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https://mathoverflow.net/questions/104667 | 9 | I found this formula for the Euler-Mascheroni constant $\gamma$.
>
> Just wondering whether such a formula already exists in literature?
> Also, wanted to know whether there are formulas that converge faster than this?
>
>
>
$$\gamma = \sum\_{k = 1}^{\infty} \frac{1}{2^k k} - \sum\_{k = 1}^{\infty}
\frac{\z... | https://mathoverflow.net/users/2865 | Series representation for Euler-Mascheroni constant | In his 1887 paper *Table des valeurs des sommes* $S\_k = \sum\_{1}^\infty n^{-k}$ (Acta Mathematica 10 (1887), 299-302; volume available [online](http://archive.org/details/actamathematica24lefgoog)), Stieltjes used almost exactly this formula to compute Euler's constant to 33 decimal places. Of course as quid points o... | 18 | https://mathoverflow.net/users/4720 | 104691 | 60,609 |
https://mathoverflow.net/questions/104685 | 4 | We know it converges for any prime $p$. I just want to know how to compute its exact value:
$$\prod\_{n=1}^{\infty} (1-p^{-n})$$
| https://mathoverflow.net/users/22928 | How to compute $\prod_{n=1}^{\infty} (1-p^{-n})$ | Wolfram MathWorld gives the expression of this product in terms of the q-Pochhammer symbol and the Jacobi theta function. See formulas (46) and (47) in
<https://mathworld.wolfram.com/InfiniteProduct.html>
| 3 | https://mathoverflow.net/users/12205 | 104693 | 60,611 |
https://mathoverflow.net/questions/104646 | 12 | At <http://en.wikibooks.org/wiki/Real_Analysis/Metric_Spaces> you can find the standard definition of a metric space: a set $X$ given with a function $d:X\times X\to\mathbb{R}$ that satisfies properties 1 through 4. Later on the page it defines open ball and open set, and proves that arbitrary unions and finite interse... | https://mathoverflow.net/users/24547 | Topological spaces determined by generalized metric spaces | Some partial answers to OP´s original question:
1) Any first-countable space (and in particular **any finite topology**) can be obtained in this way.
Let $X$ be first-countable, and for each $x \in X$ fix a local neighborhood base $\{B\_n^x:n \in \omega \}$ such that $X=B\_0^x$ and $B\_{n+1}^x \subseteq B\_n^x$. We... | 13 | https://mathoverflow.net/users/17836 | 104694 | 60,612 |
https://mathoverflow.net/questions/104699 | 14 | Tarski's Theorem on the undefinability of truth gives me a bit of a headache, and as a beginner I am still trying to grapple with its consequences. Here's a question.
Let $T$ be the set of Godel numbers of sentences $\sigma$ such that $V \models \sigma$.
Now, I know that by Tarski's theorem, there is no formula th... | https://mathoverflow.net/users/25648 | The set of Godel numbers of true sentences. | $\let\eq\leftrightarrow\def\god#1{\ulcorner#1\urcorner}$The question as such is not formally precise. The existence of a truth set is not expressible in the language of ZFC, so one cannot assert it from inside $V$, only from outside. My reading of the question is as follows. Let $\mathcal M=(M,E)$ be a model of ZFC, an... | 15 | https://mathoverflow.net/users/12705 | 104701 | 60,613 |
https://mathoverflow.net/questions/104507 | 2 | I am interested in proving existence/uniqueness to: find $u(x,t)$, $v(x,t)$ such that
$$u\_t - a\_1u\_{xx} - a\_2u\_x - a\_3u -a\_4v = f$$
$$v\_t - a\_5u\_{xx} - a\_6u\_x - a\_7u - a\_8v\_{xx} - a\_9v\_x - a\_{10}v = g$$
where the coefficients $a\_i(x,t)$ are in a parabolic version of Holder space (say $C^{k, \alpha}([... | https://mathoverflow.net/users/25650 | Linear coupled parabolic PDE system with Holder continuous coefficients | There is a classical counterexample due to Plis of an elliptic differential operator with Hölder continuous coefficients without Cauchy uniqueness. This was refined with a counterexample in divergence form by Miller in a 1974 Arch. Rat. Mech.(vol. 54) article for the elliptic and parabolic case.
Hölder continuity is ... | 2 | https://mathoverflow.net/users/21907 | 104708 | 60,617 |
https://mathoverflow.net/questions/104680 | 6 | Let $G$ be a finite group. I will say that a set of a subgroups $H\_1,\ldots ,H\_k$ defines a basis for a group $G$ if any subgroup $H$ of $G$ there exists $S\subset [k]$ such that $H=\cap\_{i\in S}H\_i$.
My question is does it possible to give any upper bounds on the size of the minimal basis for $G$.
For example d... | https://mathoverflow.net/users/4246 | Basis of a group | This is an extended comment. An element of a lattice is called meet-irreducible if it cannot be expressed as a meet of two elements containing it. For a finite lattice, the meet irreducible elements are the unique minimal generating set for the lattice under meet. Your bases are exactly meet generating subsets of the s... | 6 | https://mathoverflow.net/users/15934 | 104720 | 60,622 |
https://mathoverflow.net/questions/104704 | 2 | In Azuma's Inequality, is the statement true when $|X\_k - X\_{k-1}| < c\_k$ almost surely rather than with probability 1? If not, is there another result which gives strong concentration when the above inequality (for each $k$) holds with high probability?
| https://mathoverflow.net/users/22236 | Azuma's Inequality when the conditions hold with high probability? | There is a large literature on variations of Azuma's inequality. One lemma that is similar to what you ask is Lemma 3.1 of [this old paper of Wormald and myself](http://cs.anu.edu.au/~bdm/papers/degseq1.pdf). It considers the case where $|X\_k-X\_{k-1}|$ is within one bound with very high probability and within some wi... | 1 | https://mathoverflow.net/users/9025 | 104727 | 60,624 |
https://mathoverflow.net/questions/104716 | 9 | If $M$ is a 3x3, real symmetric matrix, then I know there are a few ways to decompose $M$ as
$M = A^T D A$,
where $D$ is a real diagonal matrix: e.g., this can always be done for some $A \in SO(3)$, or for some lower triangular $A$. Can it always be done for some $A \in SO(2,1)$?
-Jeanne
| https://mathoverflow.net/users/18048 | Unusual decomposition of 3x3 real symmetric matrices - is this possible? | Unfortunately, the answer is 'no'. You are basically asking whether you can simultaneously diagonalize two quadratic forms in three variables, and the answer is that, 'generically' you can (and you always can if some linear combination of the two is definite), but there are special pairs that cannot be simultaneously d... | 12 | https://mathoverflow.net/users/13972 | 104729 | 60,626 |
https://mathoverflow.net/questions/104698 | 2 | If I randomly sample with replacement $P$ times from a set of all possible binary strings of length $L$, what is a good lowerbound on the expected minimum Hamming distance between any two of my $P$ strings? Can we generalize this for larger ternary/etc. string alphabets?
| https://mathoverflow.net/users/24543 | The expected minimum Hamming distance within a set of randomly selected binary strings | The expected value of the distance is the sum of the probabilities that the distance is greater than $d$ for $d=0,1,2, ...$. Unfortunately, it's hard to determine the exact value of this probability. Whether this is greater than $0$ determines whether a code of that size exists.
There are some easy bounds on the prob... | 5 | https://mathoverflow.net/users/2954 | 104730 | 60,627 |
https://mathoverflow.net/questions/104711 | 12 | So, when people say, "the moduli problem of classifying elliptic curves over $\mathbb{C}$ with level $N$ structure", there are usually two associated functors I've seen:
1. $P\_N : \textbf{Ell}\rightarrow\textbf{Sets}$, where $\textbf{Ell}$ is the category of elliptic curves $E\rightarrow S$ over $S$ and morphisms ar... | https://mathoverflow.net/users/15242 | what exactly is the moduli functor for classifying elliptic curves with (full) level N structure? | Your $F\_N$ is the functor people would usually mean when they talk about the functor classifying elliptic curves with full level N structure (though it's a bit nicer if you replace $(Z/nZ )^2$ with $\mu\_n \times Z/nZ$, so that the determinant takes values in $\mu\_n$ on both sides.)
Wikipedia is not wrong. (Wikiped... | 13 | https://mathoverflow.net/users/431 | 104732 | 60,628 |
https://mathoverflow.net/questions/104741 | 6 | If $\mathcal F^\bullet$ is a chain complex of sheaves, to compute the hypercohomology, you take the cohomology of an injective resolution of $\mathcal F^\bullet$, i.e., a chain complex of injectives $\mathcal I^\bullet$ which is quasi-isomorphic to $\mathcal F^\bullet$.
It seems to be a well-known fact that for the a... | https://mathoverflow.net/users/2615 | Why does the de Rham double complex compute the algebraic de Rham cohomology ? | This has nothing to do with the details of the problem; it is a generalization of [Leray's theorem](http://math.berkeley.edu/~gasparim/charclass/leray2.pdf) to hypercohomology. If $\mathcal{F}\_{\bullet}$ is a complex of sheaves, and $U\_i$ is an open cover of $X$ such that $H^q(U\_{i\_1} \cap \cdots \cap U\_{i\_r}, \m... | 5 | https://mathoverflow.net/users/297 | 104743 | 60,637 |
https://mathoverflow.net/questions/104756 | 8 | Let $V$ be a $\Bbbk$-variety such that $\Bbbk^\times$ (as an algebraic group) acts algebraically on $V$. Given any $f\in\Bbbk[V]$, let us call $f$ **homogeneous of degree $d$** if for all $v\in V$ and all $\lambda\in\Bbbk^\times$, we have $f(\lambda.v)=\lambda^d f(v)$.
My question is: Does this define a grading on $... | https://mathoverflow.net/users/9947 | Action of k* on a variety induces grading? | Turning the action map of varieties into a map of rings, we get a ring map $\phi$ from $k[V]$ to $k[V][t,t^{-1}] $, the coordinate ring with an extra invertible variable (the coordinate on $k^\*$) adjoined. Now, for any function $\phi(f)=\sum\_{i\in \mathbb{Z}}f\_it^i$ for some $f\_i$'s, almost all of which are 0. Note... | 16 | https://mathoverflow.net/users/66 | 104758 | 60,643 |
https://mathoverflow.net/questions/104751 | 10 | $\newcommand{\eS}{\mathscr{S}}$ $\DeclareMathOperator{\SO}{SO}$ $\newcommand{\eP}{\mathscr{P}}$ $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\tr}{tr}$ Let $m>1$ be an integer and denote by $\eS\_m$ the vector space of symmetric $m\times m$ real matrices. The group $\SO(m)$ acts by conjugation on $\eS\_m$. The s... | https://mathoverflow.net/users/20302 | Invariants of symmetric matrices | This is a trick question, right? It's not true when $m=2$ because, then $\mathrm{SO}(m{-}1)=\mathrm{SO}(1)$ is trivial, so that all polynomials on $2$-by-$2$ symmetric matrices are invariants, and the four quadratics you mention clearly don't span the the six-dimensional space of all quadratics.
Moreover, for $m>2$,... | 12 | https://mathoverflow.net/users/13972 | 104762 | 60,644 |
https://mathoverflow.net/questions/104759 | 16 | Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ is at most $|G|^{100}$?
One can easily find that any subgroup is generated by at most $\log|G|$ elements thus the numb... | https://mathoverflow.net/users/4246 | Maximal number of maximal subgroups | Another extended comment. The best bound on the number of subgroups is $|G|^{(1/4+o(1))\log\_2 |G|}$ proved in <http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.100.913&rep=rep1&type=pdf> by Borovik, Pyber and Shalev.
For the number of maximal solvable subgroups they get $|G|^c$ for some constant. They don't... | 15 | https://mathoverflow.net/users/15934 | 104763 | 60,645 |
https://mathoverflow.net/questions/104757 | 17 | Motivation $\newcommand{\T}{\mathscr{T}}$
-----------------------------------------
I have many times found myself saying some variant of the following. Let $\T\_g$ be the Teichmüller space of a surface of genus $g$, and $\Gamma\_g$ its mapping class group. The quotient $\T\_g/\Gamma\_g$ is the moduli space of curves... | https://mathoverflow.net/users/1310 | Homotopy theory of topological stacks/orbifolds | Here is a simple way to talk about the homotopy type of a stack. Let $\mathfrak{X}$ be a stack and $f: U \to \mathfrak{X}$ a representable surjective submersion (an atlas) from a space $U$ (e.g., the map from the Teichmuller space to the moduli stack.) Now, form the pullback of $f$ along itself: $U\times\_{\mathfrak{X}... | 19 | https://mathoverflow.net/users/4910 | 104767 | 60,649 |
https://mathoverflow.net/questions/104775 | 1 | I have a function of the form:
$f(n) = \prod\_{i=1}^{n-1} (1-ai)$
Here, $a \geq 0$ and $(a\*i) < 1$. For $n > 10^5$ or $10^6$, what is the best possible analytic approximation for $f(n)$ that will allow me compute values for the function with reasonable computational resources? What bounds on the error of the appro... | https://mathoverflow.net/users/24543 | Approximating $\prod_{i=1}^{n-1} (1-ai)$ for large $n$ | $$f(n) = \dfrac{(-a)^{n-1} \Gamma(n - 1/a)}{\Gamma(1 - 1/a)}$$
If $a n < 1$, you'll want to use the reflection principle
$$\Gamma(1-z) \; \Gamma(z) = {\pi \over \sin{(\pi z)}}$$
so
$$ f(n) = \dfrac{a^{n-1} \Gamma(1/a)}{\Gamma(1-n+1/a)}$$
Use Stirling's series for the asymptotic approximation of $\ln(\Gamma(1-n+ 1/a... | 5 | https://mathoverflow.net/users/13650 | 104780 | 60,654 |
https://mathoverflow.net/questions/104750 | 42 | In Cushman and Bates, Global Aspects of Classical Integrable Systems, 1997, I have read
>
> In a widely circulated but unpublished letter in 1965, Palais explained the symplectic formulation of Hamiltonian mechanics.
>
>
>
I would like to know if, in the meanwhile, this letter was made available.
| https://mathoverflow.net/users/12617 | About a letter by Richard Palais of 1965. | I haven't thought about that letter for a very long time, but as far as I can recall I didn't ever make it publicly available, and I don't think any of the friends to whom I sent it did either. However, I am a bit of a pack-rat, so after Ryan Budney alerted me that this question had appeared on MO I did some searching ... | 114 | https://mathoverflow.net/users/7311 | 104781 | 60,655 |
https://mathoverflow.net/questions/104400 | 11 | Let $G=\langle H, t; K^t=K^{\prime}\rangle$ be an HNN-extension of $H$, with $t$ inducing the isomorphism $\phi: K\rightarrow K^{\prime}$. I was wondering if the following question can be answered, and if so what is the answer,
>
> When is $G$ finitely presented?
>
>
>
It seems "obvious" that the answer should... | https://mathoverflow.net/users/6503 | When is an HNN-extension finitely presented? | Here are my comments combined into an answer.
For ascending HNN extensions, i.e. $H=K$, $\phi\colon H\to K'$ an injective endomorphism (as in Baumslag-Remeslennikov case (see above), in the Grigorchuk case, and many others) one needs, as Ben Steinberg ponted out that $H$ has a finite L-presentation (named after Igor ... | 8 | https://mathoverflow.net/users/nan | 104785 | 60,657 |
https://mathoverflow.net/questions/104681 | 4 | There is a comparison theorem for spectral sequnces in Weibel's book (5.2.12) stating;
Assume $E\_{p,q}$ and $\bar E\_{p,q}$ converge to $H\_\* $ $\bar H\_\*$ respectively. Furthermore we have given a map $h: H\_{\*} \to \bar H\_{\*} $ compatible with a morphism $f$ of spectral sequences.
If $f^r: E^r\_{p,q} \to \... | https://mathoverflow.net/users/25696 | Comparing Spectral Sequences | The cokernel is entirely due to $\bar{E}^\infty\_{\*,0}$ but the kernel is more mysteriuous.
First observe that if $g:C\_\cdot\to D\_\cdot$ is a chain map and $i$ is such that $g\_i$ is surjective and $g\_{i-1}$ is injective, then
$H\_i(g)$ is surjective and $H\_{i-1}(g)$ is injective. (Exercise.)
Using this, one ... | 1 | https://mathoverflow.net/users/4794 | 104786 | 60,658 |
https://mathoverflow.net/questions/104788 | 2 | Let $(n\_1,\ldots, n\_i,\ldots)$ be an infinite tuple of nonnegative integers. Is there an abstract number ring $D$ of a given characteristic $p>0$ and $I\_1,\dots, I\_n , \ldots$ its nonzero ideals (by assumption $D/I\_i$ are finite) such that #$D/I\_i=n\_i$ for each $i$?
| https://mathoverflow.net/users/nan | Abstract number ring of any characteristic | I'm not sure what you mean by an abstract number ring; nevertheless, here's an attermpt at an answer. If $D$ is to have characteristic $p$, then each of the quotients $D/I\_i$ will be a vector space over $\mathbb Z/p$, so each $n\_i$ will have to be a power of $p$. Conversely, if all the $n\_i$ are powers of $p$, say $... | 1 | https://mathoverflow.net/users/6794 | 104792 | 60,662 |
https://mathoverflow.net/questions/101989 | 11 | In my course notes for an undergraduate course "Algebra I", I wrote at the point when I'm introducing the notion of divisibility in rings (in a section on unique factorization):
>
> We want to study factorization in arbitrary integral domains$^1$,...
>
>
> $^1$ (footnote) The study of factorization in arbitrary c... | https://mathoverflow.net/users/12858 | Divisibility and factorization in rings that are not integral domains | The biggest problem complicating factorization when there are zero-divisors present in my experience stems from the fact that there are several ways to define "associate" that are no longer equivalent.
For instance, the reference given by Guntram provides a good introduction to the issues:
$a$ and $b$ are associate... | 16 | https://mathoverflow.net/users/25730 | 104795 | 60,664 |
https://mathoverflow.net/questions/104797 | 4 | Let $\mathcal{A}$ be an admissible set, or for simplicity a model of ZF without the power set axiom.
Let $X \subset \mathcal{A}$. Is there an easy way to show $X$ is $\Sigma$ on $\mathcal{A}$?
PS: $X$ is $\Sigma$ on $\mathcal{A}$ if it is definable in $\mathcal{A}$ by a $\Sigma$ formula i.e.
a formula built from at... | https://mathoverflow.net/users/nan | How to know that a set is \Sigma? | A frequently useful way to show that a function $F$ is $\Sigma$ definable is to give a definition of it by recursion, in the form $F(x)=G(x,F\upharpoonright H(x))$ where $G$ and $H$ are already known to be $\Sigma$-definable and $H(x)$ is the set of predecessors of $x$ in a well-founded relation $\prec$ (so that the re... | 5 | https://mathoverflow.net/users/6794 | 104800 | 60,666 |
https://mathoverflow.net/questions/104673 | 0 | Let's consider a (partial) differential equation with analytic coefficients. The initial conditions may be non-analytic\*.
Is it possible a solution $f$, which is non-analytic at any point of the domain?
---
\*Even if the initial conditions are analytic, it is possible that the solution is not [[1](http://www.s... | https://mathoverflow.net/users/10095 | Analyticity of the solutions of PDE | Points off for the rude response to Otis' sincere attempt to help you!
Deanne is correct, but you can see this simply even with the operator $\partial\_{x\_1}$ in $\mathbb R^n$. If $u(0, x\_2, \ldots, x\_n)$ is nowhere analytic, then the
solution of $\partial u/ \partial\_{x\_1} = 0$ is constant in $x\_1$ but never ... | 5 | https://mathoverflow.net/users/17969 | 104801 | 60,667 |
https://mathoverflow.net/questions/102372 | 2 | I need to find an example of a cartesian closed category whose functor category (from some diagram $J$) is not cartesian closed. I thought of one possible solution: let $A$ be a commutative square (which is a poset, hence ccc) and let $J$ be the category $\bullet \to \bullet$. Then $A^J$ is the arrow category, and I cl... | https://mathoverflow.net/users/25153 | Show that $A$ cartesian closed need not imply $A^J$ is cartesian closed. | I think it will not be easy to find an example with $J=2$. Since a cartesian closed category has finite products, the result cited in David White's answer shows the category $A$ for such an example must not have equalizers. So it cannot be a preorder, and so (having binary products) it cannot be finite. It is not hard ... | 5 | https://mathoverflow.net/users/38783 | 104806 | 60,669 |
https://mathoverflow.net/questions/104808 | 4 | The definition of the Workday Number of a finite graph is given on page 14 in [<http://www.arml.com/2012_contest/2012_Contest_Final_Version.pdf>](http://www.arml.com/2012_contest/2012_Contest_Final_Version.pdf) and the rest of the problem statement is given at the top of page 12 (omitted here for brevity). Is the gener... | https://mathoverflow.net/users/25733 | Graph Theory: 2012 ARML Power Question - references? | Maybe a good place to start is Anthony Bonato and Richard J. Nowakowski, The game of cops and robbers on graphs, which is Volume 61 in the Student Mathematical Library, published by the American Mathematical Society, Providence, RI, 2011, ISBN: 978-0-8218-5347-4.
| 4 | https://mathoverflow.net/users/3684 | 104811 | 60,671 |
https://mathoverflow.net/questions/104810 | 3 | How stable is [Levinson-Durbin](http://en.wikipedia.org/wiki/Levinson_recursion) method for solution of systems of linear equations ?
I mean if condition number of matrix is $k$, does intermidiate steps involve matrixes
with higher condition number ?
For example QR is easy to see preserve condition number so it is ... | https://mathoverflow.net/users/10446 | Stability of Levinson-Durbin method for Toeplitz system solutions ? | It can be unstable, and the condition number is not an adequate measure to tell when it fails. You may want to check the numerical experiments in <http://www.jstor.org/stable/2153371> for specific examples and discussion.
If I remember correctly, you can prove stability only for symmetric matrices whose principal min... | 2 | https://mathoverflow.net/users/1898 | 104818 | 60,672 |
https://mathoverflow.net/questions/104796 | 5 | My question is basically given in the title: Are there any references for a generalization of algebraic K-theory to the scenario where the domain of the functors consists of commutative semirings (provided this has been done at all)?
| https://mathoverflow.net/users/8590 | Algebraic K-theory with commutative semirings? | There are a few papers out there dealing with a slightly different focus - algebraic K-theory over the "field with one element" $\mathbb{F}\_1$. Some of the frameworks for $\mathbb{F}\_1$ algebra include semirings, so these papers might contain material that covers what you are interested in as well.
The thesis of Ni... | 9 | https://mathoverflow.net/users/4910 | 104820 | 60,673 |
https://mathoverflow.net/questions/104783 | 2 | Let $G=(V,E)$ be a (simple) finite graph such that every vertex has degree at least 1. Then it is easy to see that there is a subset $E'$ of $E$ such every vertex in $G'=(V,E')$ still has degree at least 1 and all paths (with no repeating edges) in $G'$ are of (edge-wise) length at most 2. (I just keep removing middle ... | https://mathoverflow.net/users/25602 | Subset of edges of graph touching all vertices such that all paths consist of at most two edges | Aaron Meyerowitz suggested to try to reduce the problem to trees and, to me, this seems to work. First we can suppose that $G$ is a connected graph, because we can solve the problem separatly for each component. It is easy to see by Zorn's Lemma, that every connected graph contains a spanning tree, i.e. a subgraph whic... | 1 | https://mathoverflow.net/users/25602 | 104828 | 60,675 |
https://mathoverflow.net/questions/104826 | 2 | Is there any known formula for the number of compositions of an integer k (partitions with considering the order of the parts) of length m (exactly m parts) where the parts do not exceed a given integer n?
Without limitation of the parts there is, of course, a well-known formula (binomial k-1 over m-1). Introducing th... | https://mathoverflow.net/users/25740 | Explicit formula for the number of compositions with m strictly positive parts bounded by n? | If I understand well, you consider the number $a(k,n,m)$ of multi-indices $a=(a\_1,\dots,a\_m)\in\{1,\dots,n\}^m$ with weight $\sum\_{i=1}^m a\_i=k$. This is therefore the coefficient of $x^k$ in $$\left (\sum \_ {j=1}^n x^j \right)^m = x^m(1-x^n)^m (1-x)^{-m}\, .$$
Since the above generating function is the product of... | 2 | https://mathoverflow.net/users/6101 | 104831 | 60,677 |
https://mathoverflow.net/questions/104777 | 34 | Let $k$ be a field, and let $\mathbf{Vect}$ denote the category of vector spaces (possibly infinite-dimensional) over $k$. Taking duals gives a functor $(\ )^\*\colon \mathbf{Vect}^{\mathrm{op}} \to \mathbf{Vect}$.
This contravariant functor is self-adjoint on the right, since a linear map $X \to Y^\*$ amounts to a ... | https://mathoverflow.net/users/586 | What are the algebras for the double dualization monad? | Tom, I believe $(-)^\ast: \mathbf{Vect}^{op} \to \mathbf{Vect}$ is monadic, essentially because all objects in $\mathbf{Vect}$, in particular $k$ as a module over $k$ as ground field, are injective.
For instance, to check that $(-)^\ast$ reflects isomorphisms, suppose $f: V \to W$ is any linear map. We have two shor... | 24 | https://mathoverflow.net/users/2926 | 104845 | 60,681 |
https://mathoverflow.net/questions/104846 | 8 | Is there an easy way to determine whether a set of elements in a field generates the whole field or only a subfield?
Specifically, I have a subfield of $k(x,y)$ described in terms of a canonical set of generators, and I have two other elements of this ring that I think should be sufficient to generate the whole thing... | https://mathoverflow.net/users/22460 | Solving the field membership problem using Grobner bases | In theory, Groebner basis software can do this. I don't know whether there is a standard package for this purpose, or what practical issues may arise. Here is the theory:
Your goal is the following. You are given rational functions $r\_1$, $r\_2$, ..., $r\_n$ in $x$ and $y$. You would like know whether there are poly... | 10 | https://mathoverflow.net/users/297 | 104853 | 60,683 |
https://mathoverflow.net/questions/104850 | 1 | Dear all,
Suppose U and V are unitary matrix, A and B are positive definite,
Does:
$UAU^{-1} < VBV^{-1}$
implies $A< B$
and vice versa?
| https://mathoverflow.net/users/25747 | Unitary matrix and matrix inequality | No: think of the case of $A$, $B$ diagonal; $U$ identity, and $V$ a permutation operator.
| 1 | https://mathoverflow.net/users/6101 | 104854 | 60,684 |
https://mathoverflow.net/questions/104803 | 16 | Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg m.$ I want to choose a subset $\{v\_1, \dots, v\_m\}$ of $S$ in such a way that the condition number of the matrix with columns $v\_1, \dotsc, v\_m$ is as small as possible -- notice that this is trivial if $S$ does not span $W,$ since the conditi... | https://mathoverflow.net/users/11142 | Optimizing the condition number | *Update*: [This recent paper](http://arxiv.org/abs/1509.00748) on this topic may also be of interest; it's quite short and claims to have a fully constructive approach.
---
I think the closest to answering your question is the following paper.
"Column subset selection, matrix factorization, and eigenvalue optim... | 12 | https://mathoverflow.net/users/8430 | 104858 | 60,687 |
https://mathoverflow.net/questions/104847 | 1 | Dear all,
I have the following problem: Consider an array of $N$ vectors $v\_{i} \ i=1...N$ of size $L$ bits, where each bit is 1/0 with equal probability. I want to find a hash function $H()$ that results in no collisions when hashing the N vectors, i.e. $H(v\_{i}) \neq H(v\_{j}) \ \forall i \neq j$, and such that ... | https://mathoverflow.net/users/20207 | Designing a Hash function that results in minimum size and no collisions | Have a look at [Perfect Hash Functions](http://en.wikipedia.org/wiki/Perfect_hash_function), especially the discussion on *minimal perfect hash functions*
| 2 | https://mathoverflow.net/users/8430 | 104859 | 60,688 |
https://mathoverflow.net/questions/104832 | 10 | Given a formal power series $f(x,y)=\sum\_{n,m\ge 0}f(n,m)\: x^n y^m$ in two variables $x$ and $y$ over some field of characteristic zero, e.g. the field of complex numbers $\mathbb C$, we define a new formal power series in one variable $t$:
$\Delta(f)(t):=\sum\_{i\ge 0} f(i,i) \:t^i$.
It is known that if $f(x,y)$... | https://mathoverflow.net/users/25743 | A diagonal operation on power series | There is a discussion of taking diagonals of rational functions in Stanley's *[Enumerative Combinatorics Volume II](http://www.cambridge.org/gb/knowledge/isbn/item1167618/?site_locale=en_GB)* (section 6.3). It contains a reasonably explicit description of how to take diagonals using Puiseux series as well as a descript... | 7 | https://mathoverflow.net/users/290 | 104861 | 60,690 |
https://mathoverflow.net/questions/104827 | 13 | Let $\mathcal C$ be the category of long exact sequences of finitely generated abelian groups almost all of whose entries vanish.
The category $\mathcal C$ is naturally additive as a subcategory of complexes of abelian groups.
>
> Question: Can we write down a complete list of **isomorphism classes** (up to trans... | https://mathoverflow.net/users/1291 | Classification of long exact sequences | In a series of recent papers, Schmidmeier and Ringel show than the classification of monomorphisms in the category of finitely generated $\mathbb Z/p^n$-modules is a wild problem of representation theory for $n>6$. Hence, your problem is also wild. There's no hope to get what you want.
| 10 | https://mathoverflow.net/users/12166 | 104862 | 60,691 |
https://mathoverflow.net/questions/104869 | 8 | William Browder showed in "Open and closed disc bundles", Ann. of Math. (2) 83 (1966), 218-230 that there are open disc bundles over some complex which cannot be isomorphic to a vector bundle.
My question is: Is there any related examples?
For example: What is the least dimensions for the base complex and the dimension... | https://mathoverflow.net/users/1190 | Examples for open disc bundle which is not vector bundle | Given a disc bundle over a space $X$, there's the classifying map
$$ X \to BDiff(D^n)$$
$Diff(D^n)$ has as a subgroup $O\_n$, the orthogonal group. The disc bundle over $X$ is a vector bundle if and only if the classifying map $X \to BDiff(D^n)$ factors (up to homotopy) through a map $X \to BO\_n$, where the other... | 9 | https://mathoverflow.net/users/1465 | 104872 | 60,695 |
https://mathoverflow.net/questions/103522 | 5 | Let $\frak{g}$ be a finite-dimensional complex nilpotent Lie algebra. Given $\xi\in\frak{g}$, what is known about the intersection of $im(ad\_{\xi})$ (the image of $ad\_{\xi}:\frak{g}\rightarrow\frak{g}$) and $z(\frak{g})$ (the centre of $\frak{g}$)? In particular, under what conditions is it true that the intersection... | https://mathoverflow.net/users/25358 | Nilpotent Lie Algebras | Whenever $ad\_{\xi}$ is an endomorphism of $\mathfrak{g}$ whose corresponding partition $\pi: 1^{s\_{1}}2^{s\_{2}} \cdots \;$ of $\dim \mathfrak{g}$ is such that $s\_{1} =0$, then we have $im\; ad\_{\xi} \cap Z(\mathfrak{g}) \neq \{0\}$.
As $\mathfrak{g}$ is nilpotent, $ad\_{\xi}$ acts as a nilpotent endomomorphism ... | 2 | https://mathoverflow.net/users/9970 | 104875 | 60,696 |
https://mathoverflow.net/questions/104877 | 5 | Quick easy question: what is the meaning of the symbol $(\space\space )$. I've seen it now in two papers, one of which is Milgram's Group Representations and the Adams Spectral Sequence, available at [here](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-41/issue-1/Group-representations-and-the... | https://mathoverflow.net/users/24021 | what do empty parens symbol mean? | In the case at hand, $v$ is defined as
$$v = \bar{u}(id \times A \times A): E\_{Z\_2} \ltimes D^{r-s} \wedge D^{s-r} \to S$$
The sequence in question, which is described as the 'restriction of $v$ to the boundary', is shorthand for
$$ \partial(E\_{Z\_2} \ltimes D^{r-s} \wedge D^{s-r}) \stackrel{\pi}{\to} S^k \vee S^... | 7 | https://mathoverflow.net/users/4177 | 104887 | 60,703 |
https://mathoverflow.net/questions/104873 | 8 | I'm learning about the Thurston norm and am trying to understand the implications that the existence of fibered faces has for the ways in which a given three-manifold $M$ can fiber over the circle. In particular, I am interested in the following question:
>
> Let $M$ be a compact, oriented three-manifold without bo... | https://mathoverflow.net/users/960 | Regarding the Thurston norm and the ways that a three-manifold can fiber over the circle | The answer is: yes if the rank of $H\_2(M;\mathbb{Z})$ is $\ge 2$; and no if the rank is $1$ because in that case there is up to isotopy a unique connected surface bundle structure on $M$. The proof uses nothing more than what is in Thurston's original article MR0823443, although there are probably multiple other ways ... | 10 | https://mathoverflow.net/users/20787 | 104888 | 60,704 |
https://mathoverflow.net/questions/102586 | 2 | Suppose $(V^{2g}, g, \omega, J)$ is an almost Kahler manifold. ie. $(V,\omega)$ is a symplectic manifold with $\omega$-compatible almost complex structure $J$ ($J$ is a symplectomorphism) and such that $\omega(\cdot, J\cdot)$ coincides with a riemannian metric $g$ on $V$.
Suppose further that $L$ is a lagrangian sub... | https://mathoverflow.net/users/20516 | How else can we describe the volume of a lagrangian submanifold in a Kahler manifold? | This is probably closer in spirit to what you're looking for than what you've received in the comments. If $(V^{2m}, J, \omega, g)$ is Calabi-Yau (which for me means that $J$ is integrable, and the first Chern class $c\_1(V) = 0$), then one can say much more. In this case there exists a holomorphic nowhere vanishing $(... | 4 | https://mathoverflow.net/users/6871 | 104918 | 60,716 |
https://mathoverflow.net/questions/104914 | 6 | D. Gibb, from the Mathematical Laboratory, University of Edinburgh,
describes a **Computer Desk** in his book **A course in interpolation and numerical integration for the
mathematical laboratory**, G. Bell & Sons, Ltd., **1915**, available [here](http://archive.org/stream/courseininterpol00gibbuoft#page/n13/mode/2up):... | https://mathoverflow.net/users/22714 | Mathematical computer desk | I will interpret this question a bit freely:
There is a long history of humans performing computational tasks (not mathematics) as a profession, and the technical tools and tables they had.
The [Computer History Museum](http://www.computerhistory.org/) has among many other things some nice pictures of mechannical... | 4 | https://mathoverflow.net/users/nan | 104921 | 60,717 |
https://mathoverflow.net/questions/104915 | 6 | It is obvious that using seed value one can easily compute next value of some (deterministic) pseudo-random algorithm - so $N$th element can be computed in $O(N)$.
But is there such PRNG that allow to compute Nth element in $O(1)$ while still preserving good periodicity and distribution?
| https://mathoverflow.net/users/25763 | pseudo-random algorithm allowing O(1) computation of Nth element | I think [linear feedback shift registers](http://en.wikipedia.org/wiki/Linear_feedback_shift_register) are examples of what you want. Essentially, these compute powers of $x$ in $F\_2[x]/p(x)$ for some polynomial $p(x)$. If $p(x)$ is chosen well, the period can be long. You can compute $x^n$ rapidly by multiplying $c \... | 5 | https://mathoverflow.net/users/2954 | 104923 | 60,718 |
https://mathoverflow.net/questions/104913 | 10 | The following assertion is trivial in ZFC, or even in much weaker theories. Is it also true in ZF?
(I couldn't find it in the Consequences site so far.)
>
> If $A$ is an infinite set such that $A$ can be mapped onto $A\times 2$ then $|A\times 2|=|A|$
>
>
>
The problem is that we cannot necessarily choose fro... | https://mathoverflow.net/users/7206 | On surjections, idempotence and axiom of choice | The answer is no.
First, I argue that it is consistent with ZF that a Dedekind
finite set $A$ can map onto $A\times 2$, and much more.
To see this, begin with any infinite Dedekind finite set $B\subset 2^\omega$, which is furthermore dense in the sense that any finite binary sequence has extensions in $B$. It is c... | 15 | https://mathoverflow.net/users/1946 | 104927 | 60,721 |
https://mathoverflow.net/questions/104908 | 8 | How to calculate the DeRham cohomology of the free loop space $LM= C^\infty(S^1,M)$ as a Frechet manifold?.
Edit: It will be enough for me to know:
When $H^1\_{DR}(LM)$ is not $\{0\}$.
Bounty is ending within 5 hours.
| https://mathoverflow.net/users/23534 | Loop space: De Rham cohomology | An equivariant de Rham theory in precisely this setting seems to be developed by Leandre in "Equivariant Cohomology, Fock Space and Loop Groups" (2006). (<http://www.mth.kcl.ac.uk/staff/fa_rogers/ECFSLG.pdf>)
| 2 | https://mathoverflow.net/users/11142 | 104928 | 60,722 |
https://mathoverflow.net/questions/104919 | 7 | $\newcommand{\lm}{\lambda}$ $\newcommand{\bR}{\mathbb{R}}$ Let $m$ be an integer $>1$. Define
$$ I\_m:\bR^m\to \bR,\;\; I\_m(\lm\_1,\dotsc, \lm\_m)=\int\_{S^{m-1}}\exp\Bigl(-\sum\_{j=1}^m \lm\_j^2x\_j^2\;\Bigr)\; |dA(x)|, $$
where $S^{m-1}$ is the unit sphere in $\bR^m$ and $|dA(x)|$ denotes the "area" element on $... | https://mathoverflow.net/users/20302 | Can one recognize this symmetric function? | Your function is a special case of the [Confluent Hypergeometric function of matrix argument](http://dlmf.nist.gov/35.6), in particular it is ${}\_1F\_1(\frac{1}{2},\frac{n}{2},L)$, where $L$ is a diagonal matrix having $-\lambda\_i^2$ ($1\le i \le n$) as its diagonal components. These functions can be more generally r... | 11 | https://mathoverflow.net/users/8430 | 104935 | 60,727 |
https://mathoverflow.net/questions/104905 | 4 | This question is related to Question 2 of [my previous posting](https://mathoverflow.net/questions/103860/the-monotone-closure-of-a-c-algebra).
> **Question.** Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded Borel measurable functions on $\... | https://mathoverflow.net/users/25499 | Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra? | No. The bounded Baire class one functions on $[0,1]$ are stable under uniform limits and hence constitute a C\*-algebra. This C\*-algebra contains every semicontinuous function on [0,1]. Every function in $L^\infty[0,1]$ is equal almost everywhere to a Baire class two function, but not a Baire class one function. (The ... | 10 | https://mathoverflow.net/users/23141 | 104946 | 60,733 |
https://mathoverflow.net/questions/104911 | 7 | Which finite groups have uniqueness for the ordered sequence of composition factors (up to isomorphism)?
| https://mathoverflow.net/users/25762 | Composition Series | Here is a characterization. A group $G$ has two different composition series if and only if it has a factor $H/K$ which is a direct product of two non-isomorphic simple subgroups, where $H$ is a subnormal subgroup of $G$, $K$ is a normal subgroup of $H$. Indeed, if such $H/K$ exists, then clearly there are two differen... | 12 | https://mathoverflow.net/users/nan | 104947 | 60,734 |
https://mathoverflow.net/questions/104932 | 7 | Let $k$ be an algebraically closed field and let $G$ be a reductive linear algebraic group over $k$ with Lie algebra $\mathfrak g$. If the characteristic of $k$ is $0$ then, by a classical result of Kostant, $k[\mathfrak g]$ is free over the subalgebra $k[\mathfrak g]^G$. Is there an analog of this result when the char... | https://mathoverflow.net/users/1528 | Kostant's theorem on invariant polynomials in positive characteristic | Roger Richardson amplified Kostant's result in characteristic 0, which in turn led Steve Donkin to work out a closely parallel version in prime characteristic: *On conjugating representations and adjoint representations of semisimple groups*, Invent. Math. 91 (1988), no. 1, 137–145. (This is available online at the GDZ... | 4 | https://mathoverflow.net/users/4231 | 104949 | 60,735 |
https://mathoverflow.net/questions/104724 | 6 | Let $X$ be a topological space. The universal coefficient theorem says that there is a short exact sequence
$$
0\rightarrow Ext(H\_{k-1}(X,\mathbb{Z}),\mathbb{Z})\rightarrow H^{k}(X,\mathbb{Z})\rightarrow Hom(H\_{k}(X,\mathbb{Z}),\mathbb{Z})\rightarrow 0
$$
for $1\le k$. In particular a torsion element in $H\_{1}(X,\ma... | https://mathoverflow.net/users/25713 | Torsion in $H^2(X,\mathbb{Z})$ induced by torsion in $H_1(X,\mathbb{Z})$ | Having the cycles and boundaries of degree one, it's possible to construct a cohomology class in $H^2(X)$ that corresponds to your torsion element. This is explained in the following:
**1. Homological algebra**
Let $(C,d)$ be a chain complex of free abelian groups, $G$ an abelian group and let
$$\varphi: Ext(H\_{... | 5 | https://mathoverflow.net/users/10194 | 104950 | 60,736 |
https://mathoverflow.net/questions/104939 | 3 | Assume I have simple group G and its cyclic Sylow subgroup H. Are there any nice properties of the induced representation ?
Any remarks "around the situation" (relaxing "simple" "cyclic" "Sylow") are also welcome.
**Example:** Consider [GL\_3(F\_2)](http://en.wikipedia.org/wiki/PSL(2,7)) which is [remarkbly isomor... | https://mathoverflow.net/users/10446 | Induction from cyclic / Sylow subgroup are there any nice properties ? | Here is a partial answer. (Your assumption that $G$ is simple is irrelevant for these remarks.) Suppose, as in your example, that the Sylow $p$-subgroup $P$ has order $p$ exactly. Then every irreducible character $\chi$ with degree divisible by $p$ appears as a constituent of $\lambda^G$ with multiplicity $\chi(1)/p$ f... | 9 | https://mathoverflow.net/users/9694 | 104952 | 60,737 |
https://mathoverflow.net/questions/104948 | 6 | Hello, I'm working with a data structure which uses a uniform distribution to bucket the inputs into $k$ buckets. The efficiency of the structure is bounded by the $\frac{k\_{max}}n$, where $n$ is the number of items. How many elements are in each bucket will follow a uniform multinomial distribution. What will the dis... | https://mathoverflow.net/users/25769 | Distribution of Maximum of a uniform multinomial distribution | The probability that there is at least one bin with at least $c$ items is less than or equal to the expected number of bins with at least $c$ items, which is $k$ times the probability that a particular bin has at least $c$ items. You can bound the probability that a particular bin contains at least $c$ items using the ... | 6 | https://mathoverflow.net/users/2954 | 104955 | 60,740 |
https://mathoverflow.net/questions/104958 | 6 | Does normal Riemann hypothesis for $\zeta\_{\mathbb{Q}}$ follows from the extended Riemann hypothesis for some $K \neq \mathbb{Q}$ (i.e. the statement that all zeroes of the Dedekind zeta function $\zeta\_K$ for $K$ a number field in the critical strip lie on the axis $\mathfrak{R}(s)=1/2$)?
| https://mathoverflow.net/users/nan | Does normal Riemann hypothesis follow from extended Riemann hypothesis | Yes, certainly: zeta functions of abelian extensions of $\mathbb Q$ factor as products of Dirichlet $L$-functions over $\mathbb Q$, including $\zeta(s)$, and those other $L$-functions have no poles (which might cancel an off-line zero of zeta), so RH for any abelian extension of $\mathbb Q$ implies that for $\mathbb Q$... | 13 | https://mathoverflow.net/users/15629 | 104959 | 60,742 |
https://mathoverflow.net/questions/102789 | 2 | Let $G$ be an algebraic group with Lie algebra $\mathfrak g$ and let $\Gamma$ be any finitely generated (discrete) group. One can consider the representation variety $\mathfrak R=\mathrm{Hom}(\Gamma,G)$. If one considers the group scheme $\mathfrak G$ associated to $G$, there is a nice argument that shows that infinite... | https://mathoverflow.net/users/11084 | Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups | This fact holds for general Lie groups $G$ (which I will equip with a real-analytic structure) and finitely-generated groups $\Gamma$. It is explained in detail in Raghunathan's book "Discrete subgroups of Lie groups", sections 6.1-6.9), but goes back to Andre Weil's paper "Remarks on cohomology of groups", Annals of M... | 3 | https://mathoverflow.net/users/21684 | 104969 | 60,747 |
https://mathoverflow.net/questions/104957 | 4 | I have a sequence of symmetric positive definite matrices in $GL(n)$ (in my case, covariance matrices of some gaussians) and vectors $R^n$ (the mean of these gaussians). I consider this sequence to be the discretization of a curve embedded in $GL(n)\times R^n$.
I would like to compute the embedding curvature of this ... | https://mathoverflow.net/users/8646 | curvature of curves in the space of gaussians measures | Your question is not specific enough about what you do not understand in the quoted paper; if you want help on this, you should at least explain what you understand and where the problem appears. Here is a little information about bibliography, that might help (or miss the point, I am not sure).
For an introduction t... | 4 | https://mathoverflow.net/users/4961 | 104975 | 60,749 |
https://mathoverflow.net/questions/104933 | 9 | In algebraic K-theory one defines $K\_0(R)$ as the result of application of the Grothendieck construction to the semigroup of isomorphism classes of left f.g. projective $R$-modules.
But we can also consider the category of left f.g. $R$-modules and apply the same construction to obtain a group (let's call it $G(R)$).
... | https://mathoverflow.net/users/21892 | f.g. modules vs. f.g. projective modules | Typically the homomorphism here $K\_0 \rightarrow G\_0$ fails to be surjective. This shows up in a wide range of examples involving group algebras of finite groups (over fields of characteristic dividing the group order), restricted enveloping algebras of modular Lie algebras, etc. The homomorphism itself is often call... | 3 | https://mathoverflow.net/users/4231 | 104980 | 60,751 |
https://mathoverflow.net/questions/104985 | 1 | Does anybody know a good reference where the invariants for plane quartic curves are developed?
| https://mathoverflow.net/users/4096 | invariants of plane quartics | Did you check "Polar Covariants of Plane Cubics and Quartics" by Dolgachev and Kanev?
| 1 | https://mathoverflow.net/users/4428 | 104986 | 60,752 |
https://mathoverflow.net/questions/104982 | 7 | First of all I am far from being an expert in representation theory, so it is possible (likely) that the following question is trivial (in fact a trivial reference question):
Let $\Gamma$ be a, let's say finite (abelian) group, and $R$ be an effective representation of $\Gamma$.
Is it true, that every irreducible $\G... | https://mathoverflow.net/users/675 | Are irreducible representations subrepresentations of a symmetric power representation? | **Edit:** The answer is yes for all finite groups $G$ (even, as far as this makes sense, independently of the ground field). A reference is Theorem 1 on page 45 of Alperin: "Local representation theory". While the claim of this theorem only refers to the tensor product
$V\otimes \ldots \otimes V$ for some faithful $kG$... | 10 | https://mathoverflow.net/users/17498 | 104987 | 60,753 |
https://mathoverflow.net/questions/104976 | 5 | At first an example which I know how to treat.
Let's say we have the following integral
$\int dx \frac{1}{x^2+a^2}$
Now we do an analytic continuation of the constant $a$ to the complex numbers: $a \rightarrow ia$. This gives
$\int dx \frac{1}{x^2-a^2}$
which now has poles at $x = \pm a$. Let's say I want to ev... | https://mathoverflow.net/users/25778 | Complex Analysis - Analytic Continuation and Residual Integration | Notice that in the picture you linked to, the poles moved from $\pm ia$ toward the real axis along *counter-clockwise arcs* that did not cross the real axis itself (where the integration contour lies). It is this path of approach of the poles toward the real axis that determines how the integration contour needs to be ... | 4 | https://mathoverflow.net/users/2622 | 104988 | 60,754 |
https://mathoverflow.net/questions/104983 | 5 | Let $G$ be a extra-special $p$-group of order $p^{1+2r}$ with exponent $p$ (p odd). I want to know if $G$ has only $3$ characteristic subgroups?
Background: From [2], if $G$ is extra-special $5$-group of order $5^5$ with exponent $5^2$, then $G$ has more than $3$ characteristic subgroup. How about the exponent of th... | https://mathoverflow.net/users/22049 | Characteristic subgroups of extra-special p-groups | An extraspecial $p$-group of exponent $p$ contains exactly three characteristic subgroups, $1$, $G$ and the center of $G$.
Let $Z$ be the center of $G$ (so $Z=[G,G]=\Phi(G)$). The elementary abelian group $G/Z$ is a vector space of dimension $2r$ over the field of order $p$. The commutator map on $G$ induces a nondeg... | 9 | https://mathoverflow.net/users/36466 | 104991 | 60,756 |
https://mathoverflow.net/questions/104977 | 3 | In knot theory,
a quandle cocycle invariant was defined.
Moreover, to virtual knot theory it was generalized by avoiding for virtual crossings.
Question
Are there many application of a quandle cocycle invariant for virtual knots?
For example, in surface knot theory, we can study an invertible, triple points number ... | https://mathoverflow.net/users/16516 | Application of a quandle cocycle invariant for virtual knots | To find the answer to your question, I would look through papers of Sam Nelson and his students, all of which are available on the arXiv. It is true that most of his work has to do with using quandles and biquandle colorings, he might have some work on cocycles.
Meanwhile, be aware that the cocycle invariants for cl... | 3 | https://mathoverflow.net/users/36108 | 104992 | 60,757 |
https://mathoverflow.net/questions/102844 | 3 | Let $G$ and $H$ be two Hopf algebras, and $\pi: G \to H$ a Hopf algebra map. We will call an algebra of the form
$$
M:= \lbrace m \in G ~ | ~ m\_{(1)} \otimes \pi(m\_{(2)}) = m \otimes 1 \rbrace
$$
a *quantum homogeneous space*.
There are many well known examples of quantum homogeneous spaces where $G$ is a [faithfu... | https://mathoverflow.net/users/3072 | Non-Faithfully Flat Quantum Homogeneous Spaces | Lets look at the commutative situation. So here is an example. I will change your notations -- otherwise I get too confused. Take $G=SL(N,\mathbb{C})$ and let $H$ be the subgroup of all those elements in $G$ for which the matrix entries strictly above the diagonal vanish and for which the diagonal entries are all $1$. ... | 2 | https://mathoverflow.net/users/25781 | 104993 | 60,758 |
https://mathoverflow.net/questions/104995 | 5 | See <http://math.uni.lu/~wiese/galois/Boeckle-Luxemburg-Notes.pdf>, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}\_\ell$-representation $V$ of $G\_F$ ($F$ a global field) ramified at a set $S$ of places where $S$ is *not finite* (for every $\dim{V} \geq 1$)?
| https://mathoverflow.net/users/nan | Is there a semisimple $\mathbf{Q}_\ell$-representation of $G_F$ ramified at an infinite set of places? | It seems that for $\dim{V} = 1$, there is no such representation <http://www.math.leidenuniv.nl/scripties/KretMaster.pdf> p. 10.
| 4 | https://mathoverflow.net/users/nan | 104998 | 60,759 |
https://mathoverflow.net/questions/104996 | 9 | Consider finite group G and its subgroup H, and representation of G in k[G/H] i.e. functions on G/H.
**Question:** What is known about the question: when k[G/H] is multiplicity free ? (Let us consider k - complex numbers for simplicity).
More generally consider $Ind^G\_H V$ some induced representation of G from $H$... | https://mathoverflow.net/users/10446 | When k[G/H] is multiplicity free G module ? | Here's a very simple elementary criterion:
>
> A $\mathbb CG$-module is multiplicity-free if and only if its endomorphism ring is commutative
>
>
>
The reason is simple: if $W\oplus W$ (for some irreducible module $W$) is a direct summand of $V$, then the non-commutative matrix ring $\textrm{End}(W\oplus W) \... | 16 | https://mathoverflow.net/users/17498 | 105003 | 60,763 |
https://mathoverflow.net/questions/105006 | 12 | Essentially as the title, but I'll give a little bit more background.
I have some finite graph $G$ with $n$ vertices and adjacency matrix $A$. Let $D$ be the $n$ by $n$ matrix with the degree of vertex $i$ at the $i,i$ entry, and 0's everywhere. Finally, let $L = D - A$ be the (unnormalized) graph Laplacian of $G$. N... | https://mathoverflow.net/users/25533 | When Do a Few Eigenvectors of Graph Laplacians Not Determine the Graph? | The experimental evidence for adjacency matrices suggests that, for a random graph, its characteristic polynomial is irreducible over the rationals. I would expect that the characteristic polynomial of the Laplacian of a random graph on $n$ vertices would have one irreducible factor of degree $n-1$. However, while it e... | 10 | https://mathoverflow.net/users/1266 | 105020 | 60,769 |
https://mathoverflow.net/questions/104974 | 4 | It is a well known fact that (isomorphism classes of) *principal* $S^1$-bundles over a base space $B$ are classified by $B$'s second integral cohomology, $H^2(B;\mathbb{Z})$.
There is always the trivial one $B\times S^1$, and for $B=S^3$ for example, these are all. The Hopf bundle is an interesting example when $B=S^... | https://mathoverflow.net/users/17083 | Nontrivial examples of non-trivial principal circle bundles | A third way to think about Anton Petrunin's example is that $S^2 \to {\mathbb R}P^2$ is a ${\mathbb Z}\_2$ principal bundle where the action of ${\mathbb Z}\_2 = \lbrace +1, -1 \rbrace $ is the obvious action on vectors in ${\mathbb R}^3$. As ${\mathbb Z}\_2$ is a subgroup of $U(1)$, the circle group of complex numbers... | 7 | https://mathoverflow.net/users/13762 | 105022 | 60,770 |
https://mathoverflow.net/questions/105023 | 3 | I'm studying information theory by myself.
I'm confused about that since we already have Huffman code, which is the optimal code method, why are Shannon code and some other code still useful?
I think maybe in some specific industrial area, Shannon code is more useful.
Can someone give me an explanation?
| https://mathoverflow.net/users/18717 | With Huffman code, why do we still need Shannon code? | I have to say that I know almost nothing about Shannon coding, which is different from, for example, Shannon-Fano coding. Shannon-Fano coding is in some sense a precursor of Huffman coding, it works in a similar way, and while it does not produce the smallest expected code word lengths, the code word lengths are within... | 4 | https://mathoverflow.net/users/7743 | 105025 | 60,772 |
https://mathoverflow.net/questions/105021 | 11 | Given a pseudo-differential operator $P$ of order zero, [Seeley](http://cdsweb.cern.ch/record/469079/) showed that the holomorphic family of operators $\lbrace P^{z} : z\in \mathbb{C} \rbrace$ of all complex powers is contained in the class of pseudo-differential operators.
Apart from knowing that we can take powers... | https://mathoverflow.net/users/7333 | Why take 'complex powers' of pseudo-differential operators? | There are many reasons to do this. A principal one (that was Seeley's motivation) is that the meromorphic operator family
$P^{-s}$ is trace-class when the real part of $s$ is greater than $-n/m$ where $n$ is the
dimension and $m$ is the order of $P$. This gives a meromorphic continuation of the spectral zeta function ... | 22 | https://mathoverflow.net/users/17969 | 105027 | 60,773 |
https://mathoverflow.net/questions/105031 | 2 | $L\_{\omega\_1}$ is the $\omega\_1$-th constructible hierarchy.
I define two binary relation on $P(\omega\_1)$ as follows:
For $X,Y\in{P}(\omega\_1)$,
$R\_1(X,Y)$ means: there is $U\subset\omega\_1\times\omega\_1$ which is $\Sigma\_1$ definable with parameters in $L\_{\omega\_1}$ such that $Y=\lbrace x\in\omega\_1\... | https://mathoverflow.net/users/22635 | About Sigma_1 definability | If I've followed your definitions, then the answer to question 1 is negative.
Specifically, if $A$ is $\Sigma\_1$ but not $\Pi\_1$, then $R\_1(A,A')$ does not hold. The reason is that when $R\_1(A,A')$ holds, we get $\Sigma\_1$ information about the complement of $A$. To see what I mean, suppose $R\_1(A,A')$ holds, ... | 3 | https://mathoverflow.net/users/1946 | 105035 | 60,777 |
https://mathoverflow.net/questions/104866 | 47 | In many places (on MO, elsewhere on the Internet, and perhaps even in some textbooks) one finds a statement of the classical Brown representability theorem that looks something like this:
>
> If $F$ is a contravariant functor from the (weak) homotopy category of topological spaces $\mathrm{Ho}(\mathrm{Top}\_\ast)$ ... | https://mathoverflow.net/users/49 | Brown representability for non-connected spaces | I was thinking more about this question and found another paper by Heller which offers an answer (unfortunately a negative one). The paper is
* Peter Freyd and Alex Heller, *Splitting homotopy idempotents. II.* J. Pure Appl. Algebra **89** no. 1-2 (1993) pp 93–106, doi:[10.1016/0022-4049(93)90088-B](https://doi.org/1... | 37 | https://mathoverflow.net/users/12547 | 105049 | 60,785 |
https://mathoverflow.net/questions/105040 | 10 | This question in stackExchange remained unanswered.
Let $\mathbb F$ be a finite field. Denote by $M\_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M\_n(\mathbb F)$ what is the cardinality of $C\_{M\_n(\mathbb F)} (A)$ , the centralizer of $A$ in $M\_n(\mathbb F)$? There are papers... | https://mathoverflow.net/users/24864 | Centralizer of a Matrix over a Finite Field | Let me add some cases in which one has a clear answer:
[R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991., Corollary 4.4.18]. Let $F$ be a field and $n$ is a natural number. If
$A\in M\_n(F)$ is a cyclic matrix, then $C\_{M\_n(F)}(A)$ is the set of all matrices which are... | 7 | https://mathoverflow.net/users/19075 | 105052 | 60,787 |
https://mathoverflow.net/questions/105048 | 22 | Let $B\_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation
$\rho: B\_{2g+1} \rightarrow Sp\_{2g}(\mathbf{Z})$
called the "hyperelliptic representation," which can be described as follows. The braid group is the fundamental group of the moduli space of configurations of 2g+1 po... | https://mathoverflow.net/users/431 | The image of the point-pushing group in the hyperelliptic representation of the braid group | It's finite index by [Margulis' normal subgroup theorem.](http://groupprops.subwiki.org/wiki/Margulis%27_normal_subgroup_theorem)
Since $H \lhd P\_{2g+1}$, then $\rho(H)\lhd \rho(P\_{2g+1})$. Since $\rho(P\_{2g+1})$ is finite index in $\rho(B\_{2g+1})=\Gamma(2)$ (I'm taking your word for this),
therefore $\rho(H)... | 15 | https://mathoverflow.net/users/1345 | 105059 | 60,792 |
https://mathoverflow.net/questions/105047 | 26 | I've heard that the étale fundamental group of the moduli stack of elliptic curves (over $\mathbb{Z}$) is trivial. Is there an easy proof of that? (Note that there are plenty of étale covers once one inverts a prime $p$, given by taking elliptic curves with some form of a level $p^n$ structure.)
More generally, I'd b... | https://mathoverflow.net/users/344 | Fundamental group of the moduli stack of elliptic curves | Yes, there is a proof which is long but one might consider to be "easy" after digesting it.
Let's first show that the question (of triviality of connected finite etale covers) for the moduli stack $M\_1$ is equivalent to its counterpart for the "Deligne-Rapoport" compactification $\overline{M}\_1$ (a regular proper ... | 34 | https://mathoverflow.net/users/22479 | 105062 | 60,794 |
https://mathoverflow.net/questions/105061 | 5 | Let $X$ be a scheme (you can assume that $X$ is proper and smooth over an algebraically closed field) and $T$ is a finite subgroup of $\text{Pic } X$ (of order prime to the characteristic). Does there exist a finite étale cover $f:Y\to X$ (possibly Galois with Galois group $T$) for which the map $f^\* :\text{Pic }X\to\... | https://mathoverflow.net/users/3847 | Picard groups of abelian étale covers | The answer is "no," even for curves. Say $X$ is a genus 2 curve over $\mathbf{C}$, and $f:Y \to X$ is any finite etale degree $d$ morphism of degree $> 1$. Then $Y$ is a smooth projective curve of genus $d+1 > 2$. In particular, $\mathrm{Pic}(Y)$ has dimension $>2$, so it cannot be a quotient of $\mathrm{Pic}(X)$.
| 8 | https://mathoverflow.net/users/25792 | 105063 | 60,795 |
https://mathoverflow.net/questions/90819 | 4 | This question is related to, but apparently not exactly the same as, [Ramified cover of the $4$-sphere.](https://mathoverflow.net/questions/8697/ramified-cover-of-4-sphere) Piergallini, *et al.* have singular points on their branch loci.
Which closed orientable $4$-dimensional manifolds can be realized as simple br... | https://mathoverflow.net/users/36108 | Branched Coverings of the $4$-sphere branched along a knotted surface | Anton's comments above answer the question.
| 0 | https://mathoverflow.net/users/36108 | 105068 | 60,797 |
https://mathoverflow.net/questions/105067 | 3 | 1. A quadrilateral can be partitioned into 2 equal-area triangles if and only if one diagonal divides equally the other diagonal.
2. It can be proved that most quadrilaterals cannot be partitioned into 3 equal-area triangles.
3. Conjecture: For any natural number $n$, there exists a quadrilateral that cannot be partiti... | https://mathoverflow.net/users/20491 | How to partition a quadrilateral into a finite number of equal-area triangles | See [this Wikipedia article.](http://en.wikipedia.org/wiki/Equidissection)
| 7 | https://mathoverflow.net/users/11142 | 105069 | 60,798 |
https://mathoverflow.net/questions/104718 | 5 | Let $\Sigma\_g$ be a surface of genus $g\ge 2$, and let $\Sigma\_k$ be an $m$-sheeted covering
space of $\Sigma\_g$. It is known that $k=m(g-1)+1$.
An example of such a covering space is a regular covering obtained by choosing one ``hole" as the center of the symmetry and take $\Sigma\_k$ to have $m$-fold rotationa... | https://mathoverflow.net/users/25711 | Covering spaces of surfaces |
>
> Given any covering map $\Sigma\_h\to\Sigma\_g$ between two surfaces, is there some kind of a ``standard" covering $M\to \Sigma\_g$, which factors through $\Sigma\_h$?
>
>
>
In brief, the answer to this part of the question is 'You can take $M\to\Sigma\_g$ to be regular, but beyond that, no.' This can alread... | 5 | https://mathoverflow.net/users/1463 | 105085 | 60,807 |
https://mathoverflow.net/questions/105082 | 2 | I've read on another topic that general interpolation result from Gagliardo-Nirenberg inequality can be read as follow :
\begin{equation}
\|D^ju\|^1\_{L^p} \leq C \|D^mu\|^a\_{L^r} \|u\|^{1-a}\_{L^q}
\end{equation}
with some relations between $a$, $r$, $q$ and $p$, $j$ and $m$.
Does this inequality stands in $\ma... | https://mathoverflow.net/users/25797 | Gagliardo-Nirenberg inequality | The mother of all Gagliardo-Nirenberg inequalities is
$$
\Vert u\Vert\_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le c\_n \Vert \nabla u\Vert\_{L^{1}(\mathbb R^n)},
\tag {GN}
$$
where $c\_n$ depends only on $n$ and $u$ runs say in $C^1\_c(\mathbb R^n)$. Applying this to $u=v^2$, you get with $p=\frac{2n}{n-1}$
$$\Vert v\Vert\_{L... | 8 | https://mathoverflow.net/users/21907 | 105091 | 60,809 |
https://mathoverflow.net/questions/105086 | 6 | Is there any known improvement on the Kahn-Kalai-Linial inequality (on the influences of boolean functions) in the special case in which $f$ is the indicator function of an *intersecting* monotonic set system? More concretely, is there an absolute constant $C>0$ such that the following statement holds:
>
> If $f:\m... | https://mathoverflow.net/users/20598 | Kahn-Kalai-Linial for intersecting upsets | This is a natural question and indeed the property of being intersecting is quite interesting also in various aspect of influences. However, you cannot improve KKL's theorem if f is intersecting: An example is this: consider your variables on a circle and let f=1 if the longest run of 1's is larger than the longest 1's... | 8 | https://mathoverflow.net/users/1532 | 105099 | 60,812 |
https://mathoverflow.net/questions/105097 | 4 | What is the state-of-the-art of the proof/counterexamples of the Hasse principle for high-dimensional hypersurfaces (say 3-folds or more)?
| https://mathoverflow.net/users/4096 | Hasse principle for high dimensional varieties | Regarding counterexamples, there is Sarnak-Wang, and then the results of Bjorn Poonen. Regarding proofs that the Hasse principle holds, I don't know the best result over ALL global fields. Of course global function fields are $C\_2$, thus the Hasse principle trivially holds over global function fields for hypersurfaces... | 9 | https://mathoverflow.net/users/13265 | 105100 | 60,813 |
https://mathoverflow.net/questions/105093 | 4 | Let $S^n$ be the $n$-dimensional round sphere (i.e. with Riemannian metric of constant curvature +1). Is there any classification result of totally geodesic embedded submanifolds? Are they all round spheres? How about general case when the curvature only assumes to be positive?
edit: To make the second question more ... | https://mathoverflow.net/users/1190 | Totally geodesic submanifold of round sphere | Contrary to another comment/answer, totally-geodesic $1$-manifolds in $S^3$ are not trivial because they can be disconnected. Great circles do not have to intersect, as can be seen by taking generic planes through the origin in $\mathbb{R}^4$.
Geodesics sometimes can't be isotoped to fibers of the same Hopf fibratio... | 5 | https://mathoverflow.net/users/2954 | 105115 | 60,822 |
https://mathoverflow.net/questions/105131 | 2 | The group is generated by a,b,c,d with relations
ab=bc=ca
ac=cd=da
ad=db=ba
bd=dc=cb
I checked using GAP that it is $A\_4$ mod its center. But what is the center? Torsion free of rank=?
| https://mathoverflow.net/users/15032 | Central extension of A4 - Is center torsion-free, and what is rank? | Try the following GAP code:
```
> g:=FreeGroup("a","b","c","d");
<free group on the generators [ a, b, c, d ]>
> h:=g/ParseRelators(GeneratorsOfGroup(g),"ab=bc=ca,ac=cd=da,ad=db=ba,bd=dc=cb");
<fp group on the generators [ a, b, c, d ]>
> z:=Centre(h);
Group(<37 generators>)
> IsAbelian(z);
true
> AbelianInva... | 4 | https://mathoverflow.net/users/1446 | 105135 | 60,833 |
https://mathoverflow.net/questions/105058 | 3 | Tate showed that the functional equation for zeta functions of number fields can be proven with fourier-analytic methods on the adele ring. Can the same be done for zeta functions of varieties over finite fields?
| https://mathoverflow.net/users/7935 | Tate's thesis for varieties over finite fields | This is done in Chapter 7.3 of Ramakrishnan and Valenza's *Fourier Analysis on Number Fields* (GTM 186) which, despite the title, describes in some details also the situation over function fields in one variable (so, for curves over finite fields).
If you want to consider curves over some global field of positive cha... | 5 | https://mathoverflow.net/users/18238 | 105140 | 60,836 |
https://mathoverflow.net/questions/11456 | 13 | The classical Brown Representability Theorem states: Denote $hCW\_\*$ the homotopy category of pointed CW-complexes. Let $F : hCW\_\* \to Set\_\*$ be a contravariant functor. Then $F$ is representable if and only if
* $F$ respects coproducts, i.e. $F(\vee\_{i \in I} X\_i) = \prod\_{i \in I} F(X\_i)$ for all families ... | https://mathoverflow.net/users/2841 | Unpointed Brown representability theorem | This is a copy of my answer to [Brown representability for non-connected spaces](https://mathoverflow.net/questions/104866/brown-representability-for-non-connected-spaces/) which I repost here per request in the comment.
A negative answer to the question can be concluded from this paper:
* Peter Freyd and Alex Hell... | 14 | https://mathoverflow.net/users/12547 | 105156 | 60,841 |
https://mathoverflow.net/questions/105160 | 1 | I've the following Problem on systems of Partial Differential Equations. I have "$ N $" Physical variables. and Finally I form the equation on a bounded domain having regular boundary in $R^d$ ($d=2$ generally)
$\mbox{div}(W\_i)=f\_i$, $i=1\cdots N$
$W\_i =\displaystyle \sum\_{i, j=1}^NA\_{ij} \cdot\nabla P\_j$ wi... | https://mathoverflow.net/users/23790 | Reference to the Existence and Uniqueness of the PDE system | A good keyword here is *strongly elliptic systems*. There is an original paper by Nirenberg. Also have a look at MacLean's book *Strongly elliptic systems and boundary integral operators*. Folland's *Introduction to PDE* has a good treatment too.
| 1 | https://mathoverflow.net/users/824 | 105167 | 60,846 |
https://mathoverflow.net/questions/105161 | 4 | Let $G$ be a topological group and let $X$ and $Y$ be connected, well-pointed $G$-spaces. Suppose $f:X\to Y$ is a pointed homotopy equivalence and a $G$-equivariant map (but not an equivariant homotopy equivalence). I know that $f$ induces a (weak) homotopy equivalence on the Borel constructions, $EG\times\_G X\to EG\t... | https://mathoverflow.net/users/25828 | Does a pointed homotopy equivalence between pointed $G$-spaces which is $G$-equivariant induce a (weak) homotopy equivalence on pointed Borel constructions? | In the pointed Borel construction, you clearly mean $\wedge$
and not $\times$. Thus $$EG\_+\wedge\_G X = EG\times\_G X/EG\times\_G\ast.$$
Out of laziness, I'll assume that your $X$ and $Y$ are of the $G$-homotopy
types of $G$-CW complexes. The
based $G$-map $id\times f\colon EG\times X\longrightarrow EG\times Y$
is... | 7 | https://mathoverflow.net/users/14447 | 105168 | 60,847 |
https://mathoverflow.net/questions/105165 | 3 | In the book by Antosik, Mikusiński and Sikorski named *Theory of Distributions, The Sequential Approach* (russian translation, page 217) one can read:
>
> Таким образом, мы видим, что для произвольного положительного целого числа $n$ не составляет труда построить преобразование $G$ простраства обобщенних функций ме... | https://mathoverflow.net/users/4925 | Fourier-Möller transform | Google "Fourier-Mehler transform"
| 4 | https://mathoverflow.net/users/11142 | 105170 | 60,849 |
https://mathoverflow.net/questions/104895 | 6 | Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a slightly more complicated but naturally-occuring map: $(g,h)\mapsto ghg^{-1}h^{-1}$, i.e. $G\times G\to [G,G]\hookrightarr... | https://mathoverflow.net/users/12310 | $\pi_1$ Sequence of Topological Groups | This is really more of a comment, but it kind of answers one of the OP's question, so I am indulging myself: It is a result of W. Browder (Annals, 1961) that $\pi\_2$ of a *finite dimensional* $H$-space is trivial, so the result holds in that setting. I learned of this (and also that this is not true without the finite... | 4 | https://mathoverflow.net/users/11142 | 105173 | 60,851 |
https://mathoverflow.net/questions/105192 | 7 | For surfaces there are many statements along the lines of: if two simple closed curves are homotopic, they are isotopic. I'm interested in such questions for families of curves.
More precisely, let $\Sigma$ be a hyperbolic surface, possibly with boundary. We fix an essential simple closed curve $\gamma$ on $\Sigma$. ... | https://mathoverflow.net/users/798 | What is the homotopy type of the space of simple closed curves isotopic to a given one? | Earlier than Grayson, the determination of the homotopy-types of these spaces was done by Gramain.
There are a few special cases, like the torus and sphere and the non-orientable analogue, the case of null curves. But if they're not null homotopic the components of the embedding space have the homotopy type of $S^1$... | 5 | https://mathoverflow.net/users/1465 | 105196 | 60,867 |
https://mathoverflow.net/questions/105191 | 8 | I know there are text books of Algebraic topology. There are books of Differential geometry. But when I read papers, for example lots of papers talking about fundamental groups or higher homotopy groups of certain manifolds, sometimes lots of terminologies from abstract algebra pop out - nilpotent, solvable or amenable... | https://mathoverflow.net/users/1190 | Any text book or lecture notes regarding the algebraic part of geometry? | What about de la Harpe's topics in geometric group theory?
| 7 | https://mathoverflow.net/users/11142 | 105199 | 60,868 |
https://mathoverflow.net/questions/105214 | 3 | Let $F:{\cal A}\to {\cal B}$ be an additive, exact and faithful functor between abelian categories. Then on the level of complexes, $F$ maps quasi-isomorphisms to quasi-isomorphisms and thus induces a functor on the derived categories $DF:D({\cal A})\to D({\cal B})$.
Is it true that $DF$ is faithful? Likewise for th... | https://mathoverflow.net/users/nan | Faithfulness of derived functor. | This will usually not be the case. For example, consider a "typical forgetful functor" for example
$$A-Mod \rightarrow k-vect$$
from the category of representations of a k-algebra $A$ to vectorspaces.
It is exact faithful, but its derived functor will only be faithful if the algebra is semi-simple, since there are... | 11 | https://mathoverflow.net/users/2837 | 105215 | 60,875 |
https://mathoverflow.net/questions/105222 | 8 | It seems that the Fulkerson prize has been attributed to Thomas Hales for this work. What is the present status of the conjecture, then?
| https://mathoverflow.net/users/17164 | Hales work on Kepler conjecture | The original 1998 proof appeared in 2005/6, in an abridged version in Annals of Mathematics:
>
> A proof of the Kepler conjecture. Ann. of Math. (2) 162 (2005), no. 3, 1065–1185.
>
>
>
and unabridged as volume 36 of Discrete & Computational Geometry (2006).
It is my understanding that meanwhile there are no... | 14 | https://mathoverflow.net/users/nan | 105230 | 60,878 |
https://mathoverflow.net/questions/105232 | 6 | Let $(X, O\_X)$ be an arbitrary scheme and $\mathcal{F}$ a presheaf of $O\_X$-modules. The presheaf $\mathcal{F}$ is said to be separated if the natural map to it's sheafification $\mathcal{F}^+$ is an injection. That is, for each open set $U$, $\mathcal{F}(U) \hookrightarrow \mathcal{F}^+(U)$.
Let $\mathcal{F}, \ma... | https://mathoverflow.net/users/25854 | Tensor product of sheaves separated? | If I understand correctly your question, then the answer is "no, the presheaf tensor product need not be separated." Let $(X,\mathcal{O}\_X)$ be the projective line $(\mathbb{P}^1,\mathcal{O}\_{\mathbb{P}^1})$ over a base ring $R$. Let $\mathcal{F}$ and $\mathcal{G}$ both be the invertible sheaf $\mathcal{O}\_{\mathbb{... | 12 | https://mathoverflow.net/users/13265 | 105235 | 60,881 |
https://mathoverflow.net/questions/105236 | 8 | Are the congruence subgroups of the modular group $\Gamma\equiv\mathrm{PSL}\left(2,\mathbb{Z}\right)$ (e.g. $\Gamma\left(n\right)$, $\Gamma\_{0}\left(n\right)$, $\Gamma\_{1}\left(n\right)$ etc.) finitely presented? If so, is there a proof of this? Assuming they *are* finitely presented, are the presentations of e.g. th... | https://mathoverflow.net/users/25565 | Are congruence subgroups of the modular group finitely presented? | A subgroup of finite index in a finitely presented group is finitely presented (see Exercise 6.1.6 in Robinson: A course in the theory of groups), so all congruence subgroups of the modular group are finitely presented. I cannot answer your second question.
| 9 | https://mathoverflow.net/users/11919 | 105237 | 60,882 |
https://mathoverflow.net/questions/105221 | 30 | Given a semisimple Lie algebra $\mathfrak g$ with Cartan matrix $a\_{ij}$, the quantum group $U\_q(\mathfrak g)$ is usually defined as the $\mathbb Q(q)$-algebra with generators $K\_i$, $E\_i$, $F\_i$ (the $K\_i$ are invertible and commute with each other) and relations
$$
\begin{split}
K\_iE\_j &K\_i^{-1}=q^{\langle\a... | https://mathoverflow.net/users/5690 | quantum groups... not via presentations | $\newcommand\g{\mathfrak{g}}$The answer to your question "is there a procedure that takes $\g$ as input, produces $U\_q(\g)$ as output, and doesn't involve the choice of a Cartan subalgebra of $\g$?" is No. Not if you want it "canonical" in any sense. (Of course, if I wanted to cheat my way to a "yes," I could make cho... | 24 | https://mathoverflow.net/users/78 | 105240 | 60,884 |
https://mathoverflow.net/questions/105226 | 2 | Let $V$ be an affine variety over an algebraically closed field $k$ and
$D \subset V$ a Cartier divisor which is normal and has an isolated singularity at $p \in D$.
Let $\mathcal{O}\_V^\*, \mathcal{O}\_D^\*$ be the sheaves of invertible functions on $V$ and $D$.
Then I think that we have an exact sequence $0 \rig... | https://mathoverflow.net/users/12390 | Lifting invertible functions on a divisor to ambient affine variety | Suppose that $\bar{f} \in H^0(D, O\_D^\*)$ and consider a corresponding $f \in H^0(V, O\_V)$ (which may or may not be invertible).
Then for every point $x \in D \subseteq V$, we let $\bar{f}'$ denote the element in the stalk $O\_{D,x}$ and $f'$ the element in the stalk $O\_{V,x}$. Since $\bar{f}'$ is not in the maxi... | 5 | https://mathoverflow.net/users/3521 | 105241 | 60,885 |
https://mathoverflow.net/questions/105244 | 1 | An operator is a bounded (i.e., continuous) linear transformation between Hilbert spaces. Let $\mathcal{B}[\mathcal{H}]$ be the set of all operators in the Hilbert space $\mathcal{H}$.
Let $\mathcal{H}$ and $\mathcal{K}$ be any two Hilbert spaces. Consider $\mathcal{C}$ be the class of all strict contractions on $\ma... | https://mathoverflow.net/users/25641 | A doubt about tensor product on Hilbert Spaces | There is no standard definition of the tensor product of subsets. If you want to take a tensor product of closed subspaces $E$ of $B(H)$ and $F$ of $B(K)$ there are a variety of choices (there's a nice survey at hrcak.srce.hr/file/1655). The simplest is probably the so-called "spatial" tensor product defined as the clo... | 2 | https://mathoverflow.net/users/23141 | 105251 | 60,890 |
https://mathoverflow.net/questions/105234 | 10 | Could a function in FOL take functions as arguments? FOL only limits on the order of the individuals being quantified, but if an expression does not involve quantifying over second-order or higher terms, would it still be valid in FOL? Say, $f(g)$ where $f : (A \to B) \to C$ and $g : A \to B$.
So another way to put m... | https://mathoverflow.net/users/25855 | Second-order term in first-order logic? | I think that the spirit of this question, combined with the clarifications in comments, is:
>
> What is it that makes first-order logic "first order"?
>
>
>
Unfortunately, the terms "first order" and "second order" get used to mean various things.
A formal but unsatisfying answer would say that first-order... | 23 | https://mathoverflow.net/users/5442 | 105259 | 60,896 |
https://mathoverflow.net/questions/105263 | 16 | **Background** See [WP-article](http://en.wikipedia.org/wiki/Field_with_one_element) on F\_1 = F\_{un} = Field with one element (and also [this MO](https://mathoverflow.net/questions/2300/what-is-the-field-with-one-element) question). Paraphrasing someone:
we do not know what is it, but it is not a field :). For this q... | https://mathoverflow.net/users/10446 | Are there F_un Lie algebras ? | I will only attempt to answer the first question.
$n$-dimensional vector space over $\mathbf{F}\_1$ is the same as a pointed set with $n+1$ elements. It is natural to call $GL\_n(\mathbf{F}\_1)$ the group of automorphisms of $\mathbf{F}\_1^n$ and $\mathfrak{gl}\_n(\mathbf{F}\_1)$ the monoid of endomorphisms. There ar... | 7 | https://mathoverflow.net/users/18512 | 105270 | 60,903 |
https://mathoverflow.net/questions/105255 | 0 | Apologies if this is elementary, but I have never heard the terminology before:
>
> What is a **"non-splitting covering"** of a finite group?
>
>
>
I encountered the term while reading [this paper](http://www.bdim.eu/item?fmt=pdf&id=BUMI_2002_8_5B_1_131_0), in which the author introduces a group $H$ with subg... | https://mathoverflow.net/users/25494 | What is a "non-splitting covering" of a finite group? | I think it is rather that $H$ has a homomorphic image $S\_4,$ but no subgroup $S\_{4}.$ It is sometimes called a double cover of $S\_{4}.$ The group $S\_{4}$ has two such non-isomorphic (proper) double covers. One is ${\rm GL}(2,3)$ which has a semidihedral Sylow $2$-subgroup.
The other is the binary octahedral group, ... | 3 | https://mathoverflow.net/users/14450 | 105271 | 60,904 |
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