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https://mathoverflow.net/questions/44057 | 6 | Divide the unit circle into three arcs, and let $z$ be a point in the open unit disk. Is there a simple formula for the probability that Brownian motion started at $z$ will hit one particular arc rather than the other two when it first hits the unit circle? It would also be nice to have a way of representing $z$ in ter... | https://mathoverflow.net/users/3621 | Coordinatizing the disk via Brownian motion | I think zhoraster's answer is on the right track, but it can be said in a simple geometric way:
You can think of the interior of the disk as the hyperbolic plane, using the Poincaré disk model. Brownian paths are the same in any conformally equivalent metric. It follows (by symmetry) that the hitting probability for ... | 13 | https://mathoverflow.net/users/9062 | 44130 | 28,029 |
https://mathoverflow.net/questions/44109 | 65 | In most basic courses on general topology, one studies mainly Hausdorff spaces and finds that they fit quite well with our geometric intuition and generally, things work "as they should" (sequences/nets have unique limits, compact sets are closed, etc.). Most topological spaces encountered in undergraduate studies are ... | https://mathoverflow.net/users/7392 | How should one think about non-Hausdorff topologies? | For a variety of reasons, it's often useful to develop an intuition for *finite* topological spaces. Since the only Hausdorff finite spaces are discrete, one will have to deal with the non-Hausdorff case almost all the time.
The fact of the matter is that the category of finite spaces is equivalent to the category o... | 53 | https://mathoverflow.net/users/2926 | 44135 | 28,033 |
https://mathoverflow.net/questions/44136 | 0 | I have three pairs of points in 3D space. These may or may not be coplanar. I want to find a point such that it is equidistant from each pair of points. I know that may or may not be possible depending on the positions of the points. What I want is the best average point, which I can take safely as the centre and draw ... | https://mathoverflow.net/users/10429 | What is the average center of six points in space | Distance from a pair of points instead of from each one of a pair of points does not seem to be well defined. However, you could simply take the centroid of all six points (add the coordinates and divide by 6 in each axis), then compute the distance to each of the six points, and use the maximum for the radius of the s... | 0 | https://mathoverflow.net/users/7408 | 44138 | 28,034 |
https://mathoverflow.net/questions/44120 | 1 | Consider the Lie algebra $sl\_2$
with the standard basis $(e,f,h),$ where
\begin{equation\*}\label{sl2}
[h,e]=2\,e, [h,f]=-2\,f,[e,f]=h.
\end{equation\*}
Let $V$ be finite-dimensional $sl\_2$-module and let we know that element $e$ acts on $V$ as linear operator with a matrix $E$ (in some fixed basis).
**Question... | https://mathoverflow.net/users/9645 | Representation of Lie algebra sl_2. | Edited in light of clarifications made by OP:
Given a nilpotent matrix $E$ acting on a finite dimensional vector space $V$, it is always possible to extend it to a representation of $sl\_2$ in such a way that it represents $e$. The extension is almost never unique: conjugating the representing matrices $F$ and $H$ by... | 3 | https://mathoverflow.net/users/8552 | 44143 | 28,038 |
https://mathoverflow.net/questions/44142 | 1 | Given two torsion free coherent sheaves $M$ and $N$ wit $rk(M)=rk(N)=r$ on an smooth projective surface $S$, by definition $det(M):=\Lambda^r(M)^{\\*\\*}$.
Is the following criterion correct?
$M\cong N$ $\Leftrightarrow$ $M \hookrightarrow N$ and $c\_i(M)=c\_i(N)$ for $i=0,1,2$
One only has to look at "$\Leftarro... | https://mathoverflow.net/users/3233 | Equality of chern classes and isomorphism | That's correct. A slightly shorter argument is: if $\mathcal{Q}$ has support in codimension $d$, then its Chern character $\mathrm{ch}\_d(\mathcal{Q})$ is non-zero and effective. So a sheaf is trivial if and onlfy if $\mathrm{ch} = 0$, which is true if and only if the rank and the Chern classes vanish. In particular, t... | 5 | https://mathoverflow.net/users/7437 | 44144 | 28,039 |
https://mathoverflow.net/questions/29970 | 53 | Grothendieck famously objected to the term "perverse sheaf" in *Récoltes et Semailles*, writing "What an idea to give such a name to a mathematical thing! Or to any other thing or living being, except in sternness towards a person—for it is evident that of all the ‘things’ in the universe, we humans are the only ones t... | https://mathoverflow.net/users/6950 | What is the etymology of the term "perverse sheaf"? | When MacPherson and I first started thinking about intersection homology, we realized that there was a number that measured the "badness" of a cycle with respect to a stratum. This number had the property that when you (transversally) intersected two cycles, their
"badness" would add. The best situation occurs for coc... | 145 | https://mathoverflow.net/users/10431 | 44149 | 28,044 |
https://mathoverflow.net/questions/44047 | 11 | Under which conditions localizing an additive category by some class S of morphisms yields and additive category? It seems easy to define certain addition on morphisms if we fix their representatives as zig-zags (i.e. compositions of 'old' morphisms with inverses of morphisms in S; here I use the fact that 'my' S is cl... | https://mathoverflow.net/users/2191 | Localizing an arbitrary additive category | This is a very elementary problem. To solve it, it is better not to try to understand the localization explicitely, but to work only with the universal property of the localization (there is no need for any calculus of zig-zags of any kind). You should also think of finite sums not as something defined on each family o... | 20 | https://mathoverflow.net/users/1017 | 44155 | 28,048 |
https://mathoverflow.net/questions/44075 | 2 | In Nakajima's [Geometric construction of algebras](http://www.kurims.kyoto-u.ac.jp/~nakajima/TeX/HongKong/hongkong.pdf)(pages 3-7), he uses the subalgebra of the convolution algebra of $Gr(k^N)\times Gr(k^N)$ invariant under $GL\_N$ action to construct $U\_q(sl\_2)$. To do this, he finds imposes relations to form some ... | https://mathoverflow.net/users/348 | Can we see the geometric realization of $U_q(sl_2)$'s relations as Schubert Conditions? | Well, it depends what you mean; there's certainly a close link.
Any $GL\_N$ invariant closed subset of the product of two Grassmannians can be described as a family of Schubert varieties. Consider the projection to the first factor. This must be surjective, since $GL\_n$ acts transitively on the Grassmannian. The fib... | 1 | https://mathoverflow.net/users/66 | 44156 | 28,049 |
https://mathoverflow.net/questions/44154 | 4 | Hi,
I'm trying to understand the group of cycles (modulo numerical equivalence) contracted by a flopping contraction $f$.
More precisely, I'm in the setup of Definition 2.12 of [this paper by Yukinobu Toda](http://arxiv.org/abs/0909.5129).
Let $f: X \to Y$ be a flopping contraction: $X$ is a smooth and projectiv... | https://mathoverflow.net/users/3701 | What is the Exceptional Locus of a flopping contraction between threefolds? | 1) $f$ is *isomorphic in codimension $d$* if it is an isomorphism near any codimension $d$ point in either $X$ or $Y$. Equivalently, there exists closed subsets $Z\subseteq X$ and $W\subseteq Y$ such that ${\rm codim}\_XZ\geq d+1$, ${\rm codim}\_YW\geq d+1$, and $f:X\setminus Z\overset{\simeq}{\longrightarrow} Y\setmin... | 5 | https://mathoverflow.net/users/10076 | 44161 | 28,053 |
https://mathoverflow.net/questions/43968 | 0 | Hi,
I have a problem
>
> (1) where I need to compute the ratio of probabilities of hitting and stopping at a positive vertical barrier x vs hitting and stopping at a negative horizontal barrier y after starting from (0,0).
>
>
>
I feel that by symmetry, the answer to this would be the same as
>
> (2) Th... | https://mathoverflow.net/users/10401 | Random walk question on 2D grid, probability of vertical line vs horizontal line hit | Problems 2 and 3 are equivalent by ignoring horizontal steps.
Problems 1 and 2 are not equivalent. I believe the probabilities are known, but I don't know them. However, if the probability of hitting the vertical barrier were really $y/(x+y)$ as in problem 2, then it would not be a martingale. So, whatever symmetry a... | 1 | https://mathoverflow.net/users/2954 | 44170 | 28,057 |
https://mathoverflow.net/questions/44165 | 1 | In my research, I work with certain finitely presented quotients of Coxeter groups. These are the automorphism groups of abstract polytopes, which are combinatorial generalizations of "usual" polytopes. (Essentially, an abstract polytope is an incidence complex.) Now, in this context, there is a useful combinatorial op... | https://mathoverflow.net/users/913 | Reference request: lattice operations on the class of finitely presented groups | What you are studying is the lattice of normal subgroups of the free group $F(X)$. The normal subgroup lattices of groups have been studied a lot. For example, this lattice is complete and modular. See the references [here.](http://en.wikipedia.org/wiki/Lattice_of_subgroups)
**Update.** About your newer, more concre... | 5 | https://mathoverflow.net/users/nan | 44189 | 28,070 |
https://mathoverflow.net/questions/35270 | 2 | If A is a commutative algebra and B is an X- algebra, then the tensor product $A \otimes B$ is an X-algebra (so for example, $Com \otimes Lie$ is a Lie algebra). This is seen using the language of operads. Let $Com$ be the commutative operad. Since $Com(n)$ is a one dimensional vector space for every $n$, tensoring $Co... | https://mathoverflow.net/users/8358 | Extending a property of commutative algebras to C infinity algebras | We can see that this is false by looking at a degenerate example. Consider any operad $P$ so that $P(n)=0$ for $n>1$. If you tensor such an operad with $C\_\infty$ so that $(P\otimes C\_\infty)(n)=P(n)\otimes C\_\infty(n)$ you get $P\otimes C\_\infty=P$, since $C\_\infty(1)$ is one dimensional and $P(n)=0$ for $n>1$. B... | 6 | https://mathoverflow.net/users/3075 | 44191 | 28,072 |
https://mathoverflow.net/questions/43638 | 9 | Let $\lambda$ denote a partition of size $n$. Let
$$d\_{\lambda}= \text{number of distinct parts of } \lambda $$
$$o\_{\lambda}= \text{number of odd parts of } \lambda $$
$$f\_{\lambda}= \text{number of standard Young tableau of shape } \lambda $$
Given an involution $\pi \in S\_{n}$, whose insertion tableau has shape ... | https://mathoverflow.net/users/10335 | A Distinct parts/Odd parts identity for standard Young tableaux | "all that remains to be shown is the nice fact that the total number of fixed points in all the involutions of $S\_n$ is the difference between the number of involutions in $S\_{n+1}$ and the number of involutions in $S\_n$."
And this is straightforward: every involution in $S\_n$ can be extended to an involution in... | 2 | https://mathoverflow.net/users/4658 | 44204 | 28,078 |
https://mathoverflow.net/questions/44203 | 6 | Initial caveat: the following question could probably be answered by Google, MathSciNet or my library, if I could find the right search terms or book... but I've not had any luck today, so I hope someone can point me to a reference.
(The question is related to some of my older questions concerning characters of finit... | https://mathoverflow.net/users/763 | Character table for the affine group of Z/p^nZ | The groups you are interested in are sometimes called false Tate extensions in number theorists' jargon. They are Galois groups of the Galois closures of extensions of $\mathbb{Q}$ obtained by adjoining the $p^n$-th roots of a $p$-th power free element. The irreducible representations are very explicitly described in [... | 6 | https://mathoverflow.net/users/35416 | 44205 | 28,079 |
https://mathoverflow.net/questions/44196 | 2 | Suppose that $(\Omega,\mathcal{A},\mu)$ is a $\sigma$-finite measure space of infinite measure and $T:\Omega\to\Omega$ a measure-preserving transformation with measurable inverse. Let be $\Omega\_k\in \mathcal{A}$ an increasing sequence such that $\Omega\_k\uparrow\Omega$ and
$\mu(\Omega\_k)<+\infty$ for all $k\in\math... | https://mathoverflow.net/users/2386 | Poincare Recurrence Theorem on Infinite Measure Space | If I remember my infinite ergodic theory correctly, any measure-preserving transformation $T$ of a $\sigma$-finite measure space $(\Omega,\mathcal{A},\mu)$ leads to a decomposition of $\Omega$ into a dissipative part $\Omega\_d$ and a conservative part $\Omega\_c$. (This notation is probably non-standard.) The dissipat... | 5 | https://mathoverflow.net/users/5701 | 44214 | 28,083 |
https://mathoverflow.net/questions/44219 | 25 | I am reading some work on Mirror Symmetry from Physics perspective,the physicists seem to use some aspects of BRST quantization and BRST cohomology. What is BRST Quantization and BRST cohomology, in realm of Mathematics. More precisely,
What are the BRST complexes, how do we get this cohomology, what is the relation... | https://mathoverflow.net/users/9534 | BRST cohomology | There is no way I will be able to answer all of your questions, so instead I will focus on just a tiny part, and try at least to explain "BRST integrals". Much of what I say is probably well-known, but I am also in the process of writing up some conversations on this and related topics with Dan Berwick Evans, and if th... | 24 | https://mathoverflow.net/users/78 | 44224 | 28,088 |
https://mathoverflow.net/questions/44208 | 88 | Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to [Wikipedia](https://en.wikipedia.org/wiki/Ultrafinitism), it has been primarily studied by Alexander Esenin-Volpin. On his [opinions page](https://s... | https://mathoverflow.net/users/1574 | Is there any formal foundation to ultrafinitism? |
>
> Wikipedia also says that Troelstra said in 1988 that there were no satisfactory foundations for ultrafinitism. Is this still true? Even if so, are there any aspects of ultrafinitism that you can get your hands on coming from a purely classical perspective?
>
>
>
There are no foundations for ultrafinitism as ... | 65 | https://mathoverflow.net/users/1610 | 44232 | 28,092 |
https://mathoverflow.net/questions/44182 | 6 | Let $f:X\longrightarrow Y$ be a finite separable morphism of smooth projective integral curves over an algebraically closed field.
Then we have a linear equivalence of Weil divisors on $X$: $$ K\_X=f^\ast K\_Y + R.$$ Here $$R=\sum \textrm{length} (\Omega\_{X/Y})\_p [p]$$ is the ramification divisor on $X$. This is th... | https://mathoverflow.net/users/4333 | What does the Riemann-Hurwitz formula tell us on the Picard variety | The answer is quite classical when $f \colon X \to Y$ is an unramified double cover.
In this case Riemann - Hurwitz formula gives
$g(X)-1 = 2g(Y)-2$.
Consider the following three natural maps:
$f^\* \colon J(Y) \to J(X)$,
$Nm \colon \textrm{Pic}^0(X) \to \textrm{Pic}^0(Y), \quad Nm(\sum a\_ip\_i):= \sum a\_if(... | 9 | https://mathoverflow.net/users/7460 | 44235 | 28,094 |
https://mathoverflow.net/questions/44187 | 11 | I am not a logician or set theorist, so hopefully this makes sense. Let $T$ be a theory which is expressive enough to make statements like "Statement $A$ has a proof in $T$"; for example, $T$ might be capable of expressing elementary arithmetic and proving certain basic arithmetic facts.
Let $\Pi^0(T)$ consist of tho... | https://mathoverflow.net/users/6950 | What is the depth of the "provability hierarchy"? | I believe the concept you are looking for is that of "iterated consistency extension." A very nice treatment is given by Torkel Franzén in his book [Inexhaustibility: a non-exhaustive treatment](http://books.google.com/books?id=6gJa6bjf374C&lpg=PP1&ots=5jv2Db6sxZ&dq=inexhaustibility&pg=PA185#v=onepage&q&f=false). See a... | 11 | https://mathoverflow.net/users/2000 | 44240 | 28,096 |
https://mathoverflow.net/questions/44242 | 2 | Let $p(n)$ denote count of lattices on finite set $G$, $|G|=n$ (without isomorphism). It's know closed formula for $p(n)$?
It's clear, that $1 \leq p(n)$ and also that $p(n-1) \leq p(n)$ for $n \geq 2$. My other estimates are $p(n) \leq 2^{\frac{(n-1)(n-2)}{2}}$ (also $p(n) \leq 2^{\frac{(n-1)}{2}}$) and $p(n-1) < p(... | https://mathoverflow.net/users/9067 | Count of lattices on finite set | 1, 1, 1, 1, 2, 5, 15, 53, 222, 1078, 5994, 37622, 262776, 2018305, 16873364, 152233518, 1471613387, 15150569446, 165269824761, ...
There is a lot of information in [The On-Line Encyclopedia of Integer Sequences.](http://oeis.org/A006966)
| 2 | https://mathoverflow.net/users/4600 | 44247 | 28,099 |
https://mathoverflow.net/questions/44243 | 6 | One defines the $H^n(G,M)$ where $M$ is a $\mathbb{Z}[G]$ module as $Ext^n\_{\mathbb{Z}[G]}(\mathbb{Z},M)$ where $\mathbb{Z}$ is viewed as a trivial $\mathbb{Z}[G]$-module.
Is this part of a general pattern for how to define cohomology for non-abelian categories?
In groups we see that we switch to the abelian categ... | https://mathoverflow.net/users/5309 | Group cohomology and cohomology in non-abelian categories | Yes, this generalizes to Hochschild cohomology and André-Quillen cohomology.
Given a category $\mathcal{C}$ with finite limits and an object $X$, you can form the category of Beck modules $\operatorname{Ab}(\mathcal{C} / X)$, abelian group objects in the slice category over $X$. When $\mathcal{C}$ is the category of ... | 7 | https://mathoverflow.net/users/396 | 44251 | 28,102 |
https://mathoverflow.net/questions/44211 | 5 | For which $n\in \mathbb{N}$, can we find (reps. find explicitly) $n+1$ integers $0 < k\_1 < k\_2 <\cdots < k\_n < q<2^{2n}$
such that
$$\prod\_{i=1}^{n} \sin\left(\frac{k\_i \pi}{q} \right) =\frac{1}{2^n} $$
P.S.: $n=2$ is obvious answer, $n=6 $ is less obvious but for instance we have $k\_1 = 1$, $k\_2 = 67$, $k\_3... | https://mathoverflow.net/users/3958 | Product of sine | Consider the identity (as quoted by drvitek):
$$\prod\_{k=1}^{n} \sin \left(\frac{(2k-1) \pi}{2n}\right) = \frac{2}{2^n},$$
This is completely correct but doesn't quite answer the question because the RHS is
not $1/2^n$ $\text{---}$ this is an issue related to the fact that $\zeta - \zeta^{-1}$ is not
a unit if $\zeta$... | 11 | https://mathoverflow.net/users/nan | 44253 | 28,103 |
https://mathoverflow.net/questions/44252 | 4 | Suppose $f(z)$ is a function analytic in the strip $|Re(z)|\leq a$. Is the fourier transform $\hat{f}(w)=o(e^{-a|w|})$?
It seems plausible but I can't seem to prove it either.
There is similar result called the Paley-Wiener Theorem that states $e^{a|w|}\hat{f}(w)\in L\_2(\mathbb{R})$, but I don't think that helps... | https://mathoverflow.net/users/2011 | Decay of the Fourier transform | This is not true without additional integrability conditions on $f(\cdot+iy)$.
$\hat{f}(w)=o(e^{-a|w|})$ implies that $e^{b|w|}\hat{f}(w)\in L\_2(\mathbb{R})$ for all $b < a$. The latter inclusion holds if and only if $f(z)$ is analytic in the strip $|\Im(z)|< a$ *and* $$ \sup\limits\_{|y|\leq b}\|f(\cdot+iy)\|\_{L^... | 7 | https://mathoverflow.net/users/5371 | 44256 | 28,106 |
https://mathoverflow.net/questions/7258 | 12 | q-Catalan numbers are defined recurrently as C0=1, $C\_{N+1}=\sum\_{k=0}^N q^k C\_k C\_{N-k}$.
What can be said about the asymptotics of Cn when `0<q<1`?
P.S. In the case q>1 it is known that as n goes to infinity, $q^{-{n\choose 2}}C\_n(q)$ tends to the partition function $\prod\_{i=1}^\infty\frac1{1-q^{-i}}$. How... | https://mathoverflow.net/users/979 | Asymptotics of q-Catalan numbers | Re Leonid's comment on a previous answer.
If the ratios $C\_{n+1}/C\_n$ converge, their limit $c(q)$ is such that $C(q,q/c(q))=c(q)$.
Equivalently, $1/c(q)$ is the radius of convergence of the series $z\mapsto C(q,z)$.
Or, writing $C(q,\cdot)$ as the ratio of two $q$-hypergeometric functions, one can show that $F(q... | 3 | https://mathoverflow.net/users/4661 | 44260 | 28,108 |
https://mathoverflow.net/questions/44125 | 40 | Hi,everyone. I am looking for an introductory textbook on moduli theory,about the background on algebraic geometry,I have read Hartshorne chapter1~4. could you please show some good books or roadmap for studying moduli theory which emphasis to arithmetic aspect?
how about Mumford's GIT? is it an introductory textbook... | https://mathoverflow.net/users/5274 | What is a good introductory text for moduli theory? | Ian Morrison wrote up some nice lectures in the book Lectures on Riemann surfaces,World Scientific publishers, Proceedings of the college of Riemann surfaces in 1987, at the ICTM in Trieste. They were intended as an informal introduction to the two detailed treatments mentioned below by Mumford (l'Enseignement) and Gie... | 21 | https://mathoverflow.net/users/9449 | 44267 | 28,112 |
https://mathoverflow.net/questions/44269 | 25 | Let $G$ be a group, $G'=[G, G]$.
>
> "Note that it is not necessarily true that the commutator subgroup
> $G'$ of $G$ consists entirely of
> commutators $[x, y], x, y \in G$ (see [107] for some finite group examples)."
>
>
>
Quoted from <http://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf> page 8.
Anybody ... | https://mathoverflow.net/users/3922 | Commutator subgroup does not consist only of commutators? | The problem is whether the commutator subgroup may contain elements that are not commutators.
One example are the free groups. For instance, in the free group of rank $4$, freely generated by $x$, $y$, $z$, and $w$, the element $[x,y][z,w]$ of the commutator subgroup cannot be written in the form $[a,b]$ for some $a,b... | 33 | https://mathoverflow.net/users/3959 | 44276 | 28,117 |
https://mathoverflow.net/questions/44268 | 19 | Throughout "curve" means smooth projective curve over an algebraically closed field.
**Motivation and Background**
I read somewhere that Atiyah has classified vector bundles on elliptic curves. My understanding is that the story is roughly: every vector bundles breaks up as a direct some of indecomposable vector ... | https://mathoverflow.net/users/7 | How do you describe vector bundles on elliptic curves? | A description via cocycle usually is not convenient to work with. From my point of view, a description as an extension is much more useful. But if you want something else, I would advice the following. Since your curve $C$ is given as a double covering of $P^1$, that is as a relative spectrum of the sheaf of algebras $... | 32 | https://mathoverflow.net/users/4428 | 44279 | 28,118 |
https://mathoverflow.net/questions/44265 | 4 | I would like to check a statement about Schauder bases in $C([0,1])$ to be sure that I don't lie to my students on Monday. The statement(s) that I would like to check are:
1. The family of monomials $\{1,t,t^2,t^3,\dots\}$ is a **topological basis** but not a **Schauder basis** in $C([0,1])$ because there's not a uni... | https://mathoverflow.net/users/45 | Question about Schauder bases in C([0,1]). | I would say, monomials are not a Schauder basis for $C[0,1]$ because functions that admits a representation are analytic functions on the unit disc. Actually, $f(t)=\sum\_k a\_k t^k$ immediately implies differentiability at $t=0,$ which is enough to conclude. So, it's more a matter of non-existence than non-unicity.
... | 11 | https://mathoverflow.net/users/6101 | 44280 | 28,119 |
https://mathoverflow.net/questions/44278 | 17 | The Artin braid groups $B\_n$ and the symmetric groups $S\_n$ are closely related by the maps $1 \to P\_n \to B\_n \to S\_n \to 1$. The infinite symmetric group has interesting interactions with homotopy theory, due to a result of Barratt-Priddy(-Segal(-Quillen(-others))) that "identifies" the sphere spectrum $QS^0$ wi... | https://mathoverflow.net/users/7867 | Loop spaces and infinite braids | Yes, $B \beta\_\infty$ is homology equivalent to $\Omega^2\_0 S^2$, the zero component of the double loop space of $S^2$. The map $B\_\infty \to S\_\infty$ induces the obvious stablisation map $\Omega^2\_0 S^2 \to Q\_0S^0$.
| 18 | https://mathoverflow.net/users/318 | 44281 | 28,120 |
https://mathoverflow.net/questions/44244 | 153 | E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what about the last one hundred years? Is it still possible for an amateur to make a significant contribution to mathematics?... | https://mathoverflow.net/users/nan | What recent discoveries have amateur mathematicians made? | About ten years ago Ahcène Lamari and Nicholas Buchdahl independently proved that all compact complex surfaces with even first Betti number are Kahler. This was known since 1983, but earlier proofs made use of the classification of surfaces to reduce to hard case-by-case verification.
At the time, Lamari was a teache... | 125 | https://mathoverflow.net/users/4054 | 44283 | 28,122 |
https://mathoverflow.net/questions/44275 | 2 | Let $V$ be a real algebraic variety and let ${\cal O}(V)$ denote its algebra of regular functions. If we put a group structure on $V$ (not necessarily an algebraic group structure) it will induce a Hopf algebra structure on ${\cal O}(V)$ in the usual manner. My question is, is there a bijective correspondence between t... | https://mathoverflow.net/users/1867 | Hopf algebra and group structure correspondence for algebraic varieties | If $V$ is an affine algebraic variety over any field $k$, then there is a bijection between the algebraic group structures on $V$ and the Hopf algebra structures on $O(V)$. The reason is that the category of affine varieties over $k$ is the just the contravariant category (the category with arrows reversed) of algebras... | 7 | https://mathoverflow.net/users/1450 | 44284 | 28,123 |
https://mathoverflow.net/questions/44289 | 4 | This is exercise 20.5 out of Jech:
>
> Let $\lambda \geq \kappa$ and let $U$ be a normal measure on $P\_{\kappa}(\lambda)$. The ultraproduct $\mathrm{Ult} \_U \{ (V \_{\lambda \_x},\in) : x \in P \_{\kappa}(\lambda) \}$ is isomorphic to $(V \_{\lambda}, \in)$
>
>
>
Here $\lambda \_x$ simply denotes the order t... | https://mathoverflow.net/users/7521 | Strong Cardinals and Supercompact Cardinals | In general, $\lambda$-supercompactness, if consistent, does
not imply $\lambda$-strongness. One can see this by
observing that the smallest cardinal $\kappa$ that is
$\kappa^+$-supercompact is never $\kappa^+$-strong, and in
fact, cannot be even $(\kappa+3)$-strong. The reason is
that $\kappa^+$-supercompactness is wit... | 8 | https://mathoverflow.net/users/1946 | 44291 | 28,127 |
https://mathoverflow.net/questions/44234 | 29 | Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
>
> **Question:** (Furstenberg) Let $\mu$ be a continuous probability measure on the circle such that
> $$\int\_{S^1} f(z) d\mu = \int\_{S^1} f(z^2) d\mu = \int\_... | https://mathoverflow.net/users/8176 | Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle | Manfred Einsiedler and Alexander Fish have a paper (arxiv.org/abs/0804.3586) showing that a multiplicative subsemigroup of $\mathbb N$ which is not too sparse satisfies the desired measure classification. The semigroups they consider are still somewhat large, and in particular not contained in finitely generated semigr... | 21 | https://mathoverflow.net/users/10457 | 44297 | 28,131 |
https://mathoverflow.net/questions/44303 | 23 | I am trying to define an embedding whose range includes classes. Is there a coherent way of assigning "cardinality" to proper classes?
| https://mathoverflow.net/users/nan | Cardinality of classes | Erin, there is no need to do this. I do not know of any practical reasons for doing it. And, of course, "cardinality" has to be properly interpreted to make some sense of the word.
In extensions of set theory where classes are allowed (not just formally as in ZFC, but as actual objects as in MK or GB), sometimes it ... | 21 | https://mathoverflow.net/users/6085 | 44305 | 28,137 |
https://mathoverflow.net/questions/44299 | 1 | Let's $C \subset X$ be a smooth curve inside a three dimensional variety with split normal bundle $N\_C^X= \nu\_1 \oplus \nu\_2$. What is a locally free resolution of $\iota\_{\*}\mathcal{O}\_{C}$ ?
| https://mathoverflow.net/users/5259 | How do we write a locally free resolution for... | In general, there is no straight way to write such a resolution (note by the way, that a resolution is in no way unique!). However, in some cases there is a distinguished resolution. For example, if $C$ is the zero locus of a global section of a rank 2 vector bundle $E$ then there is a Koszul resolution
$$
0 \to \det E... | 7 | https://mathoverflow.net/users/4428 | 44306 | 28,138 |
https://mathoverflow.net/questions/44309 | 7 | Problem
-------
I'm looking for an upper bound for the number $k(G)$ of a finite group $G$, defined as follow:
>
> Let $\mathcal{F}\_k$ be the family of subsets of $G$ with size $k$, and we
> define $k(G)$ be the minimum $k$ such that every subset $X \in \mathcal F\_k$
> contains a non-empty sum-full set $S$, w... | https://mathoverflow.net/users/4248 | Upper bound for size of subsets of a finite group that contains a sum-full set | First of all the $k(G)$ cannot be smaller than the size of any proper subgroup of $G$, because if $H$ is a proper subgroup, $gH$ is a coset, $g\not\in H$, then $gHgH$ does not intersect $gH$ (if $ghgh'=gh''$, then $g\in H$). For example, if $G$ is Abelian, $|G|$ is not prime (i.e. $G$ is not cyclic of prime order), the... | 4 | https://mathoverflow.net/users/nan | 44325 | 28,149 |
https://mathoverflow.net/questions/44332 | 1 | is there an English translation of the book by Guy Barles, "Solutions de viscosite des equations de Hamiltion-Jacobi"? Springer-Verlag
| https://mathoverflow.net/users/5896 | is there an English translation of the book by Guy Barles? | It's extremely unlikely that this 1994 Springer book has been formally translated into English, since neither Springer's site nor MathSciNet lists a translation. In fact it's uncommon for most French mathematics books to be translated "officially" into English, a subject which has come up in many other posts on MO.
By... | 2 | https://mathoverflow.net/users/4231 | 44336 | 28,157 |
https://mathoverflow.net/questions/44323 | 3 | I'm reading the book about moduli spaces by Huybrechts and Lehn, and i'm stuck understanding a proof, it is Theorem 6.1.8.:
Given a K3-surface $X$ and a 2-dimensional space $M$, coherent and torsion free sheafs $F$ on $X$ and $E$ on $M\times X$. We have projections $p,q$ from $M\times X$ to $M$ and $X$ resp.
They c... | https://mathoverflow.net/users/3233 | Chern character of Hom-sheaves | If you apply ${\mathcal H}om(-,E)$ to a resolution of a sheaf $G$, you obtain a complex, the cohomology of which are ${\mathcal E}xt^i(G,E)$, hence by additivity of the Chern character, the alternating sum of Chern characters of the terms of the complex equals the alternating sum of Chern characters of the Ext sheaves.... | 2 | https://mathoverflow.net/users/4428 | 44338 | 28,158 |
https://mathoverflow.net/questions/44358 | 9 | Is there a natural reason for defining the compact-open topology on the set of continuous functions between two locally compact spaces. For example "to make ... functions continuous". Or in another way of asking this, is there an adjoint functor of the functor, say F, which assigns the topological space $F(X,Y):=Hom\_C... | https://mathoverflow.net/users/10469 | compact-open topology | In regard to your question I recommend *[Topologies on spaces of continuous functions,](http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf) Topology Proceedings, volume 26, number 2, pp. 545-564, 2001-2002* by Martin Escardo and Reinhold Heckmann.
| 9 | https://mathoverflow.net/users/1176 | 44365 | 28,179 |
https://mathoverflow.net/questions/44249 | 2 | I essentially understand (I think) how this ought to be done. Algebras in a monoidal 2-category $\mathcal{C}$, on the level of 0-cells and 1-cells, should appear as algebras in the 1-category truncation of $\mathcal{C}$. To lift these 1-level algebras we must of course weaken the usual diagrams and then describe (a zoo... | https://mathoverflow.net/users/8157 | How to explicitly describe algebras in a monoidal 2-category? | In a category $\mathscr{C}$ by finite limits you can have inside it all algebraic classical structures, for example about “Monoids” there is a category called “Monoid theory” $Mnd$ (see pioneristic works of Lawvrere) and the category of $\mathscr{C}$- monoids is the category of finite limits preserving $F: Mnd\to \math... | 0 | https://mathoverflow.net/users/6262 | 44367 | 28,181 |
https://mathoverflow.net/questions/43326 | 1 | Are there known sharp upper bounds (in terms of $k$ or $\omega(k)$, the number of distinct prime divisors of $k$) for sums of the form $\sum\_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum\_{p \mid k} \frac{1}{p+1} < 1$? (This would be a special case of the general result of O. Izhboldin and L. K... | https://mathoverflow.net/users/10280 | Sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ | If we are allowed to consider somewhat weaker bounds, both answers depend only on the number of unitary divisors of $k$, which is $2^{\omega(k)}$. By the quoted result of O. Izhboldin and L. Kurliandchik (see Fedor's and Myerson's comments), for any set of $n$ positive integers {$a\_{1}, \dots, a\_{n}$} such that $\sum... | 1 | https://mathoverflow.net/users/10280 | 44401 | 28,208 |
https://mathoverflow.net/questions/44397 | 11 | The standard examples of complete but not model-complete theories seem to be:
- Dense linear orders with endpoints.
- The full theory $\mathrm{Th}(\mathcal{M})$ of $\mathcal{M}$, where $\mathcal{M} = (\mathbb{N}, >)$ is the structure of natural numbers equipped with the relation $>$ (and nothing else, i.e. no add... | https://mathoverflow.net/users/362 | What is a good example of a complete but not model-complete theory, and why? | For the second example, let $M$ be the natural numbers and let $N$ be the
integers greater than or equal to -1. Then $M$ is a substructure of $N$
but $M\models$ ``0 is the least element", while this is false in $N$. Thus the
theory is not model complete.
The first example is similar, let $M$ be $[0,1]$ and let $N$ b... | 16 | https://mathoverflow.net/users/5849 | 44405 | 28,212 |
https://mathoverflow.net/questions/27990 | 8 | My question is related to this one: [Computing the Galois group of a polynomial](https://mathoverflow.net/questions/22923/computing-the-galois-group-of-a-polynomial).
I was wondering if there is a faster algorithm just to compute the order of the group rather than the group itself.
Also, has anybody compared the pe... | https://mathoverflow.net/users/6776 | Computing only the order of Galois group (not the group itself). | I am actually one of the authors of the Galois package in Magma. Firstly, the "too much looping" error does not happen anymore (for this example at least) in the current Magma version (2.16-13). Secondly, the way Sn/An is recognized in general is through the use of
factorisation as suggested. More precisely, the polyno... | 9 | https://mathoverflow.net/users/10478 | 44414 | 28,217 |
https://mathoverflow.net/questions/44428 | 5 | In Bourbaki's Commutative Algebra we have the following theorem:
II.5.2 Let $A$ ba a ring and $P$ an $A$-module. TFAE:
(i) $P$ is a f.g. projective module\
(ii) $P$ is a finitely presented module and, for every maximal ideal $m$ of $A$, $P\_m$ is a free $A\_m$-module.\
(iii) $P$ is a f.g. module, for all $p \in... | https://mathoverflow.net/users/10483 | On Bourbaki's characterization of projectives... | Dear fishibones, an important point is that at the beginning of Chapter II, Bourbaki states that all the rings he will consider are commutative. This implies that if a free module has finite dimension, this dimension is unambiguously defined. This property is called the Invariant Basis Number property (IBN) and may fai... | 4 | https://mathoverflow.net/users/450 | 44434 | 28,229 |
https://mathoverflow.net/questions/44202 | 3 | In the language of modules, it suffices to restrict our view to *positive-primitive* formulas - that is to say, formulas with one existential quantifier and no negation.
And I mean *existentially closed* in the sense that any witness to a particular positive-primitive formula over the module is already in the module ... | https://mathoverflow.net/users/9015 | Example or classification of existentially closed modules | Do primitive positive formulas only allow conjunctions, not disjunctions? If so, then injective modules are existentially closed, but are a strictly stronger concept in that the equivalent for infinite conjunctions are allowed.
The concept you're after is that of [algebraically compact](http://en.wikipedia.org/wiki/A... | 2 | https://mathoverflow.net/users/3711 | 44442 | 28,235 |
https://mathoverflow.net/questions/44443 | 7 | Is there any known result about the necessary and sufficient conditions for the existence of zeros for a function $f(x)=\sum\_{n=1}^{N} a\_n e^{b\_n x}$, where $a\_n,b\_n \in \mathbb{R}\, \forall n=1,2,\cdots,N$, $a\_1,a\_N >0$, $b\_1 < b\_2 < \cdots < b\_N $ and $x \in \mathbb{R}$?
It is known (see "Problem and The... | https://mathoverflow.net/users/6162 | Zeros of a combination of exponentials | Note that we can assume wlog that $b\_n\geq 0.$ In the case they are rationals, writing $b\_n=p\_n/q$, with $p\_n\in\mathbb{N},\\ $ $q\in\mathbb{N}\_+,\\ $ and $t:=e^{x/q},\\ $ puts everything into the case of positive roots of a real polinomial, with not more, nor less generality. The book by Pólya and Szegő has a sec... | 3 | https://mathoverflow.net/users/6101 | 44447 | 28,237 |
https://mathoverflow.net/questions/44408 | 13 | Does the support of a Borel probability measure always have [full measure](https://en.wikipedia.org/wiki/Almost_everywhere) in a metric space?
I know this is true for separable metric spaces, and locally compact metric spaces. Is it true in general?
| https://mathoverflow.net/users/10476 | Measure of the support of a Borel probability on a metric space | Following Pietro's lead, let me observe that if there is a
[measurable cardinal](http://en.wikipedia.org/wiki/Measurable_cardinal), then there is a counterexample.
Suppose that $\kappa$ is a measurable cardinal. Then there
is a $\kappa$-additive 2-valued measure $\mu$, measuring
all subsets of $\kappa$, giving them e... | 11 | https://mathoverflow.net/users/1946 | 44448 | 28,238 |
https://mathoverflow.net/questions/44452 | -2 | I know the following result is true in the case of strong convergence. But I don't know whether it is true in the case of weak convergence also.
Let $p>1$. Suppose that each $x\_n$ is a non negative sequence such that $\|x\_n\|\_p=1$ and $\stackrel{w}{x\_{n}\rightarrow x}$ in $\ell^p$. Is it true then that $\stackrel{w... | https://mathoverflow.net/users/7699 | weak convergence | No. Consider the unit vector basis of $\ell\_2$.
| 5 | https://mathoverflow.net/users/2554 | 44456 | 28,242 |
https://mathoverflow.net/questions/44468 | 23 | In Andrew Gleason's interview for More Mathematical People, there is the following exchange concerning Gleason's work on Hilbert's fifth problem on whether every locally Euclidean topological group is a Lie group (page 92).
>
> **MP:** Is there some "human" story you can tell us about the breakthrough when it came?... | https://mathoverflow.net/users/2926 | Monotone functions are differentiable a.e. and Hilbert's Fifth Problem: what's the connection? | Well, I cannot say for certain, but I did know Gleason well (he was my thesis advisor, and we wrote a paper together after that) and I have written an essay about Gleason's work on the Fifth Problem (in the Gleason Memorial article in the AMS Notices --- <http://www.ams.org/notices/200910/rtx091001236p.pdf> ) and based... | 23 | https://mathoverflow.net/users/7311 | 44473 | 28,249 |
https://mathoverflow.net/questions/44438 | 2 | I believe to have found a typo in Griffiths & Harris.
In the chapter on surfaces, section Rational Surfaces 1, I am trying to read the result that a holomorphic vector bundle over $\mathbb{P}^1\_{\mathbb{C}}$ is a sum of invertible bundles.
What is the exact sequence that shows up at the start of his argument? Mine... | https://mathoverflow.net/users/8867 | Help with Griffiths & Harris, Surfaces | Let $H$ be the divisor corresponding to the point $x\in \mathbb{P}^1$. Tensoring the exact sequence
$$
0\to O\_{\mathbb{P}^1}(-H)\to O\_{\mathbb{P}^1}\to O\_x\to 0.
$$ with $E\otimes H^k$, gives
$$
0\to O\_{\mathbb{P}^1}(E\otimes H^{k-1})\to O\_{\mathbb{P}^1}(E\otimes H^{k})\to E\_x\otimes H\_x^{k}\to 0.
$$Here GH wri... | 5 | https://mathoverflow.net/users/3996 | 44479 | 28,254 |
https://mathoverflow.net/questions/44474 | 6 | (1) Let $M$ be a complex manifold of real dimension $2n$, and denote the line bundle of complex $(N,0)$-forms by $\Omega^{(N,0)}(M)$. When $M = CP^N$, the line bundles are indexed by the integers, and so, $\Omega^{(N,0)}(CP^N)$ must correspond to a integer. What is this integer? In the $N=1$ case, the corresponding int... | https://mathoverflow.net/users/1648 | Complex Projective Space Spin and Dirac: Part II | ad 1.): No, the answer is that $\Omega^{1,0} CP^N$ corresponds to $N+1$. Proof sketch:
As complex vector bundles, $TCP^N \oplus C= C^{N+1} \otimes L$, $L$ the tautological line bundle (the one with holomorphic sections). Thus $L^{\otimes (N+1)} = \Lambda^{N+1} (TCP^N \oplus C) = \Lambda^N TCP^N$. Dualizing gives the an... | 3 | https://mathoverflow.net/users/9928 | 44484 | 28,259 |
https://mathoverflow.net/questions/44463 | 5 | A lax limit is [defined](http://ncatlab.org/nlab/show/2-limit) to be a 2-limit, except that the cone is only required to commute up to specified transformations, not up to isomorphism. In particular, the limit is defined up to isomorphism, and on examples such as product where there are no 2-cells in the cone, lax limi... | https://mathoverflow.net/users/10491 | Lax universality for lax limits | Adjunctions 'up to adjointness' have been considered before. Marta Bunge (*Coherent extensions and relational algebras*, Trans. AMS 197, 1974) called them 'lax adjunctions', John Gray (*Formal category theory: Adjointness for 2-categories*, LNM 391, 1974) called them 'quasi-adjunctions' (of some sort) and Barry Jay (*L... | 2 | https://mathoverflow.net/users/4262 | 44487 | 28,261 |
https://mathoverflow.net/questions/43961 | 8 | Let $T$ be a finite symmetric set generating a Zariski dense subset of an algebraic group $G$ (specifically, $PSL\_2(\mathbb{C})$ or its subgroups). Is there an $\alpha>0$ such that the set $T^{\leq n}$ of words of length at most $n$ is not in any codimension-1 subvariety of degree $n^{\alpha}$?
"Escape from subvarie... | https://mathoverflow.net/users/408 | Degree of balls in finitely-generated subgroups of SL_2(C) | In the case of $\text{PSL}(2,\mathbb{C})$, the set of words of length $O(n^3)$ is not contained in a subvariety of degree $n$. A polynomial of degree $n$ is a linear combination of matrix entries of the irreps of highest weight $\le n$. Let $A\_n$ be the direct sum of the corresponding matrix algebras; its dimension is... | 4 | https://mathoverflow.net/users/1450 | 44488 | 28,262 |
https://mathoverflow.net/questions/44207 | 5 | I am looking for the asymptotic growth of product of consecutive primes. Is there anything that is known about this growth?
| https://mathoverflow.net/users/10035 | Asymptotics of Product of consecutive primes | Denote by $$\Pi(x)=\prod\_{p\leqslant x}p,$$ thus
$$\log\Pi(x)=\sum\_{p\leqslant x}\log p:=\theta(x)\sim x,$$ which is known as the Prime Number Theorem. You may find further information in <http://en.wikipedia.org/wiki/Prime_number_theorem>
| 7 | https://mathoverflow.net/users/9944 | 44498 | 28,269 |
https://mathoverflow.net/questions/42629 | 27 | Fix a dimension $n\geqslant 2$. Let $S= \{M\_1,\ldots, M\_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is *generated* by $S$ if it may be obtained by gluing some copies of elements in $S$ via some arbitrary diffeomorphisms of their boundaries.
For i... | https://mathoverflow.net/users/6205 | Can all n-manifolds be obtained by gluing finitely many blocks? | Thanks to Ian Agol for pointing out this question and a related one on levels of Morse functions - [/30567/](https://mathoverflow.net/questions/30567/). In both case, for smooth manifold of dim $> 3$, as expected, there is no finite list of blocks ( or regular level components.) The idea is that one may define the "wid... | 23 | https://mathoverflow.net/users/1643 | 44500 | 28,270 |
https://mathoverflow.net/questions/44183 | 26 | Let $(X,d)$ be a metric space, and let $C\_u(X)$ be the Banach space of bounded *uniformly* continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C\_u(X)^\*$?
I would guess it can be described as some space of measures. I would even be interested in the case $X=\mathbb{R}$.
Obv... | https://mathoverflow.net/users/4832 | Dual of bounded uniformly continuous functions | $C\_u(\mathbb R)^\*$ is essentially the space of complex measures on $\beta \mathbb Z\coprod (\beta\mathbb Z\times(0,1)).$ Here $\beta \mathbb Z$ is the Stone-Čech compactification of $\mathbb Z,$ and the $\coprod$ denotes disjoint union.
One can identify $C\_u(\mathbb R)$ with $C\_0(\beta \mathbb Z \coprod (\beta \m... | 18 | https://mathoverflow.net/users/10500 | 44508 | 28,273 |
https://mathoverflow.net/questions/44516 | 9 | What is an example of a functor $F : \mathbb{C}\text{-Sch.} \to \text{Sets}$ with the property that
the restriction of $F$ to locally Noetherian $\mathbb{C}$-schemes can be represented by a locally Noetherian $\mathbb{C}$-scheme, but that scheme does not represent $F$.
I'd be particularly nice to see a "real-world" e... | https://mathoverflow.net/users/5337 | Non-representable functor, representable on locally Noetherian schemes? | Define $F(X) = {\rm{Hom}}\_{\mathbf{C}}(X,{\rm{Spec}}(R/tR))$ where $R$ is the valuation ring of an algebraic closure of $\mathbf{C}((t))$. Note that every element of the maximal ideal of $R/tR$ is nilpotent yet also an $N$th power for arbitrarily large $N$. For any noetherian $\mathbf{C}$-algebra $A$, every $\mathbf{C... | 18 | https://mathoverflow.net/users/3927 | 44518 | 28,278 |
https://mathoverflow.net/questions/44507 | 3 | A profinite group is said to be projective if its cohomological dimension is $\leq 1$. Is this related to some other notion of "projective"? How so?
| https://mathoverflow.net/users/5309 | What's "projective" about "projective pro-finite groups"? | A profinite group $P$ is projective if and only if any continuous group homomorphism from it to a profinite quotient group $G/H$ lifts to a continuous group homomorphism to the profinite group $G$.
| 4 | https://mathoverflow.net/users/2106 | 44531 | 28,282 |
https://mathoverflow.net/questions/44533 | 4 | Lets consider this method of finding inverse function:
$$f^{-1}(x) = \sum\_{k=0}^\infty A\_k(x) \frac{(x-f(x))^k}{k!}$$
where coefficients $A\_k(x)$ recursively defined as
$$\begin{cases} A\_0(x)=x \\ A\_{n+1}(x)=\frac{A\_n'(x)}{f'(x)}\end{cases}$$
It is evident that for some classes of functions starting fro... | https://mathoverflow.net/users/10059 | For which classes of functions this inverse function formula gives a closed form expression? | The answer is not that difficult. Assume that $A\_{n+1}\equiv0$. This means $A\_n={\rm cst}$, that is $A\_{n-1}'=cf'$. Integrating, $A\_{n-1}=cf+d$, that is $A\_{n-2}'=(cf+d)f'$. Integrating again, $A\_{n-2}=\frac{c}{2}f^2+df+e$. And so on. By induction, we find that $A\_{n-k}=P\_k(f)$, where $P\_\ell$ is a polynomial ... | 6 | https://mathoverflow.net/users/8799 | 44539 | 28,286 |
https://mathoverflow.net/questions/44541 | 2 | Given a point $x$ in a topological space $X$. I was wondering, whether one can always find a [local basis](http://en.wikipedia.org/wiki/Neighbourhood_system) at $x$, which is [totally ordered](http://en.wikipedia.org/wiki/Totally_ordered_set) (a chain) under inclusion. For example this is true for spaces, which have co... | https://mathoverflow.net/users/3969 | chains and countability | The space $\omega\_1+1$ under the order topology, where $\omega\_1$ refers to the first uncountable ordinal, has a
linearly ordered local basis at the point $\omega\_1$ (and indeed at every point),
consisting of the intervals $(\alpha,\infty)$, but there is
no countable local basis at that point, because every
countabl... | 3 | https://mathoverflow.net/users/1946 | 44546 | 28,291 |
https://mathoverflow.net/questions/41143 | 4 | From the Incompleteness theorems, if ZF is consistent, one knows there are models of ZF satisfying ¬Con(ZF). These models must be non-standard (in the sense of being models whose ordinals are not well-ordered), and so must be the proof of an inconsistency from the axioms of ZF in them.
Now, ¬Con(ZF) is a very special... | https://mathoverflow.net/users/6466 | Statements that require the existence of non-standard models to hold | Here's another example. By a "computable well ordering" I will mean an index for a well-founded computable (total) linear order on $\omega$. Because ZFC is an effective theory, there must be some computable well ordering $\zeta$ that ZFC does not prove is a well ordering. This is because:
* The set of indices of comp... | 4 | https://mathoverflow.net/users/5442 | 44556 | 28,298 |
https://mathoverflow.net/questions/44563 | 6 | Non-quasi-coherent sheaves of $\mathcal O\_X$ modules on a scheme seem like a wild concept to me; are they actually used for something?
| https://mathoverflow.net/users/8363 | When are non-quasi-coherent sheaves used? | One can think the adeles on a curve (or higher adeles on other spaces) as a sheaf of $\mathcal O$-algebras. That is, consider the sheaf $B(U)=\prod\_{x\in U}\mathcal O\_x$, where $\mathcal O\_x$ is the completion of $\mathcal O$. Then the sheaf $A=B\otimes K$, where $K$ is the sheaf of rational functions, has the adele... | 5 | https://mathoverflow.net/users/4639 | 44584 | 28,307 |
https://mathoverflow.net/questions/44569 | 5 | This question is with regards to terminology. I am writing a journal paper that describes the software I develop at my workplace call [Navigator](http://ophid.utoronto.ca/navigator/). The software is used for visualization of networks/graphs. At work, we commonly interchange the use of the words 'networks' and 'graphs'... | https://mathoverflow.net/users/10517 | Can 'network' and 'graph' be used interchangably? | If someone says "graph" without further specification I consider it to mean an unweighted undirected graph without loops or multiple edges. I consider "network" to be a more general term. Especially if the audience of the article is more on the applied side, I would go with network.
But really in your situation I wou... | 2 | https://mathoverflow.net/users/4580 | 44586 | 28,308 |
https://mathoverflow.net/questions/44581 | 7 | Leonardo of Pisa is best known as Fibonacci; various stories found in books and on the web claim that the name Fibonacci was invented by Edouard Lucas or Guillaume Libri in the 19th century, and that it means "son of Bonacci" (Leonardo's father was apparently called
Guglielmo Bonaccio). Heinz
[Lueneburg](http://mathf... | https://mathoverflow.net/users/3503 | Fibonacci = Leonardo Pisano? | From [*The Fabulous Fibonacci Numbers*](http://books.google.co.uk/books?id=-O4ZAQAAIAAJ&q=The+Fabulous+Fibonacci+Numbers&dq=The+Fabulous+Fibonacci+Numbers&hl=en&ei=jETQTIS3LY2-4garhLmwBg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC8Q6AEwAA) by A.S. Posamentier and I. Lehmann (Prometheus Books, New York (2007), pp. 17... | 8 | https://mathoverflow.net/users/5371 | 44587 | 28,309 |
https://mathoverflow.net/questions/44512 | 26 | A philosopher asked me an interesting math question today! We know that there are sets S of integers which don't have a "natural" or "naive" density -- that is, the quantity (1/n)|S intersect [1..n]| doesn't approach a limit. The "analytic" or "Dirichlet" density exists whenever the naive density does, and is equal to ... | https://mathoverflow.net/users/431 | Is there any sense in which Dirichlet density is "optimal?" | The best way to impress a philosopher is to tell him/her about ultrafilters. A (non-principal) filter on $\mathbb N$ is a set of infinite subsets of $\mathbb N$ closed under intersections and taking super-sets. A maximal filter (under inclusion) is called an ultrafilter. There are plenty of those but nobody saw them si... | 15 | https://mathoverflow.net/users/nan | 44590 | 28,310 |
https://mathoverflow.net/questions/44577 | 8 | Let $E$, $F$ be two complex elliptic curves, and $A=E \times F$. Let us denote by
$\pi\_E \colon A \to E, \quad \pi\_F \colon A \to F$
the natural projections. For all $p \in F$ let us write $E\_p$ instead of $\pi\_F^\*(p)$.
Now let us fix $p \in F$ and consider the unique indecomposable rank $2$ vector bundle $\... | https://mathoverflow.net/users/7460 | Rank 2 vector bundle on a product of elliptic curves | It's the nontrivial extension. Here is why:
Let $\pi=\pi\_F$ and consider
$$
0\to \pi\_\*\mathscr O\_A \to \pi\_\*\mathscr F \to \pi\_\*\mathscr O\_A(-E\_p)\to R^1\pi\_\*\mathscr O\_A \to \dots
$$
Now
1) $\pi\_\*\mathscr O\_A\simeq \mathscr O\_F$ and $\pi\_\*\mathscr O\_A(-E\_p)\simeq\mathscr O\_F(-p)$
2) $... | 4 | https://mathoverflow.net/users/10076 | 44591 | 28,311 |
https://mathoverflow.net/questions/44417 | 2 | Let $\mathbb Z^2$ denote the two-dimensional integer lattice with norm of $i=(i\_1,i\_2)$ given by $\|i\|=|i\_1|+|i\_2|$.
For each $x\in\mathbb Z^2$, we assign a uniform random variable, $\sigma\_x$ taking values on the set $\{-1,1\}$.
Fix $\omega\in\{-1,0,1\}^{\mathbb{Z}^2}$ and $\beta>0$. For each finite $\Lambd... | https://mathoverflow.net/users/2386 | On generalisation of Aizenman-Higuchi Theorem | Concerning questions 1 and 2, if I understand it correctly (i.e., you're simply looking at the finite-range ferromagnetic Ising model with free, resp + or -, b.c.), then the sequences actually converge and the limits are translation invariant. This follows from monotonicity in the volume (GKS for free, GKS of FKG for +... | 3 | https://mathoverflow.net/users/5709 | 44599 | 28,317 |
https://mathoverflow.net/questions/44596 | 5 | The definition of a (geometric) vector bundle over a scheme $X$ can be rewritten as follows in terms of 'not-so-geometrical algebra' if $X=Spec R$ is affine and if I am not missing something.
A *vector bundle* of rank $n$ over $R$ is an $R$-algebra $A$ such that
* for every $p\in Spec R$ there is a isomorphism (bel... | https://mathoverflow.net/users/2625 | The correspondence between affine vector bundles and f.g. projective modules | Given any $R$-module $M$, there is a scheme which corresponds to the 'total space' of $M$, given by
$$ Tot(M):=Spec( Sym\_R(M\*))$$
where $M\*$ is the dual module $Hom\_R(M,R)$ and $Sym\_RM\*$ is the symmetric algebra of $M^\*$ over $R$. If $M$ happens to a free rank $n$ $R$-module, then $Sym\_RM\simeq R[X\_1,...,X\_n... | 4 | https://mathoverflow.net/users/750 | 44628 | 28,337 |
https://mathoverflow.net/questions/44630 | 0 | I expect the following relation to be vanishing. But it seems not that obvious.
$\Gamma\_{ab}^{\lambda}t^at^b \Gamma\_{\lambda c(d)}t^c=0$
where $t^a$ are even ghosts, "$ab$" are indices for matrix element, and $\lambda$ denote different Gamma matrices. The Einstein summation convention is used above, i.e. we will ... | https://mathoverflow.net/users/6577 | Clifford Algebra and Gamma matrices: is this relation generally true for any dimension? | It's not clear to me what you mean by "even ghosts". Do you mean perhaps that $t^a t^b = t^b t^a$?
If so, then you will find that the identity is only valid in 3, 4, 6 and 10 dimensions and with lorentzian signature. Indeed, this identity is essentially the condition for the vanishing of a fermionic trilinear which a... | 4 | https://mathoverflow.net/users/394 | 44633 | 28,339 |
https://mathoverflow.net/questions/20688 | 14 | I know that there are analytic functions whose composition with itself is the exponential function, the so-called functional square root of the exponential function, with the additional property that it is real on the real line.
Is a similar property possible for a holomorphic function that interpolates the tower fu... | https://mathoverflow.net/users/4923 | What's a natural candidate for an analytic function that interpolates the tower function? | The function you want grows too fast to be interpolated by usual method, but there exists an [iterative solution with Cauchy integrals by Dmitry Kouznetsov and Henryk Trappmann](http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/)
If you relax the condition so to find a solution for $f(x+1)=a^{f(x)}$ ... | 12 | https://mathoverflow.net/users/10059 | 44646 | 28,344 |
https://mathoverflow.net/questions/44514 | 5 | Consider the following situation:
Let $M$ and $M'$ be two closed manifolds and suppose $f:M\to \mathbb{R}$ and $f':M'\to \mathbb{R}$ are smooth morse functions on $M$ and $M'$ respectively. We say the pair $(M,f)$ and $(M',f')$ are *equivalent* if there is a smooth diffeomorphism $\phi:M\to M'$ so that $f'\circ \phi=... | https://mathoverflow.net/users/26801 | Pair consisting of a compact manifold and Morse function | You might have a look at Hatcher and Thurston's paper "[A presentation for the mapping class group of a closed orientable surface](http://www.ams.org/mathscinet-getitem?mr=579573)". They use Morse functions on the surface to facilitate the derivation of their presentation. On page 223 and following, they discuss how to... | 3 | https://mathoverflow.net/users/1345 | 44648 | 28,345 |
https://mathoverflow.net/questions/44661 | 3 | Assume that $(P,\le)$ is a notion of forcing. There are several ways to define what it means for $P$ being proper and I would like to know: What is the complexity (in terms of the Levy-Hierarchy) of the statement 'P is proper'?
| https://mathoverflow.net/users/4753 | Complexity of the statement 'P is proper' | Properness is observable in any sufficiently large $V\_\alpha$, and therefore has complexity $\Sigma\_2$. In oktan's answer, it suffices to consider sufficiently large $\lambda$, rather than all $\lambda$. I think this is proved in some of the standard accounts of proper forcing.
| 5 | https://mathoverflow.net/users/1946 | 44667 | 28,358 |
https://mathoverflow.net/questions/44666 | 0 | A simple question from someone new to the field:
In a metric space, the [Hausdorff dimension](http://en.wikipedia.org/wiki/Hausdorff_dimension) of a subset is defined by covering the subset with $\epsilon$-balls and looking at how the number of required balls grows as a power or $\epsilon$ in the limit $\epsilon \to ... | https://mathoverflow.net/users/10535 | Does the Hausdorff dimension depend on the L^p-norm? | Let $B\_p$ denote the 1-ball with centre 0 with respect to the
$l^p$ norm. For any $p$ and $q$ there is a number $N$ such that $B\_p$
is covered by $N$ translates of $B\_q$. Then any $\epsilon$-ball in
the $l^p$ norm is covered by $N$ $\epsilon$-balls in the $l^q$ norm.
Thus within a constant factor, the number of $\ep... | 3 | https://mathoverflow.net/users/4213 | 44668 | 28,359 |
https://mathoverflow.net/questions/44673 | 18 | Four Color Theorem is equivalent to the statement: "Every cubic planar bridgeless graphs is 3-edge colorable". There is computer assisted proof given by Appel and Haken. Dick Lipton in of his [beautiful blogs](http://rjlipton.wordpress.com/2009/04/24/the-four-color-theorem/) posed the following open problem:
>
> Ar... | https://mathoverflow.net/users/8784 | Human checkable proof of the Four Color Theorem? | This is too long for a comment, so I am placing it here.
In this [article of the Notices of the AMS](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.141.714&rep=rep1&type=pdf), Gonthier describes a full formal proof of the four-color theorem, which makes explicit every logical step of the proof.
Although ... | 30 | https://mathoverflow.net/users/1946 | 44681 | 28,364 |
https://mathoverflow.net/questions/44677 | 7 | I'm working on Markov Decision Process and I have not found yet an example of MDP that has a stochastic (non deterministic) optimal policy. Is there MDPs that have a stochastic optimal policy or is it shown that an optimal policy is always deterministic ?
If a stochastic policy exist, is it shown that some algorithms... | https://mathoverflow.net/users/10537 | Is there MDPs (Markow Decision Process) which have a non deterministic optimal policy ? | If there is an optimal policy, there is a deterministic optimal policy. Here is a sketch of the argument:
Start with an optimal policy within the class of deterministic optimal policies. By the one-deviation-principle, you only have to check whether you can gain by randomizing after a certain history of the process. ... | 4 | https://mathoverflow.net/users/35357 | 44685 | 28,367 |
https://mathoverflow.net/questions/44445 | 6 | Why it is true that, over an algebraically closed field, any abelian variety is isogenous to a principally polarized abelian variety?
| https://mathoverflow.net/users/5329 | About isogenies of abelian varieties | This is to fill some prerequisites to BCnrd's comment-answer.
First of all, there are several definitions of a polarization on an abelian variety, and the most "coordinate-free" one is that it is a homomorphism $\lambda:A\to A^t = Pic^0(A)$ given by some (non-unique) ample divisor $D$, so that $\lambda(a) = \mathcal ... | 9 | https://mathoverflow.net/users/1784 | 44686 | 28,368 |
https://mathoverflow.net/questions/44576 | 7 | Let A be a matrix with entries in Z/nZ. (n is not assumed to be prime.) Then the size of the row span is the size of the column span. All computations are mod n, so both these numbers are finite.
I believe you can prove this using Smith normal form: both the size of the row span and the size of the column span will ... | https://mathoverflow.net/users/5399 | "Linear algebra" over Z/nZ - reference please! | Your idea of using Smith normal form leads directly to a solution: But you need to verify that for every matrix $M$ with entries in $\mathbb{Z}/n\mathbb{Z}$ there are invertible matrices $A$ and $B$ with entries in $\mathbb{Z}/n\mathbb{Z}$ such that $AMB$ is in Smith normal form. It is essential that $A$ and $B$ be inv... | 4 | https://mathoverflow.net/users/5229 | 44691 | 28,372 |
https://mathoverflow.net/questions/44692 | 25 | If ${L}$ is a line bundle over a complex manifold, what does the square root line bundle $L^{\frac{1}{2}}$ mean?
After some google, I got to know that there are certain conditions for the existence of square root line bundle. In particular,I've following questions :
1. What is the square root of a line bundle, what a... | https://mathoverflow.net/users/9534 | What is a square root of a line bundle? | If $L$ is any line bundle over a compex manifold $X$, a square root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. So your guess in part (2) is correct.
This square root (if it exists) is not unique in general, and two of them will differ by a $2$-torsion line bundle, that is a line bundle $\eta$ such that $... | 37 | https://mathoverflow.net/users/7460 | 44700 | 28,379 |
https://mathoverflow.net/questions/44657 | 5 | What does the principal L-functions on GL(n), $n \geq 3, n \in \mathbb{Z}$, look like?
Where can I find materials about principal L-functions on GL(n)?
| https://mathoverflow.net/users/1930 | Principal L-functions on GL(n) | There are several ways to attack standard L-functions (I prefer "standard" over "principal" because it is associated to the standard representation of GL(n) on an n-dimensional space).
I'm going to assume that by "look like" you mean the formula for the local factors as a function of the local data. At primes where t... | 18 | https://mathoverflow.net/users/6753 | 44701 | 28,380 |
https://mathoverflow.net/questions/44703 | 1 | Is the following a standard problem in combinatorics? Where can I find reference for it?
Consider $n$ particles in a circle, $k$ white and $n-k$ black, otherwise indistinguishable so that the number of dispositions is $n!/(n-k)!k!$. Different dispositions will have a different number of white/black/white/black... clu... | https://mathoverflow.net/users/3441 | clusters of coloured particles | We will let $k\_1=k$ and $k\_2=n-k$, and $\tilde{c}=c/2$. I believe the answer is then
$\frac{k\_1 + k\_2}{\tilde{c}} {k\_1-1 \choose \tilde{c}-1} {k\_2-1 \choose \tilde{c}-1}$.
I have no idea whether there is a reference for this anywhere.
Here's how it works. First we solve (nearly) the same problem on a line.... | 4 | https://mathoverflow.net/users/2294 | 44711 | 28,385 |
https://mathoverflow.net/questions/44680 | 6 | This question is slightly related to a popular one with the same title (see [here](https://mathoverflow.net/questions/27345)).
Let $k$ be a field with characteristic zero. It is known (see [Exercise 310](http://umpa.ens-lyon.fr/~serre/DPF/exobis.pdf)) that a matrix $A\in M\_n(k)$ is nilpotent if and only if it is a c... | https://mathoverflow.net/users/8799 | Norm of commutators (bis) | Consider the matrix $$\left\[ \begin{array}{ccc} 0 & 1 & k \\\ 0 & 0 & 1\\\ 0 & 0 & 0\end{array}\right\].$$ Then the norm of $B$ depends on $k$ (just solve the system of linear equations $AB-BA=A$). So the answer to your question seems to be "no".
| 6 | https://mathoverflow.net/users/nan | 44715 | 28,388 |
https://mathoverflow.net/questions/44705 | 55 | It seems that in most theorems outside of set theory where the size of some set is used in the proof, there are three possibilities: either the set is finite, countably infinite, or uncountably infinite. Are there any well known results within say, algebra or analysis that require some given set to be of cardinality st... | https://mathoverflow.net/users/6856 | Cardinalities larger than the continuum in areas besides set theory | The Zariski tangent space at any point of a positive dimensional $C^1$-manifold $X$ has dimension $2^{2^{\aleph\_0}}= 2^{\frak c}$.
Let me explain in the case when $X=\mathbb R$.
Consider the ring $C^1\_0$ of germs of $C^1$- functions at $0\in \mathbb R$ and its maximal ideal $\frak m $ of germs of functions vanishin... | 68 | https://mathoverflow.net/users/450 | 44733 | 28,400 |
https://mathoverflow.net/questions/44652 | 1 | Recall:
Let $FU\_\bullet:Cat\to Cat\_\Delta$ be the bar construction assigned to the comonad $FU$ determined by free-forgetful adjunction $F:Quiv\rightleftarrows Cat:U$. The restriction of $FU\_\bullet$ to the full subcategory $\Delta$ (which is isomorphic to the category of finite nonempty ordinals) naturally determ... | https://mathoverflow.net/users/1353 | Reducing straightening over an interval to straightening over a point | (Community Wiki Answer)
Urs Schreiber answered this question over at the [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=2055&page=1#Item_4), so I'll reproduce his answer here (since I don't think he's going to add it himself)
---
Since $\mathfrak{C}$ is left adjoint we can e... | 0 | https://mathoverflow.net/users/1353 | 44763 | 28,416 |
https://mathoverflow.net/questions/44722 | 1 | In a discussion today on the Shafarevich-Tate group of an elliptic curve, the following structure and question came up. I will abuse many notations and be very vague about some things, but am very open to suggestions for clarification. Not to mention that some, if not all, of the following is incorrect.
A key ingredi... | https://mathoverflow.net/users/2024 | Is the direct limit of Weil restriction of an elliptic curve a scheme? | I agree with Adam Smith that the question seems a bit misguided, but let me show anyway that the answer is negative away from certain silly cases. Well, first to make a more well-posed question, one first has to adjust the definition of the functor so that it is at least a Zariski-sheaf (ideally without changing the "v... | 7 | https://mathoverflow.net/users/3927 | 44776 | 28,421 |
https://mathoverflow.net/questions/44774 | 37 | Let $L: C^\infty(\mathbb{R}) \to C^\infty(\mathbb{R})$ be a linear operator which satisfies:
$L(1) = 0$
$L(x) = 1$
$L(f \cdot g) = f \cdot L(g) + g \cdot L(f)$
Is $L$ necessarily the derivative? Maybe if I throw in some kind of continuity assumption on $L$? If it helps you can throw the "chain rule" into the li... | https://mathoverflow.net/users/1106 | Do these properties characterize differentiation? | Yeah, these force it to be ordinary differentiation. We have to show that for each fixed $x\_0 \in \mathbb{R}$, the composite
$$C^\infty(\mathbb{R}) \stackrel{L}{\to} C^\infty(\mathbb{R}) \stackrel{ev\_{x\_0}}{\to} \mathbb{R}$$
is just the derivative at $x\_0$. For each $f \in C^\infty(\mathbb{R})$, there is a $C... | 56 | https://mathoverflow.net/users/2926 | 44778 | 28,422 |
https://mathoverflow.net/questions/44777 | 2 | I derive this question while trying to prove the monotonicity of a differentiable vector function $f(x)$ that maps from $X\subset R^n$ to $R^n$ (Here function $f(x)$ is called monotone if $(x-y)'(f(x)-f(y))\geq 0$, $\forall x,y\in X$). The domain $X$ only consists of vectors $x$ such that $1'x=0$, here $1$ is the vecto... | https://mathoverflow.net/users/7595 | How to characterize real square matrices A, such that v'Av >= 0, for all real vectors v with 1'v=0 (1 is the vector of all ones)? | If $A$ is symmetric, then the matrices that you mention are called:
**Conditionally positive definite** (CPD) --- these are intimately related to the venerable *infinitely divisible matrices*
There is a vast amount of literature on these matrices, some useful pointers can already be found in R. Bhatia's wonderful b... | 8 | https://mathoverflow.net/users/8430 | 44789 | 28,430 |
https://mathoverflow.net/questions/44735 | 3 | By a Classification of Dickson everysubgroup of PSL(2,p) has index at least p+1
is there an easy proof with out this classification??
What can be said about the minimal index of subgroups PSL(r,q)??
There is a classification of subgroups of PSL(3,p) by Bloom , even for this
list i could not calculate all the index... | https://mathoverflow.net/users/10551 | minimal index of proper subgroups of PSL(r,q) | The first published proof that the index of a subgroup of PSL$(2,p)$ is at least $p+1$ for primes $p \ge 13$ is in:
C. Jordan, "Note sur les equations modulaires", C.R. Acad. Sci. Paris 66 (1868), 308-312,
a long time before the classification!
The minimal indexes of subgroups of classical simple groups are deter... | 5 | https://mathoverflow.net/users/35840 | 44794 | 28,433 |
https://mathoverflow.net/questions/44738 | -1 | Is finite dual of an algebra morphism a morphism of coalgebras? Does taking finite dual preserves exactness of an exact sequence of algebra morphisms? When is this possible?
| https://mathoverflow.net/users/9492 | Finite dual of an algebra morphism. | Gee, I have not a clue what is up, doc! I have never seen an exact sequence of algebra morphisms because usual linear kernels are not subalgebras. Having said that, algebra morphisms may conceivably have kernels but you need to expand on that
BTW, the answer to the first question is yes! All it is kinda saying that $... | 1 | https://mathoverflow.net/users/5301 | 44795 | 28,434 |
https://mathoverflow.net/questions/44582 | 11 |
>
> **Quick version of the question**. Let $(X, \mu)$ be a probability measure space and let $Z$, the group of integers, act on $X$ in a measure preserving way. How can I decompose $X$ into ergodic componenets? More precisely, can $X$ be equivariantly decomposed into a countable union of subspaces $U\_i$, each of whi... | https://mathoverflow.net/users/2631 | Decomposition of a dynamical system into ergodic componenents | Answer to the quick version. Yes it is true as soon as $(X,\mu)$ is a Lebesgue space. Beware that the transformation on the product $A\_i\times B\_i$ is not necessarily a true product, but instead it is a skew-product of the form $(x,y)\mapsto (T\_x(y),y)$. This follows from the ergodic decomposition theorem, together ... | 4 | https://mathoverflow.net/users/6129 | 44806 | 28,440 |
https://mathoverflow.net/questions/44801 | 34 | So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and solvable extensions unramified everywhere for his students to play with and that he had find this easy to construct with class ... | https://mathoverflow.net/users/2284 | $A_5$-extension of number fields unramified everywhere | If you take the splitting field of $x^5+ax+b$ and consider it as an extension of its quadratic subfield, then it will be unramified with Galois group contained in $A\_5$ whenever $4a$ and $5b$ are relatively prime. This is a result of [Yamamoto](http://www.ams.org/mathscinet-getitem?mr=266898). For almost all $a$ and $... | 24 | https://mathoverflow.net/users/297 | 44812 | 28,444 |
https://mathoverflow.net/questions/44804 | 5 | Consider the following system of equations:
$$
\sum\_{i=1}^{2n}a\_i=0
$$
$$
\sum\_{i=1}^{2n}\frac{1}{a\_i}=0
$$
Where for each $i$ $a\_i$ is an odd integer and the $a\_i$ are not necessarly distinct. A solution $(a\_1,\dots,a\_{2n})$ is trivial
if (after some permutation of the coefficients) for each $i$ we have
$$... | https://mathoverflow.net/users/5001 | A Diophantine problem related to egyptian fractions | Well, -1,3,3,5,5,-15 comes to mind. This is 2n variables for n=3. But you said you know there are non-trivial solutions for n>2. Did you mean "for every n>2" ? If so, what are you asking? Also, why not consider the case of an odd number of integers (some of which would be even)?
| 4 | https://mathoverflow.net/users/8008 | 44820 | 28,447 |
https://mathoverflow.net/questions/44684 | 3 | Consider the following counting problem (or the associated decision problem): Given two positive integers encoded in binary, compute their greatest common divisor (gcd). What is the smallest complexity class this problem is contained in? Can you provide a reference?
In this question I am not primarily interested in a... | https://mathoverflow.net/users/10539 | complexity of greatest common divisor (gcd) | I cross-posted this question on [stackexchange](https://cstheory.stackexchange.com/questions/2708/complexity-of-greatest-common-divisor-gcd%20%22here%22) and John Watrous posted an answer. The gist was that it is not known whether gcd is in NC or P-complete. See, e.g., "J. Sorenson. Two fast GCD algorithms. Journal of ... | 2 | https://mathoverflow.net/users/10539 | 44824 | 28,449 |
https://mathoverflow.net/questions/44823 | 3 | Let $f:E\to F$ be a morphism of vector bundles on an irreducible algebraic variety $X$. Does anybody know any results about the irreducibility or smoothness of the degeneracy locus of $f$? I know only the Connectedness Theorem due to Fulton.
| https://mathoverflow.net/users/33841 | Irreducibility\smoothness of the degeneracy locus | The degeneracy locus can be reducible, and even non-reduced. For instance, take $X= \mathbb{P}^2$ and consider a morphism
$\mathcal{O}(-1)^2 \stackrel{f} \to \mathcal{O}^2$.
$f$ is given by a $2 \times 2$ matrix of linear forms, so its degeneracy locus is a conic. For a general choice of $f$ this conic will be smoo... | 9 | https://mathoverflow.net/users/7460 | 44830 | 28,452 |
https://mathoverflow.net/questions/44831 | 2 | Given two smooth projective schemes $X$ and $Y$ over some algebraically closed fields $k$, one has the product $X\times Y$ with the projections $\pi\_X$ and $\pi\_Y$.
Now i have a coherent sheaf $M$ on $X$ and a coherent sheaf $N$ on $Y$, and i have locally free resolutions $M\_{\\*}\rightarrow M$ of length m and $N\... | https://mathoverflow.net/users/3233 | Flatness of the canonical projections | Yes. If $f: X\to S$ is a flat morphism and $S'\to S$ is an arbitrary morphism, then the projection $f\_{S'}: X\times\_S S'\to S'$ is flat. A reference is EGA IV.2.1.4.
| 2 | https://mathoverflow.net/users/8680 | 44836 | 28,454 |
https://mathoverflow.net/questions/44798 | 8 | Assume the topological group $\mathbb{R}$ acts properly on a space $X$. Does then the projection map $p:X\rightarrow \mathbb{R}\backslash X$ have local sections ?
(for every $\mathbb{R}x\in \mathbb{R}\backslash X$, there is a open neighbourhood $U \subset \mathbb{R}\backslash X$) and a section of $p|\_{p^{-1}(U)}:p^{-... | https://mathoverflow.net/users/3969 | local structure of free $\mathbb{R}$ actions | Such a theorem is proved for completely regular spaces $X$ in my article:
On the Existence of Slices for Actions of Non-Compact Lie Groups, Richard S. Palais, The Annals of Mathematics, Second Series, Vol. 73, No. 2 (Mar., 1961), pp. 295-323 .
Actually, that paper considers more general groups than just $\mathbb{R}... | 10 | https://mathoverflow.net/users/7311 | 44842 | 28,456 |
https://mathoverflow.net/questions/44845 | 8 | Let $X$ and $Y$ be projective varieties. I am assuming that there is some construction of a "moduli space" parametrizing the morphisms $X \to Y$. For instance, if one identifies such morphisms with their graphs in $X \times Y$, one can at least hope that these graphs would correspond to a locally closed subscheme of th... | https://mathoverflow.net/users/5094 | What is known about the "moduli space of morphisms" $X \to Y$? | It is called the Hom scheme, and I think it's defined in Grothendieck's [Bourbaki 221 paper](http://www.numdam.org/item?id=SB_1960-1961__6__249_0), where he constructs the Hilbert and Quot schemes. For any $S$-schemes $X$ and $Y$, $\underline{Hom}\_S(X,Y)$ assigns to any $S$-scheme $T$ the set $Hom\_T(X\_T,Y\_T)$. Osse... | 9 | https://mathoverflow.net/users/121 | 44847 | 28,458 |
https://mathoverflow.net/questions/44833 | 5 | Let $S$ be a semigroup. If $S$ is abelian, then it follows that the semigroup algebra $k[S]$ is finitely generated if and only if $S$ is.
What if we relax the condition on $k[S]$, so that $k[S]$ is only noetherian. Does it in this case follow that $S$ is finitely generated?
| https://mathoverflow.net/users/3996 | If $k[S]$ is noetherian, is S finitely generated? | It is an open problem (or was, last time I checked!) whether the noetherianity of $k[S]$ implies finite generation of $S$, when $S$ is not abelian.
This is discussed in chapter 5 of *Noetherian semigroup algebras* by Eric Jespers and Jan Okniński, along with various cases where we know that $S$ is finitely generated.... | 9 | https://mathoverflow.net/users/1409 | 44870 | 28,469 |
https://mathoverflow.net/questions/44859 | 25 | It seems like when we assume "niceness" in homotopy theory we assume that $X$ has the homotopy type of a CW complex, and in fiber bundle theory we assume that $X$ is paracompact. How do these two interact? Is any space with the homotopy type of a CW complex paracompact? (In particular, is $I^I$ paracompact?)
(CW comp... | https://mathoverflow.net/users/1874 | CW complexes and paracompactness | $I^I$ is paracompact. It is a theorem of [O'Meara](http://www.jstor.org/pss/2037695) that for $X$ a separable metric space, and $Y$ a metric space, then $Y^X$ - with the compact-open topology - is paracompact. $I$ is certainly a separable metric space, so the result holds.
As to your first question, I doubt that ever... | 10 | https://mathoverflow.net/users/4177 | 44872 | 28,470 |
https://mathoverflow.net/questions/44827 | 2 | I know my question is very imprecise. I am trying to understand Tate-Farrell cohomology of the infinite Lie group $S^1$ (say, with coefficients in $\mathbb C$). I would expect that the answer is something like the space of Laurent polynomials $\mathbb C[t^{-1},t]$. Is there any geometric intuition for this? What would ... | https://mathoverflow.net/users/6772 | Tate-Farrell cohomology of a circle | **Comment:** Farrell-Tate cohomology as defined in Brown's book "Cohomology of Groups"
requires the group to be of finite virtual cohomological dimension (i.e. the group has a finite index subgroup which has a finite projective resolution). But $S^1$ doesn't have finite virtual cohomological dimension because it has f... | 3 | https://mathoverflow.net/users/10194 | 44875 | 28,473 |
https://mathoverflow.net/questions/44877 | 12 | Suppose we are talking about graphs with $n$ labeled vertices. Which graphs are more common: connected or disconnected?
| https://mathoverflow.net/users/979 | Are there more connected or disconnected graphs on $n$ vertices? | Connectedness wins, since the complement of any disconnected graph is connected.
EDIT: Perhaps you'd like a proof of this. Let G be a disconnected graph, G' its complement. If v and u are in different components of G, then certainly they're connected by an edge in G'. And if they're in the same component of G, then t... | 63 | https://mathoverflow.net/users/1060 | 44889 | 28,481 |
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