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 |
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https://mathoverflow.net/questions/31237 | 15 | **Background:**
Let $G$ be a linear algebraic group over an algebraically closed field $k$ and let $I \subseteq k[G]$ be the ideal of the identity element. The hyperalgebra $U(G)$ of $G$ is defined to be the subspace of the linear dual of $k[G]$ consisting of all $f$ such that $f(I^n) = 0$ for some $n > 0$; then ther... | https://mathoverflow.net/users/1528 | Hopf algebra duality and algebraic groups | In prime characteristic (or for algebraic groups rather than Lie algebras in general), the comments already posted indicate a need for caution. Jantzen's
Part I covers a lot of the ground, but he refers back at a few delicate points to Demazure-Gabriel.
Duality for general Hopf algebras is discussed in section 3.5 o... | 8 | https://mathoverflow.net/users/4231 | 31302 | 20,342 |
https://mathoverflow.net/questions/31332 | 7 | Let f(n) be a space-constructible superpolynomial function. Then BQP $\subseteq$ PSPACE $\subset$ SPACE(f(n)), so in particular, SPACE(f(n)) $\not\subseteq$ BQP. Let L be a problem such that every problem in SPACE(f(n)) is BQP-reducible to L. Then L $\notin$ BQP.
Are there any problems that have been proven to not be... | https://mathoverflow.net/users/nan | Provably intractable problems | There are few complexity class separations known which do not follow from some type of diagonalization (a complexity hierarchy theorem of some kind). I know of none for $\mathbf{BQP}$. One canonical example of a separation that doesn't seem to follow from a diagonalization argument is $\mathbf{AC}^0 \subsetneq \mathbf{... | 5 | https://mathoverflow.net/users/2618 | 31344 | 20,371 |
https://mathoverflow.net/questions/31268 | 4 |
>
> **Important Edit**: I e-mailed Jacob Lurie, and he said that the statement of condition (\*) is incorrect as printed.
>
>
> Here is the correct statement of (\*):
>
>
> For any cofibration $f:A\to B$ and any *trivial* fibration $g:X\to Y$ in $C$, the induced morphism:
>
>
> $$\operatorname{Map}(B,X)\to \ope... | https://mathoverflow.net/users/1353 | Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories | Assuming I understand the question, isn't the following a counterexample? Let $\cal C$ be sSet with the usual model structure, but with the trivial simplicial enrichment in which the simplicial set $Map(A,X)$ is the discrete (constant) set of sSet morphisms $A\to X$. So $A\otimes K$ is coproduct of $\pi\_0(K)$ copies o... | 5 | https://mathoverflow.net/users/6666 | 31346 | 20,373 |
https://mathoverflow.net/questions/31271 | 10 | Is it true that for every function $\mathbb{R} \to \mathbb{R}$ there exists a sequence of continuous functions $f\_n(x): \mathbb{R} \to \mathbb{R}$ such that for any $x \in \mathbb{R}$ $f\_n(x)$ converges to $f(x)$?
I started with characteristic function of rationals and tried to find corresponding sequence and got s... | https://mathoverflow.net/users/7079 | Approximation with continuous functions | Quick answer:
What you want to do is look up ``Baire function'' (in Wikipedia, for example). The Baire class of functions is the least collection of functions $f:\mathbb R\to\mathbb R$ that contains the continuous functions and is closed under limits.
Here is a simple way of seeing that the answer is negative: Any... | 10 | https://mathoverflow.net/users/6085 | 31371 | 20,387 |
https://mathoverflow.net/questions/31372 | 7 | Recently on glancing through Hartshorne's description
of Cartier divisors I started pondering the definition of
sheafification which led me to a question I can't answer. Neither
can I find a discussion in the standard texts.
First let's set up some notation.
Let $\mathcal{F}$ be a presheaf (let's say of sets)
on the ... | https://mathoverflow.net/users/4213 | Describing global sections of sheafifications | The following is a counterexample for $\mathcal{F}$ a presheaf of abelian groups (or sets, if you like).
Let $X=\lbrace a,b,c,d\rbrace$ with nontrival opens given by $\lbrace a \rbrace,\lbrace b \rbrace,U=\lbrace a,b,c \rbrace,V=\lbrace a,b,d \rbrace, U\cap V$.
Define the presheaf $\mathcal{F}$ by
$\mathcal{F}(... | 7 | https://mathoverflow.net/users/5513 | 31386 | 20,398 |
https://mathoverflow.net/questions/31391 | 4 | The classical problem regarding the action of symplectic group on its Lie algebra gives rise to the following question in the finite field case.
Let $\mathbb F\_p$ be a finite field. Then the symplectic group over $\mathbb F\_p$ acts by conjugation on the set of matrices over $\mathbb F\_p$ that satisfy $\Omega A + A... | https://mathoverflow.net/users/7386 | Orbits of a symplectic group on its Lie algebra in the finite field case | This problem is answered in a paper by Burgoyne and Cushman. I don't have the reference to hand.
This also came up in [Classification of adjoint orbits for orthogonal and symplectic Lie algebras?](https://mathoverflow.net/questions/25901/)
| 6 | https://mathoverflow.net/users/3992 | 31393 | 20,403 |
https://mathoverflow.net/questions/31376 | 13 | Given a real oriented vector bundle E over the base space B of rank n, such that the Euler characteristic class in the n-th cohomology group of B vanishes, is it true that there exists a global nowhere-vanishing section of the bundle?
Any idea where to find a proof or a counterexample?
Thanks!
| https://mathoverflow.net/users/7499 | Vanishing of Euler class | Hi Dima,
I think that the answer to your question is no: as it is pointed out in the book "Differential Forms in Algebraic Topology" of R. Bott and W. Tu, cohomological invariants are too coarse to ensure the existence of geometrical objects.
More precisely, Example 23.16 of the book of Bott and Tu shows that $S^4$... | 12 | https://mathoverflow.net/users/1568 | 31394 | 20,404 |
https://mathoverflow.net/questions/31406 | 13 | Take a natural number's prime factors and list them increasingly and repeating them according to multiplicity. Concatenate their decimal (or in any base) representation to get a new number and repeat the process. Does this always end in a prime number for any input?
| https://mathoverflow.net/users/5506 | Does listing the prime factors always stop? | It's open problem, sequence [A037274](http://www.research.att.com/~njas/sequences/A037274) from OEIS, so-called "[home primes](http://mathworld.wolfram.com/HomePrime.html)". Hm, the value for n=77 is even unknown.
P.S. [On-Line Encyclopedia of Integer Sequences](http://www.research.att.com/~njas/sequences/) definitel... | 17 | https://mathoverflow.net/users/1888 | 31411 | 20,416 |
https://mathoverflow.net/questions/31414 | 0 | Let T be a two-dimensional torus and Y be the one point
compactification of a two dimensional sphere ($S^2$) minus three points.
I have to prove:
1)they have the same fundamental group
2)they are homotopically equivalent
This is what i thought: i can see Y this way, let A,B,C three
distinct points on the sphere ... | https://mathoverflow.net/users/4971 | Question about the fundamental group and homotopy equivalence | I think Y is homotopy equivalent to $S^2\vee S^1\vee S^1$. Proof: $Y$ is homotopy equivalent to the homotopy cofiber or the map from $3$ point to $S^2$. This map is null-homotopic, so $Y$ is equivalent to the wedge sum of $S^2$ with the suspension of three points. Suspension of three points is equivalent to $S^1\vee S^... | 5 | https://mathoverflow.net/users/6668 | 31415 | 20,419 |
https://mathoverflow.net/questions/27663 | 1 | Hi,
I need an inequality similar to this one for bounded domain [0,L].
<http://img94.imageshack.us/img94/3166/screenshot1qy.png>
My u(x) is not 0 on the boundary.
I will appreciate if you can help me about this question,
Edit: More precisely, is the following the statement true?
For $F\in C^1[0,L]$ with $F(0)... | https://mathoverflow.net/users/6722 | Weighted Hardy Inequality for bounded domains | I don't think what you want to prove is true just assuming $f(0) = 0$. Let $f\_\sigma(x) = x^{\sigma}$. Then $x f\_\sigma'(x) = \sigma f\_\sigma(x)$, and $f(0) = 0$. (Okay, so $f$ is not strictly in $C^1$, but you can chomp off a bit to smooth it out and take limits.) Now the left hand side is equal to $\int\_0^1 f\_\s... | 2 | https://mathoverflow.net/users/3948 | 31421 | 20,425 |
https://mathoverflow.net/questions/31427 | 2 | I'm trying to follow an argument in Lück's "Algebraische Topologie: Homologie und Mannigfaltigkeiten" (to which there apparently doesn't exist an english translation). The aim is to check homotopy invariance of cellular homology by constructing a chain homotopy.
Let me sketch the argument. Let $h\colon (X,A)\times [0... | https://mathoverflow.net/users/1291 | Proving homotopy invariance of cellular homology by constructing a chain homotopy | Your calculation is almost correct. I don't have a copy of the book you are referring to, but if it says what you claim it does, then this is a typo. This clearly must be, since the matrix in question is supposed to be a differential yet only squares to zero if the mysterious map "(-1)^n" is zero.
However your matrix... | 6 | https://mathoverflow.net/users/184 | 31438 | 20,436 |
https://mathoverflow.net/questions/31436 | 23 | What is the largest $m$ such that there exist $v\_1,\dots,v\_m \in \mathbb{R}^n$ such that for all $i$ and $j$, $1\leq i< j\leq m$, we have $v\_i \cdot v\_j < 0$.
Also, the preview screen is not displaying set braces for LaTeX. Is that just the preview, or does it mean the site wouldn't display them after the questio... | https://mathoverflow.net/users/nan | Largest number of vectors with pairwise negative dot product | You can have $m=n+1$. Take the vertices of a regular simplex
with centre at the origin.
You can't have $m=n+2$. There is at least a two-dimensional space
of vectors $(a\_1,\ldots,a\_{n+2})$ such that
$$\sum\_{i=1}^{n+2} a\_i v\_i=0.$$
This gives enough room for manoeuvre
to ensure some $a\_i>0$ and some $a\_j<0$. Thu... | 40 | https://mathoverflow.net/users/4213 | 31440 | 20,438 |
https://mathoverflow.net/questions/30527 | 18 | I am trying to understand Deligne's proof of the Ramanujan conjecture and more generally how one associates geometric objects (ultimately, motives) to modular forms.
As the first step, which I understand more or less, one identifies the space of cusp forms $S\_k(\Gamma)$ with the first cohomology group $W$ of $X(\Gam... | https://mathoverflow.net/users/2260 | Deligne's proof of Ramanujan's conjecture | Deligne's construction works as follows. He identifies the space $GL\_2(\mathbb{A})/GL\_2(\mathbb{Q})$ with the $\mathbb{C}$-points of the variety\* $\mathcal{M}\_{ell,level}$ parameterizing elliptic curves with complete level structure, or equivalently, the moduli space of elliptic curves up to isogeny with complete (... | 11 | https://mathoverflow.net/users/307 | 31462 | 20,455 |
https://mathoverflow.net/questions/31464 | 15 | In statistical mechanics, the Boltzmann distribution gives the probability of a system being in state $i$ as
$$\displaystyle \frac{e^{- \beta E\_i}}{\sum\_i e^{-\beta E\_i}}$$
where $E\_i$ is the energy of state $i$. I have generally seen this demonstrated, starting with some reasonable physical assumptions, via a ... | https://mathoverflow.net/users/290 | Can I derive the Boltzmann distribution by an invariance argument? | Like Andreas, I find a maximum entropy argument to be intellectually appealing. However,
he says the solution can be found by Lagrange multipliers and I don't know the justification for using Lagrange multipliers. That is, in the space of all probability distributions on the particles, how do you know the maximum entr... | 6 | https://mathoverflow.net/users/3272 | 31469 | 20,458 |
https://mathoverflow.net/questions/31460 | 13 | By a theorem of Ganesan, if a commutative ring not a domain has only finitely many zero-divisors, then the ring must be finite. (There are analogous results for non-commutative rings.)
There are plenty of examples of infinite rings with a finite number of nonzero nilpotents. There are also plenty of examples of infin... | https://mathoverflow.net/users/4087 | Can an infinite commutative ring have a finite (but nonzero) number of non-nilpotent zero-divisors? | No, there is no such example.
Recall that the nilradical $N$ of $R$ is the ideal of nilpotent elements. It equals the intersection of all prime ideals of $R$.
On the other hand, the set $D$ of zero-divisors of $R$ can be expressed as the union of the radicals of the annihilators of individual nonzero elements of $R... | 12 | https://mathoverflow.net/users/7399 | 31471 | 20,460 |
https://mathoverflow.net/questions/31336 | 5 | Let $F$ be a smooth foliation of a torus. Assume that $F$ can be mapped by a homeomorphism to an irrational-straight-line foliation $L$. Does it follow that $F$ can be mapped to $L$ by a diffeomorphism?
I am interested in 2 dimensional case and higher dimensional case.
| https://mathoverflow.net/users/7492 | A question about regularity of foliations | As far as I understand, the answer to the question is no, you can check this in
Handbook of Dynamical Systems, Volume 1, Part 1 By Boris Hasselblatt, Anatole Katok, page 173. This question is identical to the following -- suppose we have a diffeo of S^1 toplogically conjugate to an irrational rotation, can we make this... | 6 | https://mathoverflow.net/users/943 | 31474 | 20,463 |
https://mathoverflow.net/questions/31476 | 12 | First a little background. In racing it is possible for a player to win a tournament without winning a single race, however, how bad can a tournament winner actually be? Can a player win a tournament without even doing better than coming third? Or even fourth? Obviously this depends on the scoring method used for award... | https://mathoverflow.net/users/3121 | The Worst Possible Winner | With the "right" scoring function, it is possible that $best(\alpha\_{i}) = p-1$: Suppose our winner is next-to-last in every race, that each of the other racers is last in at least one race, and the scoring function awards $100^{p}-k+1$ (or some other large enough number) points for position $k=1,\ldots, p-1$ and $0$ ... | 14 | https://mathoverflow.net/users/5883 | 31489 | 20,473 |
https://mathoverflow.net/questions/31499 | 12 | There's a chestnut about 100 prisoners, labeled 1 through 100, and 100 boxes, each with a number 1 through 100 in it. Each prisoner, completely independently of the others, tries to find the box which has their label in it. If they all find their label, they win.
They each get to sequentially look inside 50 boxes, me... | https://mathoverflow.net/users/7508 | 100 Prisoners, 100 Boxes: Proof of Optimality | I'm not sure this is at an appropriate level for Math Overflow, but while the question is open... Yes, there is a proof that the strategy is optimal, and it's in [this paper](https://doi.org/10.1007/BF02986999):
* Eugene Curtin and Max Warshauer, *The locker puzzle*, The Mathematical Intelligencer, Volume 28, Number ... | 18 | https://mathoverflow.net/users/111 | 31501 | 20,483 |
https://mathoverflow.net/questions/31475 | 31 | This is a follow-up question to [this one](https://mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums) about eigenvalues of matrix sums. Suppose you have matrices $A$ and $B$, and know their singular values. What can you say about the singular values of $A+B$?
For Hermitian matrices and eigenvalues, this ques... | https://mathoverflow.net/users/2294 | Singular values of matrix sums | The singular values of a $n \times m$ matrix A are more or less the eigenvalues of the $n+m \times n+m$ matrix $\begin{pmatrix} 0 & A \\\ A^\* & 0 \end{pmatrix}$. By "more or less", I mean that one also has to throw in the negation of the singular values, as well as some zeroes. Using this, one can deduce inequalities ... | 28 | https://mathoverflow.net/users/766 | 31503 | 20,484 |
https://mathoverflow.net/questions/31337 | 119 | Paper A is in the literature, and has been for more than a decade.
An error is discovered in paper A and is substantial in that many
details are affected, although certain fundamental properties
claimed by the theorems are not. (As a poor analogue, it would be
like showing that certain solutions to the Navier-Stokes... | https://mathoverflow.net/users/3402 | How do I fix someone's published error? | **UPDATE 07.24** : The set of answers for this question seem to have
stabilized. I encourage all who visit this question to review all of the answers and comments
posted here and posted behind the meta.mathoverflow link in the question.
This answer has an incomplete summary; you might find what you need in one of th... | 31 | https://mathoverflow.net/users/3402 | 31516 | 20,491 |
https://mathoverflow.net/questions/31426 | 12 | Given a threefold $Y$ containing a surface $S$. Under which conditions can I contract $S$ so that I still end up with a smooth variety?
In other words
>
> what are the conditions for the
> existence of a smooth variety $X$ and
> a morphism $Y\rightarrow X$ such that
> the image of $S$ under the morphism is
> a ... | https://mathoverflow.net/users/4046 | When is a surface in a threefold contractible to a curve? | You want a divisorial contraction $Y \to X$ on a smooth 3-fold $Y$. *Extremal* divisorial contractions (i.e., contractions associated to a $K\_Y$-negative extremal ray in the Mori cone of $Y$) have been classified by S. Mori in his paper
[*3-folds whose canonical bundles are not numerically effective*](http://www.js... | 11 | https://mathoverflow.net/users/7460 | 31518 | 20,492 |
https://mathoverflow.net/questions/30769 | 5 | A quasitopological group is a group $G$ with topology such that multiplication $G\times G\rightarrow G$ is continuous in each variable (i.e. all translations are continuous) and inversion $G\rightarrow G$ is continuous. Sometimes these are called semitopological or semicontinuous groups. What (if it exists) is an examp... | https://mathoverflow.net/users/5801 | Example of a quasitopological group with discontinuous power map | Maybe, the following topology on the plane works: a base at 0 is formed by the usual neighborhoods at 0 in the plane minus a convenient subset of the diagonal, e.g. the sequence 1/3^n (and -1/3^n).
| 2 | https://mathoverflow.net/users/7516 | 31523 | 20,495 |
https://mathoverflow.net/questions/31355 | 12 | This maybe a very general question.
If we have a group given by its presentation only, what kind of properties could be proven about it?
I know examples about non-amenability of some Burnside groups.
What kind of examples are there in literature where one proves some property "just" from a presentation?
| https://mathoverflow.net/users/7307 | What can be said about a group from its presentation? | Quite a lot, I believe (although `a lot' is subjective).
A neat example of a property which can be read immediately off of a (finite) presentation $\langle X; R \rangle$ is the Deficiency of said presentation. This is defined to be $|X|-|R|$. Now, this contain some intriguing properties. For example, every group of d... | 2 | https://mathoverflow.net/users/6503 | 31534 | 20,499 |
https://mathoverflow.net/questions/31507 | 16 | Main Question:
Does ZF (no axiom of choice) prove that every Principal Ideal Domain is a Unique Factorization Domain?
The proofs I've seen all use dependent choice.
Minor Questions:
Does ZF + Countable Choice prove all PIDs are UFDs?
Does ZF prove "If all PIDs are UFDs, then [some choice principle]"?
(If an... | https://mathoverflow.net/users/nan | Does ZF prove that all PIDs are UFDs? | ZF alone does not prove that every PID is a UFD, according to [this paper](http://journals.cambridge.org/action/displayAbstract?aid=2076124): Hodges, Wilfrid. *Läuchli's algebraic closure of $Q$.* Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 2, 289--297. [MR 422022](http://www.ams.org/mathscinet-getitem?mr=422022)... | 24 | https://mathoverflow.net/users/6401 | 31557 | 20,513 |
https://mathoverflow.net/questions/31553 | 15 | For every even $n>4$ there exists nonabelian group. As example of such group we can take dihedral group.
The question is about odd $n$. For some of them there are no nonabelian groups of order $n$ (for example, if $n$ is prime then the group of order $n$ is cyclic and hence abelian).
For what odd $n$ are there know... | https://mathoverflow.net/users/7079 | Finite nonabelian groups of odd order | It's well-known that for a natural number $n$ with prime factorization
$n=\prod\_i p\_i^{r\_i}$, all groups of order $n$ are abelian if and only if
all $r\_i\le 2$ and
$\gcd(n,\Phi(n))=1$ where $\Phi(n)=\prod\_i (p\_i^{r\_i}-1)$.
(See <http://groups.google.co.uk/group/sci.math/msg/215efc43ebb659c5?hl=en>)
For other $... | 37 | https://mathoverflow.net/users/4213 | 31558 | 20,514 |
https://mathoverflow.net/questions/31551 | 10 | One way to think of a manifold is as a family of of open subsets $U\_i \subset \mathbb{R}^n$, together with distinguished subsets $V\_{ij} \subset U\_i$ and isomorphisms $\psi\_{ij}: V\_{ij} \to V\_{ji}$ that satisfy the cocycle condition. This may not be useful practically, but occasionally it might be an intuitive cr... | https://mathoverflow.net/users/344 | Can a scheme be defined by gluing open affines such that the intersections are affine? | You can, if you use a slightly more general notion of gluing. (The notion of gluing you present is "wrong", or at least simplistic, in roughly the same way that it is "wrong" to require that a basis for a topology be closed under intersections. E.g., if you do this, then the set of open balls in $\mathbb{R}^n$ for $n >... | 6 | https://mathoverflow.net/users/5094 | 31559 | 20,515 |
https://mathoverflow.net/questions/31571 | 14 | In his paper "Smooth models for elliptic threefolds" (In: The Birational Geometry of Degenerations, Progress in Mathematics, v. 29, Birkhauser, (1983), 85-133), Rick Miranda mentions in the example of section 8 (page 101-102) that it is an unfortunate fact of life that there are no small resolutions for the singularity... | https://mathoverflow.net/users/4046 | Non-existence of small resolutions for the singularity $y^2=u^2+v^2+w^3$ | A $3$-dimensional hypersurface singularity of type $$y^2=u^2+v^2+w^k$$ admits a small resolution if and only if $k$ is even.
If $k$ is odd the corresponding singularity is factorial, so there is no small resolution.
See the paper by R. Friedman [Simultaneous resolution of 3-fold double points](http://link.springer.c... | 10 | https://mathoverflow.net/users/7460 | 31575 | 20,521 |
https://mathoverflow.net/questions/31576 | 1 | The category of $\mathcal{O}\_X$ modules on a scheme $X$ has enough injectives, every sheaf can be inbedded in an injective sheaf. Now if I take a quasi-coherent sheaf, is this hull again quasi-coherent, and how does one go about proving this? I did not find this fact in Hartshorne.
| https://mathoverflow.net/users/2300 | The injective hull of a quasi-coherent sheaf. | It is an exercise in Hartshorne that every quasi-coherent sheaf in a noetherian scheme can be embedded in an injective quasi-coherent sheaf (see Hartshorne, Chapter III, exercise 3.6).
EDIT As Brian points out below, this doesn't answer the question since the author is looking for an injective object in the category... | 2 | https://mathoverflow.net/users/3521 | 31579 | 20,522 |
https://mathoverflow.net/questions/31584 | 1 | In EGA IV2, Def. 3.2.4, Grothendieck defines a quasicoherent sheaf over a locally Noetherian scheme to be "*irredondant*" if it has a unique associated point. Presumeably, a module over a Noetherian ring is *irredondant* if it has a unique associated prime. However, googling gives no relevant results for "irredundant s... | https://mathoverflow.net/users/5094 | Name for a module with only one associated prime | [Wikipedia](http://en.wikipedia.org/wiki/Lasker-Noether_theorem) calls a module over a commutative
Noetherian ring with only
one associated prime a *coprimary* module. I don't recall hearing
this terminology elsewhere, but it is certainly common to call
a submodule $N$ of $M$ a *primary* submodule if $M/N$ is coprimary... | 1 | https://mathoverflow.net/users/4213 | 31586 | 20,524 |
https://mathoverflow.net/questions/31585 | 9 | On the Wikipedia page of Goldbach's conjecture, a [heuristic justification](http://en.wikipedia.org/wiki/Goldbach%27s_conjecture#Heuristic_justification) is given, which did not completely satisfy me. It roughly goes as follows:
>
> * randomly define a subset integers in accordance with the *prime number
> theorem... | https://mathoverflow.net/users/1229 | Heuristic justification for Goldbach's conjecture | I'm not even sure that your heuristic is as easy as Goldbach. On one hand it allows exceptions, but on the other it requires that only the density be used, not other properties of the primes.
I prefer to justify the conjecture by looking at the expected number of exceptions (again, using only the density of the prime... | 11 | https://mathoverflow.net/users/6043 | 31590 | 20,528 |
https://mathoverflow.net/questions/28764 | 10 | The Beal, Granville, Tijdeman-Zagier Conjecture, i.e.
If $A^x+B^y=C^z$ , where $A, B, C, x, y,z$ are positive integers and $x,
y$ and $z$ are all greater than $2$, then $A, B$ and $C$ must have a common prime
factor.
... and its associated [$1,000,000 prize](http://ns3.ams.org/bealprize.html) for proof or disproof ... | https://mathoverflow.net/users/1320 | Status of Beal, Granville, Tijdeman-Zagier Conjecture | The sci.math discussions linked to above suggest that Andrew Granville suggested the problem in 1992 and that it was discussed as early as 1985.
I have in my notes:
T-Z predates Beal; see Frits Beukers, "The Diophantine equation $Ax^p+By^q=Cz^r$", *Duke Math. J.* **91**:1 (1998), pp. 61-88.
This kind of informal ... | 6 | https://mathoverflow.net/users/6043 | 31596 | 20,532 |
https://mathoverflow.net/questions/31568 | 32 | There is a situation that comes up regularly in algebraic topology when giving proofs of facts about manifolds, like Poincare duality and the like. The typical sequence goes like this:
* Prove something for $\mathbb{R}^n$.
* Then it follows for open disks.
* Use a Mayer-Vietoris argument to prove it for finite unions... | https://mathoverflow.net/users/360 | "Affine communication" for topological manifolds | There are piecewise linear counterexamples in dimension $2$.
Arrange $2m$ evenly spaced rays $R\_i$ around the origin, $m\ge 3$. If $C$ is a convex neighborhood of the origin, let $r\_i$ be the reciprocal of the length of the portion of $R\_i$ in $C$. For some number $K>0$ (depending on $m$), convexity implies $r\_{... | 26 | https://mathoverflow.net/users/6666 | 31597 | 20,533 |
https://mathoverflow.net/questions/31594 | 0 | Does anyone know what a 3x3x3 Laplacian kernel looks like? I realize that might be an open-ended question, but I need to apply a Laplacian convolution using a 3x3x3 Laplacian kernel, and frankly I don't know what it looks like...
**edit:** and by what it "looks like" I'm hoping someone can just tell me in the form of... | https://mathoverflow.net/users/7445 | 3x3x3 Laplace Kernel? | I think (perhaps?) you are looking for the [discrete Laplacian operator](https://en.wikipedia.org/wiki/Discrete_Laplace_operator). That Wikipedia page lists the $3 \times 3 \times 3$ convolution kernels explicitly.
| 5 | https://mathoverflow.net/users/6094 | 31601 | 20,535 |
https://mathoverflow.net/questions/31603 | 165 | I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability space rather than Lebesgue measurable functions. This is so in every textbook on probability theory which I consulted. I... | https://mathoverflow.net/users/7392 | Why do probabilists take random variables to be Borel (and not Lebesgue) measurable? | One should be careful with the definitions here. Notation: Given measurable spaces $(X, \mathcal{B}\_X), (Y, \mathcal{B}\_Y)$, a measurable map $f : X \to Y$ is one such that $f^{-1}(A) \in \mathcal{B}\_X$ for $A \in \mathcal{B}\_Y$. To be explicit, I'll say $f$ is $(\mathcal{B}\_X, \mathcal{B}\_Y)$-measurable.
Let $... | 182 | https://mathoverflow.net/users/4832 | 31609 | 20,539 |
https://mathoverflow.net/questions/31607 | 4 | The discussion at [Decision problem restricted to inputs that satisfy some necessary condition.](https://mathoverflow.net/questions/31577) got me thinking about specific promises on a graph that would reduce the complexity of the coloring problem from NP-complete to some (presumably) tighter class; in particular, are t... | https://mathoverflow.net/users/7092 | Is there a promise version of 3-coloring equivalent to Graph Isomorphism? | Hi Steven,
(1) To start with the "duh" observation, you could define an artificial class, namely "those 3-coloring instances that are obtained by starting from Graph Isomorphism and then applying a standard NP-completeness reduction." That would indeed give you a subclass of 3-coloring instances that are provably pol... | 5 | https://mathoverflow.net/users/2575 | 31612 | 20,542 |
https://mathoverflow.net/questions/31338 | 6 | Let $A$ and $B$ be two Young tableaux, i. e. Young diagrams filled with the numbers $1$, $2$, ..., $n$ for some $n$ (not necessarily the same $n$). (They need not be semistandard.)
**(a)** ([Etingof's *Lectures on Representation Theory*](http://www-math.mit.edu/~etingof/replect.pdf), proof of Lemma 4.40): If $A$ and ... | https://mathoverflow.net/users/2530 | What is the most general "two in one row for A & in one column for B" theorem? | I can't give you your desired "most general" theorem, but I can say a little about this. In (b), the condition "shape(A) is lexicographically larger than shape(B)" is much stronger than it needs to be: "shape(A) is not dominated by shape(B)" will yield the same conclusion (recall the dominance order on partitions: $\la... | 8 | https://mathoverflow.net/users/6771 | 31615 | 20,545 |
https://mathoverflow.net/questions/31621 | 0 | What is the rank of $A^{n}$ if A is the zero ring? It's clearly not $n$ as many careless authors claim, since it's not even invariant. I don't think it's 0 either because it does have a linearly independent element(0, the only element).
| https://mathoverflow.net/users/5292 | Useless question on rank | As far as I'm concerned, "free $A$-module of rank $n$" means "$A$-module isomorphic to $A^n$. You just have to remember that a free module of rank $n$ is sometimes also a free module of rank $m$ even when $n$ is different from $m$. This happens for all $m$ and $n$ in the case of the $0$ ring. It also happens for some p... | 11 | https://mathoverflow.net/users/6666 | 31624 | 20,550 |
https://mathoverflow.net/questions/31502 | 7 | This is probably a trivial question, but I don't see the answer, and I haven't found it on [Wikipedia](http://en.wikipedia.org/wiki/Cartesian_closed_category), [nLab](http://ncatlab.org/nlab/show/cartesian+closed+category), nor [MathOverflow](https://mathoverflow.net/questions/19004/is-the-category-commutative-monoids-... | https://mathoverflow.net/users/78 | Is the category of affine schemes (over a fixed field) Cartesian closed? | The existence of such an adjunction implies that $B \otimes -$ preserves limits, which doesn't seem very likely.
Here is a counterexample, though probably not the simplest one. Set $B = k[y]$ and consider the inverse limit of $k[x]/(x^{n+1})$. If we take the tensor products first, then we get $k[y][[x]]$ while if we ... | 9 | https://mathoverflow.net/users/373 | 31634 | 20,557 |
https://mathoverflow.net/questions/31622 | 1 | Ok, I understand and am convinced by the standard solution of the Monte Hall Problem, i.e. it is better to switch doors after Monte opens one, and improve one's probability of winning from 1/3 to 2/3. If I had remaining doubts, they were removed by the many computer simulations, e.g. see <http://demonstrations.wolfram.... | https://mathoverflow.net/users/7002 | Assume the standard (better to switch) solution of the Monte Hall problem. Then there's the 3-card Monte problem | The following sentence is false:
"
If the card turned over by the dealer is not the card you secretly chose mentally, then you are apparently playing the classic Monte Hall game, i.e. you should secretly mentally switch card positions to increase your prob of winning from 1/3 to 2/3.
"
In the classic Monty (note sp... | 5 | https://mathoverflow.net/users/4658 | 31636 | 20,558 |
https://mathoverflow.net/questions/31437 | -1 | Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X\*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. I am of course interested in the case that g is NOT positive definite, because this is the situation when g represent... | https://mathoverflow.net/users/7466 | The lie algebra of the orthogonal group of an arbitrary space time metric | You don't need to diagonalize. You are looking at the group of those $A$ such that $A g A^T = g$. Putting $A=1+ \epsilon B$ for some small $\epsilon$, you want $(1+\epsilon B)g(1+\epsilon B^T) = g$ or $\epsilon(Bg+g B^T) = O(\epsilon^2)$. So the Lie algebra you want is $\{ B : Bg+gB^T=0 \}$. Since $g=g^T$, this can als... | 4 | https://mathoverflow.net/users/297 | 31640 | 20,561 |
https://mathoverflow.net/questions/31629 | 19 | Many topologists express a clear preference for working with CW complexes instead of simplicial sets.
One of the reasons is that the cellular chain complex of a CW complex is often easier to work with than a simplicial chain complex. However, simplicial sets have many nice features that spaces do not. The category o... | https://mathoverflow.net/users/1353 | Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets? | I think there are many times that simplicial sets are preferable (e.g for classifying spaces the simplicial construction is often advantageous), but to answer the stated question:
* CW complexes connect more immediately to manifold theory (Morse functions give CW structures; a finite CW complex is homotopy equivalent... | 22 | https://mathoverflow.net/users/4991 | 31641 | 20,562 |
https://mathoverflow.net/questions/30874 | 34 | I want to understand the idea of the proof of the artihmetic fixed point theorem. The theorem is crucial in the proof of Gödel's first Incompletness theorem.
First some notation: We work in $NT$, the usual number theory, it has implemented all primitve recursive functions. Every term or formula $F$ has a unique Gödel... | https://mathoverflow.net/users/2841 | Arithmetic fixed point theorem | The fixed point lemma is profound because it reveals a
surprisingly deep capacity in mathematics for
self-reference: when a statement $A$ is equivalent to
$F(A)$, it effectively asserts "$F$ holds of me". How
shocking it is to find that self-reference, the stuff of
paradox and nonsense, is fundamentally embedded in our... | 43 | https://mathoverflow.net/users/1946 | 31649 | 20,566 |
https://mathoverflow.net/questions/31635 | 19 | After a previous question that I asked <https://mathoverflow.net/questions/31565/request-for-comments-about-a-claimed-simple-proof-of-flt-closed> was closed, someone suggested in the comments that I ask another question that is more suited for MO. That question is as follows:
Are there any nontrivial theorems of the ... | https://mathoverflow.net/users/7089 | Theorems which say "such and such method cannot possibly prove FLT" | For a while at the end of undergrad and beginning of graduate school I made some money correcting an enthusiastic amateur mathematician's incorrect proofs of FLT. (It was a good experience, he was an academic in another field so was professional and was willing to pay what my time was worth.) When I first read his argu... | 26 | https://mathoverflow.net/users/22 | 31654 | 20,570 |
https://mathoverflow.net/questions/31626 | 3 | In euclidean n-space, it's easy to show that given a set $S$ of radius $< r$, the $a$-neighbourhood of $S$ is a ball, for any $a \geq 2r$.
>
> **Proof**: Let $S$ be contained in $B\_r(y)$, $y \in \mathbb{R}^n$.
> Note that if $a \ge 2r$ then $ B\_r(y) \subset Nbd\_a(S)$.
> Let $z\in Nbd\_a(S) \backslash B\_r(y)$.... | https://mathoverflow.net/users/3 | When is the neighbourhood of a set a ball? | If I understand the question right, the answer is no. Make a triangulated $2$-manifold with Euclidean metrics on the simplices, such that the total angle around some vertex is very small. Let $S$ consist of two points, both of which are the same small distance $D$ from that vertex and (subject to that) as far from each... | 3 | https://mathoverflow.net/users/6666 | 31658 | 20,573 |
https://mathoverflow.net/questions/31631 | 9 | **Background**
Let $f: M \to N$ be a smooth map between smooth manifolds.
Two vector fields $X$ in $M$ and $Y$ in $N$ are said to be **$f$-related** if for all $p \in M$, $(f\_\*)\_p(X\_p) = Y\_{f(p)}$; equivalently, if for every smooth function $g: N \to \mathbb{R}$, one has
$$(Yg) \circ f = X(g \circ f).$$
One ... | https://mathoverflow.net/users/394 | Lie group actions and f-relatedness | I hate to throw cold water on the party, but surprisingly, the formula that the OP was trying to prove ($[X',Y']=[X,Y]'$) is actually *false* when $G$ acts on the left. This formula is correct if $G$ acts on the *right* on $M$; but if $G$ acts on the left, then the correct formula is $[X',Y']=-[X,Y]'$.
Here's how to... | 17 | https://mathoverflow.net/users/6751 | 31664 | 20,576 |
https://mathoverflow.net/questions/31681 | 5 | In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things as simple as possible. Is there any good reason why not to define it as $0$ at $0$?
| https://mathoverflow.net/users/5034 | Euclidean function of Euclidean domain defined at 0 | You'll find your answer and much more in the little-known paper [1] which surveys all of the dozen known ways of axiomatizing Euclidean rings (including those of Nagata and Samuel), and explores in-depth all of their logical interrelations. It's a convenient reference to have at hand when you're comparing texts which u... | 8 | https://mathoverflow.net/users/6716 | 31704 | 20,601 |
https://mathoverflow.net/questions/23331 | 1 | I have one continuous variable, Variable, and two categorical
variables, Factor1 and Factor2, each comprising two levels. What does
it mean if
1) According to a t-test, the difference in Variable between the two levels in
Factor2 are statistically significant only for a subset of Factor1
(only within one of its leve... | https://mathoverflow.net/users/5282 | Interpretations among t-test, ANOVA, Tukey HSD results? | It is very often the case that some subset of your data will come out to be statistically significant by random chance. If you are running t-tests among the levels given the level of each other factor, that's four tests. Your chance of one of those four comparisons being significant at $\alpha=0.05$ is $1 - (0.95)^4$, ... | 2 | https://mathoverflow.net/users/7056 | 31708 | 20,605 |
https://mathoverflow.net/questions/31690 | 27 | Given a smooth manifold $M$, we define the differentiable structure on $TM$ in the usual way.
I would like to know examples of two smooth manifolds which are non-diffeomorphic, but their tangent bundles are.
Which is the smallest dimension in which one can find such examples?
What if I ask the same question for $... | https://mathoverflow.net/users/47274 | Examples of non-diffeomorphic smooth manifolds with diffeomorphic tangent bundle | Here are examples of non-diffeomorphic **closed** manifolds with diffeomorphic
tangent bundles:
1. 3-dimensional lens spaces have trivial tangent bundles, which are diffeomorphic if and only if the lens spaces are homotopy equivalent, e.g. $L(7,1)$, $L(7,2)$ are not homeomorphic, but their tangent bundles are diffeom... | 43 | https://mathoverflow.net/users/1573 | 31717 | 20,610 |
https://mathoverflow.net/questions/31714 | 10 | **Question:** Is the following statement true?
>
> Let $R$ be an associative, commutative, unital ring. Let $M$ and $N$ be $R$-modules. Let $n\geq 1$. Then $Tor\_n^R(M,N)$ is torsion.
>
>
>
By " $Tor\_n^R(M,N)$ is torsion" I mean that every of its elements is a torsion element. Maybe I want to assume that $R$ ... | https://mathoverflow.net/users/1291 | Is Tor always torsion? | $Tor$ commutes with extension of scalars, hence (if $R$ is an integral domain and $K$ is its field of fractions), we have
$$
Tor\_n^R(M,N) \otimes\_R K = Tor\_n^K(M\otimes\_R K,N\otimes\_R K).
$$
The right-hand-side vanishes for $n\ge 1$, because $K$ is a field. Hence $Tor$ vanishes after tensoring with $K$, which mean... | 17 | https://mathoverflow.net/users/4428 | 31725 | 20,618 |
https://mathoverflow.net/questions/31650 | 54 | Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. Would you need to be familiar with something like the contents of Eisenbud's *Commutative Algebra: With a View Toward Algebraic Geometry*, or is less nee... | https://mathoverflow.net/users/4842 | Modern algebraic geometry vs. classical algebraic geometry | I agree with Donu Arapura's complaint about the artificial distinction between modern and classical algebraic geometry. The only distinction to me seems to be chronological: modern work was done recently, while classical work was done some time ago. However, the questions being studied are (by and large) the same.
A... | 92 | https://mathoverflow.net/users/2874 | 31730 | 20,621 |
https://mathoverflow.net/questions/31716 | 4 | Given $k,n\in\mathbf{N}$ with $n\ge k$, define the set $\mathcal{F}(k,n)$ of $(k,n)$-forests of binary rooted trees (where a $(k,n)$-forest is a collection of $k$ rooted trees, which have a totality of $n$ leaves).
My aim is to count the cardinality of $\mathcal{F}(k,n)$.
For example it is well known that $|\mathca... | https://mathoverflow.net/users/47274 | Counting $(n,k)$-forests of binary trees | The number of $(k,n)$-binary forests is the $(n-1,n-k)$ entry of [Catalan's triangle](http://mathworld.wolfram.com/CatalansTriangle.html). Thus the formula is:
$$
f\_{k,n} \:=\: \frac{\:k\:}{n}\binom{2n-k-1}{n-1}.
$$
Given this formula, you can use [Stirling's approximation](http://en.wikipedia.org/wiki/Stirling%27s_ap... | 8 | https://mathoverflow.net/users/6514 | 31735 | 20,624 |
https://mathoverflow.net/questions/31733 | 3 | Every deterministic context free grammar can be represented by a LR(1) grammar, so this question can be rephrased as: can I build an equivalent LL(k) grammar from every LR(k) grammar? Can I have an example of deterministic context free language that can not have an LL(k) grammar?
| https://mathoverflow.net/users/7562 | Can I have an LL grammar for every deterministic context free language? | I’m not an expert on this topic, but I found these [course notes](http://www.gdi.uni-bamberg.de/teaching/SS10/GdI-GTI-B/tomlect8.pdf) (including some bibliographical references) which state that the language *L* = {*xn* : *n* ∈ ℕ} ∪ {*xnyn* : *n* ∈ ℕ} has no LL(*k*) parser, while being deterministic context-free (see p... | 4 | https://mathoverflow.net/users/5304 | 31746 | 20,630 |
https://mathoverflow.net/questions/31742 | 9 | This is a follow up to my question [What is the precise relationship between groupoid language and noncommutative algebra language?](https://mathoverflow.net/questions/31248/). I will briefly review some definitions; for details, a good place to look is [Christian Blohmann, Alan Weinstein. Group-like objects in Poisson... | https://mathoverflow.net/users/78 | Is a groupoid determined by its Hopfish algebra? | I believe this is the subject of Tannakian reconstruction (as in [this question](https://mathoverflow.net/questions/3446/tannakian-formalism))? i.e. if I understand correctly the Hopfish algebra attached to a groupoid is built so that its category of modules as a tensor category is the category of vector bundles on the... | 3 | https://mathoverflow.net/users/582 | 31754 | 20,635 |
https://mathoverflow.net/questions/31694 | 2 | I consider the singular fourfold $X$ defined as follows:
$$X: \quad x\_1 x\_2 x\_3 -y\_1 y\_2=0\quad \text{in}\quad \mathbb{C}^5.$$
Its singular locus is a bouquet of three planes meeting at the origin:
$$Sing(X):\quad y\_1=y\_2=x\_1 x\_2=x\_2 x\_3=x\_1 x\_3=0.$$
How can I described the small resolution of th... | https://mathoverflow.net/users/4046 | How can I get a small resolution for the binomial fourfold $x_1 x_2 x_3- y_1 y_2=0$ in $\mathbb{C}^5$? | As Alex Woo says, this is a toric example, and hence can be solved with toric methods. Your variety is $\mathrm{Spec} \ \mathbb{C}[S]$ where $S$ is the semigroup ring generated by $(1,0,0,1)$, $(0,1,0,1)$, $(0,0,1,1)$, $(0,0,0,1)$ and $(1,1,1,2)$. (These correspond to the variables $x\_1$, $x\_2$, $x\_3$, $y\_1$ and $y... | 7 | https://mathoverflow.net/users/297 | 31761 | 20,638 |
https://mathoverflow.net/questions/31554 | 7 | I have a graph-theoretical conjecture which I think would have been studied before, but for which I cannot find anything in the literature.
Let G be a finite, simple, connected graph. Let the feedback vertex number $FVS(G)$ be the minimum number of vertices that have to be deleted from $G$ to break all cycles, so the... | https://mathoverflow.net/users/5200 | Is the feedback vertex number bounded by the maximum number of leaves in a spanning tree? | Bill Waller and I proved the stronger statement that for G a graph on n(G) > 1 vertices, the order of a largest induced linear forest is at least one plus the connected domination number. See our [preprint](http://cms.dt.uh.edu/faculty/delavinae/research/DelavinaWaller2009.pdf). A linear forest is a forest in which eac... | 6 | https://mathoverflow.net/users/7575 | 31763 | 20,640 |
https://mathoverflow.net/questions/31238 | 6 | Given two real symmetric matrices $A$ and $B$ of common square size $n$ with no strictly negative eigenvalues, can the symmetric matrix $AB+BA$ have strictly more than $n/2$ eigenvalues which are strictly negative?
The answer to this question is yes, thanks to Junkie. My random examples did not hit a counterexample s... | https://mathoverflow.net/users/4556 | A signature inequality? | This [review](http://www.ams.org/mathscinet-getitem?mr=393156) seems to imply that any symmetric real matrix $C$ with positive trace is the Jordan product $(AB+BA)/2$ of two positive definite real matrices $A,B$. If so, then the maximum number of negative eigenvalues of $(AB+BA)/2$ for $n\times n$ symmetric positive de... | 3 | https://mathoverflow.net/users/6451 | 31764 | 20,641 |
https://mathoverflow.net/questions/31713 | 1 | I'm trying to find a closed form solution to the following probability given two random values $a$ and $b$:
$P(a \mod{p} < b \mod{p}~|~a \mod{q} > b \mod{q},~p \lt q)$
Ideas?
| https://mathoverflow.net/users/7551 | Modular Inequality | I think one can hack out an expression, although it's not particularly beautiful.
I'm assuming that by "counting measure" you mean that a and b are independent and are, say, uniform in the set $\{0,1,2,...pq-1\}$ (one could replace $pq-1$ by $\text{lcm}(p,q)-1$ or whatever).
Let $g=\text{gcd}(p,q)$. Suppose we ar... | 1 | https://mathoverflow.net/users/5784 | 31766 | 20,643 |
https://mathoverflow.net/questions/31769 | 1 | Given an indefinite integral quadratic form $Q(x,y)=ax^2 +bxy + cy^2$ with $b^2-4ac=d>0$, is there an easy way to count the number of integers in $t \in (-\sqrt{d}/2, \sqrt{d}/2)$ such that there exists $(m,n)\in Z \times N$ with $Q(m,n)=t$, ie. m any integer, n a positive integer? Good upper bounds would also be accep... | https://mathoverflow.net/users/695 | Number of integers $<\sqrt{d}/2$ represented by an indefinite quadratic form | First reference is "Binary Quadratic Forms: Classical Theory and Modern Computations" by Duncan A. Buell.
From this, and from a much older book, "Introduction to the Theory of Numbers" by Leonard Eugene Dickson: every form is "equivalent" to a reduced form, that is it represents the same numbers and primitively repre... | 3 | https://mathoverflow.net/users/3324 | 31776 | 20,648 |
https://mathoverflow.net/questions/31513 | 6 | Let $H$ be a discrete hypergroup. Suppose I have a matrix $A=(A\_{x,y})$ indexed over $H$ with nonnegative entries which defines a bounded operator on $\ell^2(H)$. When does there exist $f\in\ell^1(H)$ such that $A\_{x,y}=\langle f\*\delta\_x,\delta\_y\rangle$, i.e., $A$ is the matrix of transition probabilities for a ... | https://mathoverflow.net/users/351 | When does a matrix define a convolution operator on a hypergroup? | In case of discrete groups, it requires amenability of $H$. Indeed, $H$ is amenable if and only if $f\in\ell^1(H)$ for all $f\geq0$ such that $[f(xy^{-1})]\_{x,y} \in B(\ell^2H)$. I just don't know what are hypergroups.
| 10 | https://mathoverflow.net/users/7591 | 31784 | 20,654 |
https://mathoverflow.net/questions/14356 | 17 | Bourbaki used a very very strange notation for the epsilon-calculus consisting of $\tau$s and $\blacksquare$. In fact, that box should not be filled in, but for some reason, I can't produce a \Box.
Anyway, I was wondering if someone could explain to me what the linkages back to the tau mean, what the boxes mean. Whe... | https://mathoverflow.net/users/1353 | Bourbaki's epsilon-calculus notation | Let me address the part of the question about "what the linkages back to the tau mean, what the boxes mean." The usual notation for using Hilbert's epsilon symbol is that one writes $(\varepsilon x)\phi(x)$ to mean "some (unspecified) $x$ satisfying $\phi$ (if one exists, and an arbitrary object otherwise)." If, like B... | 27 | https://mathoverflow.net/users/6794 | 31787 | 20,656 |
https://mathoverflow.net/questions/31789 | 4 | Is there any connection between statistical physics and string theory, or a statistical interpretation of string theory, perhaps? I mean, the way electromagnetic forces and thermodynamic laws are described as emerging are all describable by statistical physics formulations. String theory accounts for gravity (I don't r... | https://mathoverflow.net/users/942 | Statistical physics of string theory | What you seem to be thinking isn't really statistical physics, but effective field theory (EFT). Loosely, EFTs take a more fundamental theory, average something out, and give you a field theory that works in some domain of applicability.
Electromagnetism in matter is an EFT because it takes electromagnetism in vacuum... | 7 | https://mathoverflow.net/users/3329 | 31791 | 20,658 |
https://mathoverflow.net/questions/31538 | 62 | Over the years, I've been somewhat in the habit of asking questions in this vein to experts in the Langlands programme.
As is well known, given an algebraic number field $K$, they propose to replace the reciprocity map
$$A\_K^\\*/K^\*\rightarrow Gal(K^{ab}/K)$$ of abelian class field theory by a correspondence betw... | https://mathoverflow.net/users/1826 | Non-abelian class field theory and fundamental groups | In response to Minhyong's request, I am reposting my comments above as an answer:
As James Newton commented, if $L/K$ is unramified, then an irreducible $n$-dimensional
representation (over $\mathbb C$) of $Gal(L/K)$ will correspond, in the Langlands paradise,
to a cuspidal automorphic representation of $GL\_n(\mathb... | 29 | https://mathoverflow.net/users/2874 | 31792 | 20,659 |
https://mathoverflow.net/questions/31783 | 4 | This question is, in a way, a follow-up of [this earlier question of mine](https://mathoverflow.net/questions/30647/fibered-products-of-cyclic-groups).
**Background**
Let $A$, $B$ and $F$ be finite groups and let $\alpha: A \to F$ and $\beta: B \to F$ be surjective homomorphisms.
Let $A \times\_F B$ denote the fi... | https://mathoverflow.net/users/394 | Isomorphism type of fibered products of groups | (1) No. Let $C\_n$ denote the cyclic group of order $n$, and since I will only consider abelian groups, I will write all groups additively. There is a surjection $C\_9 \times C\_3 \to C\_3 \times C\_3$ given on generators by $(1,0) \mapsto (1,0)$ and $(0,1) \mapsto (0,1)$. Then all preimages of $(1,0) \in C\_3 \times C... | 3 | https://mathoverflow.net/users/78 | 31798 | 20,661 |
https://mathoverflow.net/questions/31797 | 14 | Consider a finite group where all elements have the same order $n$.
What could be said about such groups?
For $n=2$ it could be proved that such group is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^k$.
Could it be somehow generalized on case $n>2$?
**EDIT:** Surely the identity has order 1, so we have to exclude it.
| https://mathoverflow.net/users/7079 | Finite groups with elements of order n | This question is closely related to the [restricted Burnside problem](http://en.wikipedia.org/wiki/Restricted_Burnside_problem): given numbers $m$ and $p$, is the restricted Burnside group $B\_0(m,p)$ finite? Every group with $m$ generators of exponent $p$ is the quotient of the Burnside group $B(m,p)=F\_m/\langle w^p\... | 18 | https://mathoverflow.net/users/5740 | 31803 | 20,664 |
https://mathoverflow.net/questions/31796 | 5 | Is there any problem such that its input size is N; its output size is polynomial in terms of N, yet its running time is super-polynomial(N) on a deterministic Turing machine?
Is there any problem such that its input size is N; its output size is polynomial in terms of N, yet its running time is super-polynomial(N) o... | https://mathoverflow.net/users/3609 | EXPTime algorithms | **Short answer:** Yes and yes. For the first question, you could take any $EXPTIME$-complete problem. For the second you could take any $NEXPTIME$-complete problem.
**Long answer:**
Your first question is answered by the problem:
*Given a deterministic Turing machine M, string x, and integer k in binary, does M ... | 18 | https://mathoverflow.net/users/2618 | 31805 | 20,666 |
https://mathoverflow.net/questions/31814 | 14 | Given a binary function $f: [1..n] \times [1..n] \to [1..n]$ how to check that this operation is a group operation on $[1..n]$?
It's obvious that this can be done in $O(n^3)$ time just by checking all group properties. The most time-expensive property is associativity. Also it's clear that it could not be done faster... | https://mathoverflow.net/users/7079 | Checking whether given binary operation is a group operation | See this entry - <http://rjlipton.wordpress.com/2010/06/03/an-amplification-trick-and-stoc-2010/> for a nice discussion on the question you have posed and other related ones. The article also has links to the original papers.
| 13 | https://mathoverflow.net/users/2878 | 31815 | 20,670 |
https://mathoverflow.net/questions/31740 | 3 | This question is quite specific, but it may admit answers in more general contexts.
Consider a subset $\Lambda \subset D^2$ where $D^2$ is the two dimensional disk.
We consider in $\Lambda$ an equivalence relation such that the equivalence class of each point is a contractible compact set.
Assume that the quoti... | https://mathoverflow.net/users/5753 | Collapsing contractible subsets of the two-disk. | I think you may find the *Bing shrinking criterion* useful.
First, assume $\Lambda$ itself be closed (hence compact) in $D$. More generally, equivalence classes can form a so-called upper semi-continuous decomposition of your compact initial space $X$, namely one such that $X/\sim$ is Hausdorff (necessary anyway), m... | 3 | https://mathoverflow.net/users/6451 | 31845 | 20,688 |
https://mathoverflow.net/questions/31849 | 14 | I am interested to what extent the famous identity
$$
\int\_a^b f'(x) \ dx=f(b)-f(a)
$$
is true for a function $f:[a,b]\to \mathbb C$ continuous on $[a,b]$ and differentiable on $(a,b)$. One famous easy case of this problem is where $f'$ is continuous. In the above identity, the integral is with respect to Lebesgue mea... | https://mathoverflow.net/users/3484 | The Fundamental Theorem of Calculus in Lebesgue Theory | See [this Wikipedia article](http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Generalizations).
Your "famous identity" may not be quite what you want it to be; the usual way of stating the FTC is to let $$F(x)=\int\_a^x f ~dx$$ for integrable $f$. Then $F'(x)=f(x)$. This is subtly different from what you ... | 12 | https://mathoverflow.net/users/6950 | 31850 | 20,690 |
https://mathoverflow.net/questions/31838 | 0 | There are many concepts in mathematics that begin with the German word "eigen": eigenvector, eigenvalue, eigenspace, eigenstate, eigenfunction, eigensystem etc. (to name just the most important (?) ones)
**My question:**
What is the most helpful intuition to get a feeling for what this "eigen" really means (in its... | https://mathoverflow.net/users/1047 | Intuitions/connections/examples for "eigen-*" | When you see the word **eigen**, replace it with the term **spectrum of an operator** (see [spectral theory](http://en.wikipedia.org/wiki/Spectral_theory)) View the matrix as a continuous or discrete linear transform acting on a vector. Similar matrices ($B = MAM^{-1}$) represent the same transform with respect to a di... | 5 | https://mathoverflow.net/users/5372 | 31862 | 20,698 |
https://mathoverflow.net/questions/31846 | 58 |
>
> 1) Can the Riemann Hypothesis (RH) be expressed as a $\Pi\_1$ sentence?
>
>
>
More formally,
>
> 2) Is there a $\Pi\_1$ sentence which is provably equivalent to RH in PA?
>
>
>
---
### Update (July 2010):
So we have two proofs that the RH is equivalent to a $\Pi\_1$ sentence.
1. Martin Davi... | https://mathoverflow.net/users/7507 | Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence? | I don't know the best way to express RH inside PA, but the following inequality
$$\sum\_{d \mid n} d \leq H\_n + \exp(H\_n)\log(H\_n),$$
where $H\_n = 1+1/2+\cdots+1/n$ is the $n$-th harmonic number, is known to be equivalent to RH. [J. Lagarias, [An elementary problem equivalent to the Riemann hypothesis](http://www.... | 42 | https://mathoverflow.net/users/2000 | 31870 | 20,701 |
https://mathoverflow.net/questions/31868 | 5 | What is reference for complex irreducible representations of Hecke algebra of finite Coxeter groups (say generic case q =1)? I am interested in knowing its Wedderburn decomposition. So want explicit information regarding the number of irreducible representations (and parameterization, if any) with their multiplicities.... | https://mathoverflow.net/users/7386 | Representations of finite Coxeter groups | There are many relevant papers, but the most convenient book to consult is:
MR1778802 (2002k:20017) 20C15 (20C08 20F55),
Geck, Meinolf (F-LYON-GD); Pfeiffer,G¨otz (IRL-GLWY)
Characters of finite Coxeter groups and Iwahori-Hecke algebras.
London Mathematical Society Monographs. New Series, 21.
The Clarendon Press, Oxfor... | 9 | https://mathoverflow.net/users/4231 | 31873 | 20,704 |
https://mathoverflow.net/questions/31865 | 4 | In the special case of affine schemes, there is an exercise on Hartshorne saying that when Spec A is a Noetherian topological space A may not be a Noetherian ring. While it is easy to find an example for that when A has nilpotent elements, e.g. $A=k[x\_1,...,x\_i,...]/(X\_1, x\_2^2,...,x\_i^i,...).$ It is not clear to ... | https://mathoverflow.net/users/1877 | Does reduced+Noetherian space imply Noetherian scheme | The answer is no, consider $k[x,xy, xy^2, xy^3, \dots]$.
Some more details. This is basically a copy of A^2 where all the points of one axis (including the generic point of that axis) are all glued together (into the obvious maximal ideal of that ring).
EDIT: Your example may be right too, I'm not quite sure I se... | 7 | https://mathoverflow.net/users/3521 | 31876 | 20,706 |
https://mathoverflow.net/questions/31811 | 43 | For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of odd order.
I am well aware of the complexity and length of the proof. However, would it be possible to provide a rough... | https://mathoverflow.net/users/4842 | Feit-Thompson theorem: the Odd order paper | During a [discussion at the n-category theory cafe](http://golem.ph.utexas.edu/category/2009/02/last_person_standing.html#more) Stephen Harris sent me this [excellent expository article by Glauberman](http://www.mat.unb.br/~matcont/16_5.ps) which goes into a bit more depth than wikipedia.
| 25 | https://mathoverflow.net/users/22 | 31877 | 20,707 |
https://mathoverflow.net/questions/31881 | 1 | I have a dense unlabeled graph ( each vertex has got at least 4 incident edges ).
Number of vertices (V) of the graph is always a perfect square.
I want to find all the meshes of $\sqrt{v} {x} \sqrt{v}$ in it.
Are there any known algorithms to accomplish this?
Any help is appreciated.
Thanks!
| https://mathoverflow.net/users/5360 | Algorithm for finding mesh subgraphs? | There could easily be an exponential number of meshes (e.g. let your graph be complete). And it's NP-complete to find even one of them (a graph G has a Hamiltonian path if and only the Cartesian product of G with a path contains a mesh).
So, yes, there are algorithms — you can just do a brute force search over all pe... | 2 | https://mathoverflow.net/users/440 | 31888 | 20,713 |
https://mathoverflow.net/questions/31893 | 17 | Inspired by the comments to [this](https://mathoverflow.net/questions/30874/arithmetic-fixed-point-theorem) question, I wonder if someone can explain the history of the fixed point combinator (often called the Y combinator) in lambda calculus.
Where did it first appear? Was it directly inspired by the Arithmetic Fi... | https://mathoverflow.net/users/4042 | What is the history of the Y-combinator? | The paper [History of Lambda-calculus and combinatory logic](http://www.users.waitrose.com/~hindley/SomePapers_PDFs/2006CarHin,HistlamRp.pdf) by F. Cardone and J.R. Hindley is a good starting point for answering such a question, and many others (it has 38 pages of bibliography). There’s a brief account on fixed-point c... | 10 | https://mathoverflow.net/users/5304 | 31901 | 20,722 |
https://mathoverflow.net/questions/29872 | 7 | When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.
One often hears (or at least I often hear) that HH^\* corresponds to A-infinity deformations. I am wondering whether ther... | https://mathoverflow.net/users/83 | Hochschild cohomology and A-infinity deformations | Well. Even in the case of a DG (or $A\_\infty$) algebra $A$, infinitesimal (i.e. 1st order) deformations are classified by $HH^2(A,A)$. Namely, the structure maps (a-k-a Taylor components) of an $A\_\infty$-algebra, viewed as elements of the Hochschild cochain complex, do have total degree $2$.
I think that one reco... | 11 | https://mathoverflow.net/users/7031 | 31903 | 20,724 |
https://mathoverflow.net/questions/31904 | 11 | Which free abelian groups can be realized as the fundamental group of a closed 3-manifold? The only one I can come up with is $\mathbb{Z}$, which is the fundamental group of $S^1 \times S^2$. For the application I have in mind, the key case is $\mathbb{Z}^2$. Here it is easy if you allow boundary (just take $T^2 \times... | https://mathoverflow.net/users/7621 | Closed 3-manifolds with free abelian fundamental groups | (I assume all occuring 3-manifolds to be orientable and closed)
A manifold with a free abelian fundamental group cannot be a connected sum of non-trivial 3-manifolds since its fundamental group is not a free product. A prime manifold is either $S^1\times S^2$ or irreducible (Hatcher's notes on 3-manifolds, 1.4). By 3... | 16 | https://mathoverflow.net/users/2039 | 31908 | 20,727 |
https://mathoverflow.net/questions/31867 | 2 | Given the decision version of the factoring problem, is there an interactive proof system with the perfect zero knowledge property? I know there is for just the zero knowledge property, but is there without the assumption of one way functions?
perfect zero knowledge property: Let P and V be randomized algorithms of ... | https://mathoverflow.net/users/7590 | Is there an interactive proof system for factoring with the perfect zero knowledge property? | If by the decision version of the factoring problem you mean: does this number have a non-trivial factorization (i.e. is this number prime?), that's primality testing, which is in P so it's automatically in PZK.
Otherwise, I think "the decision version of the factoring problem" is ambiguous. You could mean questions... | 5 | https://mathoverflow.net/users/2294 | 31910 | 20,729 |
https://mathoverflow.net/questions/31036 | 1 | Suppose we call a knot an equivalence class of embeddings of S1 --> R3 under ambient isotopy, a knot representative a particular such embedding, and a knot diagram the "2 1/2 dimensional" shadow of such a knot representative on S2 from a particular vantage point P, i.e. the light source for the shadow is at P, and the ... | https://mathoverflow.net/users/7002 | Can you characterize the group of transformations of knot diagrams which preserve the knot embedding? | I remember attending a talk by Barbara Jablonska at Knots in Washington (2009) in which she studied a knot in a geometric, rigid fashion. As the direction of projection varies over $S^2$, she obtained interesting surfaces by looking at the locus of a particular crossing (if memory serves). Here's an abstract of the tal... | 2 | https://mathoverflow.net/users/813 | 31917 | 20,734 |
https://mathoverflow.net/questions/31912 | 2 | Is there a centered-hexagonal, triangular, square (apart from 0 and 1)?
In other words, is there a positive integer that is simultaneously
(1) a perfect square, $n^2$, $n \ge 2$,
(2) a triangular number, $\frac{m(m+1)}{2}$, $m$ an integer,
and (3) a centered-hexagonal number, $(p+1)^3 - p^3$, $p$ an integer?
... | https://mathoverflow.net/users/7625 | Centered-hexagonal triangular squares | The only solution is 1 - this was a question asked in a book by Gardner and proved by Charles Grinstead, *On a Method of Solving a Class of Diophantine Equations* , Mathematics of Computation, Vol. 32, No. 143 (Jul., 1978), pp. 936-940.
See <http://www.jstor.org/pss/2006498>.
(N.B. The hexagonal numbers you are us... | 9 | https://mathoverflow.net/users/3143 | 31923 | 20,737 |
https://mathoverflow.net/questions/31905 | 14 | Let the Chern-Simons lagrangian for a group $G$ be,
$$L= k \epsilon^{\mu \nu \rho} Tr[A\_\mu \partial \_ \nu A\_\rho + \frac{2}{3} A\_\mu A\_\nu A\_\rho]$$
Then it is claimed that on "infinitesimal" variation of the gauge field ("connection") the lagrangian changes by,
$$\delta L = k \epsilon^{\mu \nu \rho} Tr[\del... | https://mathoverflow.net/users/2678 | Some basic questions about Chern-Simons theory | The secret to understanding the Chern-Simons functional over a 3-manifold is to realise that it's a 4-dimensional functional in disguise.
If $X$ is a closed, oriented 4-manifold, and $P \to X$ a principal $SU(n)$-bundle, one has the Chern-Weil formula for the second Chern number,
$$ c\_2(P)[X] = \frac{1}{8 \pi^2} \in... | 24 | https://mathoverflow.net/users/2356 | 31926 | 20,739 |
https://mathoverflow.net/questions/31932 | 19 | The Hessian matrix $\{\partial\_i \partial\_j f \}$ of a function $f:\mathbb{R}^n \to \mathbb{R}$ depends on the coordinate system you choose. If $x\_1,\cdots,x\_n$ and $y\_1,\cdots,y\_n$ are two sets of coordinates (say, in some open neighborhood of a manifold), then $\frac{\partial f(y(x))}{\partial x\_i} = \sum\_{k}... | https://mathoverflow.net/users/1355 | Hessian as a tensor, multi-dimensional taylor series, and generalizations | No, no, no! You left out a term involving $\frac{df(y(x))}{dy}\frac{d^2y}{dx^2}$. This term vanishes at critical points -- points where $df=0$ -- so that indeed at such a point the Hessian define a tensor -- a symmetric bilinear form on the tangent space at that point -- independent of coordinates. Paying attention to ... | 23 | https://mathoverflow.net/users/6666 | 31935 | 20,743 |
https://mathoverflow.net/questions/31897 | 3 | Let $T$ be the set of pythagorean triples, that is, triples of integers (a,b,c) satisfying a2 + b2 = c2. We think of $T$ as the set of right angles triangles with integer lengths. And let $f : T \rightarrow \mathbb{Z}$ be the function $(a,b,c) \mapsto \frac{ab}{12}$ which computes the area of a triangle (divided by 6, ... | https://mathoverflow.net/users/401 | Does the function which sends a right angled triangle to its area produce infinitely many numbers having hardly any prime factors? | Presumably, there are infinitely many primes $p$, $q$, such that $p+q$ is 6 times a prime and $p-q$ is 4 times a prime (e.g., $73+5=6\times13$, $73-5=4\times17$). This ought to follow from the prime $k$-tuples conjecture. Proving it is another matter.
Slightly simpler (but still out of reach), there should be infini... | 6 | https://mathoverflow.net/users/3684 | 31939 | 20,746 |
https://mathoverflow.net/questions/31172 | 19 | In Langlands' notes "On the classification of irreducible representations of real algebraic groups", available at the Langlands Digital Archive page [here](http://sunsite.ubc.ca/DigitalMathArchive/Langlands/representation.html#classification), Langlands gives a construction which is now referred to as "the local Langla... | https://mathoverflow.net/users/1384 | Uniqueness of local Langlands correspondence for connected reductive groups over real/complex field. | Dear Kevin,
Here are some things that you know.
(1) Every non-tempered representation is a Langlands quotient of an induction of a non-tempered twist of a tempered rep'n on some Levi, and this description is canonical.
(2) Every tempered rep'n is a summand of the induction of a discrete series on some Levi.
(3)... | 12 | https://mathoverflow.net/users/2874 | 31946 | 20,751 |
https://mathoverflow.net/questions/31944 | 14 | Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?
See <http://en.wikipedia.org/wiki/Constructible_number> and <http://en.wikipedia.org/wiki/Neusis>.
| https://mathoverflow.net/users/nan | Neusis constructions | Just as straightedge and compass constructions give the numbers in the closure
of the rationals under square roots, neusis gives the closure of the rationals
under square roots *and* cube roots.
For more details, also for an alternate characterization in terms of
origami, see [this paper](http://www.math.sjsu.edu/~al... | 12 | https://mathoverflow.net/users/1587 | 31951 | 20,754 |
https://mathoverflow.net/questions/31600 | 3 | I'm wondering if the following can be true:
Let Y be a second countable space and
$\pi\_2:Y \times \mathbb{R}\rightarrow\mathbb{R}$ ($\mathbb{R}$ with its usual topology and
$\pi\_2$ the projection onto the second factor)
be a closed map: do these assumptions imply
that Y is compact? (There is no assumption $T\_0$, $T\... | https://mathoverflow.net/users/4971 | Question about closed projection | Let $(y\_n)$ be a sequence in $Y$. Let $A$ be the subset of $Y \times \mathbf{R}$ of all points $(y\_n, \frac{1}{n})$ for $n \in \mathbf{N}$, and let $B$ be its closure.
Then $\pi\_2[B]$ is closed in $\mathbf{R}$, and contains all points $\frac{1}{n}$, so it contains $0$. So for some $y \in Y$, $(y,0) \in B$. Using the... | 3 | https://mathoverflow.net/users/2060 | 31957 | 20,759 |
https://mathoverflow.net/questions/31842 | 6 | I'm trying to understand the definition of tensor product of two vector spaces. So far, I've read the one using free vector spaces and a quotient space ([here](http://en.wikipedia.org/wiki/Tensor_product#Tensor_product_of_vector_spaces)), and I think I understand it well. However, I want to understand the other definit... | https://mathoverflow.net/users/7607 | Tensor product and category theory | Just some definitions, in case you're unfamiliar with them: Let $\hat{V}$ denote the vector space of linear functions from a vector space $V$ to the scalar field. Remember, a multilinear map is one of the form $V \times V \times \cdots \times V \to W$ (with $n$ copies of $V$), where $W$ is another vector space, such th... | 5 | https://mathoverflow.net/users/1355 | 31985 | 20,776 |
https://mathoverflow.net/questions/31990 | 5 | I solved a problem in analysis and i was thinking of generalizing this question which i couldn't succeed.
If $f:\mathbb{R} \to \mathbb{R}$ is a continuous function which satisfies $f(x)=f(2x+1)$, for all $x \in \mathbb{R}$ then prove that $f$ is constant. I was able to prove it considering $g(x)=f(x-1)$ and showing ... | https://mathoverflow.net/users/1483 | Continuous functions remaining constant | I'll assume, per Pete's comment above, that you are looking for polynomials $p(x)$ with real coefficients such that $f(p(x)) = f(x)$ implies $f$ is constant.
It suffices (though I am not sure if this is necessary) then that $p(x)$ has a fixed point that is strictly unstable or strictly stable in the strict sense. In... | 3 | https://mathoverflow.net/users/3948 | 31994 | 20,782 |
https://mathoverflow.net/questions/31971 | 4 | Let $K=F\_q$ and $F=F\_{q^3}$, define the set A={$x \in F$ : $Tr\_{F/K} (x)=0$}.
Is it true that for every $x \in A$ there are $y,z \in A$ such that $x=yz$?
| https://mathoverflow.net/users/3461 | Elements of trace zero in a field extension | In characteristic $3$, yes.
Proof: Let $y=1$.
In characteristic $2$, yes. In all other cases, no.
Proof: The symmetric bilinear form $Tr(yz)$ on the vector space $F$ is nondegenerate (for any finite separable field extension). Its restriction to the codimension one subspace $A$ is also nondegenerate, since the or... | 7 | https://mathoverflow.net/users/6666 | 31999 | 20,786 |
https://mathoverflow.net/questions/32011 | 35 | There are plenty of simple proofs out there that $\sqrt{2}$ is irrational. But does there exist a proof which is not a proof by contradiction? I.e. which is **not** of the form:
Suppose $a/b=\sqrt{2}$ for integers $a,b$.
[*deduce a contradiction here*]
$\rightarrow\leftarrow$, QED
Is it impossible (or at lea... | https://mathoverflow.net/users/7647 | Direct proof of irrationality? | Below is a simple direct proof that I found as a teenager:
**THEOREM** $\;\rm r = \sqrt{n}\;$ is integral if rational, for $\;\rm n\in\mathbb{N}$.
Proof: $\;\rm r = a/b,\;\; {\text gcd}(a,b) = 1 \implies ad-bc = 1\;$ for some $\rm c,d \in \mathbb{Z}$, by Bezout
so: $\;\rm 0 = (a-br) (c+dr) = ac-bdn + r \implies r... | 53 | https://mathoverflow.net/users/6716 | 32017 | 20,797 |
https://mathoverflow.net/questions/32019 | 0 | Let G and H be affine algebraic groups defined over a field k of characteristic zero, with H a closed subgroup of G. Suppose they have the same k-points. Have they to be equal?
| https://mathoverflow.net/users/7614 | Same rational points | No: Take $k$ to be the rational numbers and $G$ to be the group of third roots of unity. Then the only rational point in $G$ is $1$. Then take $H$ to be the component of the identity. This satisfies your conditions but $H \neq G$.
The problem here is that the groups are no connected.
edit: I notice now after postin... | 0 | https://mathoverflow.net/users/5101 | 32022 | 20,799 |
https://mathoverflow.net/questions/32026 | 1 | The other day, I asked this question [3x3x3 Laplace Kernel?](https://mathoverflow.net/questions/31594/3x3x3-laplace-kernel), regarding what the 3x3x3 kernel was for applying a Laplacian convolution.
On that page, it mentions the kernels were "deduced by using discrete differential quotients." Does anyone know how th... | https://mathoverflow.net/users/7445 | How to "fill in" 3-dimensional Laplacian kernels | If you want the "5x5x5" kernel, then it is no longer the Laplacian in the usual sense.
Let me quickly describe where that kernel you saw on the Wikipedia page comes from. The Laplacian, as a differential operator, is $\sum\_i (\partial\_i)^2$. Now if we discretize the space into a grid, we can approximate the partia... | 2 | https://mathoverflow.net/users/3948 | 32037 | 20,809 |
https://mathoverflow.net/questions/32033 | 10 | This is an idle question motivated by two comments I made to a previous MO question (which I just searched for, unsuccessfully). That question asked if the characteristic function of the rationals is a pointwise limit of continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$.
My first reply was that the answer ... | https://mathoverflow.net/users/1149 | Points of continuity of Baire class one functions | This is impossible. Baire proved that if a function defined on $\mathbb R$ is of Baire class 1, then it is continuous everywhere except, possibly, for a [meagre set](http://en.wikipedia.org/wiki/Meagre_set). And by another Baire's theorem a complement of a meagre set in $\mathbb R$ is dense.
An elementary exposition ... | 13 | https://mathoverflow.net/users/5371 | 32039 | 20,811 |
https://mathoverflow.net/questions/32009 | 4 | Has the problem of factoring (over the rationals) the general trinomial $ax^n+bx^k+c$ with $a,b,c\in\mathbb{Z}$, $n,k\in\mathbb{N}, n>k>1$ been solved? By *solved* I mean a classification theorem which will either give the factors or give a certificate that it is irreducible.
One could ask the same question, but abou... | https://mathoverflow.net/users/3993 | Factoring and solving trinomials | One can of course apply general algorithms for irreducibility testing and factorization, so I presume you are asking if there is something more efficient or more explicit that can be said in the case of trinomials. Except for special cases I don't believe that is the case.
While it is known that every binomial in $\... | 9 | https://mathoverflow.net/users/6716 | 32049 | 20,821 |
https://mathoverflow.net/questions/32054 | 8 | I can imagine a map $f: X\to Y$ which is a homotopy equivalence of unpointed spaces, but which is not a homotopy equivalence of pointed spaces, no matter what basepoint is chosen. That being the case, I don't see why $f$ would have to be a weak homotopy equivalence.
More detail: by choosing $x\in X$, and its image $y... | https://mathoverflow.net/users/3634 | Are there homotopy equivalences that are not weak homotopy equivalences? | f is always a weak homotopy equivalence. This is related to the following assertion: Let $g\colon X\to X$ be homotopic to the identity. Then for any $x\_0\in X$. $g$ induces an isomorphism $g\_\*\colon\pi\_\*(X,x\_0)\to \pi\_\*(X, g(x\_0))$. This is so because $g\_\*$ is the same as conjugation by the path $H(x\_0\time... | 15 | https://mathoverflow.net/users/6668 | 32056 | 20,824 |
https://mathoverflow.net/questions/32055 | 5 | First of all, my knowledge of both GR and differential geometry is quite weak, so forgive me if the physics here doesn't make much sense.
Let $(M, g)$ be a smooth, connected Lorentzian manifold of dimension $n$. Let $f: \mathbb{R}\to M$ be a smooth curve such that the pullback of $g$ through $f$ is everywhere negativ... | https://mathoverflow.net/users/6950 | When can we factor out the time dimension? | This sounds to me like you're asking that your spacetime admit a family of [Cauchy surfaces](http://en.wikipedia.org/wiki/Cauchy_surface) (modulo annoyances like having $f$ be closed and acausal). There's a theorem of Geroch which guarantees that this is equivalent to [global hyperbolicity](http://en.wikipedia.org/wiki... | 5 | https://mathoverflow.net/users/35508 | 32059 | 20,827 |
https://mathoverflow.net/questions/32021 | 7 | I am looking for a reference talking about the complete(or not)description of t-structures in bounded derived category of $R-mod$, i.e. $D^b(R-mod)$.where $R$ is commutative ring, in particular, polynomial ring, say $C[x]$.
Thanks!
| https://mathoverflow.net/users/1851 | Classification of t-structures in derived category of R-mod? | Provided $R$ has a dualizing complex an answer is given in [this (very nice) preprint](http://arxiv.org/abs/0706.0499) of Alonso, Jeremias, and, Saorin in terms of certain filtrations on the spectrum of $R$. Corollary 6.11 is the result you are after.
| 5 | https://mathoverflow.net/users/310 | 32064 | 20,829 |
https://mathoverflow.net/questions/32050 | 10 | It is well known that for $K=\mathbb{Q}(\sqrt{D})$, $D < 0$, the non-maximal order of squarefree conductor $f$, relatively prime to $D$, has class number $$h\_K \prod\_{p|f} (p-(\frac{D}{p}))$$
What is the class number of a non-maximal order in an imaginary quadratic extension of $\mathbb{F}\_p[t]$? Is it proven usin... | https://mathoverflow.net/users/2024 | Class number of non-maximal order in imaginary quadratic function field? | There is a formula that works in all degrees, not just imaginary quadratic. In a global field $K$, let $O$ be integral over ${\mathbf Z}$ or ${\mathbf F}[t]$ (${\mathbf F}$ a finite field) and be "big", i.e., it has fraction field $K$. Let $\mathfrak c$ be the conductor ideal of $O$ in its integral closure $R$. Then
$... | 19 | https://mathoverflow.net/users/3272 | 32078 | 20,835 |
https://mathoverflow.net/questions/32070 | 0 | Since the continuum has to be a regular/singular uncountable cardinal, and since by Easton Theorem it can be anything we want it to be, what is the forcing that makes the continuum a weakly compact cardinal?
If we can do that then $2^{\aleph\_0}$ will have the tree property. But I have to find out first what is the ... | https://mathoverflow.net/users/3859 | How can I force the continuum to be weakly compact? | The continuum can't be weak compact. The continuum can't be a strong limit (basically by definition of a strong limit) and weak compact cardinals are always strong limits.
You could start with a weakly compact cardinal in the ground model and make it the continuum by your favorite way of changing the continuum, but b... | 8 | https://mathoverflow.net/users/27 | 32080 | 20,837 |
https://mathoverflow.net/questions/31949 | 4 | A result of Schreier and Ulam from their 1935 paper "Sur le nombre des g$\acute{\textrm{e}}$n$\acute{\textrm{e}}$rateurs d'un groupe topologique compact et connexe" says that if $G$ is a connected compact second countable group then the set $D=\{ (g,h)\in G^2 : \overline{\langle g,h\rangle} = G\}$ of pairs generating a... | https://mathoverflow.net/users/1243 | Why does the generic pair generate a dense subgroup of a connected compact Polish group? (cf. Schreier and Ulam) | The paper "Dense embeddings of surface groups" by Emmanuel Breuillard, Tsachik Gelander, Juan Souto, and Peter Storm (Geometry & Topology 10 (2006) 1373–1389) gives a proof in section 8.
| 1 | https://mathoverflow.net/users/1243 | 32084 | 20,839 |
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