://mathoverflow.net/users/1729 | 136064 | 74,827 |
https://mathoverflow.net/questions/136029 | 2 | I have the following question. Let's denote by $\Omega$ the cobar functor, by $C$ some DG-coalgebra (not co-commutative) over a field $k$. Also suppose $V=k\times\dots\times k$ is just the product of $n$ copies of the base field $k$, and $V^\*$ is the linear dual of $V$, which is a coalgebra in a natural way.
**Q:** ... | https://mathoverflow.net/users/32741 | cobar construction of a direct sum of coalgebras | I am sorry, I guess I was endeed overlooking some simple things.
The cobar functor is left adjoint, so it commutes with colimits. In particular it commutes with coproducts. The part I was missing was that in the category of coalgebras the direct sum of underlying vector spaces gives the coproduct. The coproduct in th... | 3 | https://mathoverflow.net/users/32741 | 136069 | 74,829 |
https://mathoverflow.net/questions/136057 | 26 | First fix the following notations:
$AF:=$ The axiom of foundation
$ZFC^{-}:=ZFC\setminus \left\lbrace AF \right\rbrace $
$G:=$ The proper class of all sets
$V:=$ The proper class of Von neumann's cumulative hierarchy
$L:=$ The proper class of G... | https://mathoverflow.net/users/nan | Is there any large cardinal beyond Kunen inconsistency? | The answer to question 2 is yes, and one can even have nontrivial automorphisms. For example, the theory $\mathit{ZFC}^-+{}$“there are two urelements (i.e., sets $x$ satisfying $x=\{x\}$) and the whole universe is obtained from them by iterated power set” is consistent relative to ZFC, and one can define uniquely in th... | 16 | https://mathoverflow.net/users/12705 | 136072 | 74,830 |
https://mathoverflow.net/questions/136021 | 12 | There is an equivalence relation between inclusion of finite groups coming from the world of [subfactors](http://en.wikipedia.org/wiki/Subfactor):
**Definition**: $(H\_{1} \subset G\_{1}) \sim(H\_{2} \subset G\_{2})$ if $(R^{G\_{1}} \subset R^{H\_{1}})\cong(R^{G\_{2}} \subset R^{H\_{2}})$ as subfactors.
Here, $R$ i... | https://mathoverflow.net/users/34538 | Is there a purely group-theoretic reformulation of an equivalence of subgroups? | For finite groups, the answer was given by Izumi in his paper "Characterization of isomorphic group-subgroup subfactors" (MR1920326). There he looks at the crossed product subfactor, but you can always take duals.
Edit after @Andre's comment:
The actual condition between the two pairs of subgroups is quite technica... | 10 | https://mathoverflow.net/users/351 | 136085 | 74,838 |
https://mathoverflow.net/questions/135889 | 9 | Let $I, J \subset S = k[x\_1,\dots,x\_n]$ be two monomial ideals and $k$ a field. If every element of $S$ which is $S/J$-regular is also $S/I$-regular is it true that depth$\_S S/I \geq$ depth$\_S S/J$ ?
| https://mathoverflow.net/users/13874 | Depth of ideals in a commutative ring | $R = k[a,b,c,d]$, $I= (a, b)\cap (c,d)$ and $J = (ac, bd) = (a, b) \cap (a, d)\cap (c, b)\cap (c, d)$. Then $\mathrm{Ass}R/I \subseteq \mathrm{Ass}R/J$. Hence if $x$ is a regular element of $R/J$ then $x$ is a regular element of $R/I$ but $\mathrm{depth}R/I = 1$ and $\mathrm{depth}R/J = 2$.
| 6 | https://mathoverflow.net/users/17901 | 136088 | 74,840 |
https://mathoverflow.net/questions/136091 | 4 | Let $M$ be a (usual, finite dimensional) smooth manifold, and $C^\infty(M)$ the algebra of (real valued) smooth functions on $M$. For each point $a\in M$ and for each subalgebra $A$ in $C^\infty(M)$ (I mean $A$ contains constants and is closed under summing and multiplication) let us denote by $I\_a[A]$ the ideal in $A... | https://mathoverflow.net/users/18943 | Ideals in a subalgebra in $C^\infty(M)$ | **Edit:** modified to satisfy the last condition.
Consider the subalgebra of $C^\infty(\mathbb{R})$ generated by two elements $x^2,xe^x$. This algebra separates points and "sees" any nonzero tangent vector. The ideal $I\_0[A]$ has the functions $x^{2k+l}e^{lx}$ where $k,l\in\mathbb{N}$ and not both $k$ and $l$ are ze... | 2 | https://mathoverflow.net/users/745 | 136093 | 74,841 |
https://mathoverflow.net/questions/134998 | 29 | The following problem arose in collaborative work with Subhro Ghosh:
**Question**: To any polynomial $P\_n(z)=\sum\_{i=0}^n a\_i z^i =a\_n \prod\_{i=1}^n(z-z\_i)$, attach the empirical measure of zeros
$L\_n=n^{-1}\sum \delta\_{z\_i}$ (a probability measure on the complex plane).
Let ${\cal M}$ denote the collection... | https://mathoverflow.net/users/35520 | Zeros of polynomials with real positive coefficients | I don't have a complete proof yet, but I have a plausible conjecture. Let $\mu$ be a probability measure in the plane, define the potential
$$u(z)=\int\log|1-z/t|d\mu(t).$$
Then I conjecture that $\mu\in M$ iff $u$ satisfies $u(z)\leq u(|z|)$ for all $z$.
It is evident that this condition is necessary.
It seems that ... | 20 | https://mathoverflow.net/users/25510 | 136099 | 74,844 |
https://mathoverflow.net/questions/136089 | 1 | Let $x$ and $y$ be two element of general linear group $SL(n,q)$, such that the orders of $x$ and $y$ be some primitive prime divisor of $q^n-1$. Is it true that if $xy\not=yx$, then $x$ and $y$ generate $SL(n,q)$ ?
Note that the order of an element like $x$ of $SL(n,q)$ is a primitive prime divisor of $q^n-1$ if an... | https://mathoverflow.net/users/36749 | The generators of special linear groups | Not necessarily. For $n$ even, they could generate one of the smaller classical groups ${\rm Sp}(n,q)$ or $\Omega^-(n,q)$.
For $n$ odd, it might be true sometimes - I think it is true in ${\rm SL}(5,2)$ for example. But there will sometimes be exceptions. For example, in ${\rm SL}(5,3)$ $x,y$ could generate $L\_2(11)... | 6 | https://mathoverflow.net/users/35840 | 136101 | 74,846 |
https://mathoverflow.net/questions/136112 | 4 | In the paper "On a problem of Zariski", David Rees presents a counterexample to the following problem of Zariski.
>
> Let $F/k$ be a f.g. field extension, $S$ a f.g. normal integral domain over $k$ such that $Q(S)\supseteq F$. Is $S\cap F$ a finitely generated $k$-algebra?
>
>
>
His construction starts with a ... | https://mathoverflow.net/users/22998 | A certain property of elliptic curves in a paper by Rees | If the line at infinity meets $C$ only at the point $O$ (the usual thing), then a curve of equation $f=0$ and degree $m$ meets $C$ only at $P$ if and only if the divisor of $f$ as a function on $C$ is $m(P-O)$, that is $P$ is torsion in the group law (which is equivalent to the "value of the parameter being a rational ... | 2 | https://mathoverflow.net/users/2290 | 136116 | 74,851 |
https://mathoverflow.net/questions/136120 | 5 | Does there exist an English translation of Steinitz' 1910 work "Algebraische Theorie der Körper"?
<http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002167042>
| https://mathoverflow.net/users/33757 | English translation of Steinitz 1910? | You may already know this, but the best reference about Steinitz $1910$ work I could find is the following summary by Peter Roquette:
<http://www.rzuser.uni-heidelberg.de/~ci3/STEINITZ.pdf>. Some English references are given there, and apparently a lot of Steinitz $1910$ work can be found in the book "Modern Algenbra"... | 4 | https://mathoverflow.net/users/32332 | 136122 | 74,854 |
https://mathoverflow.net/questions/136134 | 10 | In Appendix B of his [Uniform Central Limit Theorems](https://rads.stackoverflow.com/amzn/click/com/0521052211) (1999), Dudley writes:
>
> It is consistent with the usual axioms of set theory (including the axiom f choice) that there are no measurable cardinals, in other words all cardinals are of measure 0; see, f... | https://mathoverflow.net/users/238 | Do Measurable Cardinals Exist? (assuming ZFC) | This is not really a problem.
If $\kappa$ is a measurable cardinal then $V\_\kappa$, or the set of sets which are hereditarily have size smaller than $\kappa$, is a model of $\sf ZFC$. This means that $\sf ZFC$ cannot even prove the consistency of $\sf ZFC+\exists\kappa\text{ measurable}$, because of the incompletene... | 17 | https://mathoverflow.net/users/7206 | 136135 | 74,861 |
https://mathoverflow.net/questions/136082 | 1 | Algebraic functions have a discrete set of singularities. [Lacunary functions](http://en.wikipedia.org/wiki/Lacunary_function), e.g. $f(z)=\sum\_{n=0}^\infty z^{2^n}$, have a continuum of singularities at every point of the boundary of their disk of convergence. The [Ostrowski–Hadamard gap theorem](http://en.wikipedia.... | https://mathoverflow.net/users/36742 | Finite construction of lacunary functions using algebraic and certain analytic operations | 1. You have to define what a "lacunary function" means. There are various lacunary conditions in the literature.
2. If you mean sufficiently large lacunas so that your series have a whole circle of
singularities, like in the example you give, such functions cannot be constructed as
a finite composition of algebraic, ex... | 3 | https://mathoverflow.net/users/25510 | 136138 | 74,862 |
https://mathoverflow.net/questions/136146 | 8 | Let $k$ be a field, and $K$ its separable closure. Consider two different $k$-schemes, $X$ and $Y$, which become isomorphic upon extension of scalars to $K$: $X\_K \cong Y\_K$. Then the etale cohomologies (and indeed, etale homotopy types) of $X\_K$ and $Y\_K$ will be equivalent, but may differ at $k$. I'd like to know... | https://mathoverflow.net/users/4649 | Etale cohomology of and forms of algebraic groups | For your first question: the two forms of $GL\_2$ differ by modifying the Galois action by an automorphism of $GL\_2$. The outer automorphism group of $GL\_2$ is cyclic of order $2$: an automorphism is inner if and only if it is trivial on the center of $GL\_2$. Since the form of $GL\_2$ you're considering has the same... | 13 | https://mathoverflow.net/users/7721 | 136153 | 74,871 |
https://mathoverflow.net/questions/136156 | 1 | I've faced the beltway reconstruction problem and I've developed a simple backtrack algorithm, what algorithms do you know for this problem?
**Beltway Reconstruction Problem**:
Assume there is a set of non-identical integers between 0 and N, we only have pairwise distances of points of that set mod N, How can w... | https://mathoverflow.net/users/36047 | What algorithms do you know for beltway reconstruction? | A relatively recent result for a sparse case can be found [here](http://arxiv.org/pdf/1212.2386v1.pdf) (see Algorithm 2). It’s a probabilistic polynomial-time algorithm.
| 1 | https://mathoverflow.net/users/34050 | 136160 | 74,874 |
https://mathoverflow.net/questions/136171 | 2 | Let us call a **subgroup** an injective homomorphism between groups.
>
> **I warn the reader that a *subgroup* designates here an inclusion $(H \subset G)$, not $H$ alone.**
>
>
>
A subgroup $H \subset G$ is **maximal** if for all intermediate subgroups $H \subset K \subset G$, then $K=H$ or $G$.
Let $\si... | https://mathoverflow.net/users/34538 | An upper bound for the maximal subgroups at fixed index? | I'm absolutely ignorant about subfactors, but if I correctly understand what you said [there](https://mathoverflow.net/q/136021/6451) (the theorem in the question), the answer to all three questions is yes, even with non maximal subgroups.
Namely, let $H\subset G$ be of finite index $n$. Since the intersection $K$ of... | 5 | https://mathoverflow.net/users/6451 | 136176 | 74,882 |
https://mathoverflow.net/questions/136196 | 4 |
>
> Let $l$ be a prime $\geq 5$. Does there exist a pair $E,E'$ of elliptic curves, both defined over the same number field $K$, which are not $l$-isogenous over $K$, but are $l$-isogenous over a *quadratic* extension?
>
>
>
I feel that the answer is **yes**, though I cannot come up with an example.
I would a... | https://mathoverflow.net/users/5744 | Elliptic Curves isogenous only over an extension? | Take an elliptic curve $E$ (say, over the rationals) with CM and $\ell$ a prime that splits in the CM field. Now take $E'$ to be a twist of $E$, so $E, E'$ are not isomorphic over the rationals and not $\ell$ isogenous either, as such an isogeny, composed with a self-isogeny will force them to be isomorphic. Now, over ... | 5 | https://mathoverflow.net/users/2290 | 136204 | 74,890 |
https://mathoverflow.net/questions/136186 | 2 | I hope this question is not too vague.
Let $G$ be a complex reductive group, $B$ a Borel subgroup of $G$, and $P$ a parabolic containing $B$.
Denote by $\pi:G/B\to G/P$ the canonical map. Consider the right derived functor $R\pi\_\*:D^b\_B(G/B)\to D^b\_B(G/P)$, $\mathcal{F}\_1$ and $\mathcal{F}\_2$ simple perverse sh... | https://mathoverflow.net/users/32972 | Derived Push-Forward of Morphism of Perverse Sheaves and Translation Functors | Consider the collection of equivariant derived categories $D^b\_{P\times Q}(G)$ for $P,Q$ parabolics corresponding to collections $I,J$ of simple roots. Between these categories we have pushforward and pullback functors.
On the other hand there is a collection of categories given by bimodules:
$R^I-Mod-R^J$, where $R... | 1 | https://mathoverflow.net/users/2837 | 136214 | 74,893 |
https://mathoverflow.net/questions/136200 | 2 | The long exact sequence associated to the Hochschild-Serre spectral sequence for extension of groups $1 \to H \to G \to G/H \to 1$ is
$$
\begin{array}[t]{lll}
1 \to & H^1(G/H, A^{G/H}) \xrightarrow{\inf} H^1(G,A)
\xrightarrow{\mathrm{res}} H^1(H, A)^{G/H} \xrightarrow{\mathrm{tr}} & \\[1ex]
& \to H^2(G/H,A^H) \xrigh... | https://mathoverflow.net/users/2234 | the sixth morphism in the long exact sequence associated to the Hochschild-Serre spectral sequence | Yes, the morhpisms have been described in detail in the paper "A seven-term exact sequence for the cohomology of a group extension" by Dekimpe, Hartl and Wauters: see <http://arxiv.org/pdf/1103.4052.pdf>.
Section $6$ gives details for $\rho$ (the sequence is slightly different, but applies more or less to your situ... | 1 | https://mathoverflow.net/users/32332 | 136221 | 74,896 |
https://mathoverflow.net/questions/136195 | 21 | A recent project has forced my colleague and me to take a rather abstract approach to dynamical systems, and the following definition arose naturally in that context.
Let $\mathcal{C}$ be a category. Its *endomorphism category* $\text{End}(\mathcal{C})$ is defined as follows. The objects are the endomorphisms $f:x \t... | https://mathoverflow.net/users/18263 | How does it End? | Note that $\DeclareMathOperator\End{End}\End(C)$ is the category of functors $\newcommand{\BN}{{\mathrm B\mathbb N}}\renewcommand\hom{\operatorname{hom}}\hom(\BN,C)$, where $\BN$ is the category with one object and $\mathbb N$ morphisms, also called the *walking endomorphism*. Thus $\End(\End(C)) = \hom(\BN^2,C)$, by t... | 14 | https://mathoverflow.net/users/78 | 136227 | 74,898 |
https://mathoverflow.net/questions/136180 | 3 | Let $M$ be a n-dimensional closed smooth manifold, $\eta\_{k}$ be the k-form on $M$, $X$ be the smooth vector field on $M$.Let $\eta=\eta\_{0}+\eta\_{1}+\cdots+\eta\_{n-1}+\eta\_{n}$ be a ($d-i\_{X}$)-closed form,
then we have $$i\_{X}\eta\_{n}=d\eta\_{n-2},$$ the relation imply that $\eta\_{n}$ is exact outside the se... | https://mathoverflow.net/users/3896 | A question about equivariant closed form | If you can find a closed 1-form $\alpha\in \Omega^1(M\setminus Z(X))$ with $i\_X\alpha=1$, then $\eta\_n=\alpha\wedge i\_X\eta\_n = \alpha\wedge d\eta\_{n-2}= -d(\alpha\wedge\eta\_{n-2})$ is exact.
Maybe, the source that you are reading has such $\alpha$.
| 3 | https://mathoverflow.net/users/26935 | 136228 | 74,899 |
https://mathoverflow.net/questions/136236 | 22 | Reverse mathematics, as I mean here, is the study of which theorems/axioms can be used to prove other theorems/axioms over a weak base theory. Examples include
* **Subsystems of Second Order Arithmetic (SOSOA).** Theorems expressible in second order arithmetic are compared over $\mathsf{RCA}\_0$ (which is roughly the... | https://mathoverflow.net/users/12978 | Prospects for reverse mathematics in Homotopy Type Theory | Before I attempt to address your specific questions, let me give a thought provoking non-answer:
>
> Reverse mathematics is impossible (and irrelevant) in HoTT!
>
>
>
This is because HoTT fully supports *proof-relevant mathematics*, so when you refer to a theorem you necessarily refer to a proof of that theor... | 23 | https://mathoverflow.net/users/2000 | 136249 | 74,908 |
https://mathoverflow.net/questions/136251 | 4 | Suppose that $x\_1,\ldots,x\_n$ are $n$ vectors in $\mathbb{R}^m$ (where $m<n\leq m^2$) such that any subset of $m$ of them are linearly independent (i.e., they are "generic"). Now, form the $m^2\times n$ matrix
$A = [x\_1\otimes x\_1, x\_2\otimes x\_2,\cdots,x\_n\otimes x\_n]$ (where $\otimes$ is the tensor product o... | https://mathoverflow.net/users/36154 | On tensor products of "generic" vectors | The result is not true and the best upper-bound on $n$ to make it work happens to
be $2m-1$.
Assume that $n<2m$. Let $(t\_1,\dots,t\_n)$ be a family of real numbers
such that $\sum\_{k=1}^n t\_k \,x\_k \otimes x\_k=0$. Assume that some $t\_k$ is non-zero.
Denote by N the number of indices $k$ such that $t\_k \neq ... | 6 | https://mathoverflow.net/users/34951 | 136254 | 74,909 |
https://mathoverflow.net/questions/136255 | 4 | Well-known Theorem:
Let $a$ be an ideal of the noetherian ring $R$ and let $M$ be a finitely generated
$R$-module. Let $i \in \Bbb N\_0$ be such that $H^j\_a
(M)$ is finitely generated for all $j < i$. Then the set
$Ass\_R((H^i\_a
(M))$ is finite.
Now my questions:
Is there an example that $H^0\_a(M), H^1\_a(M), ... | https://mathoverflow.net/users/36801 | finiteness of local cohomology | Ok, let $X$ by a smooth Abelian surface (for example, the segre product of elliptic curves) and let $R\_0$ be section ring of some projectively normal embedding.
For example, to form this ring $R\_0$, take the Segre product of the two standard graded rings
$$
k[x,y,z]/\langle x^3 + y^3 + z^3 \rangle \text{ with } k[a... | 1 | https://mathoverflow.net/users/3521 | 136259 | 74,912 |
https://mathoverflow.net/questions/136215 | 24 | Consider all of your basic constructions/tools/theorems for manifolds: fundamental group, Euler characteristic, triangulations, orientation, smoothness, bundle structure, cobordisms, etc.. Viewing orbifolds as the natural generalization of manifolds by quotients of group actions (or just locally $\mathbb{R}^n/G\_i$), i... | https://mathoverflow.net/users/12310 | What tools cannot work for orbifolds? | I'm not an expert and this might be wrong, but I think that [Cerf theory](http://en.wikipedia.org/wiki/Cerf_theory) should be impossible for orbifolds, and therefore all that comes from it, *e.g.* Kirby Calculus. Could somebody who knows please confirm this? I'm guessing that this comes **not** from orbifolds having si... | 12 | https://mathoverflow.net/users/2051 | 136267 | 74,915 |
https://mathoverflow.net/questions/135746 | 8 | It is known that given a set of Areas $A\_f$ and normals $\vec{n}\_f$ if $\sum\_f A\_f \vec{n}\_f=0$ exist a unique convex polyhedron with given face areas and normals. (Minkowski theorem - See Alexandrov book on Convex Polyhedra).
Obviously here I'm identifying all the isometric polyhedra.
In principle with the sa... | https://mathoverflow.net/users/36506 | Volumes of convex vs non-convex polyhedra with prescribed facets areas | The convex polytope has the largest volume. This was proved in
K. Boroczky, I. Bárány, E. Makai Jr. & J. Pach: Maximal volume enclosed by plates and proof of the chessboard conjecture, Discrete Math. 69 (1986) 101–120.
For measures and convex bodies, a proof is provided by
Zhang, Gaoyong: The affine Sobolev inequ... | 3 | https://mathoverflow.net/users/36865 | 136271 | 74,917 |
https://mathoverflow.net/questions/136263 | 3 | The fundamental group of a knot $K$ (otherwise known as the knot group) is the fundamental group of the knot complement $S^{3} \backslash K $ in $S^{3} $.
In "Virtual Knots: The State of the Art" (<http://books.google.com.au/books?id=WaCJ_-MpdBYC>) on page 7, it says that the fundamental group of a knot recognizes pr... | https://mathoverflow.net/users/36858 | The knot group of a prime knot | The statement on page 7 is referring to Corollary 2.1 of Gordon and
Luecke's "[Knots are determined by their complements](http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0965210-7/S0894-0347-1989-0965210-7.pdf)."
Also a prime knot can not be an amalgamated free product of
non-trivial knot groups by Coroll... | 6 | https://mathoverflow.net/users/27453 | 136272 | 74,918 |
https://mathoverflow.net/questions/136286 | 2 | I am interested in some results on equations of the form
$$\displaystyle a\_0 x\_0^{\alpha\_0} + a\_1 x\_1^{\alpha\_1} + a\_2 x\_2^{\alpha\_2} = 0,$$
with $a\_0, a\_1, a\_2 \in \mathbb{Z}$ and $\alpha\_0, \alpha\_1 | \alpha\_2$. I was told that in the case $a\_0 = a\_1 = -a\_2 = 1$ and $\alpha\_0 = 2, \alpha\_1 = 3, \a... | https://mathoverflow.net/users/10898 | Reference request: a class of diophantine equations | I just want to make a comment, that one can *in general* find a lot of information on the (primitive) integer solutions of the generalized Fermat equation $Ax^p+By^q=Cz^r$ (depending on the three cases $\delta<0$, $\delta=0$ and $\delta>0$, where $\delta=1-1/p-1/q-1/r$).
For example, in the survey "The ABC's of Num... | 5 | https://mathoverflow.net/users/32332 | 136295 | 74,927 |
https://mathoverflow.net/questions/133464 | 3 | There is a known phenomenon in integer partition theory that almost all integer partitions, after a normalization ($\pi/\sqrt{6n}$ where $n$ is the norm of the partition), have young diagrams which are approximated by the curve $$e^{-x}+ e^{-y} = 1. $$ This curve is called the limit shape. I was wondering "how large mu... | https://mathoverflow.net/users/12337 | Rate of Convergence for Limit Shape with Integer Partitions | It sounds like you are asking about large deviations.
For partitions, there is a paper by Boris Pittel called "On a Likely Shape of the Random Ferrers Diagram." (1997). This paper contains a proof of the formula in your question, along with big-O estimates about the variability around that curve. I am not aware of an... | 3 | https://mathoverflow.net/users/36883 | 136300 | 74,930 |
https://mathoverflow.net/questions/135758 | 24 | Let $m>2$ be an integer and $k=\varphi(m)$ be the number of $m$-th primitive roots of the unity. Let $\Phi = \{ \xi\_1, \ldots, \xi\_{k/2}\} $ be a set of $k/2$ pairwise distinct primitive $m$-th roots of the unity such that $\Phi \cap \overline{\Phi} = \emptyset$, where
$\overline\Phi = \{ \overline \xi\_1, \ldots, \o... | https://mathoverflow.net/users/605 | Products of primitive roots of the unity | **Theorem** If $m$ is relatively prime to $30$, then $N=3$ works.
I'll use additive notation. The cyclic group of order $n$ is denoted $C(n)$, the generators of this cyclic group are denoted $U(n)$.
**Lemma** Let $p \geq 7$ be prime and let $X$, $Y$ and $Z$ be subsets of $C(p)$ of size at least $(p-1)/2$. Then $0 ... | 12 | https://mathoverflow.net/users/297 | 136310 | 74,935 |
https://mathoverflow.net/questions/136296 | 2 | What is a good test for identifying cubic non-residues/residues and higher power non-residues/residues modulo a prime $R$ in terms of computational complexity?
Given $M$ and $N$, is there a good way to find a prime $R$ such that $M$ is cubic non-residue modulo $R$ and $N$ is cubic residue modulo $R$?
Update after D... | https://mathoverflow.net/users/10035 | On Cubic Non-Residues Modulo a Prime | Assuming that $M$, $N$, $M N$ and $M N^2$ are noncubes, for a random prime $p$, the probability that $p \equiv 1 \bmod 3$ and $M$ is a cube mod $p$ and $N$ is not a cube is $1/9$. Trying some random primes and checking as Igor Rivin describes is probably faster than trying to be clever.
Proof of the probability claim... | 2 | https://mathoverflow.net/users/297 | 136312 | 74,937 |
https://mathoverflow.net/questions/133855 | 8 | We are interested in a sum-product type estimate. Let $p$ be an odd prime, and let $A$ be the order $p-1$ subgroup of $(\mathbb{Z}/p^2\mathbb{Z})^\times$. That is, let $A = \langle g^p \rangle$, where $(\mathbb{Z}/p^2\mathbb{Z})^\times = \langle g \rangle$. We seek an estimate of the size of the sum set
$$|A + A|.$$
Th... | https://mathoverflow.net/users/35014 | A sum-product estimate in Z/p^2Z | Heath-Brown and Konyagin stop just short of proving that $|A+A|\gg |A|^{3/2}$ in [their paper](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.155.3891) on Gauss sums and Heilbronn's sum.
(Here $A$ is the same subgroup as above; since $A\cup\{0\}=\{0^p,\ldots,(p-1)^p\}$, we may write Heilbronn's sum as $H\_p(... | 2 | https://mathoverflow.net/users/36862 | 136315 | 74,938 |
https://mathoverflow.net/questions/135875 | 13 | We are interested in estimating the size of a certain sumset in $\mathbb{Z}/p^2\mathbb{Z}$. Let $p$ be an odd prime, $g$ a primitive root modulo $p^2$, and $A=\langle g^p\rangle$ the unit subgroup of order $p-1$. Experimentation suggests strongly that if $p \geq 61$, then the size of the sumset $3A = A + A + A$ approxi... | https://mathoverflow.net/users/35014 | Size of a certain sumset in $\mathbb{Z}/p^2\mathbb{Z}$ | Using some results of Shkredov et al (following Heath-Brown and Konyagin), it's possible to show that $|A+A+A|\gg |A|^{11/6}/(\log|A|)^{1/4}$. I posted more details in the old thread. Shkredov's results are rather complicated, so it would be really nice to see a slick proof.
| 2 | https://mathoverflow.net/users/36862 | 136318 | 74,941 |
https://mathoverflow.net/questions/136314 | 29 | As part of a more complex algorithm, I need a fast method to find random points of the n-sphere, $S^n$, starting with a RNG (random number generator). A simple way to do this (in low dimensions at least) is to select a random point of the (n+1)-ball and normalize it. And to get a random point of the (n+1)-ball select a... | https://mathoverflow.net/users/7311 | What is a good method to find random points on the n-sphere when n is large? | The usual approach is to generate $n+1$ i.i.d. mean zero Gaussian random variables $X\_1, \dotsc, X\_{n+1}$ to get a random point $X$ in $(n+1)$-space with rotationally invariant distribution and normalize.
Incidentally, if you ever actually need to generate a random point in an $n$-ball, the best way is probably to ... | 31 | https://mathoverflow.net/users/1044 | 136319 | 74,942 |
https://mathoverflow.net/questions/136194 | 8 | Let $D\_2$ be the topological operad of little disks. This operad can be modelled "combinatorially" in terms of an operad of groupoids called $\newcommand{\PaB}{\mathbf{PaB}}\PaB$, the operad of *parenthesized braids*. An object of the groupoid $\PaB(n)$ is a complete parenthesization of a permutation of the symbols $\... | https://mathoverflow.net/users/1310 | Identifying the little disk operad with parenthesized braids | It seems to me that the operad morphism from **PaB** to $\Pi\_1(D\_2)$ works in positive arity but does not respect composition with the $0$-ary operation on the object level.
| 4 | https://mathoverflow.net/users/36894 | 136321 | 74,944 |
https://mathoverflow.net/questions/136325 | 9 | Let $G$ be a compact Lie group and let $R(G)$ denote its complex representation ring. If $G$ is simply connected, such as $G\_2$, $F\_4$ or $E\_8$, then it is known that $R(G)$ is a polynomial ring [F. Adams, *Lectures on Lie groups* ('69), Theorem 6.41].
**Question:** Are the representation rings of the adjoint grou... | https://mathoverflow.net/users/1291 | Representation rings of exceptional Lie groups | As Victor points out in his comment, your formulation is somewhat out of focus. The general fact is that the representation ring of a connected semisimple compact Lie group is a polynomial algebra iff each simple factor of the group is simply connected or is an odd dimensional special orthogonal group. It's useful to c... | 12 | https://mathoverflow.net/users/4231 | 136329 | 74,949 |
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