parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/26917 | 2 | Let $G$ be a finite group acting on an associative $k$-algebra with unity ($k$ algebraically closed of characteristic zero). The action is called ergodic if $ A^G = \{ a \in A | g \cdot a = a, \forall g \in G \} = k$.
Anyone know an example of a non-semisimple finite dimensional algebra with an ergodic action of a f... | https://mathoverflow.net/users/6517 | A question on group action on algebras | Let $A = k[x,y]/(x^2,xy,y^2)$ and let the generator of the two-element group act by $x \mapsto -x$, $y \mapsto -y$.
More generally, take any finite group acting on $R = k[x\_1, \dots, x\_n]$ with invariant subring $S = k[f\_1, \dots, f\_m]$, and set $A = R/(f\_1, \dots, f\_m)$.
| 8 | https://mathoverflow.net/users/460 | 26921 | 17,605 |
https://mathoverflow.net/questions/25933 | 3 | For a monad $(T,\mu,\eta)$ if $T(A) = T(B)$, does this imply that $\mu\_A = \mu\_B$? I want to know because in the bijection between Kleisli triples and monads, given a monad, we define $f^\* := T(f) ; \mu\_B$ if $f : A\to T(B)$ (c.f. [Prop 1.6](http://www.google.com/url?sa=t&source=web&ct=res&cd=1&ved=0CBgQFjAA&url=ht... | https://mathoverflow.net/users/6082 | Kleisli Monad bijection | In fact, the Kleisli star is a partial map ${}^\* : \mathrm{Mor}(\mathbf{C}) \times \mathrm{Obj}(\mathbf{C}) \times \mathrm{Obj}(\mathbf{C}) \to \mathrm{Mor}(\mathbf{C})$. So there is no problem!
| 0 | https://mathoverflow.net/users/6082 | 26923 | 17,607 |
https://mathoverflow.net/questions/26927 | 13 | Let $X$ be a (singular) projective variety, in other words something given by a collection of polynomial equations in $\mathbb CP^n$ or $\mathbb RP^n$. How can one prove it is a finite $CW$ complex?
Similar question: Suppose that $X$ affine (i.e. given by polynomial equations in $\mathbb C^n$, or $\mathbb R^n$). How... | https://mathoverflow.net/users/943 | How to prove that a projective variety is a finite CW complex? | The Lojasiewicz theorem says that every semi-algebraic subset of $\mathbf{R}^n$ can be triangulated. Moreover, there is a similar statement for pairs of the form (a semi-algebraic set, a closed subset). See e.g. Hironaka, Triangulations of algebraic sets, Arcata proceedings 1974 and references therein (including the or... | 16 | https://mathoverflow.net/users/2349 | 26929 | 17,611 |
https://mathoverflow.net/questions/26919 | 13 | I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications to other fields of science. None. It is pure mathematics." This seems like a false statement. My first thought was of pro... | https://mathoverflow.net/users/343 | Real analysis has no applications? | As it happens, I just finished teaching a quarter of undergraduate real analysis. I am inclined to rephrase Pugh's statement into a form that I would agree with. If you view analysis broadly as both the theorems of analysis and methods of calculation (calculus), then obviously it has a ton of applications. However, I m... | 25 | https://mathoverflow.net/users/1450 | 26936 | 17,615 |
https://mathoverflow.net/questions/26932 | 3 | I am following the explicit construction of homotopy colimits as described by Dugger in the paper: "Primer on homotopy colimits", which can be found here: <http://www.uoregon.edu/~ddugger/hocolim.pdf>
As described in the appendix of Topological hypercovers and A1-realizations, Mathematische Zeitschrift 246 (2004) in th... | https://mathoverflow.net/users/6499 | Simple examples of homotopy colimits | Here's an answer to question 2: A sufficient condition for $\mathrm{colimit}(X \leftarrow A\rightarrow Y)$ to be weakly equivalent to the homotopy colimit, is (a) for one of the maps (say $A\to X$) to have the homotopy extension property. Another sufficient condition is that the diagram $(X\leftarrow A\rightarrow Y)$ i... | 7 | https://mathoverflow.net/users/437 | 26937 | 17,616 |
https://mathoverflow.net/questions/26897 | 21 | I have enough fears that this question might get struck down. Still let me try.
I shall restrict myself to $\frac{\lambda \phi^4}{4!}$ perturbed real scalar quantum field theory and call as "symmetry factor" of a Feynman diagram to be the eventual number by which the power of $\lambda$ is divided in the final integr... | https://mathoverflow.net/users/2678 | How to count symmetry factors of Feynman diagrams? | Exactly how you count symmetries depends on your normalizations, and in particular on whether you use divided (i.e. "exponential") power series or ordinary power series. I believe strongly that divided powers are the way to go. To establish notation, I will first review the preliminaries, that I'm sure you already know... | 19 | https://mathoverflow.net/users/78 | 26938 | 17,617 |
https://mathoverflow.net/questions/26931 | 2 | Suppose we have a finite square n x n matrix of complex numbers H that is Hermitian and skew-symmetric:
$H^\dagger = H$ and $H^T = -H$.
(T denotes transpose, $\dagger$ denote conjugate transpose. I know that these conditions mean that H is a purely imaginary skew-symmetric matrix.)
It is a textbook result that t... | https://mathoverflow.net/users/4673 | Infinite hermitian matrix | Not an answer really, but a collection of several comments.
1. The "skew-symmetric" condition is not really natural for an operator on a complex Hilbert space, since it isn't preserved by unitary transformations.
2. Do you have a reference for the statement that Q is the closest Hermitian positive semidefinite matrix... | 1 | https://mathoverflow.net/users/4832 | 26948 | 17,623 |
https://mathoverflow.net/questions/26612 | 5 | Let $(B\_i)$ be a collection of i.i.d. random variables taking values 0 or 1.
Suppose $0 < x^- < x^+ < 1$. Consider two different "success probabilities" for the i.i.d. collection $B\_i$: under measure $P^-$, each $B\_i$ takes value 1 with probability $x^-$, while under measure $P^+$, each $B\_i$ takes value 1 with p... | https://mathoverflow.net/users/5784 | probabilities of increasing events under different product measures. | I managed to hack out an answer to this question, and get the following result:
For any $k$ and $\epsilon$, if
$
x^+\geq 1-(1-x^-)^{1+1/k}
$
and
$
P\_{x^-}(A)\in (k\epsilon^{1/2}, 1-k\epsilon^{1/2})
$
then
$
P\_{x^+}(A)-P\_{x^-}(A)\geq \epsilon.
$
Given $u^-$, $x^-$ and $x^+$, it's easy to use this to g... | 1 | https://mathoverflow.net/users/5784 | 26956 | 17,628 |
https://mathoverflow.net/questions/26959 | 8 | Martin's Axiom implies that $2^{\aleph\_0}$ is a regular cardinal. But can $2^{\aleph\_0}$ be a singular cardinal?
By Konig's Lemma, it can never be $\aleph\_{\omega}$ since cf($2^{\aleph\_0}$)>$\aleph\_0$ but under what conditions can it be $\aleph\_{\omega\_1}$? It is even possible it can be $\aleph\_{\omega\_1}$?... | https://mathoverflow.net/users/3859 | Can the continuum be a singular cardinal? | Yes, but it must have uncountable cofinality. So if it is to be singular, the smallest possibility is $\aleph\_{\omega\_1}$.
The basic fact is that if $\kappa$ is any cardinal such that $\kappa^\omega=\kappa$, then there is a forcing extension $V[G]$ in which $2^\omega=\kappa$. The forcing to achieve this is $\text{... | 13 | https://mathoverflow.net/users/1946 | 26960 | 17,631 |
https://mathoverflow.net/questions/26942 | 55 | Part of what I do is study typical behavior of large combinatorial structures by looking at pseudorandom instances. But many commercially available pseudorandom number generators have known defects, which makes me wonder whether I should just use the digits (or bits) of $\pi$.
A colleague of mine says he "read somewh... | https://mathoverflow.net/users/3621 | Is pi a good random number generator? | Strictly speaking, there are some known patterns in the digits of $\pi$. There are some known results on how well $\pi$ can be approximated by rationals, which imply (for example) that we know *a priori* that the next $n$ as-yet-uncomputed digits of $\pi$ can't all be zero (for some explicit value of $n$ that I'm too l... | 55 | https://mathoverflow.net/users/3106 | 26970 | 17,639 |
https://mathoverflow.net/questions/26983 | 0 | I want to know which isomorphism of vector bundles is the following?
[alt text http://2.bp.blogspot.com/\_uGcgLiQvkI8/TAgpIivl6oI/AAAAAAAAAKk/DCg9iK2W3rA/s1600/Capture-52.png](http://2.bp.blogspot.com/_uGcgLiQvkI8/TAgpIivl6oI/AAAAAAAAAKk/DCg9iK2W3rA/s1600/Capture-52.png)
where $G/H$ is the quotient of group $G$ by ... | https://mathoverflow.net/users/2597 | A metric on T(G/H) | There is an obvious map $G\times\mathfrak g\to G\times\_H\mathfrak g/\mathfrak h$, and an isomorphism $TG\to G\times\mathfrak g$. On the other hand, the projection $G\to G/H$ gives a map $TG\to T(G/H)$. Now you can construct a map $T(G/H)\to G\times\_H\mathfrak g/\mathfrak h$ as the composition $$T(G/H)\leftarrow TG\to... | 5 | https://mathoverflow.net/users/1409 | 26986 | 17,650 |
https://mathoverflow.net/questions/26979 | 30 | Is there a finite group such that, if you pick one element from each conjugacy class, these don't necessarily generate the entire group?
| https://mathoverflow.net/users/799 | Generating a finite group from elements in each conjugacy class | No, this is impossible. This is a standard lemma, but I'm finding it easier to give a proof than a reference: Let $G$ be your finite group. Suppose that $H$ were a proper subgroup, intersecting every conjugacy class of $G$. Then $G = \bigcup\_{g \in G} g H g^{-1}$. If $g\_1$ and $g\_2$ are in the same coset of $G/H$, t... | 68 | https://mathoverflow.net/users/297 | 26993 | 17,655 |
https://mathoverflow.net/questions/26991 | 4 | JS Milne has a page about common errors in mathematical papers, and one of them is the usage of "verify" to mean "satisfy".
>
> Improper usage: "The set $A$ verifies the condition."
>
>
> Proper usage: "The set $A$ satisfies the condition."
>
>
> Proper usage: "We verify that $A$ satisfies the condition."
>
> ... | https://mathoverflow.net/users/1353 | The origin of the satisfy-verify mixup | Dear Harry, in Serre's collected papers, vol.1, page 183 [or Annals of Math.58(1953) page 270]
you'll find (line -5)
"Soit $\mathcal C$ une classe **vérifiant** (II\_A)..."
and many such examples on the same page, corroborating your testimony on papers and books you read in French. I recoil in horror at the though... | 2 | https://mathoverflow.net/users/450 | 26996 | 17,658 |
https://mathoverflow.net/questions/26985 | 7 | I have a curiosity on the Ergodic decomposition given by the von Neumann's theorem:
$$L^2(X,\Sigma,\mu)=L^2(X,\Sigma\_T,\mu)\oplus\overline{\{f-f\circ T\ :\ f\in L^2(X,\Sigma,\mu)\}},$$
that occurs for a measure-preserving map $T$ of a probability space $(X,\Sigma,\mu)$, $\Sigma\_T$ being the sub-σ-algebra of all $... | https://mathoverflow.net/users/6101 | Ergodic splitting in L_p | The mean ergodic theorem on L^p spaces is due to F. Riesz (1938) and S. Kakutani (1938). For p in $[1,\infty[$, this is theorem 1.2 ff in the book of Krengel, "Ergodic theorems".
If $p=\infty$, you get the splitting only on the set of functions for which the Birkhoff sums are converging, which, I think is not everyth... | 2 | https://mathoverflow.net/users/6129 | 27019 | 17,670 |
https://mathoverflow.net/questions/27016 | 12 | In non-relativistic quantum mechanics, what are the necessary conditions on the potential (or on the hamiltonian in general) for the ground state to be non-degenrate?
| https://mathoverflow.net/users/6547 | Non-degeneracy of ground state in quantum mechanics | If a finite number of non-relativistic particles are moving in an infinite potential well, then the combined system has a nondegenerate ground state, regardless of the symmetry of the hamiltonian. I remember this from a long time ago, and I always thought it was impressive. I also remember I was always annoyed that I d... | 3 | https://mathoverflow.net/users/799 | 27030 | 17,677 |
https://mathoverflow.net/questions/27033 | 3 | Is a subgroup of a finite group uniquely determined, up to conjugation, by the subset of conjugacy classes of the larger group that it intersects?
| https://mathoverflow.net/users/799 | Determining conjugacy class of a subgroup from intersection with conjugacy classes | Let $G$ be the group of affine linear maps over the Galois field $k=GF(16)$
of order $16$. The elements of $G$ are maps from $k$ to itself of the form
$x\mapsto ax+b$ where $a\in k^\*$ and $b\in G$. Those with $a=1$ form
a normal elementary abelian subgroup~$H$. All nontrivial elements of $H$
are conjugate. Then $H$ co... | 5 | https://mathoverflow.net/users/4213 | 27034 | 17,678 |
https://mathoverflow.net/questions/27032 | 14 | Given two convex bodies $A$ and $B$, in $\mathbb R^3$ let's say. We define $A(t)$ and $B(t)$ as $A+xt$ and $B+yt$ where $x,y$ are two arbitrary points. (That is the Minkowski sum, so the two bodies are moving at constant velocity in the $x$ and $y$ directions, and $t$ is the time variable.) Can one show that the functi... | https://mathoverflow.net/users/2384 | Pushing convex bodies together | The sets $\{ (A(t),t)|t\in \mathbb{R} \} \subset \mathbb{R}^4$ and $\{ (B(t),t)|t\in \mathbb{R} \} \subset \mathbb{R}^4$ are convex, their intersection $K$ is a bounded convex set, and $f(t)$ is the volume of the slice of $K$ at height $t$. By Brunn-Minkowski inequality, this is log-concave, so definitely unimodal.
| 16 | https://mathoverflow.net/users/2368 | 27038 | 17,681 |
https://mathoverflow.net/questions/27042 | 4 | Let $W$ be a standard Brownian motion under given probability space.
For a given constant $a$, $W^a$ is a truncated Brownian motion by stopping time
$T^a = \inf(t>0:W(t) = a)$. That is, $W^a(t) = W(t \wedge T^a)$.
We want to consider the following question: Is the process $W^1$ of [the class DL](http://almostsure.... | https://mathoverflow.net/users/5656 | Is the truncated Brownian motion of the class DL? | $W^a$ is in fact a martingale. To see this, write $W^a(t) = W(t \land T\_a)$. See also Theorem 3.39 [here](http://books.google.com/books?id=JYzW0uqQxB0C&pg=PA85).
When you write an expression like $\mathbb{E}(W^a(T^a))$ you are implicitly assuming that $W^a(T^a)$ is measurable. This requires $t \ge T\_a$ (and trivial... | 3 | https://mathoverflow.net/users/1847 | 27046 | 17,684 |
https://mathoverflow.net/questions/27039 | 8 | Let $X$ be a stone space, i.e. a compact, totally disconnected hausdorff space. Then $H^1(X,\mathbb{Z}/2)=0$. Here's one way of proving this: $X$ with $\mathbb{Z}/2$ (the constant sheaf) is an affine scheme, now use the vanishing result for quasicoherent sheaves on affine schemes.
What happens if we also allow locall... | https://mathoverflow.net/users/2841 | Sheaf Cohomology on a Stone Space | A search brought up [Sheaf Cohomology of Locally Compact Totally Disconnected Spaces](https://doi.org/10.2307/2035693) by R. Wiegand. [Some topological invariants of Stone spaces](https://doi.org/10.1307/mmj/1029000311) by the same author might also be of interest.
| 3 | https://mathoverflow.net/users/2384 | 27048 | 17,686 |
https://mathoverflow.net/questions/27044 | 2 |
>
> **Possible Duplicates:**
>
> [Free, high quality mathematical writing online?](https://mathoverflow.net/questions/1722/free-high-quality-mathematical-writing-online)
>
> [Most helpful math resources on the web](https://mathoverflow.net/questions/2147/most-helpful-math-resources-on-the-web)
>
>
>
A lot... | https://mathoverflow.net/users/5627 | Websites hosting free math ebooks. | I have found some list of math books here, though not so much advanced.
[onlinecomputerbook](http://www.onlinecomputerbooks.com/free-math-books.php) The name is misleading(There are math books)
| 1 | https://mathoverflow.net/users/5627 | 27053 | 17,690 |
https://mathoverflow.net/questions/27040 | 7 | I have to work with the following variation of Minkowski sum:
>
> Let $\mathbb E$ be a Euclidean space and $K$ be a convex set in $\mathbb E\times \mathbb E$.
> Set
> $$K^+=\{\\,x+y\in\mathbb E\mid(x,y)\in K\\,\}.$$
>
>
>
Note that if $K=K\_x\times K\_y$ for some convex sets $K\_x$ and $K\_y$ in $\mathbb E$... | https://mathoverflow.net/users/1441 | A variation of Minkowski sum | In additive combinatorics, we call the Minkowski sum the sumset, and write it as ${\mathbb E}+{\mathbb E}$. We call what you're talking about the "sumset along a graph", and write it as ${\mathbb E}+\_K{\mathbb E}$, where $K$ is any graph (you call it a subset of ${\mathbb E}\times {\mathbb E}$ and I call it a graph, b... | 5 | https://mathoverflow.net/users/935 | 27064 | 17,696 |
https://mathoverflow.net/questions/27004 | 20 | Each of n players simultaneously choose a positive integer, and one of the players who chose [the least number of [the numbers chosen the fewest times of [the numbers chosen at least once]]] is selected at random and that player wins.
For n=3, the symmetric Nash equilibrium is the player chooses m with probability 1/... | https://mathoverflow.net/users/nan | Lowest Unique Bid | There is some published literature on this problem. See for example the following papers and the references therein.
Baek and Bernhardsson, [Equilibrium solution to the lowest unique positive integer game](http://arxiv.org/pdf/1001.1065v1)
Rapoport et al., [Unique bid auctions: Equilibrium solutions and experimenta... | 2 | https://mathoverflow.net/users/3106 | 27069 | 17,701 |
https://mathoverflow.net/questions/27059 | 1 | Does the function $f(x,y) = ((x-1) \mod y)+1$ have an existing name?
`f(1,5) = 1`
`f(2,5) = 2`
`f(3,5) = 3`
`f(4,5) = 4`
`f(5,5) = 5`
`f(6,5) = 1`
`f(7,5) = 2`
| https://mathoverflow.net/users/2644 | What is the name of the function f(x,y) = ((x-1) mod y)+1 ? | In math, as opposed to in computer science, when you apply "mod y" you land in the integers modulo y, denoted Z/yZ, not back in the integers. This means that mod 7, the symbols 1 and 8 *denote the same thing*, i.e. the equivalence class {...,-13,-6,1,8,15,...}.
A more computer-y way to say this is that for mathemati... | 7 | https://mathoverflow.net/users/22 | 27070 | 17,702 |
https://mathoverflow.net/questions/27071 | 2 | Can someone lead me to a method for calculating the number of sequences of length $n$ if the terms of the sequence are chosen (with replacement) from a set with $k$-elements under the condition that all $k$-elements are chosen at least once? It's not my field.
I lead myself to consider the recurrence relation
$$ R(k... | https://mathoverflow.net/users/6556 | Counting sequences - a recurrence relation. | These are related to Stirling numbers. These don't have a "closed form".
Your $R(k,n)=k!S(n,k)$ where $S(n,k)$ is the Stirling number of
the second kind as defined at
<http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind> .
| 7 | https://mathoverflow.net/users/4213 | 27072 | 17,703 |
https://mathoverflow.net/questions/27075 | 106 | What is the oldest open problem in mathematics? By old, I am referring to the date the problem was stated.
Browsing Wikipedia list of open problems, it seems that the *Goldbach conjecture* (1742, every even integer greater than 2 is the sum of two primes) is a good candidate.
The *Kepler conjecture* about sphere p... | https://mathoverflow.net/users/6129 | What is the oldest open problem in mathematics? | Existence or nonexistence of odd perfect numbers.
Update: Goes back at least to Nicomachus of Gerasa around 100 AD, according to [J J O'Connor and E F Robertson](https://web.archive.org/web/20100209232200/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html). Nichomachus also asked about infin... | 89 | https://mathoverflow.net/users/22 | 27080 | 17,708 |
https://mathoverflow.net/questions/27089 | 2 | Off hand, does anyone know of some useful conditions for checking if a ring (or more generally a semiring) has non-trivial derivations? (By non-trivial, I mean they do not squish everything down to the additive identity.) Part of the motivation for this is that I was thinking about it the other day, and had trouble fin... | https://mathoverflow.net/users/4642 | Necessary and sufficient criteria for non-trivial derivations to exist? | There is a notion of a universal derivation for an algebra. I'll assume
everything is commutative for simiplcitity. If $A$ is a $k$-algebra
($k$ a commutative ring) then there is an $A$-module $\Omega\_{A/k}$,
the module of *Kahler differentials* of $A$ over $k$ and a $k$-derivation
$d:A\to\Omega\_{A/k}$ which is unive... | 4 | https://mathoverflow.net/users/4213 | 27093 | 17,716 |
https://mathoverflow.net/questions/27076 | 14 | Often undergraduate discrete math classes in the US have a calculus prerequisite.
Here is the description of the discrete math course from my undergrad:
>
> A general introduction to basic
> mathematical terminology and the
> techniques of abstract mathematics in
> the context of discrete mathematics.
> Topic... | https://mathoverflow.net/users/6555 | Why does undergraduate discrete math require calculus? | A significant portion (my observation was about 20-30% at Berkeley, which means it must approach 100% at some schools) of first year students in the US do not understand multiplication. They do understand how to calculate $38 \times 6$, but they don't intuitively understand that if you have $m$ rows of trees and $n$ tr... | 50 | https://mathoverflow.net/users/3077 | 27097 | 17,719 |
https://mathoverflow.net/questions/27090 | 6 | I have a stupid question about the [Metropolis-Hastings sampling algorithm](http://en.wikipedia.org/wiki/Metropolis%25E2%2580%2593Hastings_algorithm).
If I got this right, for every variable $X$ in turn, which currently has value $x\_{old}$, you generate a new sample $x\_{new}$. To do that, you draw $x\_{new}$ from ... | https://mathoverflow.net/users/6561 | Question about this ratio in Metropolis-Hastings MCMC algorithm | From what you're saying, I'm not sure if you want a proof or intuition. As the proof is written up in many places, I'll just guess that you want intuition.
Very informally: the algorithm allows you to, in effect, sample from distribution P using samples from distribution Q. So in a sense we want to take the samples f... | 3 | https://mathoverflow.net/users/1233 | 27101 | 17,721 |
https://mathoverflow.net/questions/27107 | 14 | I heard that the following problem lead to determine the rational points of an elliptic curve:
For which integers $n$ there are integers $x,y,z$ such that $x/y+y/z+z/x=n$. Could anyone show me why this question leads to the theory of elliptic curves?
| https://mathoverflow.net/users/6266 | Transforming a Diophantine equation to an elliptic curve | Nearly 10 years ago, I gave a talk at Wesleyan, and a gentleman named Roy Lisker asked me the same question: Fix an integral solution $(x, \ y, \ z)$ and make the substitution
$$u = 3 \ \frac {n^2 z - 12 \ x}z \qquad v = 108 \ \frac {2 \ x \ y - n \ x \ z + z^2}{z^2}$$
Then $(u, \ v)$ is a rational point on the ell... | 36 | https://mathoverflow.net/users/6563 | 27113 | 17,730 |
https://mathoverflow.net/questions/27106 | 4 | Given a set family, what is the best way (empirically) to check whether the set family is equivalent to set of independent sets of some matroid. The input can be either the set family explicitly or bunch of cardinality constraints that must be satisfied. For example, given a universe E, subsets $A\_1,..,A\_k$ of E and ... | https://mathoverflow.net/users/1720 | Checking whether a set family forms a matroid. | One way is to delete an element $x$ from your alleged matroid $M$, recursively check that the smaller structure $M'$ is a matroid, and then check that adding back $x$ gives you a single-element extension of $M'$. An algorithm for testing for single-element extensions is explained in [this paper by Mayhew and Royle](htt... | 4 | https://mathoverflow.net/users/3106 | 27117 | 17,732 |
https://mathoverflow.net/questions/27087 | 3 | I am trying to learn a little Lagrangian Floer theory and I was hoping someone could explain the following calculation. Consider CP^n x CP^n with the form (omega,-omega) and the diagonal Lagrangian L. Now the FH\*(L,L) is isomorphic to the quantum cohomology of CP^n as a ring. How about the higher A-infinity structure ... | https://mathoverflow.net/users/6986 | The higher structure of the Floer cochains of the diagonal in CP^ x CP^n | Here's an argument that the diagonal Lagrangian correspondence $\Delta$ in $\mathbb{C}P^n \times \mathbb{C}P^n$ is *formal*. That is, its Floer cochains $CF^\ast(\Delta,\Delta)$, as an $A\_\infty$-algebra over the rational Novikov field $\Lambda=\Lambda\_\mathbb{Q}$ (say), are quasi-isomorphic to the underlying cohomol... | 6 | https://mathoverflow.net/users/2356 | 27122 | 17,734 |
https://mathoverflow.net/questions/27118 | 3 | I'd be surprised if somebody proved the inverse Galois problem for $\mathbb{C}(x,y)$, but I wanted to make sure.
| https://mathoverflow.net/users/5309 | Is the (regular) inverse Galois problem known for the field C(x,y)? | Surely the inverse Galois problem is known over $\mathbf{C}(x)$: The Galois group of the maximal extension of $\mathbf{C}(x)$ unramified away from $n+1$ given primes of $\mathbf{C}[x]$ is the free profinite group on $n$ generators. Any finite group $G$ is a quotient of such a group, so there exists a finite Galois exte... | 13 | https://mathoverflow.net/users/1114 | 27123 | 17,735 |
https://mathoverflow.net/questions/27120 | 1 | This is the first time I post a question on MO, so I'm shy a liite bit. Can you give a "non-trivial" example of a finite dimensional hereditary algebra which is quotient of an infinite dimensional algebra ?
By "non-trivial" I mean not by killing loops in the path algebra of some quiver, for example
$k[X\_1] \times\ldo... | https://mathoverflow.net/users/6565 | Hereditary algebras as quotient algebras | If you use generators and relations, then any algebra is a quotient of an infinite-dimensional algebra, i.e., a quotient of the free associative algebra corresponding to the generators you pick.
| 2 | https://mathoverflow.net/users/321 | 27128 | 17,738 |
https://mathoverflow.net/questions/27099 | 8 | There appear to be a number of rational canonical forms. The best thing about standards is how many there are to choose from. However, the standard I choose seems to have a centralizer that is difficult to describe.
>
> 1) Is there a reference that chooses a specific (hopefully pretty) rational canonical form, and ... | https://mathoverflow.net/users/3710 | Centralizers in GL(n,p) | For a start the accepted usage for "rational canonical form" in the literature
is for a diagonal sum $C(f\_1)\oplus C(f\_2)\oplus\cdots\oplus C(f\_k)$ where
$C(f\_i)$ is the companion matrix for a monic polynomial $f$ and
$f\_1\mid f\_2\mid\cdots\mid f\_k$. That said, if I needed to find a centralizer
explcitly it isn'... | 7 | https://mathoverflow.net/users/4213 | 27136 | 17,741 |
https://mathoverflow.net/questions/27134 | 4 | In mathematics, it is common that theorems/results and problems appearing dull in one generation get revitalized and become the center of research in another one.
Sometimes conjectures that are thought to be untouchable become resolved within years.
I would like to know if there were conjectures that did not appeal... | https://mathoverflow.net/users/5627 | Less-known conjectures of significant influence and the contrary | An example of an important solution to a little-known problem might be
Frank P. Ramsey's "On a problem of formal logic" in Proc. London Math.
Soc. 30 (1930) 264-286. The problem was in logic and not well-known even
to logicians, but Ramsey's solution was taken up by combinatorialists
(notably Erdős and Szekeres) and i... | 12 | https://mathoverflow.net/users/1587 | 27139 | 17,742 |
https://mathoverflow.net/questions/27131 | 1 | A friend of mine and I were trying to answer a question related to his research and he couldn't remember whether or not the special linear group over the complex numbers, SLn(C),was simply connected. (It IS,of course.)
This got me wondering:What are all the simply connected topological subgroups of the general linea... | https://mathoverflow.net/users/3546 | The Simply Connected Subgroups of GLn(C)? | There can be no such classification except for small n because that would imply the classification of nilpotent Lie algebras up to isomorphism, which is a well-known wild problem.
By Lie-Engel's theorem, any nilpotent Lie algebra of $n$ by $n$ matrices is a direct sum of a central ideal and a Lie algebra of strictly ... | 14 | https://mathoverflow.net/users/5740 | 27142 | 17,745 |
https://mathoverflow.net/questions/27129 | 13 | Is there a classification of involutions in $\text{GL}\_n(\mathbb{Z})$?
Here's some more details about what I mean. Consider $f \in \text{GL}\_n(\mathbb{Z})$ such that $f^2=1$. Regard $f$ as an automorphism of $\mathbb{Z}^n$. Extend $f$ to an automorphism $g$ of $\mathbb{Q}^n$. Then we can write $\mathbb{Q}^n = E\_1 ... | https://mathoverflow.net/users/6567 | Involutions in GL_n(Z) | The problem is equivalent to classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C\_2$ of order 2, or $\mathbb{Z}[C\_2]$-modules on $\mathbb{Z}^n$. This group has exactly 3 isomorphism classes of indecomposable free $\mathbb{Z}$-modules:
(1) trivial
(2) sign representa... | 16 | https://mathoverflow.net/users/5740 | 27145 | 17,746 |
https://mathoverflow.net/questions/27133 | 4 | I'm studying Farb and Margalit's A primer on mapping class groups and trying to understand Wajnryb's finite presentation of Mod(S). I understand that There exists a finite presentation, but I can't understand how they got explicit Wajnryb's finite presentation. More precisely, I want to know how they knew that those 5 ... | https://mathoverflow.net/users/6569 | About the proof of Wajnryb's finite presentation of Mod(S) | It is still an open problem to find a short and simple way to derive a finite presentation for the mapping class group. The book by Farb and Margalit (in the recent preliminary version 4.00) gives a clear sketch of the known derivations, which are rather long and complicated. For the details one must consult the origin... | 13 | https://mathoverflow.net/users/23571 | 27148 | 17,749 |
https://mathoverflow.net/questions/26911 | 5 | A subcategory $\mathcal{C}$ of the category $Top$ of topological spaces is a [reflective subcategory](http://en.wikipedia.org/wiki/Reflective_subcategory) if the inclusion functor $i:\mathcal{C}\hookrightarrow Top$ has a left adjoint $R:Top\rightarrow \mathcal{C}$. In other words, for every space $X$, there is a space ... | https://mathoverflow.net/users/5801 | How do you know when a reflective subcategory of Top is quotient-reflective? | The following theorem (16.8 in Abstract and Concrete Categories, Adamek-Herrlich-Strecker) could be of use.
Let E and M be subclasses of epis and monoes respectively, closed under composition with isomorphisms.
If A is a full subcategory of an (E,M)-factorisable category B, then the following conditions are
equivalen... | 4 | https://mathoverflow.net/users/2300 | 27160 | 17,758 |
https://mathoverflow.net/questions/27144 | 91 | As you all probably know, Vladimir I. Arnold passed away yesterday. In the obituaries, I found the following statement (AFP)
>
> In 1974 the Soviet Union opposed Arnold's award of the Fields Medal, the most prestigious recognition in work in mathematics that is often compared to the Nobel Prize, making him one of t... | https://mathoverflow.net/users/4925 | Why didn't Vladimir Arnold get the Fields Medal in 1974? | Pontryagin wrote a book "Biography of Lev Semenovich Pontryagin, a mathematician, composed by himself". It is available online at <http://www.ega-math.narod.ru/LSP/book.htm>, in the original Russian. Google does a fairly good job of translation, although it refuses to translate the individual chapters completely becaus... | 31 | https://mathoverflow.net/users/1784 | 27171 | 17,764 |
https://mathoverflow.net/questions/27175 | 9 | In his answer to my question [here](https://mathoverflow.net/questions/27129/involutions-in-gl-nz), Victor Protsak quoted the following result:
Let $C\_2$ be a finite cyclic group of order $2$. Then every $\mathbb{Z}[C\_2]$ structure on $\mathbb{Z}^n$ is isomorphic to a direct sum of representations of the following ... | https://mathoverflow.net/users/6567 | Representation theory over Z | See Curtis–Reiner's textbook on the Representation Theory of Finite Groups and Associative Algebras ([MR 144979](http://www.ams.org/mathscinet-getitem?mr=144979)), Theorem 74.3, page 507, and especially the introduction starting on page 493.
The result for cyclic groups of prime order, and for order 4 was originally ... | 17 | https://mathoverflow.net/users/3710 | 27178 | 17,769 |
https://mathoverflow.net/questions/27176 | 13 | The thought process that led me to this question is that the identity
$$ \left(\prod\_i \frac1{1-x\_i}\right)\left(\prod\_i {1-x\_i}\right)=1$$
can be understood as expressing exactness of the Koszul complex.
This identity is rewritten by taking $\left(\prod\_i \frac1{1-x\_i}\right)$
as the generating function for the... | https://mathoverflow.net/users/3992 | Can the Jacobi-Trudi identity be understood as a BGG resolution? | Look at the short paper MR902299 (89a:17012) 17B10 (20C30)
Zelevinski˘ı, A.V. [Zelevinsky, Andrei] (2-AOS-CY),
Resolutions, dual pairs and character formulas. (Russian)
Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 74–75, as well as the independent work by Kaan Akin (a former student of David Buchsbaum) including M... | 13 | https://mathoverflow.net/users/4231 | 27180 | 17,771 |
https://mathoverflow.net/questions/27163 | 26 | It's well known that the nilradical of a commutative ring with identity $A$ is the intersection of all the prime ideals of $A$.
Every proof I found (e.g. in the classical "Commutative Algebra" by Atiyah and Macdonald) uses Zorn's lemma to prove that $x \notin Nil(A) \Rightarrow x \notin \cap\_{\mathfrak{p}\in Spec(A)... | https://mathoverflow.net/users/6382 | Nilradicals without Zorn's lemma | Since you asked for a proof, let me complement Chris Phan's answer by outlining a proof that relies only on the [Compactness Theorem](http://en.wikipedia.org/wiki/Compactness_theorem) for propositional logic, which is yet another equivalent to the [Ultrafilter Theorem](http://en.wikipedia.org/wiki/Boolean_prime_ideal_t... | 16 | https://mathoverflow.net/users/2000 | 27187 | 17,777 |
https://mathoverflow.net/questions/23975 | 17 | I'm looking for some examples of actions on Sylow p-groups, and often those actions appear in the case of finite almost simple groups. Given a finite almost simple group, I understand in principle how to calculate its Sylow p-subgroups (here p is usually 2 or 3), but perhaps I am just too slow at doing it. In particula... | https://mathoverflow.net/users/3710 | How to find more (finite almost simple) groups with a given Sylow subgroup | If you are only interested in actions on the Sylow p-subgroup, you may not need to find all groups, but just all so-called p-fusion patterns, as formalized in the theory of fusion systems due to Puig, and developed further by Broto-Levi-Oliver and others.
See Broto, Carles; Levi, Ran; Oliver, Bob The homotopy theory ... | 10 | https://mathoverflow.net/users/6574 | 27189 | 17,779 |
https://mathoverflow.net/questions/27164 | 33 | I have just completed an introductory course on analysis, and have been looking over my notes for the year. For me, although it was certainly not the most powerful or important theorem which we covered, the most striking application was the [Fourier analytic proof of the isoperimetric inequality](https://mathoverflow.n... | https://mathoverflow.net/users/1106 | Why is Fourier analysis so handy for proving the isoperimetric inequality? | Experience with Fourier analysis and representation theory has shown that every time a problem is invariant with respect to a group symmetry, the representation theory of that group is likely to be relevant. If the group is abelian, the representation theory is given by the Fourier transform on that group.
In this ca... | 61 | https://mathoverflow.net/users/766 | 27196 | 17,785 |
https://mathoverflow.net/questions/27197 | 29 | The following definitions are standard:
An affine variety $V$ in $A^n$ is a complete intersection (c.i.) if its vanishing ideal can be generated by ($n - \dim V$) polynomials in $k[X\_1,\ldots, X\_n]$. The definition can also be made for projective varieties.
$V$ is locally a complete intersection (l.c.i.) if the l... | https://mathoverflow.net/users/6576 | Local complete intersections which are not complete intersections | (To supplement Alberto's example)
If $V$ is projective, then the gap between being locally c.i and c.i is quite big. In particular, any *smooth* $V$ would be locally c.i., but they are not c.i. typically. For instance, take $V$ to be a few points in $\mathbb P^2$ would give simple examples. In higher dimensions, by G... | 33 | https://mathoverflow.net/users/2083 | 27203 | 17,790 |
https://mathoverflow.net/questions/27181 | 7 | I am currently reading Vickers' text "topology via logic" and Peter Johnstone's "stone spaces", and I understand the material in both of these texts to pertain directly to constructions in elementary topos theory (by which I do not mean 'the theory of elementary topoi). However, these things do not seem to be mentioned... | https://mathoverflow.net/users/6131 | Stone Spaces, Locales, and Topoi for the (relative) beginner | Many good books have already been mentioned; I like MacLane+Moerdijk as an introduction, and after that both books by Johnstone (in particular, Part C of the Elephant does a good job of connecting locale theory with topos theory). But I also wanted to mention Vickers' paper "Locales and Toposes as Spaces," which I thin... | 8 | https://mathoverflow.net/users/49 | 27214 | 17,795 |
https://mathoverflow.net/questions/27198 | 4 | I just want to consider the simplest case:
Let S=[0,0,0,0,1], how to derive the general formula for $Sym^k$ S?
My conjectured formula based on the results of LiE program for finite k values is:
$Sym^k(S)=\overset{\left[k/2\right]}{\underset{i=0}{\oplus}}[i,0,0,0,k-2i]$
But I have no clue how to prove it or even... | https://mathoverflow.net/users/6577 | How to calculate symmetric tensor products of SO(10) representations? | Based on comments from Eric Rowell and José Figueroa-O'Farrill, I assume that your $V$ is a half-spinor representation of $Spin\_{10}$. The key issue is that your hypothetical decomposition of $S(V)$ is *multiplicity-free*, i.e. any module that appears in it has multiplicity one. There is a beautiful theory of multipli... | 9 | https://mathoverflow.net/users/5740 | 27215 | 17,796 |
https://mathoverflow.net/questions/22934 | 36 | In his book *Metric Number Theory*, Glyn Harman mentions the following problem he attributes to Erdős:
Let $f(\alpha)$ be a bounded measurable function with period 1. Is it true that
$$\lim\_{N\rightarrow\infty} \frac{1}{\log N} \sum\_{n=1}^N \frac{1}{n}f(n\alpha) = \int\_0^1 f(x) dx$$
for almost all $\alpha$,
... | https://mathoverflow.net/users/5621 | A question of Erdős on equidistribution | The statement was shown to be false by J. Bourgain in a paper published in 1988 (Almost Sure Convergence and Bounded Entropy, [doi:10.1007/BF02765022](http://dx.doi.org/10.1007/BF02765022)). Well before either Harman's 1997 book and the 2003 paper Gjergji mentioned in the comment, which both say that it is an open prob... | 23 | https://mathoverflow.net/users/1004 | 27216 | 17,797 |
https://mathoverflow.net/questions/27229 | 26 | Long time ago there was a [question](https://mathoverflow.net/questions/21003/)
on whether a polynomial bijection $\mathbb Q^2\to\mathbb Q$ exists. Only one attempt
of answering it has been given, highly downvoted by the way. But this answer isn't obviously
unsuccessful, because the following problem (for case $n=2$) r... | https://mathoverflow.net/users/4953 | Polynomials with rational coefficients | Let $f(x)=x^3-5x/4$. Then for $x\neq y$, $f(x)=f(y)$ iff $x^2+xy+y^2=5/4$ or $(2x+y)^2+3y^2=5$. The last equation clealy have real solutions. But if there are rational solutions, then there are integers $X,Y,N$ such that $(2X+Y)^2+3Y^2=5N^2$. This shows $X,Y,N$ all divisible by $5$, ...
| 32 | https://mathoverflow.net/users/2083 | 27231 | 17,803 |
https://mathoverflow.net/questions/27244 | 17 | Call a tack the one point union of three open intervals. Can you fit an uncountable number of them on the plane? Or is only a countable number?
| https://mathoverflow.net/users/6610 | How many tacks fit in the plane? | First of all, the one-point compactification of three open intervals is not a "tack", it's a three-leaf clover. I think that you mean a one-point union of three closed intervals; of course it doesn't matter if the other three endpoints are there or not. This topological type can be called a "Y" or a "T" or a "simple tr... | 45 | https://mathoverflow.net/users/1450 | 27248 | 17,811 |
https://mathoverflow.net/questions/27235 | 1 | Let $k$ be an arbitrary field, $d\in k$, and $d$ is not a square in $k$.
Consider the binary quadratic form $f(x,y)=x^2-d y^2$
(it is the norm from $k(\sqrt{d})$ to $k$).
I am looking for a reference to a proof of the following fact:
There exist a field extension $K/k$ and a non-zero element $a\in K$
such that $f$ do... | https://mathoverflow.net/users/4149 | Non-representability by a binary quadratic form | Take $K = k((t))$ (formal Laurent series field) and apply Proposition V.2.3 of Serre's *Local Fields* to the unramified quadratic extension $Kl/K$. (Note that separability of $l/k$ is needed so that $Kl/K$ is unramified.) Then the image of the norm map consists precisely of elements of even $t$-adic valuation: in parti... | 3 | https://mathoverflow.net/users/1149 | 27256 | 17,815 |
https://mathoverflow.net/questions/27271 | 31 | There is a lot of good stuff contained in the Problems section of the *American Mathematical Monthly*. One difficulty with extracting that information, however, is that if I see an old *Monthly* problem, it is not always so easy to locate the solution. Sometimes the solution does not appear until many years later, and ... | https://mathoverflow.net/users/3106 | Is there an index for solutions to American Mathematical Monthly problems? | This is somewhat redundant with Igor Pak's P.S., but I found an index for the 1918-1950 range in *The Otto Dunkel memorial problem book*, starting on page 80 of [this pdf](https://drive.google.com/file/d/0B_pEp00B111JNjZjZjE1NGUtN2YwOC00OTUyLWE4NmYtNzEzY2VhMzk5Y2Iz/view?usp=sharing&resourcekey=0-YwwNCk5wkCydkdlbCw9vBQ)... | 22 | https://mathoverflow.net/users/1119 | 27277 | 17,832 |
https://mathoverflow.net/questions/27219 | 17 | Suppose $\mathcal{C}$ is a category with small colimits, and $G \in \mathrm{Ob}(\mathcal{C})$ is a strong generator. (This means that $f: X \to Y$ is an isomorphism iff $f\_{\*}: \mathrm{Hom}(G,X) \to \mathrm{Hom}(G,Y)$ is a bijection.)
>
> How does one prove that every object $X$ is a colimit of copies of $G$ (i.e... | https://mathoverflow.net/users/2536 | Proof that objects are colimits of generators | I think there are actually *three* possible things that you might be asking, but the answer to all of them is no. Suppose that G is a strong generator in a cocomplete category C. Then you can ask:
1. Is every object X of C the colimit of G over the canonical diagram of shape $(G\downarrow X)$? (If so, then G is calle... | 19 | https://mathoverflow.net/users/49 | 27287 | 17,838 |
https://mathoverflow.net/questions/27258 | 15 | I recently read the statement "up to conjugacy there are 4 nontrivial finite subgroups of ${\rm SL}\_2(\mathbb{Z})$." They are generated by
$$\left(\begin{array}{cc} -1&0 \\\ 0&-1\end{array}\right),
\left(\begin{array}{cc} -1&-1 \\\ 1&0\end{array}\right),
\left(\begin{array}{cc} 0&-1 \\\ 1&0\end{array}\right),
\l... | https://mathoverflow.net/users/2669 | Finite subgroups of ${\rm SL}_2(\mathbb{Z})$ (reference request) | A finite subgroup will map to a finite subgroup of $PSL\_2(\mathbb Z)$, which is a free product $Z\_2 \* Z\_3$. I believe I have been told that finite subgroups of free products of finite groups are conjugate to subgroups of the factors being free-producted together.
| 23 | https://mathoverflow.net/users/391 | 27290 | 17,841 |
https://mathoverflow.net/questions/27284 | 11 | It is widely believed that mathematicians have a uniform standard of what constitutes a correct proof. However, this standard has, at minimum, changed over time. What are some striking examples where controversies have arisen over what constitutes a correct proof?
Examples of this include:
1. The acceptability of ... | https://mathoverflow.net/users/6309 | The definition of "proof" throughout the history of mathematics | Paolo Ruffini's work on the impossibility of solving the quintic by radicals did meet a strong passive resistance. Around 1800 he proved the theorem up to a minor gap, that himself or somebody else could have fixed soon, had his book met the attention that deserved. But times were not ready for a such a revolutionary i... | 12 | https://mathoverflow.net/users/6101 | 27304 | 17,852 |
https://mathoverflow.net/questions/27318 | 5 | I have following two questions
1) Let $S$ be a compact oriented surface with (non-empty) boundary. Also assume that the Euler characteristic of $S$ is negative (Thus, $S$ is not disk or annulus). Then is it possible to put a hyperbolic metric on $S$ such that every boundary component becomes a geodesic?
2) In abov... | https://mathoverflow.net/users/5538 | Hyperbolic structure on surfaces with boundary | The answers to both questions are positive.
1) If the Euler characteristic of $S$ is negative, then you can find a pants decomposition (where some of the pants boundary components are glued together, while the others are boundary components of $S$). Then you can put hyperbolic metrics on each pair of pants, while pr... | 9 | https://mathoverflow.net/users/4961 | 27323 | 17,864 |
https://mathoverflow.net/questions/27316 | 21 | I have no intuition for field theory, so here goes. I know what the algebraic and separable closures of a field are, but I have no feeling of how different (or same!) they could be.
So, what are the differences between them (if any) for a perfect field? A finite field? A number field?
Are there geometric parallels?... | https://mathoverflow.net/users/4177 | Separable and algebraic closures? | Geometrically there is a very big difference between separable and algebraic
closures (in the only case where there is a difference at all, i.e., in positive
characteristic $p$). Technically, this comes from the fact that an algebraically
closed field $k$ has no non-trivial derivations $D$; for every $f\in k$ there is
... | 48 | https://mathoverflow.net/users/4008 | 27332 | 17,871 |
https://mathoverflow.net/questions/22536 | 8 | For a finite dimensional $k$-algebra $A$, each $A^{\otimes n}, n \geq 0$ is a $k$-algebra ($A^0 = k $). Let $T= \prod\_{n \geq 0} A^{\otimes n}$. This is a $k$-alegbra with unit $(1,1,\dots)$ and multiplication is component-wise. Let $\Delta^{(n)} : A^{\otimes n} \to T \otimes T$ be the deconcatenation map
$$ \Delta^{(... | https://mathoverflow.net/users/3318 | Hopf algebra structure on $\prod_n A^{\otimes n}$ for an algebra $A$ | You can indeed complete the tensor product and get a good comultiplication, but it's not strictly speaking a Hopf algebra. An algebraic geometer would call it an affine formal group. If you think of the infinite product $T=\prod\_n A^{\otimes n}$ as a pro-object indexed by $\mathbb{N}$ with $T(n) = \prod\_{i=0}^n A^{\o... | 9 | https://mathoverflow.net/users/4183 | 27335 | 17,873 |
https://mathoverflow.net/questions/27240 | 14 | It seems to me (at least according to books and papers on the subject I read) that the field of automated theorem proving is some sort of art or experimental empirical engineering of combining various approaches, but not a science which tries to explain WHY its methods work in various situations and to find classes of ... | https://mathoverflow.net/users/6307 | Reasons for success in automated theorem proving | You may be interested in the wonderful little book ``The Efficiency of Theorem Proving Strategies: A Comparative and Asymptotic Analysis'' by David A. Plaisted and Yunshan Zhu. I have the 2nd edition which is paperback and was quite cheap. I'll paste the (accurate) blurb:
``This book is unique in that it gives asympt... | 12 | https://mathoverflow.net/users/4915 | 27340 | 17,876 |
https://mathoverflow.net/questions/19404 | 9 | I am reviewing and documenting a software application (part of a supply chain system) which implements an approximation of a normal distribution function; the original documentation mentions the same/similar formula quoted [here](http://mathworld.wolfram.com/NormalDistributionFunction.html)
$$\phi(x) = {1\over \sqrt{... | https://mathoverflow.net/users/4927 | Approximation of a normal distribution function | This is the (divergent) asymptotic development for $
f(x)=\int\_x^{\infty} e^{-{1\over 2}t^2} \ dt
$ given by
$$f(x) \sim
e^{-{x^2\over 2}}\ (\ {1\over x}\ + \
\Sigma\_{k=1}^{\infty}\ {(-1)^k(2k-1)!\over 2^{k-1}(k-1)!}\ {1\over x^{2k+1}}\ )
$$
or
$$f(x) \sim
e^{-{x^2\over 2}}\ (\ {1\over x}\ - {1\over x^3} + {3\over ... | 7 | https://mathoverflow.net/users/6129 | 27341 | 17,877 |
https://mathoverflow.net/questions/27297 | 8 | Let's start with a picture: <https://i.stack.imgur.com/mEFct.png>
What you see here are boxes and circles inside the boxes. Each circle is connected to zero or more boxes. One box is the primary box, it's the grey one. Here is another example (the top box is the primary box): <https://i.stack.imgur.com/V0mQS.png> (I ap... | https://mathoverflow.net/users/6210 | Is this problem solvable in polynomial time? | There is a straightforward encoding of 3-SAT into this problem, which means that unless P=NP, no, there cannot be a polynomial-time algorithm that solves it. The encoding can be constructed as follows. Given a formula $\phi=\psi\_1\wedge\dots\wedge\psi\_k$ in conjunctive normal form over propositional variables $v\_1,\... | 12 | https://mathoverflow.net/users/6634 | 27342 | 17,878 |
https://mathoverflow.net/questions/27357 | 19 | When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand.
There are many convenient examples of commutative rings, both finite and infinite, and there are many convenient examples of infinite non-commutative rings. But I don't have a good collection of finite... | https://mathoverflow.net/users/4087 | What are your favorite finite non-commutative rings? | 2 families of examples that are sometimes useful to have in mind:
(1) The group ring of a non-abelian finite group over a finite commutative ring.
and
(2) the incidence algebra of a finite poset over a finite commutative ring (the ring of upper triangular matrices is a basic example of this).
Of course, both ... | 15 | https://mathoverflow.net/users/nan | 27363 | 17,889 |
https://mathoverflow.net/questions/27375 | 20 | For a group $G$, is there an interpretation of $\mathbb C[G]$ as functions over some noncommutative space?
If so, what does this space "look like"? What are its properties? How are they related to properties of $G$?
| https://mathoverflow.net/users/2837 | Geometric interpretation of group rings? | The noncommutative space defined by $C[G]$ is (by definition) the dual $\widehat{G}$ of G.
There are as many ways to make sense of this space as there are theories of noncommutative geometry.
(Edit: In particular if G is not finite you have lots of possible meanings for the group ring, depending on what kind of reg... | 26 | https://mathoverflow.net/users/582 | 27383 | 17,902 |
https://mathoverflow.net/questions/27382 | 10 | Let me begin by noting that I know quite little about category theory. So forgive me if the title is too vague, if the question is trivial, and if the question is written poorly.
Let $\mathcal{C}$ and $\mathcal{D}$ be categories. Say that a functor $T: \mathcal{C} \to \mathcal{D}$ has property X (maybe there is a rea... | https://mathoverflow.net/users/4362 | Category Theory / Topology Question | The property $X$, as you call it, is well-known. A functor with this propery is said to "reflect isomorphisms". Another example of such a functor is the geometric realization functor from simplicial sets to compactly generated Hausdorff spaces. There are all sorts of ways of building a category $D$ and a functor $T$ wi... | 6 | https://mathoverflow.net/users/4528 | 27390 | 17,907 |
https://mathoverflow.net/questions/27406 | 9 | Is there a notion of a dimension associated to free resolutions like projective and injective dimensions associated to projective and injective resolutions? My guess is that it coincides with projective dimension but couldn't find any reference talking about this. Any help or tips will be appreciated!
| https://mathoverflow.net/users/5292 | Free resolution dimension? | I'm sort of stealing the idea from t3suji, but here goes:
If the module is projective, i.e. $PD = 0$, then $FD \leq 1$.
If the module is not projective, i.e. $PD > 0$, then $PD = FD$.
The first statement is t3suji's comment. The second statement follows by first observing that if $F$ is the free module on infinit... | 11 | https://mathoverflow.net/users/5513 | 27415 | 17,923 |
https://mathoverflow.net/questions/27305 | 35 | This is an extract from [Apéry's biography](http://peccatte.karefil.com/PhiMathsTextes/Apery.html)
(which some of the people have already enjoyed in
[this answer](https://mathoverflow.net/questions/25630/major-mathematical-advances-past-age-fifty/25631#25631)).
>
> During a mathematician's dinner in
> Kingston, Ca... | https://mathoverflow.net/users/4953 | A binomial generalization of the FLT: Bombieri's Napkin Problem | Some solutions for $n=3$ can be found at <http://www.oeis.org/A010330> where there is also a reference to J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780, MR 19, 837f (but from the review it seems that paper deals with ${x\choose n}+{y\choose n}={z\choose n}+{w\choose n}$).... | 12 | https://mathoverflow.net/users/3684 | 27425 | 17,930 |
https://mathoverflow.net/questions/27423 | 3 | Let $U : \mathbb R^d \to \mathbb R^d$ be a smooth vector field, and let $F\_t : \mathbb R \times \mathbb R^d \to \mathbb R^d$ be the corresponding smooth flow, defined by the differential equation $$\tfrac{d}{dt} F\_t(x) = U(F\_t(x)).$$ Let $f : \mathbb R^d \to \mathbb R$ be an integrable function, and consider the int... | https://mathoverflow.net/users/238 | A formula for the Jacobian of a flow | This is [the Liouville formula](http://eom.springer.de/l/l059660.htm). It is explained nicely in *Ordinary Differential Equations* by Arnold.
| 4 | https://mathoverflow.net/users/5371 | 27426 | 17,931 |
https://mathoverflow.net/questions/27438 | 1 | In section 1.1 under the subtitle system of classical particles with potential, the authors claim that
"for a system of classical particles with rigid constraints, the configuration space is a Riemannian manifold X with Riemannian structure given by twice the kinetic energy."
I don't quite how the configuration spac... | https://mathoverflow.net/users/4923 | Clarification of classical field theory lecture notes by P. Deligne and D. Freed | *Configuration* space is, by definition, the position space of your particles. *Phase* space, on the other hand, is the space of pairs (position, momentum). The latter has a symplectic structure; the former has a Riemannian structure.
Regarding the relationship between kinetic energy and the Riemannian structure: You... | 8 | https://mathoverflow.net/users/83 | 27443 | 17,943 |
https://mathoverflow.net/questions/26608 | 13 | Let $\tau$ be $(-1+\sqrt{5})/2$, let $f(x)$ be $\lfloor (x+1)\tau \rfloor$, let
$s\_n$ be $\tau n (n+1) (n+2) / 6$, and
let $S\_n$ be $$\sum\_{k=0}^{n} (n−2k)f(n−k) = n f(n) + (n-2) f(n-1)
+ (n-4) f(n-2) + \dots - (n-2) f(1) - n f(0).$$
Is $(S\_n-s\_n)/(n \log n)$ bounded for $n > 1$?
(It stays between -0.35 and +0.3... | https://mathoverflow.net/users/3621 | A specific Dedekind-esque sum | I can't help thinking that the answer ought to come out of a combination of standard techniques. First, indeed, reduce to fractional parts: but with a caveat, that it would be sensible to do some manipulation first. I'm thinking about "summation by parts".
**Edit**: There is a classic reference: G. H. Hardy & J. E. ... | 3 | https://mathoverflow.net/users/6153 | 27448 | 17,947 |
https://mathoverflow.net/questions/27401 | 16 | This is an extremely elementary question but I just can't seem to get things to work out. What I am looking for is a natural definition of the quotient bundle of a subbundle $E'\subset E$ of $\mathbb R$ (say) vector bundles over a fixed base space $B$.
Every source I find on this essentially leaves the construction to ... | https://mathoverflow.net/users/5323 | Defining Quotient Bundles | (I was going to leave this as a comment but decided that it's a bit long for that)
A couple of remarks:
1. You express an aversion to Riemannian metrics because you want to be able to apply this in the *topological* category. That's fine, except for two things: firstly, *Riemannian* metrics would not be explicitly ... | 11 | https://mathoverflow.net/users/45 | 27450 | 17,948 |
https://mathoverflow.net/questions/27461 | 5 | The title sounds tautological, right? Perhaps I'm missing something completely trivial here ...
Assume $X$ is a compact totally disconnected hausdorff space. It is known that $X$ can be written as directed inverse limit of finite discrete spaces $X\_i$ with surjective transition maps (i.e. $X$ is profinite). How do y... | https://mathoverflow.net/users/2841 | profinite spaces are the pro-completion of finite sets | Let $f:X \to Z$ be a map to a finite discrete space. Note that each fiber, $f^{-1}(z)$, is both open and closed in $X$. Let $p\_i: X \to X\_i$ be the projection maps.
Fix some $z \in Z$. Since $f^{-1}(z)$ is open, and the fibers of the maps are a basis, there is an open cover of $f^{-1}(z)$ by sets of the form $p\_i... | 7 | https://mathoverflow.net/users/297 | 27466 | 17,958 |
https://mathoverflow.net/questions/26831 | 4 | Hi everybody,
I am dealing with the O' Nan Scott theorem, the classification of finite primitive groups (I am reading "Classes of Finite Groups", by Adolfo Ballester-Bolinches and Luis M. Ezquerro). Here for "primitive group" I mean a finite group $G$ endowed with a core-free maximal subgroup $U$. I say that a primit... | https://mathoverflow.net/users/5710 | Monolithic primitive groups without diagonals | You have left out a possbility for a monolithic action. It is possible to have $U\cap N=1$. This can occur when either S is abelian (here G is a subgroup of AGL(d,p) in its usual action on $p^d$ points) or nonabelian.
As for your question, in the case where $S$ is nonabelian, a monolithic primitive group G is a subg... | 5 | https://mathoverflow.net/users/3214 | 27468 | 17,960 |
https://mathoverflow.net/questions/27454 | 4 | Let $G$ be a complex semisimple group, $B$ a Borel subgroup of $G$ and $X=G/B$ the flag variety of $G$. If $G$ is simply connected, then every line bundle $L$ on $X$ can be made $G$-equivariant (see the proof of Theorem 1 [here](http://www.math.harvard.edu/~lurie/papers/bwb.pdf)). Is this also true if $G$ isn't simply ... | https://mathoverflow.net/users/430 | Equivariance of vector bundles over G/B | A vector bundle does not have to be $G$-equivariant. For instance, Horrock-Mumford bundle on $P^4$ is equivariant only with respect to Heisenberg group and not full $SL\_5$. Pull it back to the flag variety of $SL\_5$ to get a bundle of rank 2, not $SL\_5$-equivariant. I am sure someone will tell you easier examples th... | 5 | https://mathoverflow.net/users/5301 | 27472 | 17,962 |
https://mathoverflow.net/questions/27361 | 36 | Here is a question in the intersection of mathematics and sociology. There is a standard way to encode a Sudoku puzzle as an integer programming problem. The problem has a 0-1-valued variable $a\_{i,j,k}$ for each triple $1 \le i,j,k \le 9$, expressing that the entry in position $(i,j)$ has value $k$. The Sudoku rules ... | https://mathoverflow.net/users/1450 | Do actual Sudoku puzzles have a unique rational solution? | I wasn't planning on answering here but since someone mentioned my paper in the long comment thread above maybe I should anyway.
When I'm solving problems by hand one of the sets of patterns I frequently use involve uniqueness: something can't happen because it would lead to a puzzle with more than one solution, but ... | 16 | https://mathoverflow.net/users/440 | 27480 | 17,968 |
https://mathoverflow.net/questions/27476 | 11 | Some background: my coauthors and I are working on a problem which deals with the exponential growth rates of certain infinite products of matrices. One of the sub-problems which arises in this project is to prove that a particular function from the unit interval to the reals, which describes the growth rates of a fami... | https://mathoverflow.net/users/1840 | Existence of a pair of matrices in SL(2,Z) satisfying certain constraints on the spectral radius | This is true: if one has two matrices $A, B \in SL\_2 C$, and one knows the three traces $tr(A),tr(B),tr(AB)$,
then this uniquely determines the matrices $A,B$ up to conjugacy *if $A$ and $B$ generate a non-elementary discrete group* (except in
a few degenerate cases which won't occur for positive matrices). See for e... | 10 | https://mathoverflow.net/users/1345 | 27488 | 17,974 |
https://mathoverflow.net/questions/27490 | 6 | Motivation
----------
The common functors from topological spaces to other categories have geometric interpretations. For example, the fundamental group is how loops behave in the space, and higher homotopy groups are how higher dimensional spheres behave (up to homotopy in both cases, of course). Even better, for ni... | https://mathoverflow.net/users/343 | Geometric interpretation of the fundamental groupoid | I'm not sure how to answer this, because it already seems pretty geometric to me. So let me
answer a slightly different question: what is the fundamental groupoid good for?
Since one knows that the fundamental group and groupoid are equivalent as categories for
path connected spaces, it's tempting to view the groupoid ... | 17 | https://mathoverflow.net/users/4144 | 27492 | 17,975 |
https://mathoverflow.net/questions/27471 | 3 | Let $L/K$ be a finite Galois extension of function fields, with Galois groups $G$. I want to look at the ramification of primes in the extension, i.e. to get $e\_p$ and $f\_p$ for a prime $p$ in the base field $K$ (since the extension is Galois, the ramification index and inertia degree are independent of the choice of... | https://mathoverflow.net/users/3849 | on the computation of decomposition groups | Henri Cohen's A Course in Computational Algebraic Number Theory contains quite a bit of information. Chapters 4.8, 6.2 and 6.3 combined result in algorithms that compute decomposition groups. Note that if you want to relate different primes you will have to first compute the galois group (6.3) and fix a presentation.
... | 2 | https://mathoverflow.net/users/2024 | 27499 | 17,980 |
https://mathoverflow.net/questions/27494 | 15 | The two definitions alluded to in the title can be found here: <http://en.wikipedia.org/wiki/Separable_sigma_algebra> (one is that the $\sigma$-algebra is countably generated, the other is pretty much the standard usage of the word separable wrt the semi-metric given by the measure). Why are they equivalent?
| https://mathoverflow.net/users/5498 | Separable sigma-algebra: equivalence of two definitions | The two notions are not equivalent. Indeed, they are not
equivalent even when one considers completing the measure
by adding all null sets with respect to any countably
generated $\sigma$-algebra. Nevertheless, the forward
implication holds.
First, let me explain the forward implication. Suppose that
$S$ is a $\sigma... | 12 | https://mathoverflow.net/users/1946 | 27502 | 17,981 |
https://mathoverflow.net/questions/27367 | 14 | The homology algebra $H\_\*( K(\mathbb{Z},2n); \mathbb{Z})$ contains a
divided polynomial algebra on a generator $x$ of dimension $2n$.
I suppose I could read through the Cartan seminar for a proof, but I'm hoping
someone knows of a nice simple argument for this fact.
| https://mathoverflow.net/users/3634 | Good reference for homology of $K(\mathbb{Z}, 2n)$? | We need to do three things: (1) show that the homology contains a polynomial algebra, (2) show that powers of the generator are sufficiently divisible, and (3) show that torsion doesn't interfere.
Let me repeat Hatcher's path through (1), since we need the particular generator to show that it is divisible. According ... | 11 | https://mathoverflow.net/users/4639 | 27506 | 17,985 |
https://mathoverflow.net/questions/27508 | 17 | Hi,
I have no idea where to look for, so I'm hoping you can give me some pointers.
I'm interested by numbers of form $p-1$ when $p$ is a prime number. Do they have a name, so that I can google them?
More precisely, I'm interested in their factors. Ok, obviously 2 is a factor, but what about the others? Are there ... | https://mathoverflow.net/users/6673 | Factors of p-1 when p is prime. | Two extremes of this problem are Fermat primes and Sophie Germain primes. If either class had infinitely many members, that would contribute to an answer of your question. There is literature about the distribution of prime factors (cf. Riesel and Knuth), but I do not know the literature regarding the restriction to nu... | 8 | https://mathoverflow.net/users/3528 | 27510 | 17,987 |
https://mathoverflow.net/questions/27481 | 4 | A balanced smooth rational curve in a calabi-Yau X is a smooth rational curve whose normal bundle is $O(-1)\oplus O(-1)$.
We usually like these curves because of their rigidity.
But, Is there any theorem that guaranty the existence of at least one such curve. For example for Quintic? Or for any other example?
| https://mathoverflow.net/users/5259 | balanced curves in Calabi-Yau 3-folds | Perhaps you already know this: but we don't even know how to show that there are finitely many rational curves of a given degree $d$ on the general quintic threefold. This was originally conjectured by Clemens. However, for low degrees (up to $d = 11$ or something close to that), the conjecture is verified in the "stro... | 5 | https://mathoverflow.net/users/397 | 27513 | 17,989 |
https://mathoverflow.net/questions/27516 | 4 | I am reading the paper:
The Hausdorff dimension of horseshoes of Diffeomorphism of surfaces, Bulletin Brazilian Mathematical Society, Ricardo Mañe,(1990). Roughly speaking the author state that, the unstable and stable foliation of horseshoe can be extended to a $C^1$ foliation on a neighborhood of the horseshoe. I can... | https://mathoverflow.net/users/2386 | Local product structure horseshoes | In page 166 of [Palis-Takens](http://books.google.com/books?id=pwydPA23KVUC&printsec=frontcover&dq=palis+takens+hyperbolicity&hl=es&cd=1#v=onepage&q&f=false) book that is also stated. In the discusion before, the outline is given (see Appendix 1).
| 2 | https://mathoverflow.net/users/5753 | 27518 | 17,993 |
https://mathoverflow.net/questions/27511 | 5 | Let $M$ be a compact Riemannian manifold with metric $g$ and let $f \in Diff(M)$.
Under what circumstances is there a natural metric $g\_f$ s.t. the associated smooth measure $\nu\_f$ is preserved by $f$, and how can such a $g\_f$ be obtained?
| https://mathoverflow.net/users/1847 | When is there a natural Riemannian metric whose measure preserves a self-diffeomorphism? | Let $\Omega$ be the standard volume on your Riemannian manifold,
and $\phi$ a smooth function on M. A quick computation shows that
$e^\phi \Omega$ is invariant by f if and only if the following cohomological equation is satisfied:
$$ \phi(f^{-1}(x))-\phi(x)=log\ Jf(x)$$
where Jf is the jacobian of f. This implies for e... | 6 | https://mathoverflow.net/users/6129 | 27520 | 17,994 |
https://mathoverflow.net/questions/21511 | 10 | ### Background
Zev Chonoles recently asked the question "[which number fields are monogenic?"](https://mathoverflow.net/questions/21267/which-number-fields-are-monogenic-and-related-questions). The answers say that for a specific number field the question is hard. So, I thought, how about looking at all of them.
##... | https://mathoverflow.net/users/2024 | Density of monogenic number fields? | A recent article of Bhargava and Shankar, "Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves" (<http://arxiv.org/abs/1006.1002>), addresses, among many other related questions, the density of monogenic cubic orders, counted by the height (slightly modified) of th... | 5 | https://mathoverflow.net/users/2024 | 27523 | 17,995 |
https://mathoverflow.net/questions/27522 | 6 | Hi,
We have a real symmetric matrix M with diagonal elements 0's, the eigenvalues and eigenvectors of M are computed.
Now we wish to change its diagonal elements arbitrarily to minimize the sum of absolute eigenvalues. Does there exist a way to find such modifications?
If we add a constraint : keep Tr(M)=0, would... | https://mathoverflow.net/users/6679 | minimize the sum of absolute eigenvalues | The sum of the absolute value of the eigenvalues is the same (since the matrix is real and symmetric) as the sum of the singular values. This sum is called the nuclear norm of the matrix. So what you are saying is that you have an affine space of matrices (a "matrix pencil") over which you would like to minimize the nu... | 9 | https://mathoverflow.net/users/5963 | 27526 | 17,998 |
https://mathoverflow.net/questions/27521 | 4 | I must be missing something trivial here.
Let $G$ be, say, a reductive Lie group (or more generally any locally compact Hausdorff unimodular topological group). A *unitary Hilbert space representation* of $G$ is a group homomorphism from $G$ to the group of unitary endomorphisms $U(H)$ of a Hilbert space $H$, with th... | https://mathoverflow.net/users/1384 | Injection between non-isomorphic irreducible Hilbert space reps? | No conditions are needed on the group $G$, or on the continuity of the representation, you do need the assumption that $i$ is continuous however. Since $i:V \to W$ is continuous (equivalent to being bounded) it has a continuous adjoint $i^\* : W \to V$ which is also $G$ equivariant, hence $i^\* i:V \to V$ is $G$ equiva... | 12 | https://mathoverflow.net/users/6460 | 27528 | 18,000 |
https://mathoverflow.net/questions/27519 | 5 | Hello **MathOverflow**, my first question,
*I apologize for the LaTeX, it works in preview but not in Safari once posted.*
There are many methods out there that generate a list of unique unit fractions that sum to some rational number. One of the simplest is called the "Splitting Algorithm" which uses the identity ... | https://mathoverflow.net/users/1150 | other examples of composition of functions | Computing a continued fraction representation for a real number x can be seen as a repeated application of two functions. Starting with some real number x in [0,1[, apply 1/x, then x+1 the correct amount of time to come back in [0,1[, then 1/x again and so on.
The theory of fuchsian groups makes use of these codings ... | 6 | https://mathoverflow.net/users/6129 | 27533 | 18,003 |
https://mathoverflow.net/questions/27536 | 10 | Recently I noticed an intriguing talk by Kollár at the [MAGIC conference](http://www.nd.edu/%7Emagic/MAGIC%2710/). The abstract says:
>
> Title: Cohomology groups of structure sheaves
>
>
> Abstract: I will discuss the behavior of cohomology groups of the structure sheaf and of the dualizing sheaf under deformati... | https://mathoverflow.net/users/2083 | A recent talk by Kollar on cohomology of structure sheaves | I would guess, he was lecturing about this article: <http://arxiv.org/abs/0902.0648>
| 7 | https://mathoverflow.net/users/3822 | 27555 | 18,016 |
https://mathoverflow.net/questions/27572 | 24 | A problem is said to be complete for a complexity class $\mathcal{C}$ if a) it is in $\mathcal{C}$ and b) every problem in $\mathcal{C}$ is log-space reducible to it. There are natural examples of NP-complete problems (SAT), P-complete problems (circuit-value), NL-complete problems (reachability), and so on.
Papadimi... | https://mathoverflow.net/users/344 | Are there complexity classes with provably no complete problems? | The [zoo of complexity classes](http://qwiki.stanford.edu/wiki/Complexity_Zoo) extends naturally into the realm of [computability theory](http://en.wikipedia.org/wiki/Computability_theory) and beyond, to [descriptive set theory](http://en.wikipedia.org/wiki/Descriptive_set_theory), and in these higher realms there are ... | 13 | https://mathoverflow.net/users/1946 | 27575 | 18,025 |
https://mathoverflow.net/questions/27579 | 15 | Let $p \ne 2$ be a prime and $a$ the smallest positive integer that is a primitive root modulo $p$. Is $a$ necessarily a primitive root modulo $p^2$ (and hence modulo all powers of $p$)? I checked this for all $p < 3 \times 10^5$ and it seems to work, but I can't see any sound theoretical reason why it should be the ca... | https://mathoverflow.net/users/2481 | Is the smallest primitive root modulo p a primitive root modulo p^2? | It is not true in general. See <http://primes.utm.edu/curios/page.php/40487.html> for the example, 5 mod 40487^2.
| 26 | https://mathoverflow.net/users/3710 | 27581 | 18,029 |
https://mathoverflow.net/questions/27592 | 25 | This question may sound ridiculous at first sight, but let me please show you all how I arrived at the aforementioned 'identity'.
Let us begin with (one of the many) equalities established by Euler:
$$ f(x) = \frac{\sin(x)}{x} = \prod\_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\pi^2}\Big) $$
as $(a^2-b^2)=(a+b)(a-b)$, ... | https://mathoverflow.net/users/93724 | Why is $ \frac{\pi^2}{12}=\ln(2)$ not true? | You cannot split
$$\left(1-\left(\frac{x}{n}\right)^2\right)\tag{1}$$
into
$$\left(1 -\frac{x}{n}\right) \left(1 + \frac{x}{n}\right)\tag{2}$$
since the products no longer converge.
| 44 | https://mathoverflow.net/users/3983 | 27596 | 18,040 |
https://mathoverflow.net/questions/27578 | 7 | In the general case, quiver cycles are of the form of orbit closures of $GL\cdot V\_{\vec{r}}$, where $GL= \prod\_{i=0}^n GL\_{r\_i}$ is the possible changes of basis on all of the vector spaces on each of the vertices and $V\_{\vec{r}}$ is any representation of the quiver with fixed dimension vector $\vec{r}$. In the ... | https://mathoverflow.net/users/6353 | Why do non-equioriented A<sub>n</sub> quivers have singularities identical to the singularities of Schubert varieties? | The relevance of hom-controlled functors comes from Zwara's paper "Smooth morphisms of module schemes" (Theorem 1.2). The definition there is that two schemes with basepoints $(X,x)$ and $(Y,y)$ have identical singularities if there is a smooth morphism $f \colon X \to Y$ such that $f(x) = f(y)$.
Let $F$ be a hom-co... | 2 | https://mathoverflow.net/users/321 | 27604 | 18,046 |
https://mathoverflow.net/questions/27606 | 3 | Suppose $f: \mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is has continuous partial derivatives and
$$4f(x,y)=f(x+\delta,y+\delta)+f(x-\delta,y+\delta)+f(x-\delta,y-\delta) + f(x+\delta,y-\delta)$$
for all $(x,y)$ in $\mathbb{R}\times\mathbb{R}$ and all $\delta$ in $\mathbb{R}$.
I don't believe that $f$ is necessarly ... | https://mathoverflow.net/users/6700 | Harmonic Functions | Yes, the Taylor series works. Actually $C^2$ suffices for the remainder term, although my sophomore calculus book gives the proof using $C^3.$ I get
$$ 4 f(x\_0, y\_0) = 4 f(x\_0, y\_0) + \left( 2 f\_{xx}(x\_0, y\_0) + 2 f\_{yy}(x\_0, y\_0) \right) \delta^2 \; + \; o( \delta^2 ) $$
and
$$ \left( 2 f\_{xx}(x\_0, y\_0) +... | 4 | https://mathoverflow.net/users/3324 | 27608 | 18,049 |
https://mathoverflow.net/questions/27424 | 21 | This question is motivated by this recent [question](https://mathoverflow.net/questions/27406/free-resolution-dimension). Suppose $R$ is commutative, Noetherian ring and $M$ a finitely generated $R$-module. Let $FD(M)$ and $PD(M)$ be the shortest length of free and projective resolutions (with all modules f.g.) of $M$ ... | https://mathoverflow.net/users/2083 | A ring such that all projectives are stably free but not all projectives are free? | The more canonical example probably is the standard universal example for such a question. So, let $R\_n=k[x\_i,y\_i]/\sum x\_iy\_i=1$ where $k$ is any field and there are $2n$ variables. By localization one easily checks that $K\_0(R\_n)=\mathbb{Z}$ for any $n$. But the projective module given by the presentation,
$$0... | 21 | https://mathoverflow.net/users/9502 | 27611 | 18,051 |
https://mathoverflow.net/questions/27586 | 4 | I need to count the number of monomials of degree $n$ in $k$ variables, $x\_1,\ldots ,x\_k$, that contain at least one variable with a power of 1. The monomials need not include all the variables. Their powers just need to some to $n$ and they must be divisible by $x\_i$, but not $x\_i^2$, for some $i$.
| https://mathoverflow.net/users/6694 | Number of A Subset of Monomials | Another formula (almost without alternating signs) can be obtained as a variation of the comment of David Speyer. Namely, for each $S\subset\{1,\ldots,k\}$ we can consider the set of all monomials that depend precisely on all $x\_k$ with $k\in S$ and \emph{do not satisfy the property we are studying}. Such a monomial i... | 5 | https://mathoverflow.net/users/1306 | 27614 | 18,053 |
https://mathoverflow.net/questions/27618 | 4 | Hello.
Looking at a set S of N points in the plane, I call a subset B of S "legal" if the set of points contained in the convex hull of B is exactly B itself. In other words, a subset B of S in legal if there exists some polygon P such that the set of points in and on P is exactly B.
Given N, can you bound from abo... | https://mathoverflow.net/users/nan | Upper bound for the number of subsets of N points which exhaust their convex hull | Why can't you just put the points in a circle? Doesn't that make all subsets legal?
| 12 | https://mathoverflow.net/users/1459 | 27620 | 18,057 |
https://mathoverflow.net/questions/27625 | 3 | P. Erdős and Leon Alaoglu proved in [**1**] that for every $\epsilon > 0$ the inequality $\phi(\sigma(n)) < \epsilon \cdot n$ holds for every $n \in \mathbb{N}$, except for a set of density $0$.
C. L. mentioned in [**2**] that as a consequence of the previous result one can ascertain that $\displaystyle \lim\_{n \to ... | https://mathoverflow.net/users/1593 | A limit involving the totient function | Everyone knows (but no one can prove) that there are infinitely many primes $p$ such that $q=2p-1$ is also prime. $\sigma(q)=q+1=2p$, $\phi(\sigma(q))=\phi(2p)=p-1$, $\phi(\sigma(q))/q=(p-1)/(2p-1)\to1/2$ as
$q\to\infty$.
**Edit**: I don't know why it didn't occur to me to look at Guy, Unsolved Problems In Number T... | 7 | https://mathoverflow.net/users/3684 | 27628 | 18,060 |
https://mathoverflow.net/questions/3697 | 14 | Is there a 'classification' of the endofunctors F: Δ --> Δ where Δ denotes the simplex category with objects [n] and the weakly monotone maps [m] --> [n] as morphisms (Actually, I don't know if 'simplex category' is the right name)?
For instance there is a shift functor S: Δ --> Δ defined by S([n])=[n+1] on objects a... | https://mathoverflow.net/users/467 | What are the endofunctors on the simplex category? | To carry Charles' train of thought further:
By an 'interval' let us mean a finite ordered set with at least two elements; let $Int$ be the category of intervals, where a morphism is a monotone map preserving both endpoints.
The simplicial set $\Delta^1$ can be viewed as a simplicial interval. That is, this functor ... | 17 | https://mathoverflow.net/users/6666 | 27638 | 18,064 |
https://mathoverflow.net/questions/27634 | 9 | This question is motived by this recent [question](https://mathoverflow.net/questions/27424/a-ring-such-that-all-projectives-are-stably-free-but-not-all-projectives-are-free/).
$K\_{0}(R)=\mathbb{Z}$ is often used as a euphemism for saying that every finitely generated projective module is stably free; however, there... | https://mathoverflow.net/users/3613 | $K_{0}(R) =\mathbb{Z}$ but some f.g. projective not stably free? | If $R$ is a commutative ring with $K\_{0}(R)=\mathbb{Z}$, then $\mathop{\rm Spec} R$ is connected, because otherwise $R$ would split as a product, and $K\_{0}(R)$ would contain a copy of $\mathbb{Z} \oplus \mathbb{Z}$. Hence there is a well defined surjective rank homomorphism $K\_{0}(R) \to \mathbb{Z}$, which must the... | 11 | https://mathoverflow.net/users/4790 | 27645 | 18,068 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.