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https://mathoverflow.net/questions/15832 | 7 | This is probably pretty basic, but as I said before I'm just beginning my way in the language of stacks.Say you have an etale cover X->Y of stacks (in the etale site). Is there a standard way to define the degree of this cover? Here's my intuition: if X and Y are schemes, we can look etale locally and then this cover i... | https://mathoverflow.net/users/3238 | Degrees of etale covers of stacks | There is a notion of degree of an integral morphism of DM stacks, and I think it is in Vistoli's "Intersection theory ... " paper [here](https://doi.org/10.1007/BF01388892). Its unrelated to Yoneda triviality.
| 5 | https://mathoverflow.net/users/2 | 15868 | 10,628 |
https://mathoverflow.net/questions/15867 | 0 |
>
> **Motivation**
>
>
> Consider the situation: You know that
> every $x$ that has property $P$ must have property $Q$. $Q$ is a
> rather strong condition but not strong
> enough to fulfill $P$. What is *missing*?
>
>
>
Consider the formulas of a first-order language, a distinguished set of axioms $S$, and... | https://mathoverflow.net/users/2672 | Deficiency of necessary conditions | Normally your "defect" is called an "additional assumption"/"extra condition"/... and the typical phrase is "the inverse implication also holds under the additional assumption that...". Yes, the search for such things is something that mathematicians do on an everyday basis trying to bridge the gap between what is nece... | 4 | https://mathoverflow.net/users/1131 | 15870 | 10,630 |
https://mathoverflow.net/questions/15872 | 13 | I thought I had heard or read somewhere that the existence of a non-principal ultrafilter on $\omega$ was equivalent to some common weakening of AC. As I searched around, I read that this is not the case: neither countable choice nor dependent choice are strong enough.
This leads me to two questions:
Where would I ... | https://mathoverflow.net/users/2143 | Non-principal ultrafilters on ω | See the [Prime ideal theorem](http://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem).
The existence of ultrafilters on every Boolean algebra (which implies non-principal ultrafilters on ω, since these come from ultrafilters on the Boolean algebra P(ω)/Fin) is a set-theoretic principle that follows from AC and is ... | 9 | https://mathoverflow.net/users/1946 | 15873 | 10,631 |
https://mathoverflow.net/questions/15850 | 2 | Is there a condition for a $G$-equivariant map $X \to Y$ to be a cofibration of $G$-spaces? Here $X$ and $Y$ are CW complexes, the group $G$ is finite, and acts by cellular maps.
I am using the model structure on CW-complexes with G-action where the fibrations and weak equivalences are those maps which are fibrations... | https://mathoverflow.net/users/1676 | characterization of cofibrations in CW-complexes with G-action | In the model structure you describe, the cofibrations should be the retracts of the free relative G-cell maps: i.e., retracts of maps obtained by attaching cells of the form $G \times S^{n} \to G \times D^{n+1}$.
One way to see this is via the following general machine: There is an adjoint pair
$$ G \times -: \mathbf... | 8 | https://mathoverflow.net/users/1921 | 15880 | 10,638 |
https://mathoverflow.net/questions/13901 | 8 | According to the Baez-Dolan cobordism hypothesis, an extended TQFT is determined by its value on a single point. This value a fully dualizable object of a symmetric monoidal $n$ category (a fully dualizable object is a higher categorical analogue of a finite dimensional vector space). The Alexander polynomial is a quan... | https://mathoverflow.net/users/2051 | What is the Alexander polynomial of a point? | Understanding the Alexander polynomial can get a bit technical, because the quantum dimension is zero. So I don't think I fully understand how you get a TQFT from the Alexander polynomial.
In a more typical situation, like the Jones polynomial (related to SL(2) instead of the more confusing GL(1|1) in the Alexander s... | 5 | https://mathoverflow.net/users/22 | 15882 | 10,640 |
https://mathoverflow.net/questions/15884 | 4 | In many places the existence of automorphism is acknowledged as one of the reasons why fine moduli spaces cannot exist. A typical example is the following.
Consider a curve $C$ with a nontrivial automorphism, for instance a hyperelliptic curve with its involution $\phi$. Now let $B$ be any scheme with a free action o... | https://mathoverflow.net/users/828 | How to use automorphisms to produce isotrivial non trivial families | Here is an explicit example. Denote by $E$ the elliptic curve $E=\mathbb C/(\mathbb Z+i\mathbb Z)$. Now let $B=C=E$. The involution that we will consider is
$$(z,w)\in (E\times E)\to (z+\frac{1}{2},-w).$$
Notice now that this quotient is not a direct product anymore. You can see this by calculating $H\_1(E\times E)/... | 7 | https://mathoverflow.net/users/943 | 15885 | 10,642 |
https://mathoverflow.net/questions/15856 | 1 | A theorem of Erdos states:
"There exists an absolute constant c such that, if n>ck, and if a1/b1, a2/b2, ... are the Farey fractions of order n, then ax/bx and ax+k/bx+k are similarly ordered."
Can someone provide a definition of "similarly ordered" as used here?
Thanks for any insight.
Cheers, Scott
@ARTICL... | https://mathoverflow.net/users/4111 | Similarly Ordered | Here is a link that gives the definition: <http://qjmath.oxfordjournals.org/cgi/pdf_extract/os-13/1/185>
It's to the paper by Mayer that Erdos refers to. (The link gives you just the first page of the paper, but fortunately the definition is on that page.)
| 3 | https://mathoverflow.net/users/1459 | 15891 | 10,646 |
https://mathoverflow.net/questions/15893 | 12 | This is a strictly technical question on peer-review systems currently employed in the mathematical literature, not a subjective discussion of merits/drawbacks of such systems, so I think/hope it's suitable for MO.
I have noticed that some journals (e.g. PNAS, CRAS, Nonlinearity...) always publish papers with the nam... | https://mathoverflow.net/users/469 | Editors in peer-review systems | Journals of scientific societies (such as PNAS, CRAS) were, once upon a time, records of meetings and a member of the society would actually present the papers he accepted to the other members, so in this case I believe it's just a tradition. In the case of Nonlinearity it's probably just an affectation. Pay no attenti... | 12 | https://mathoverflow.net/users/2290 | 15894 | 10,648 |
https://mathoverflow.net/questions/15888 | 3 | Hi,
I have a superspace spanned by 4 commuting coordinates + 2 anti-commuting ones $\{x^\mu,\theta^\alpha\}$, I have to do the change of coordinates $dx^\mu\to dy^\mu= dx^\mu+d\theta^\alpha \eta\_\alpha^{\;\:\mu}$ where $\eta$ have to be a local function i.e: $\eta\equiv\eta(x)$, and leave the $d\theta$'s unchanged, ... | https://mathoverflow.net/users/2597 | Change of coordinates introduced through dx | First of all, I would not call $dx^\mu \mapsto dy^\mu$ a change of coordinates. Not every transformation on 1-forms is going to be the pullback by a smooth map. In fact, your question is whether the transformation $(dx^\mu,d\theta^\alpha) \mapsto (dy^\mu,d\theta^\alpha)$ comes from a change of coordinates.
Pullbacks ... | 5 | https://mathoverflow.net/users/394 | 15898 | 10,651 |
https://mathoverflow.net/questions/993 | 70 | I was trying to explain finite groups to a non-mathematician, and was falling back on the "they're like symmetries of polyhedra" line. Which made me realize that I didn't know if this was actually true:
Does there exist, for every finite group G, a positive integer n and a convex subset S of R^n such that G is isomor... | https://mathoverflow.net/users/625 | Is every finite group a group of "symmetries"? | The permutohedron may have additional symmetries. For example, the order 3 permutohedron $\{(1,2,3),(1,3,2),(2,1,3),(3,1,2),(3,2,1)\}$ is a regular hexagon contained in the plane $x+y+z=6$, which has more than 6 symmetries.
I think we can solve it as follows:
Let $G$ be a group with finite order $n$ thought via Cay... | 28 | https://mathoverflow.net/users/4118 | 15903 | 10,652 |
https://mathoverflow.net/questions/15897 | 17 | **Background/motivation**
One of the main reason to introduce (algebraic) stacks is build "fine moduli spaces" for functors which, strictly speaking, are not representable. The yoga is more or less as follows.
One notices that a representable functor on the category of schemes is a sheaf in the fpqc topology. In pa... | https://mathoverflow.net/users/828 | In what topology DM stacks are stacks | The rule of thumb is this: Your DM (or Artin) stack will be a sheaf in the fppf/fpqc topology if the condition imposed on its diagonal is fppf/fpqc local on the target ("satisfies descent").
In other words, in condition 2 you asked that the diagonal be a relative scheme/relative algebraic space perhaps with some ext... | 23 | https://mathoverflow.net/users/1921 | 15910 | 10,656 |
https://mathoverflow.net/questions/15863 | 5 | Let's say $f$ is a Dirichlet series which converges on the half-plane $\text{Re }s>\sigma$ to a function $f(s)$. Suppose further that $f(s)$ admits an analytic continuation to an entire function, together with the standard sort of functional equation. Let $g\_n$ be a sequence of Dirichlet series, also convergent on $\t... | https://mathoverflow.net/users/271 | Convergence of a sequence of continuable Dirichlet series | (This answer is a community wiki version of a comment above by FC which answered the question.)
For any integer M, there exists a prime p such that chi\_p(n) = (n/p) = 1 for all n = 1...M. This means that the Dirichlet series L(s,V chi\_p) (for any representation V) "converges" in your sense to L(s,V). but they do no... | 2 | https://mathoverflow.net/users/22 | 15920 | 10,663 |
https://mathoverflow.net/questions/15759 | 11 | I have recently had occasion to investigate the Fourier series of the function $f(x,y)=\log({2+\cos 2\pi x} +\cos{2\pi y})$. Accordingly, define
$I(m,n)=\int\_{0,0}^{1,1}f(x,y)\cos{2\pi mx}\cos{2\pi ny}dxdy$
which is the $(m,n)$th Fourier coefficient. If there is some easy change of variables or flick of the wrist... | https://mathoverflow.net/users/1464 | A two-variable Fourier series and a strange integral | Here is a proof of the conjecture, a proof that also shows how to compute the integrals explicitly.
The proof is somewhat similar to David Speyer's approach, but instead of using multivariable residues, I will just shift a one-variable contour. Without loss of generality, $m>0$. Eliminating the trigonometric function... | 21 | https://mathoverflow.net/users/2757 | 15924 | 10,667 |
https://mathoverflow.net/questions/15925 | 3 | In *Wiersema: Brownian Motion Calculus* on p. 205 (in an Annex on Moment Generating Functions (mgf)) the following equation is being presented $${d^k \over d\theta^k} \left ({1\over k!}\theta^k\mathbb{E}[X^k]\right)={1\over k!}\theta^k{d^k \over d\theta^k} \mathbb{E}[X^k]$$ with $X$ being a random variable, $\theta$ th... | https://mathoverflow.net/users/1047 | Moment Generating Function: Pulling a term out of k-times differentiation | That's clearly a typo. Unless I misunderstand the point of this, the right formula should be
$${d^k \over d\theta^k} \left ({1\over k!}\theta^k\mathbb{E}[X^k]\right)={1\over k!}\mathbb{E}\left[{d^k \over d\theta^k} \theta^k X^k\right]$$
In fact, this is what is said in words before the formula in the book you mention... | 3 | https://mathoverflow.net/users/394 | 15929 | 10,671 |
https://mathoverflow.net/questions/15904 | 7 | Let $(X, x\_0)$ be a pointed space. Then we can define the homotopy groups $\pi\_i(X, x\_0)$ for $i \geq 1$. They are abelian groups for $i \geq 2$. It is well-known that the fundamental group $\pi\_1(X, x\_0)$ acts on each of the higher groups $\pi\_i(X, x\_0)$, and that this action generalizes to the [Whitehead Produ... | https://mathoverflow.net/users/184 | Whitehead Products without Base Points? | As I posted in my comment, I think Paul's suggestion does work. Here's a (sloppy) description of how I think things will work:
The local systems you describe can be obtained, by passing to homotopy groups, from a "local system of loop spaces" $$ \Omega: \Pi\_{\leq \infty} X \to \Omega\mathbf{Spaces}$$
One can imagin... | 6 | https://mathoverflow.net/users/1921 | 15934 | 10,675 |
https://mathoverflow.net/questions/15902 | 22 | I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G\_{Q\_p}$ over $F\_p$ which are in correspondence with 1-dimensional etale (phi,Gamma)-modules over $F\_p((T))$.
There are finit... | https://mathoverflow.net/users/86179 | One dimensional (phi,Gamma)-modules in char p | OK...I think I see how to do this now. In the end, I am seeing $(p-1)^2$ distinct $(\phi,\Gamma)$-modules which matches well with the Galois side.
To do this, let $D$ be any 1-dimensional etale $(\phi,\Gamma)$-module. Let $e$ be a basis, and set $\phi(e)=h(T)e$ with $h(T) \in F\_p((T))^\times$. Write $h(T) = h\_0 T^a... | 13 | https://mathoverflow.net/users/86179 | 15940 | 10,680 |
https://mathoverflow.net/questions/15944 | 2 | I call a set X of positive integers *strongly lcm-closed* if a,b ∈ X if and only if lcm(a,b) ∈ X. In the finite case X is the set of divisors of lcmx ∈ Xx. But in the infinite case it is more interesting, for example, $\{a \geq 1: a \not\equiv 0 \pmod p\}$ and $\{p^a:a \geq 0\}$ for any prime p, are strongly lcm-closed... | https://mathoverflow.net/users/2264 | Classification of strongly lcm-closed sets | Given a [supernatural number](http://en.wikipedia.org/wiki/Supernatural_numbers) $N$, the set of positive integer divisors of $N$ is a strongly lcm-closed set. And any nonempty strongly lcm-closed set $X$ arises in this way, with $N$ equal to the supernatural lcm of the $x$ in $X$. (See Serre, *Galois cohomology* for t... | 5 | https://mathoverflow.net/users/2757 | 15947 | 10,685 |
https://mathoverflow.net/questions/15951 | 23 | I've been reading up a bit on the fundamentals of formal logic, and have accumulated a few questions along the way. I am pretty much a complete beginner to the field, so I would very much appreciate if anyone could clarify some of these points.
1. A complete (and consitent) propositional logic can be defined in a num... | https://mathoverflow.net/users/602 | Propositional Logic, First-Order Logic, and Higher-Order Logics | This is a long list of questions! These are all related to a certain extent, but you might consider breaking it up into separate questions next time.
1. Proof theorists tend to prefer systems with many rules and few axioms such as natural deduction systems and Gentzen systems. The reason is that these are much easier... | 35 | https://mathoverflow.net/users/2000 | 15954 | 10,690 |
https://mathoverflow.net/questions/15955 | 8 | Here's a mathematical modeling problem I came across while working on a hobby project.
I have a website that presents each visitor with a list of movie titles. The user has to rank them from most to least favorite. After each visit, I want to create a cumulative ranking that takes into account each visitor's individu... | https://mathoverflow.net/users/4135 | Defining "average rank" when not every ranking covers the whole set | There are a few different ways of approaching the problem. A good reference for this precise problem is '[Rank Aggregation methods For The Web](http://www.eecs.harvard.edu/~michaelm/CS223/rank.pdf)', by Dwork, Kumar, Naor and Sivakumar from the WWW conference in 2001. It's not the most recent work, but it lays out the ... | 5 | https://mathoverflow.net/users/972 | 15958 | 10,691 |
https://mathoverflow.net/questions/15909 | 6 | I need to calculate (or bound) the probability that one binomial variable is greater than other. Specifically, if $x \leftarrow B(n,p)$ and $y \leftarrow B(n,q)$, what is the probability that $y \geq x$? Informally, I will flip two bent coins with known biases $n$ times each, and I would like a convenient expression/bo... | https://mathoverflow.net/users/4120 | Probability of one binomial variable being greater than another. | Edit: I've filled in a few more details.
The Hoeffding bound from expressing $Y-X$ as the sum of $n$ differences between Bernoulli random variables $B\_q(i)-B\_p(i)$ is
$$Prob(Y-X \ge 0) = Prob(Y-X + n(p-q) \ge n(p-q)) \le \exp\bigg(-\frac{2n^2 (p-q)^2}{4n}\bigg)$$
$$Prob(Y-X \ge 0) \le \exp\bigg(-\frac{(p-q)^2}... | 5 | https://mathoverflow.net/users/2954 | 15961 | 10,693 |
https://mathoverflow.net/questions/11396 | 5 | Let $F$ be a local field of characteristic zero (for simplicity), $\overline{F}$ an algebraic closure of $F$ and $L/F$ a fixed finite Galois extension. If $G$ is a linear algebraic group defined over $F$, then the Galois cohomology group $H^1(F,G)$ can be defined as a direct limit of $H^1(K/F,G)$, where $K$ runs throug... | https://mathoverflow.net/users/1832 | Galois cohomology of linear groups over local fields | As Hunter and Sean noted, since the inflation map ${\rm{H}}^1(L/F,G(L)) \rightarrow {\rm{H}}^1(F,G)$ is
injective and ${\rm{H}}^1(F,G)$ is always finite (Borel-Serre), such an $L$ always exists. Below we give an explicit sufficient condition on $L$ (often satisfied) when $G$ is connected. (One could probably do better ... | 14 | https://mathoverflow.net/users/3927 | 15963 | 10,695 |
https://mathoverflow.net/questions/15087 | 35 | Are there any general methods for computing fundamental group or singular cohomology (including the ring structure, hopefully) of a projective variety (over C of course), if given the equations defining the variety?
I seem to recall that, if the variety is smooth, we can compute the H^{p,q}'s by computer -- and thus ... | https://mathoverflow.net/users/83 | Computing fundamental groups and singular cohomology of projective varieties | This is an interesting question. To repeat some of the earlier answers, one should be able to get one's hands on a triangulation algorithmically using real algebro-geometric methods, and thereby compute singular cohomology and (a presentation for) the fundamental group. But this should probably be a last resort in prac... | 25 | https://mathoverflow.net/users/4144 | 15974 | 10,703 |
https://mathoverflow.net/questions/15979 | 14 | In the search for a Weil cohomology theory $H$ over a field $K$ (with $\text{char}(K)=0$) for varieties in characteristic $p$, a classical argument by Serre shows that the coefficient field cannot be a subfield of $\mathbb{R}$ or of $\mathbb{Q}\_p$; an obvious choice is to take $\mathbb{Q}\_\ell$ for a prime $\ell \neq... | https://mathoverflow.net/users/362 | Motivation for the étale topology over other possibilities | As observed, the Zariski topology has too few open subsets to compute cohomology with constant coefficients. It had already been observed by Serre that etale covers were enough to trivialize principal bundles for many algebraic groups. That suggested using etale covers. The etale "topology" is the coarsest for which th... | 14 | https://mathoverflow.net/users/930 | 15980 | 10,708 |
https://mathoverflow.net/questions/15981 | 1 | I found the following closed form solution for the abovementioned problem:
$${1\over k^n}\cdot{k!\over (k-m)!}\cdot{\{{n\over m}\}}$$ with ${\{{n\over m}\}}$ being the Stirling Number of the second kind.
Although it seems to have some intuition and seems to work for a sample problem for which I have the solution th... | https://mathoverflow.net/users/1047 | Probability of n k-sided dice showing exactly m different faces | Applied probability by Kenneth Lange deals with this problem on page 74. It is on Google books, here is the [URL](http://books.google.com/books?id=otp4TDnz6FwC&pg=PA74&lpg=PA74&dq=Stirling+Number+of+the+second+kind+dice&source=bl&ots=YHHbtcknAO&sig=sGd1_d1UXtlNkWGYINVZH6Tadu0&hl=en&ei=boiBS-CpDImKsgOq6en5Aw&sa=X&oi=boo... | 5 | https://mathoverflow.net/users/1098 | 15983 | 10,710 |
https://mathoverflow.net/questions/15987 | 10 | I was feeling a bit rusty on my field theory, and I was reviewing out of McCarthy's excellent book, *Algebraic Extensions of Fields*. Out of Chapter 1, I was able to work out everything "left to the reader" or omitted except for one corollary, stated without proof ([see here](http://books.google.com/books?id=dC5HvUrfy4... | https://mathoverflow.net/users/1916 | If K/k is a finite normal extension of fields, is there always an intermediate field F such that F/k is purely inseparable and K/F is separable? | Edit: My mistake, I misread your post. Here's the correct answer.
<http://books.google.com/books?id=FJmiSW1KRBAC&lpg=PP1&ots=k1ecm3FdbZ&dq=lang%20algebra&pg=PA251#v=onepage&q=&f=false>
Proposition 6.11
| 2 | https://mathoverflow.net/users/1353 | 15988 | 10,713 |
https://mathoverflow.net/questions/15907 | 19 | It is known that there exists a fine moduli space for marked (nonalgebraic) K3 surfaces over $\mathbb{C}$. See for example the book by Barth, Hulek, Peters and Van de Ven, section VIII.12. Of course the marking here is necessary, otherwise the presence of automorphisms can be used to construct isotrivial non trivial fa... | https://mathoverflow.net/users/828 | Moduli space of K3 surfaces | I think you are looking for this
<http://arxiv.org/pdf/math/0506120>
which is the same as:
Rizov, Jordan Moduli stacks of polarized $K3$ surfaces in mixed characteristic. Serdica Math. J. 32 (2006), no. 2-3, 131--178.
It builds on an earlier result:
Olsson, Martin C. Semistable degenerations and period spaces... | 13 | https://mathoverflow.net/users/404 | 15997 | 10,718 |
https://mathoverflow.net/questions/15967 | 4 | I have a camera matrix $P$ which defines a projective transformation $\mathbb{P}^3 \rightarrow \mathbb{P}^2$. In the former space there is a plane $[ x|\pi^Tx=0 ]$. The image of the plane under $P$ does not preserve angles.
How can I find a transformation $H : \mathbb{P}^2 \rightarrow \mathbb{P}^2$ such that a right a... | https://mathoverflow.net/users/818 | Rectifying texture from image | By picking orthogonal coordinates in the given plane you can make an angle preserving projective map $\mathbb{P}^2\to\mathbb{P}^3$ whose image is the given plane. Composing with your camera mapping, you now have a mapping $G\colon\mathbb{P}^2\to\mathbb{P}^2$ that does not preserve angles. Let $H=G^{-1}$. The compositio... | 2 | https://mathoverflow.net/users/802 | 16006 | 10,725 |
https://mathoverflow.net/questions/15991 | 3 | In general, the étale topology does not form a topology in the strict sense. However, is there any subcategory of $Sch$ where we can realize the étale topology as an honest topology on some scheme?
| https://mathoverflow.net/users/1353 | Can the étale topology ever be realized as an "honest" topology? | I think Senor Borger speaks truth.
Maybe I am talking non-sense, but maybe this could be a sketch of some way to proceed (maybe not....):
Key feature of a site which we cannot have in a classical topology is that there may be several distinct "inclusion" morphisms from a smaller open into a bigger one.
Hence, we ... | 4 | https://mathoverflow.net/users/3888 | 16008 | 10,727 |
https://mathoverflow.net/questions/16010 | 1 | The terminology would suggest that a separable field extension is so because the resulting field extension has some sort of separable topology, and that a normal extension corresponds to one with a normal topology.
I imagine this is true, or else they wouldn't have named them in such a way.
Also, I'm not sure what ... | https://mathoverflow.net/users/96 | Do separable and normal have topological meanings for fields? | This is incorrect. Words like "separable" and "normal" occur in unrelated ways in various parts of mathematics. (Normal subgroup, separable differential equations...)
Other words, too. Like "regular", "perfect" ...
"Separable" in toplogy... Does it mean something can be "separated"? What? I believe it goes back to Fr... | 4 | https://mathoverflow.net/users/454 | 16015 | 10,731 |
https://mathoverflow.net/questions/15990 | 18 | My question points in a direction similar to [Qiaochu's](https://mathoverflow.net/questions/10947/whats-the-analogue-of-the-hilbert-class-field-in-the-following-analogy), but it's not the same (or so I think). Let me provide you with a little bit of background first.
Let E be an elliptic curve defined over some numbe... | https://mathoverflow.net/users/3503 | What's the Hilbert class field of an elliptic curve? | **EDIT**: This is a completely new answer.
I will prove that your specific suggestion of defining a Hilbert class field of an elliptic curve $E$ over $K$ does not work. I am referring to your proposal to take the smallest field $L$ such that the corestriction (norm) map $\operatorname{Sha}(L) \to \operatorname{Sha}(K... | 15 | https://mathoverflow.net/users/2757 | 16018 | 10,734 |
https://mathoverflow.net/questions/13856 | 7 | I'm interested in results about functorially-defined subgroups (in a loose sense), especially in the non-abelian case, and would like to know about references I may have missed.
The question, it seems, comes up in its simplest form when noticing a number of common subgroups (the center, commutator subgroup, Frattini ... | https://mathoverflow.net/users/1307 | References on functorially-defined subgroups | After some research, I asked on the [Group-pub mailing list](http://people.bath.ac.uk/masgcs/gpf.html), where [very knowledgeable people roam](http://people.bath.ac.uk/masgcs/folk/folk.html) (the University of Bath, which hosts the mailing-list, also hosted the ['Groups St Andrews' conference in 2009](http://www.groups... | 3 | https://mathoverflow.net/users/1307 | 16046 | 10,754 |
https://mathoverflow.net/questions/16048 | 5 | I found myself "naturally" dealing with an object of this form:
X is a complex vector space, with a "product" (a,b) → {aba} which is quadratic in the first variable, linear in the second, and satisfies some associativity conditions. These conditions are actually complicated, but more or less say that {aba} looks like... | https://mathoverflow.net/users/1626 | Is there a name for this algebraic structure? | In a Jordan algebra with product $\cdot$, a triple product is defined by $$\{abc\}=(a\cdot b)\cdot c+(b\cdot c)\cdot a-(a\cdot c)\cdot b.$$
In a special Jordan algebra (constructed by symmetrising an associative product) one has $\{aba\}=aba$, and it is easy to show that in such algebras one always has the identity $$\... | 26 | https://mathoverflow.net/users/1409 | 16057 | 10,762 |
https://mathoverflow.net/questions/16047 | 7 | I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the étale fundamental group in positive characteristics is a very difficult object, especially so in the non-projective case d... | https://mathoverflow.net/users/259 | Simply connected quasi-projective varieties in positive characteristic | This is an answer to Pete's question on simply connected affine varieties (I can not put it in a comment because of space limitation).
I think that in positive characteristic $p$, no affine **irreducible** variety $X$ of positive dimension is simply connected. We can assume $X=\operatorname{Spec}(A)$ integral becaus... | 10 | https://mathoverflow.net/users/3485 | 16060 | 10,765 |
https://mathoverflow.net/questions/16063 | 2 | Given a scheme $X$ with generic point p and a quasi-coherent sheaf $F$ on $X$.
Viewing $X$ as a scheme over $Spec(\mathbb{Z})$, let us assume
$f: X \rightarrow Spec(\mathbb{Z})$ is a proper map.
What conditions have $X$ and $F$ to satisfy, so that one can embed the $\mathbb{Z}$-module $F(X)=H^0(X,F)$ in $F\_p$, resp... | https://mathoverflow.net/users/3233 | When is the restriction map on global sections an embedding | If $X$ is integral and $F$ is torsion-free, then for any non-empty affine open subset $U$ of $X$, the canonical map $F(U)\to F\_p$ is injective. So $F(X)\to F\_p$ is injective. You don't need hypothesis on $X \to Spec(\mathbb Z)$. If $X$ is not necessarily reduced, then the flatness of $F$ over $X$ is also enough (same... | 5 | https://mathoverflow.net/users/3485 | 16065 | 10,768 |
https://mathoverflow.net/questions/16066 | 1 | In one paper I saw this equality:
$$\sum\_{\eta=-\infty}^{\infty}\frac{z}{(z+\eta)}=\pi z\cot(\pi z)$$
which is the same as
$$\sum\_{\eta=-\infty}^{\infty}\frac{1}{(z+\eta)}=\pi \cot(\pi z)$$
where summation is understood in the sense of a principal value. What does it mean?
In another paper I found the next expres... | https://mathoverflow.net/users/3589 | what is summation in the sense of a principal value? | A principal-value sum (or integral) is usually one in which unconditional summation (or integration) does not converge, so one needs to sum in a particular way to achieve convergence. I suspect that, in this case, the necessary summation is symmetric, so that we consider $\lim\_{N \to \infty} \sum\_{n = -N}^{n = N} f(n... | 2 | https://mathoverflow.net/users/2383 | 16067 | 10,769 |
https://mathoverflow.net/questions/14764 | 16 | I just started to collect the papers of this field and know little things. So if I make stupid mistake, please correct me.
It seems that there are several approaches to localize Kac-Moody algebra(in particular, affine Lie algebra). I just took look at several papers by
**Kashiwara-Tanisaki:(1989)**
They construct... | https://mathoverflow.net/users/1851 | What is the recent development of D-module and representation theory of Kac-Moody algebra? | I am not an expert in this but I would of course expect something like ind-scheme approach to be natural. Gerd Faltings used I think ind-schemes to treat Sugawara construction, algebraic loop groups and Verlinde's conjecture in
Gerd Faltings, Algebraic loop groups and moduli spaces of bundles.
J. Eur. Math. Soc. (J... | 7 | https://mathoverflow.net/users/35833 | 16075 | 10,775 |
https://mathoverflow.net/questions/16095 | 1 | From wikipedia [quantification](http://en.wikipedia.org/wiki/Quantification) has meaning:
>
> In logic, quantification is the
> binding of a variable ranging over a
> domain of discourse
>
>
>
**Is there any formal "definition" of universal quantifier for example using definition of domain of discourse?**
... | https://mathoverflow.net/users/3811 | Is there formal definition of universal quantification? | There is a definition in terms of $\varepsilon$-operator of Hilbert. See [wikipedia](http://en.wikipedia.org/wiki/Epsilon_calculus). If not, either universal quantification or existential quantification is taken as primitive in classical logic, for in classical logic, one is derivable from the other. This is not true i... | 3 | https://mathoverflow.net/users/1353 | 16096 | 10,788 |
https://mathoverflow.net/questions/16107 | 20 | I have been working on this problem for several months now but have not made much progress. It concerns the set of all [integer partitions](https://en.wikipedia.org/wiki/Integer_partitions) of n.
Let the vertices of the graph G=G(n) denote all the p(n) integer partitions of n. There is an edge between two partitions ... | https://mathoverflow.net/users/4176 | Hamiltonian paths where the vertices are integer partitions | What you want is known as a Gray code for integer partitions.
It exists.
See C. D. Savage, [Gray code sequences of partitions](https://doi.org/10.1016/0196-6774(89)90007-2), *Journal of Algorithms* 10 (1989) 577-595. Disclaimer: I haven't actually read this paper. Other sources such as [this 1997 survey by Savage](... | 18 | https://mathoverflow.net/users/143 | 16112 | 10,798 |
https://mathoverflow.net/questions/16105 | 7 | The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi\_1(F)\to SU(n)$. We can define the homology with local coefficients $H\_\*(F,\rho)$ straightforwardly as the homology of the twisted complex $$C\_\*(F,\rho):=C\_\*(\widetilde... | https://mathoverflow.net/users/492 | Intersection form in twisted homology (homology with local coefficients) | For me it is easier to work with cohomology (just for psychological reasons). Also, I will distinguish the representation $\rho$ from the local system $V$ with fibres ${\mathbb C}^2$ that it gives rise to. So where you would write $H^1(F,\rho)$ I will write $H^1(F,V)$.
I will let $\overline{V}$ denote the complex conju... | 5 | https://mathoverflow.net/users/2874 | 16119 | 10,803 |
https://mathoverflow.net/questions/15957 | 22 | We say that a group *G* is in the class *Fq* if there is a CW-complex which is a *BG* (that is, which has fundamental group *G* and contractible universal cover) and which has finite *q*-skeleton. Thus *F0* contains all groups, *F1* contains exactly the finitely generated groups, *F2* the finitely presented groups, and... | https://mathoverflow.net/users/4133 | Is any interesting question about a group G decidable from a presentation of G? | It seems to me that the analogue of Rice's theorem fails for finitely presented
groups $G$ because of questions like: is the abelianization of $G$ of rank 3?
The rank of the abelianization of any finitely presented $G$ can be computed
by reducing the abelianization to normal form, so this (slightly) interesting
questio... | 12 | https://mathoverflow.net/users/1587 | 16122 | 10,805 |
https://mathoverflow.net/questions/16121 | 13 | Can someone give a really concrete example of such a sequence? I am looking at several notes related with such things, but haven't seen any well-calculated example. And I'm really confused at this point.
Besides asking for a good example, I am also wondering about the following two things:
1. There is an exact sequ... | https://mathoverflow.net/users/1238 | Example of connected-etale sequence for group schemes over a Henselian field? | Regarding 1. : If you pass to the $n$-torsion parts of the members of this exact sequence,
you will get the $R$-valued points
of the connected-etale sequence for $E[n]$.
Regarding 2. : If $E$ has good reduction, then $E[n]$ is a finite flat group scheme.
If the residue char. $p$ of $R$ does not divide $n$ then it is ... | 8 | https://mathoverflow.net/users/2874 | 16130 | 10,810 |
https://mathoverflow.net/questions/16085 | 12 | I recently worked through most of the proof of the rationality of the moduli of genus 3 curves, which seemed to have the following structure:
1. Every nonhyperelliptic genus 3 curve is a smooth plane quartic.
2. The plane quartics form a projective space.
3. Apply GIT to this projective space and the $PGL(3)$ action.... | https://mathoverflow.net/users/622 | Rationality of GIT quotients | A useful general result is the 'no-name lemma' stating that when a reductive group $G$ acts linearly on two vector spaces $V$ and $W$ 'almost freely' (that is, the stabilizer subgroup of a general point is trivial), then the GIT-quotients $V/G$ and $W/G$ are stably rational (that is, $V/G \times \mathbb{C}^m$ and $W/G ... | 10 | https://mathoverflow.net/users/2275 | 16133 | 10,811 |
https://mathoverflow.net/questions/16104 | 16 | A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is:
1. Which Fréchet manifolds have a smooth partition of unity?
2. How is the existence of smooth partitions of unity on Fréchet manifolds related to paracompactness of the underlying topology?
... | https://mathoverflow.net/users/3473 | Which Fréchet manifolds have a smooth partition of unity? | Use the [source](https://ncatlab.org/nlab/show/The+Convenient+Setting+of+Global+Analysis), Luke.
Specifically, chapters 14 (Smooth Bump Functions) to 16 (Smooth Partitions of Unity and Smooth Normality). You may be particularly interested in:
>
> **Theorem 16.10** If $X$ is Lindelof and $\mathcal{S}$-regular, the... | 14 | https://mathoverflow.net/users/45 | 16143 | 10,818 |
https://mathoverflow.net/questions/16146 | 7 | Fix an algebraically closed **†** ground field $k$ of any characteristic. I want to use the classical definition of projective $n$-space $\mathbb{P}^n$ as set quotient of $\mathbb{A}^{n+1}\setminus 0$ by the action of $k^\*$, and for its Zariski topology the definition that closed sets are the "zero sets" of homogeneou... | https://mathoverflow.net/users/84526 | Elementary proof that projective space is a quotient | Let $f$ be a polynomial which vanishes on $\hat{S}$. Write $f=\sum f\_i$, where $f\_i$ is homogenous of degree $i$. The set $\hat{S}$ is homogenous so, for any $\lambda \in k^\*$, the polynomial $f(\lambda \cdot x) = \sum \lambda^i f\_i$ also vanishes on $f$.
Since $k$ is infinite, we can find more equations of the f... | 7 | https://mathoverflow.net/users/297 | 16147 | 10,820 |
https://mathoverflow.net/questions/16145 | 34 | Let $K$ be a field and $G:=SL\_2(K)$, then $G$ is a $K-$split reductive group (to use some big words). These groups are classified by a based root datum $(X,D,X',D')$. Let $G'$ be group associated to $(X',D',X,D)$, the so called dual group.
Is it correct that $G'=PGL\_2(K)$?
I"m wondering how $PSL\_2(K)$ fits into t... | https://mathoverflow.net/users/3380 | What is the difference between PSL_2 and PGL_2? | Yes, the dual of $SL\_2$ is $PGL\_2$.
But you're not going down the right track with $PSL\_2$. The problem with $PSL\_2$ is that it's not a variety at all! You can quotient out the variety $SL\_2$ by the subgroup $\pm1$ but the quotient is the variety $PGL\_2$ (recall that quotients in the category of sheaves (for th... | 51 | https://mathoverflow.net/users/1384 | 16150 | 10,823 |
https://mathoverflow.net/questions/16141 | 40 | I was looking at Wilson's theorem: If $P$ is a prime then $(P-1)!\equiv -1\pmod P$. I realized this
implies that for primes $P\equiv 3\pmod 4$, that $\left(\frac{P-1}{2}\right)!\equiv \pm1 \pmod P$.
Question: For which primes $P$ is $\left(\frac{P-1}{2}\right)!\equiv 1\pmod P$?
After convincing myself that it's no... | https://mathoverflow.net/users/4181 | Primes P such that ((P-1)/2)!=1 mod P | I am a newcomer here. If p >3 is congruent to 3 mod 4, there is an answer which involves only $p\pmod 8$ and $h\pmod 4$, where $h$ is the class number of $Q(\sqrt{-p})$ .
Namely one has $(\frac{p-1}{2})!\equiv 1 \pmod p$ if an only if either (i) $p\equiv 3 \pmod 8$ and $h\equiv 1 \pmod 4$ or (ii) $p\equiv 7\pmod 8$ and... | 37 | https://mathoverflow.net/users/4186 | 16154 | 10,827 |
https://mathoverflow.net/questions/16131 | 12 | My primary reference for this question is the very good book *Quantum Groups and Knot Invariants* by C. Kassel, M. Rosso, and V. Turaev. I'm also drawing from P. Etingof and O. Schiffmann, *Lectures on quantum groups*, another very good book. If you want to see pictures of Lie bialgebras and quasitriangular structures,... | https://mathoverflow.net/users/78 | Comparing two similar procedures for quantizing a Casimir Lie algebra | The second construction (Lie bialgebra quantization) in fact also uses a Drinfeld associator. The braided tensor categories obtained in these two ways are equivalent, since the quasitriangular QUE algebra produced by the second construction is obtained by twisting the quasitriangular quasiHopf QUE algebra produced by t... | 7 | https://mathoverflow.net/users/3696 | 16155 | 10,828 |
https://mathoverflow.net/questions/16074 | 49 | This is maybe not an entirely mathematical question, but consider it a pedagogical question about representation theory if you want to avoid physics-y questions on MO.
I've been reading Singer's [Linearity, Symmetry, and Prediction in the Hydrogen Atom](http://www.springer.com/mathematics/algebra/book/978-0-387-2463... | https://mathoverflow.net/users/290 | How is the physical meaning of an irreducible representation justified? | Invariant states are *not* the only meaningful ones. Even in classical mechanics, a baseball traveling 90 mph toward my head is quite meaningful to me, even though it is of no consequence to my fellow mathematician a mile away.
The focus on invariant subspaces comes not from an assumption, but from the way physicist... | 62 | https://mathoverflow.net/users/4188 | 16156 | 10,829 |
https://mathoverflow.net/questions/16134 | 7 | In the paper "Complete topoi representing models of set theory" by Blass and Scedrov, they consider a general notion of Boolean-valued model of set theory, and one of the conditions they impose is that the model contain "no extra ordinals after those of V", i.e. that for all z in the model we have
$$\Vert z \text{ is... | https://mathoverflow.net/users/49 | Can models of set theory contain extra ordinals? | I have two answers.
First, the standard method of building B-valued models of set
theory, where B is any complete Boolean algebra, always
satisfies your condition.
Suppose that B is any complete Boolean algebra, and denote
the original set-theoretic universe by V. One constructs
the B-valued universe VB by buildin... | 4 | https://mathoverflow.net/users/1946 | 16157 | 10,830 |
https://mathoverflow.net/questions/16024 | 16 | What kind of role do quantum groups play in modern physics ?
Do quantum groups naturally arise in quantum mechanics or quantum field theories?
What should quantum symmetry refer to ?
Can we say that the "symmetry" of a noncommutative space (quantum phase space) should be a quantum group?
Do quantum groups describe "ex... | https://mathoverflow.net/users/4155 | What is the relation between quantum symmetry and quantum groups? | Yes, quantum groups naturally arise in many physics problems. E.g. solutions of the quantum Yang-Baxter equation appear as scattering matrices of integrable 2-dimensional quantum field theories (see "Quantum fields and Strings: a course for Mathematicians", p.1179). Also, quantum groups appear in the description of mon... | 16 | https://mathoverflow.net/users/3696 | 16158 | 10,831 |
https://mathoverflow.net/questions/16163 | 11 | Could anyone give me some references where I could find
(a) discrete version(s) of [Ito's lemma](http://en.wikipedia.org/wiki/Ito%27s_lemma)
(b) a proof how it converges to the continuous form in the limit
(c) its usage within stochastic difference equations
(d) a deduction of a discrete version of the Blac... | https://mathoverflow.net/users/1047 | Discrete version of Ito's lemma | 1. [*Stochastic Calculus for Finance II: Continuous-Time Models*](http://rads.stackoverflow.com/amzn/click/0387401016) by Shreve or
2. Shreve or Øksendal's [*Stochastic Differential Equations*](http://rads.stackoverflow.com/amzn/click/3540047581)
3. Øksendal
4. Williams' [*Probability with Martingales*](http://books.go... | 11 | https://mathoverflow.net/users/1847 | 16166 | 10,837 |
https://mathoverflow.net/questions/8993 | 5 | This is a question about a name of a very useful lemma,
that permits one in particular to show that smooth birational complex projective
varieties have isomorphic fundamental groups.
If this lemma has no name, I would like at least to have a reference (if it exits).
The lemma can be seen as a
truncated version of the... | https://mathoverflow.net/users/943 | Truncated exact sequence of homotopy groups | Check out the paper "A Vietoris Mapping Theorem for Homotopy," by S. Smale, Proc. Amer. Math. Soc. 8 (1957), 604-610, available at <http://www.jstor.org/stable/2033527> .
Paraphrase of the main theorem: If $f:X\to Y$ is a proper, onto map of 0-connected, locally compact, separable metric spaces, X is $LC^n$, and eac... | 2 | https://mathoverflow.net/users/1822 | 16181 | 10,848 |
https://mathoverflow.net/questions/16178 | 13 | We are interested in a solution to the following scheduling problem, or any information about how to find it or its existence. This one comes from real life, so you will not only be helping a mathematician quench his thirst of knowledge!
>
> We have 18 players playing a certain sport (let's say curling) on 3 differ... | https://mathoverflow.net/users/4189 | Is there a tournament schedule for 18 players, 17 rounds in groups of 6, which is balanced in pairs? | A collection of $6$-tuples on 18 points with the property that each pair is covered $5$ times is a balanced incomplete block design with $(v,k,\lambda) = (18,6,5)$ and $t=2$. The condition that you can schedule the matches to occur simultaneously in $17$ rounds is that the design is resolvable.
[This article](http:/... | 7 | https://mathoverflow.net/users/2954 | 16182 | 10,849 |
https://mathoverflow.net/questions/16187 | 5 | Let $X \text{~} \text{Binomial}(n, p)$.
What is $\text{P}[X \mod 2 = 0]$? Is it of the form $1/2 + O(1/2^n)$?
| https://mathoverflow.net/users/4197 | Binomial distribution parity | This probability is in fact $1/2 + (1-2p)^n/2$.
Here's a proof. The probability generating function of $X$ is $f(z) = (q+pz)^n$, where $q = 1-p$. That is, the coefficient of $z^k$ in $(q+pz)^n$ is the probability $P(X=k)$.
Now, consider $(f(z)+f(-z))/2$. This polynomial contains just the terms of $f(z)$ which conta... | 7 | https://mathoverflow.net/users/143 | 16188 | 10,852 |
https://mathoverflow.net/questions/15509 | 10 | Let $\Delta\_k$ be the k-simplex and $\mu$ a non-negative measure over $\Delta\_k$. I want to know if there exists a function $u : \Delta\_k \to \mathbb{R}$ such that $u$ is convex, $u(e\_i) = 0$ for all vertices $e\_i$ of $\Delta\_k$, and $M[u] = \mu$ where
$M[u] = \det\left(\frac{\partial^2 u}{\partial x\_j \partial... | https://mathoverflow.net/users/1915 | Solutions to a Monge-Ampère equation on the simplex | Any set of values of $u$ at the vertices of $\Delta\_k$ can be attained just by adding an affine function to $u$, which does not change $M[u]$. To see that the solution of your problem is not unique, consider $u(x,y)=ax^2+a^{-1}y^2+\mathrm{(affine\ terms)}$ with $a>0$. Clearly $M[u]=4$ for any $a$.
On the other hand... | 6 | https://mathoverflow.net/users/2912 | 16206 | 10,864 |
https://mathoverflow.net/questions/16183 | 17 | I read several times that $(\infty,1)$-categories (weak Kan complexes, special simplicial sets) are a generalization of the concept of model categories. What does this mean? Can one associate an $(\infty,1)$-category to a model category without losing the information on the co/fibrations? How? Why is the $(\infty,1)$-c... | https://mathoverflow.net/users/4011 | $(\infty,1)$-categories and model categories | Mostly I refer you to my answer [here](https://mathoverflow.net/questions/2185/how-to-think-about-model-categories/2317#2317) and also [this question](https://mathoverflow.net/questions/8663/infinity-1-categories-directly-from-model-categories).
To answer the question about (co)fibrations: No, there is no notion corr... | 9 | https://mathoverflow.net/users/126667 | 16209 | 10,866 |
https://mathoverflow.net/questions/10056 | 20 | Given a topological space **X** and a finite cover **X** = $\cup X\_i$, one can define Cech cohomology of a sheaf of abelian groups **F** with respect to the cover $\{X\_i\}$ in two different ways:
1. (Ordered): The kth term of the Cech
complex is $\bigoplus\_{i\_1 < \ldots
< i\_k} \Gamma(X\_{i\_1} \cap \ldots
\c... | https://mathoverflow.net/users/2 | Equivalence of ordered and unordered cech cohomology. | I wrote it up for my algebraic geometry course as a 2-page [handout](http://math.stanford.edu/~conrad/papers/cech.pdf), inspired by EGA $0\_{\rm{III}}$, 11.8.7 (which isn't to say this is a canonical reference; just some written reference...).
| 17 | https://mathoverflow.net/users/3927 | 16213 | 10,870 |
https://mathoverflow.net/questions/16216 | 9 | I am writing an undergraduate thesis on local and global class field theory from a classical (i.e., non-cohomological) approach and am hoping to obtain copies of the early groundbreaking publications in the field. I am primarily interested in finding English translations of articles from Weber, Hasse, Hilbert, Kronecke... | https://mathoverflow.net/users/4204 | Where can I find online copies of class field theory publications by Kronecker, Weber, Chevalley, Hasse, Hilbert, Takagi, etc? | First, there's no need to focus on online copies, as asked for in the question. We used to have things called libraries which contain journal articles in them. :) Try looking there.
More seriously, I think your task is to a large extent hopeless. Most of those works were never translated into English. But there are ... | 22 | https://mathoverflow.net/users/3272 | 16222 | 10,876 |
https://mathoverflow.net/questions/15127 | 14 | For several years people have tried to characterize finite groups of maximal order in $\mathrm{GL}(n,\mathbb{Q})$ (or $\mathrm{GL}(n,\mathbb{Z})$) and their orders.
It appear in many articles a reference to an "preprint" article from Walter Feit in 1995 that gave full characterization. And I read a quote that Feit's pa... | https://mathoverflow.net/users/3958 | The maximum order of finite subgroups in $GL(n,Q)$ | Feit published his paper in the proceedings of the first Jamaican conference, [MR1484185](http://www.ams.org/mathscinet-getitem?mr=1484185). He defines M(n,K) to be the group of monomial matrices whose entries are roots of unity. M(n,Q) is the group of signed permutation matrices.
Theorem A: A finite subgroup of GL(n... | 13 | https://mathoverflow.net/users/3710 | 16223 | 10,877 |
https://mathoverflow.net/questions/16224 | 7 | Can we construct the category of spectra (or maybe just its homotopy category) from the category of pointed topological spaces
using some kind of localization combined with other categorical constructions?
[The first part of the original question was wrong for a trivial reason pointed out by Reid Barton.]
| https://mathoverflow.net/users/402 | Spectra and localizations of the category of topological spaces | [Removed a paragraph relating to an earlier version of the question]
You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top as a presentable (∞,1)-category. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming ℤ[1/p] ... | 9 | https://mathoverflow.net/users/126667 | 16231 | 10,881 |
https://mathoverflow.net/questions/16243 | 26 | I am looking for an analogue for the following 2 dimensional fact:
Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is spherical/euclidean/hyperbolic, iff the angle sum is smaller than /equal to/bigger than $\pi$. And the length of the sides (resp. their rati... | https://mathoverflow.net/users/3969 | Tetrahedra with prescribed face angles | The short answer is no - there is no single inequality criterion. Already in $\mathbb{R}^3$ everything is much more complicated. Let me give a sample of inequalities the angles should satisfy. Denote by $\gamma\_{ij}, 1\leq i < j \leq 4$ the six dihedral angles of a Euclidean tetrahedron. Then:
$$
\gamma\_{12}+\gamma\_... | 19 | https://mathoverflow.net/users/4040 | 16250 | 10,896 |
https://mathoverflow.net/questions/16254 | 1 | This question is not homework, just asked out of curiosity. I wondered how many zeroes could be found at the end of $1990!$ . I computed something that seemed to work and found out 439. So I computed it in Python, and it returned 494 zeroes, so I'm 55 short.
My reasoning went like this : I get 2 zeroes by series of 1... | https://mathoverflow.net/users/2446 | Counting trailing zeros for factorials | Take any integer $x$, and let $t,f$ represent the highest integers such that $2^t | x$ and $5^f | x$. Then the number of trailing zeros in the base 10 representation of $x$ is $z := \min\{t,f\}$. (One way to see this is to note it must be at least z since you have $(2\*5)^z | x$, so you can write $x = 10^z \* y$ where ... | 4 | https://mathoverflow.net/users/2621 | 16255 | 10,900 |
https://mathoverflow.net/questions/16180 | 7 | Goguen has popularized the initial algebra view of semantics via his "no junk, no confusion" slogan. By "no junk", he means that models of a theory presentation should not have unnecessary elements, and "no confusion" that terms should not be mapped to equal values unless they are provably equal. Sometimes, "no junk" i... | https://mathoverflow.net/users/3993 | Formalizing "no junk, no confusion" | The way I understand each of the slogans is as follows:
1. "No junk" I just take to mean that an appropriate induction principle is valid -- that is, we should look for initial models in the appropriate category of algebras for the theory. This also implies that every element of the model is in the image of the inte... | 7 | https://mathoverflow.net/users/1610 | 16263 | 10,905 |
https://mathoverflow.net/questions/16261 | 6 | I need a construction in linear algebraic groups which uses taking quotient by a central finite group subscheme.
My question is, whether it goes through in ``bad'' characteristics, when this group subscheme is not smooth.
First I write this construction in a special case, and then in the general case.
Let $G$ be a co... | https://mathoverflow.net/users/4149 | Quotient of a reductive group by a non-smooth central finite subgroup | This is an instance of what I believe is called the $z$-construction, and it is a very useful trick in the arithmetic theory of algebraic groups. (Small correction: your diagonal embedding should really be "anti-diagonal". You are really computing a "central pushout".) However, you may need to restrict your ground fiel... | 7 | https://mathoverflow.net/users/3927 | 16273 | 10,911 |
https://mathoverflow.net/questions/16265 | 14 | In one of the exercises in McDuff and Salamon, they mention homology with compact supports. I know how to define \*co\*homology with compact supports, but I can't picture the homology version. How do I say that a chain has compact support? If I use singular chains, don't they all have compact support anyway?
Google i... | https://mathoverflow.net/users/3909 | homology with compact supports | To elaborate on Ekedahl's comment:
It will be easiest to describe things for a triangulated space, so I can work with simplicial chains and cochains. (But my space could be infinitely triangulated; e.g. think of Escher's famous picture of the infinitely triangulated hyperbolic plane. I will also assume that my triang... | 13 | https://mathoverflow.net/users/2874 | 16281 | 10,916 |
https://mathoverflow.net/questions/16312 | 62 | So, I can understand how [non-standard analysis](https://en.wikipedia.org/wiki/Non-standard_analysis) is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta arguments (both these points are debatable).
However, although many th... | https://mathoverflow.net/users/2233 | How helpful is non-standard analysis? | From the [Wikipedia article](https://en.wikipedia.org/wiki/Non-standard_analysis#Applications):
>
> the list of new applications in
> mathematics is still very small. One
> of these results is the [theorem proven
> by Abraham Robinson and Allen
> Bernstein](https://projecteuclid.org/journals/pacific-journal-of-math... | 33 | https://mathoverflow.net/users/1847 | 16314 | 10,937 |
https://mathoverflow.net/questions/16310 | 17 | Morel and Voevodsky construct the motivic stable homotopy category, a category through which all cohomology theories factor and where they are representable, by starting with a category of schemes, Yoneda-embedding it into simplicial presheaves, endowing those with the $\mathbb{A}^1$-local model structure, and then pas... | https://mathoverflow.net/users/733 | Why does one invert $G_m$ in the construction of the motivic stable homotopy category? | I guess there are "internal" and "external" motivations. External for instance -- most natural examples of functors we have from the stable motivic homotopy category to some other category invert G\_m (e.g. any of the usual realizations, or K-theory). Internal for instance -- we want the suspension spectra of varieties... | 12 | https://mathoverflow.net/users/3931 | 16320 | 10,941 |
https://mathoverflow.net/questions/16302 | 4 | Does anyone know of the existence of an archive of the work of J Sutherland Frame?
The Briscoe Center for American History maintains about 100 archives of American mathematics and I have found the folks there to be quite helpful.
Cheers, Scott
| https://mathoverflow.net/users/4111 | Archive of the Work of J Sutherland Frame | The short answer is "no". Frame did interesting work, though usually
outside the conceptual mainstream of representation theory. Some of his
calculations of character tables (such as that of the most exceptional
Weyl group) have permanent value, I think.
| 1 | https://mathoverflow.net/users/4231 | 16322 | 10,943 |
https://mathoverflow.net/questions/16292 | 11 | I was reading Mumford again and I noticed a comment in the beginning of the book:
"Finally, in any study of general preschemes, the varieties are bound, for many reasons which I will not discuss here, to play a unique and central role."
My question is: is there a particular theorem (or series of theorems) Mumford mig... | https://mathoverflow.net/users/3701 | The central role of varieties (a comment from Mumford's Red Book) | Here is a really cool illustration of the principle which Emerton was [outlining](https://mathoverflow.net/a/16294). We know that the Picard group of projective $(n-1)$-space over a field $k$ is $\mathbf{Z}$ ($n \ge 2$), generated by $\mathcal{O}(1)$. This underlies the proof that the automorphism group of such a proje... | 21 | https://mathoverflow.net/users/3927 | 16324 | 10,945 |
https://mathoverflow.net/questions/16214 | 35 | Recall that a function $f\colon X\times X \to \mathbb{R}\_{\ge 0}$ is a *metric* if it satisfies:
* definiteness: $f(x,y) = 0$ iff $x=y$,
* symmetry: $f(x,y)=f(y,x)$, and
* the triangle inequality: $f(x,y) \le f(x,z) + f(z,y)$.
A function $f\colon X\times X \to X$ is *associative* if it satisfies:
* associativity... | https://mathoverflow.net/users/1079 | Is there an associative metric on the non-negative reals? | Seems that this is possible. Here is a (non-constructive) proof.
Suggestions are welcome.
The proof is inspired by [Mazurkiewicz's argument](http://www.mathnerds.com/best/mazurkiewicz/index.aspx). This is second version
of the proof: it includes improvements in
the set-theoretic argument suggested by Joel David Hamk... | 23 | https://mathoverflow.net/users/2653 | 16328 | 10,948 |
https://mathoverflow.net/questions/16257 | 49 | What I am talking about are reconstruction theorems for commutative scheme and group from category. Let me elaborate a bit. (I am not an expert, if I made mistake, feel free to correct me)
**Reconstruction of commutative schemes**
Given a quasi compact and quasi separated commutative scheme $(X,O\_{X})$ (actually, ... | https://mathoverflow.net/users/1851 | How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers) | The Tannakian reconstruction for schemes and more generally, geometric stacks from QCoh(X) with its tensor structure (due to Jacob Lurie) is explained in my answer to [Tannakian Formalism](https://mathoverflow.net/questions/3446/tannakian-formalism/3467#3467) . One can (and one has) try to extend this to the derived se... | 25 | https://mathoverflow.net/users/582 | 16334 | 10,951 |
https://mathoverflow.net/questions/15519 | 19 | Here's a question that has come up in a couple of talks that I have given recently.
The 'classical' way to show that there is a knot $K$ that is locally-flat slice in the 4-ball but not smoothly slice in the 4-ball is to do two things
1. Compute that the Alexander polynomial of $K$ is 1, and so by results of Freedm... | https://mathoverflow.net/users/3923 | topological "milnor's conjecture" on torus knots. | Related to the early investigation of the Thom conjecture,the G-signature thm was used circa 1970 to give 4-ball genus bounds for torus knots which asyptocically (in some cases) were a fixed fraction of what we now know to be the smooth category answer. I belive Larry Tayor observed (in the '70s or early 80s) that thes... | 11 | https://mathoverflow.net/users/1643 | 16337 | 10,953 |
https://mathoverflow.net/questions/16330 | 7 | For any finite group, G, we can find a cover of ℙ1ℂ which is G-Galois. The regular inverse Galois problem is equivalent to there existing such a cover that descends with action to ℚ. My question is about the easier problem: given a finite group G, can we find a cover of ℙ1ℂ such that it descends to ℚ as a mere cover (m... | https://mathoverflow.net/users/2665 | For a given finite group G, is there a cover of P^1 over Q s.t. over C it's G-Galois? | I don't know about *every* finite group $G$ (I'll guess no), but there are definitely infinitely many finite groups $G$ for which the situation you describe obtains: the extension $K/\mathbb{C}(t)$ has a model over $\mathbb{Q}$ but is not Galois over $\mathbb{Q}$. (And for most of these groups, we do not know how to re... | 4 | https://mathoverflow.net/users/1149 | 16344 | 10,956 |
https://mathoverflow.net/questions/16342 | 5 | I was reading HTT 2.1.4, and I just totally lost what was going on. Could someone provide some motivation for this section? Why do we want another model structure?
I'm sorry for not providing motivation, but as you can see, I'm unable to do so.
Here's a link to the ArXiv version: <http://arxiv.org/pdf/math/0608040... | https://mathoverflow.net/users/1353 | Motivation for the covariant model structure on SSet/S | If you have a category $C$, then you can consider the category of functors $Func(C,Sets)$ from $C$ to sets. This comes with the Yoneda functor, $C^{op}\to Func(C,Sets)$, and is a generally useful thing to think about.
In $(\infty,1)$-category land, you want to start with an $(\infty,1)$-category $S$, and build the $(... | 8 | https://mathoverflow.net/users/437 | 16345 | 10,957 |
https://mathoverflow.net/questions/16331 | 3 | Does anyone know of generalizations on what Mumford (Red Book) calls "uniformizing parameters"? For example, given a regular quasi-projective scheme over a *finite* field $\mathbb{F}$, is there an etale morphism into affine space over $\mathbb{F}$?.
| https://mathoverflow.net/users/4235 | When does a projective morphism give an etale morphism (into affine space)? (Finite field) (normalization) | [This paper](http://arxiv.org/abs/math/0303382) by Kedlaya might be what you want, since it contains some rearrangement of the words you used, but I can't really tell from the question. If you want a proper F-scheme to have an etale map to affine space, it has to be a disjoint union of finite F-schemes, and the affine ... | 3 | https://mathoverflow.net/users/121 | 16346 | 10,958 |
https://mathoverflow.net/questions/16360 | 5 | I'm a 2nd year grad student and I'm looking for conferences/summer schools to attend this summer. I checked out the AMS calendar but couldn't find anything I found relevant there. Anyone have any suggestions?
| https://mathoverflow.net/users/4239 | Looking for good conference this summer for homotopy theory | The Georgia Topology Conference will be largely focused on a geometric end of algebraic topology this year. As least, that what I suspect. The homotopy-type of embeddings of smooth manifolds, Goodwillie calculus, configuration spaces, etc, should be a major feature:
<http://www.math.uga.edu/~topology/>
I think ther... | 5 | https://mathoverflow.net/users/1465 | 16364 | 10,969 |
https://mathoverflow.net/questions/16335 | 11 | The following simple theorem is known as Cauchy's mean value theorem. Let $\gamma$ be an immersion of the segment $[0,1]$ into the plane such that $\gamma(0) \ne \gamma(1)$. Then there exists a point such that the tangent line at that point is parallel to the line passing through $\gamma(0)$ and $\gamma(1)$. So the bou... | https://mathoverflow.net/users/2823 | A generalization of Cauchy's mean value theorem. | The answer is no for $n=2$. It is sufficient to construct 3 surfaces with common boundary (say $\Sigma\_i$, $i\in\{1,2,3\}$)
such that there is no choice of points $p\_i\in\Sigma\_i$ with pairwise parallel tangent planes.
Let us take a smooth function $f:S^1\to \mathbb R$, $f(t)\approx\sin(2\cdot t)$ with one little ... | 5 | https://mathoverflow.net/users/1441 | 16366 | 10,970 |
https://mathoverflow.net/questions/15243 | 8 | **Note.** This question had a bounty, so at the end I accepted the best (and only) answer. However, a solution is implied by the answer to [this question](https://mathoverflow.net/questions/15204/space-bounded-communication-complexity-of-identity).
**Question.**
Fix n. We are interested in the biggest t for which the... | https://mathoverflow.net/users/955 | Two [n] to [n] function families | Okay, so I tried to see how this could possibly work. After some thinking I decided that one may as well take $P\_i=Q\_i$, so that the orbit of 3 (under the action of $P\_i$) is a cycle containing 1. If you take the length of this cycle to be roughly $n/2$, send $2\to 3$ and everything else to 2, that's not a bad idea ... | 4 | https://mathoverflow.net/users/4040 | 16371 | 10,973 |
https://mathoverflow.net/questions/16386 | 6 | I read some time ago some papers about proof formalization. Typically, I began whith [this one](http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf), from Lamport.
Are there more recent works in this field ?
| https://mathoverflow.net/users/2446 | Proof formalization | In general, "formalising mathematical vernacular" is a good source for work relevant to your question. The field generally started in earnest with de Bruijn's Automath project.
Harvey Friedman has done some nice work on leveraging his theory of explicit definitions for set theory (cf. [The Logical Strength of Mathema... | 4 | https://mathoverflow.net/users/3154 | 16387 | 10,981 |
https://mathoverflow.net/questions/16368 | 7 | Frequently it is useful do deal with countable transitive models M of ZFC, for example in forcing constructions.
The notion of being an ordinal is absolute for any transitive model, so certainly if ($\alpha$ is an ordinal)M then also $\alpha$ is an ordinal. For the same reason, M will contain successors of every ord... | https://mathoverflow.net/users/4241 | Least ordinal not in a countable transitive model of ZFC | The least ordinal not in any transitive model of ZFC can also be described as the supremum of the heights of transitive models of ZFC. It is natural here to consider the class S consisting of all ordinals λ for which there is a transitive model of ZFC of height λ. Thus, the ordinal of your title, when it exists, is sim... | 11 | https://mathoverflow.net/users/1946 | 16394 | 10,985 |
https://mathoverflow.net/questions/16237 | 4 | Part of one of my calculations involves (the innocent looking) expression
$\sum\_{\alpha\in\Sigma} (\alpha,\alpha)$
for simple Lie algebras.
I have two methods of calculating it -- which don't agree. I'm pretty sure that the first one is wrong, but I don't know why. Any help is welcome (which is why I posted here)!
... | https://mathoverflow.net/users/358 | Sum of all root lengths in simple Lie algebra | It turns out that I was wrong and both methods give correct results. In fact using the two results you can reproduce the scaling parameters for the simple Lie algebras given in the [Broughton paper](http://www.rose-hulman.edu/~brought/Epubs/preprints/killing.pdf) linked to by David.
My mistake was in assuming that th... | 3 | https://mathoverflow.net/users/358 | 16402 | 10,991 |
https://mathoverflow.net/questions/16403 | 6 | How many ways is there to build an arithmetic expression with fixed number of terms and fixed order? Let’s assume we have only one distinct operation that is neither commutative nor associative. The problem can be reduced to the question of how many different, full, binary trees could be constructed with a fixed number... | https://mathoverflow.net/users/4247 | Count of full, binary trees with fixed number of leaves | There is a general algorithm that solves this kind of problem.
1. Calculate the first terms of your sequence by hand.
2. Plug them into Sloane's and see if it is a known sequence.
<http://oeis.org?q=1%2C1%2C2%2C5%2C14&sort=0&fmt=0&language=english&go=Search>
Here you see that your numbers are called Catalan numbe... | 20 | https://mathoverflow.net/users/1310 | 16405 | 10,993 |
https://mathoverflow.net/questions/16399 | 6 | What is the relation between characters of a group and its lie algebra?
Roughly,I know that there is a one to one correspondence between representations of a lie algebra and its simple connected lie group by the exp map,and two irreducible representations of a lie group are unequivalent if and only if their character... | https://mathoverflow.net/users/4155 | What is the relation between characters of a group and its lie algebra? | The primary reason for studying Lie algebras is the following fundamental fact: the representation theory of a Lie algebra is the same as the representation theory of the corresponding connected, simply connected Lie group.
Of course, the representation theory of a Lie group in general is very complicated. First of a... | 14 | https://mathoverflow.net/users/78 | 16415 | 11,001 |
https://mathoverflow.net/questions/16401 | 3 | Since a pullback of two functions f and g with common codomain into **Set** category is just a subset of cartesian product like this: {(x,y)/f(x)=g(y)} (with two more functions not important here) could this pullback set be the empty set in some cases (for exemple in the case of constant functions)?
My question is r... | https://mathoverflow.net/users/3338 | Pullbacks for primitive recursive functions. | I'm not sure if this is what you want, but it is not difficult to prove that if f and g are primitive recursive functions, then the set A = { (x,y) | f(x) = g(y) } is a primitive recursive subset of the natural number plane. That is, the characteristic function of this set is primitive recursive. The (trivial) reason i... | 2 | https://mathoverflow.net/users/1946 | 16418 | 11,002 |
https://mathoverflow.net/questions/16423 | 0 | A vertex operator is a linear map associating every state to a operator-valued distributions (quantum field) on a algebra curve, which is also called operator-state correspondence.
Chose a local complex coordinate, we can locally expand quantum fields as operator valued formal Laurent series, this process is called ... | https://mathoverflow.net/users/4155 | About vertex algebra, mode expansion | If you expand a meromorphic function in a Laurent series about $z=0$ and now take $z$ on the unit circle in the complex plane, so that $z= e^{i\theta}$, then the Laurent series is a Fourier series.
| 5 | https://mathoverflow.net/users/394 | 16432 | 11,012 |
https://mathoverflow.net/questions/16427 | 14 | How do we define quasi-coherent sheaves on schemes?
Say we start by defining the category of affine schemes Aff as CRing$^{op}$ (the opposite category of unitary commutative rings).
In this context we have an obvious way to define quasi-coherent sheaves:
A quasi-coherent sheaf on an affine scheme X=Spec A is just a... | https://mathoverflow.net/users/3701 | Quasi-coherent sheaves in the Functor-of-points approach | The corresponding [nlab page](http://ncatlab.org/nlab/show/quasicoherent+sheaf) has several approaches to the definition of quasicoherent sheaves of O-modules including some in functor of points approach, in various degrees of abstractness. All these definitions while simultaneously applicable define equivalent categor... | 10 | https://mathoverflow.net/users/35833 | 16438 | 11,016 |
https://mathoverflow.net/questions/16445 | 5 | Hello everyone, could anyone please tell me where can I find information about the Elton–Odell theorem?
It states:
For any infinite dimensional Banach space $X$ there is a $q > 1$ so that $X$ contains a sequence $(x\_n)$ with $\|x\_n\|=1$ and $\|x\_n-x\_m\|\ge q$, whenever $m\ne n$.
Thanks
| https://mathoverflow.net/users/2737 | Info about Elton–Odell theorem | J. Diestel, Sequences and series in Banach spaces, Springer-Verlag, New York, 1984.
(Chapter XIV, page 241)
| 4 | https://mathoverflow.net/users/3536 | 16447 | 11,022 |
https://mathoverflow.net/questions/16434 | 46 | There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let $\mathcal{E}$ be a vector bundle of rank $r$, with $\mathcal{O}\_X$ the usual sheaf of functions on $X = \mathbb{P}^1$, then w... | https://mathoverflow.net/users/1828 | Using linear algebra to classify vector bundles over ℙ¹ | I must admit I have never read this reference, but I remember it from a similar discussion on a German forum, according to which there is a simple proof in
[Michiel Hazewinkel and Clyde Martin, *A short elementary proof of Grothendieck's theorem on algebraic vectorbundles over the projective line*, Journal of pure an... | 13 | https://mathoverflow.net/users/2530 | 16450 | 11,025 |
https://mathoverflow.net/questions/15913 | 13 | I'm looking for either a few precise mathematical statements about Wiener integrals, or a reference where I can find them.
Background
----------
The Wiener integral is an analytic tool to define certain "integrals" that one would like to evaluate in quantum and statistical mechanics. (Hrm, that's two different mech... | https://mathoverflow.net/users/78 | Which functions are Wiener-integrable? | Hi Theo,
0) Your definition is roughly correct, yes. For Wiener measure on paths in vector spaces, see Chapter 3 + Appendix A of the 2nd edition of Glimm & Jaffe. On curved targets, I think Bruce Driver has some good lecture notes. One warning: the rough definition of Wiener measure is misleading in one way: Wiener m... | 6 | https://mathoverflow.net/users/35508 | 16459 | 11,031 |
https://mathoverflow.net/questions/16460 | 33 | Two things today motivated this question.
First, the professor said that in a lecture Thurston mentioned
>
> Any manifold can be seen as the configuration space of some physical system.
>
>
>
Clearly we need to be careful here, so the first question is
>
> 1) What is a precise formulation and an argumen... | https://mathoverflow.net/users/348 | How to see the Phase Space of a Physical System as the Cotangent Bundle | Let's start by answering the first question.
Let $M$ be any manifold. Consider a physical system consisting of a point-particle moving on $M$. What are the configurations of this physical system? The points of $M$. Hence $M$ is the configuration space.
Typically one takes $M$ to be riemannian and we may add a poten... | 30 | https://mathoverflow.net/users/394 | 16462 | 11,033 |
https://mathoverflow.net/questions/16393 | 24 | Is there any result about the time complexity of finding a cycle of fixed length $k$ in a general graph?
All I know is that [Alon, Yuster and Zwick](http://www.tau.ac.il/~nogaa/PDFS/col5.pdf) use a technique called "color-coding",
which has a running time of $O(M(n))$, where $n$ is the number of vertices of the input g... | https://mathoverflow.net/users/4248 | Finding a cycle of fixed length | Finding a cycle of any *even* length can be found in $O(n^2)$ time, which is less than any known bound on $O(M(n))$. For example, a cycle of length four can be found in $O(n^2)$ time via the following simple procedure:
>
> Assume the vertex set is $\{1,...,n\}$. Prepare an $n$ x $n$ matrix $A$ which is initially a... | 27 | https://mathoverflow.net/users/2618 | 16464 | 11,035 |
https://mathoverflow.net/questions/16381 | 4 | I'm reading Knutson's book on algebraic spaces, and I stumbled over the quasi-separatedness axiom in his definition of algebraic spaces (Definition 1.1, Chapter II). He defines an algebraic space A as a sheaf on the site of schemes with the étale topology satisfying:
I) Local representability. There exists a represen... | https://mathoverflow.net/users/1084 | Quasi-separatedness for Algebraic Spaces | One can certainly make the basic definitions, and the real issue is to show that the definition "works" using any etale map from a scheme. More precisely, the real work is to show that a weaker definition actually gives a good notion: rather than assume representability of the diagonal, it suffices that $R := U \times\... | 11 | https://mathoverflow.net/users/3927 | 16483 | 11,046 |
https://mathoverflow.net/questions/16487 | 6 | I am trying to factorize $\sin(x)\over x$ which by [Taylor series expansion](http://en.wikipedia.org/wiki/Taylor_theorem) and using the roots is $$a \cdot \left(1 - \frac{x}{\pi} \right) \left(1 + \frac{x}{\pi} \right) \left(1 - \frac{x}{2\pi} \right) \left(1 + \frac{x}{2\pi} \right) \left(1 - \frac{x}{3\pi} \right) \l... | https://mathoverflow.net/users/1047 | Using Weierstrass’s Factorization Theorem | The value of this product for small x's is the product of $(1-x^2/(n \pi)^2)$ which, when you take logs (and due to the second power in x), behaves like the sum over n of $-x^2/(n\pi)^2$, which approaches 0 as x approaches 0.
| 5 | https://mathoverflow.net/users/404 | 16493 | 11,053 |
https://mathoverflow.net/questions/16507 | 5 | I have often come across this implicit translation of the classical field of a particle of a given spin into a specific tensor field. But I could not locate any literature from which I could learn this.
In a paper of Avrimidi I found this statement,
"*The tensor fields describe the particles with integer spin while... | https://mathoverflow.net/users/2678 | Spins as tensor fields | In a nutshell, particles "are" unitary irreducible representations of the Poincaré group, which is the isometry group of Minkowski spacetime on which it acts transitively. Such representations can be constructed using the method of induced representations (cf. Wigner, Bargmann, Mackey,...) as classical fields on Minkow... | 4 | https://mathoverflow.net/users/394 | 16510 | 11,063 |
https://mathoverflow.net/questions/16466 | 3 | This is may be obvious, but I am not comfortable with ind-schemes.
I have an ind-scheme $X$ over $\mathbb{C}$. Every point has a neighborhood which can be written as an ascending union of regular varieties, which is about as smooth as an ind-scheme can be.
I have an unipotent ind-group $U$. More precisely, $U$ is ... | https://mathoverflow.net/users/297 | Principal bundle for contractible group is weak homotopy equivalence for ind schemes | My recollection is that when you turn these into analytic spaces you get something which is locally contractible topologically. In this case what you are describing is a principal bundle for locally contractible spaces in which the fiber is contractible. If the base is paracompact then this will indeed be a weak equiva... | 2 | https://mathoverflow.net/users/184 | 16514 | 11,066 |
https://mathoverflow.net/questions/16495 | 14 | The study of the homotopy groups of spheres $\pi\_i(S^n)$ is a major subject in algebraic topology. One knows for example that nearly all of them are finite groups. Some are explicitly known. There is a 'stable range' of indices which one understands better than the unstable part.
I think that there is an analogy (h... | https://mathoverflow.net/users/4011 | Applications of homotopy groups of spheres | A few comments on applications that aren't covered by the above Wikipedia article.
I don't know any applications to cryptography. Most cryptosystems require some kind of one-way lossless function and it's not clear how to do that with the complexity of the homotopy groups of spheres. Moreover, the homotopy-groups of ... | 6 | https://mathoverflow.net/users/1465 | 16515 | 11,067 |
https://mathoverflow.net/questions/16471 | 18 | Consider the set of random variables with zero mean and finite second moment. This is a vector space, and $\langle X, Y \rangle = E[XY]$ is a valid inner product on it. Uncorrelated random variables correspond to orthogonal vectors in this space.
Questions:
(i) Does there exist a similar geometric interpretation f... | https://mathoverflow.net/users/4267 | A geometric interpretation of independence? | There is a Hilbert space interpretation of independence, which follows from the interpretation of conditional expectation as an orthogonal projection, though it may be more complicated than you had in mind.
Say your underlying probability space is $(\Omega, \mathcal{F}, \mathbb{P})$, and write $L^2(\mathcal{F})$ for... | 11 | https://mathoverflow.net/users/1044 | 16517 | 11,069 |
https://mathoverflow.net/questions/16422 | 4 | Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x\_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P\_n$ and $\mathbb P$ be Gaussian measures on $X$ with covariance $K$ and means $x\_n$ and $x$, respectively.
**Question:** How do I show that $\mathbb P... | https://mathoverflow.net/users/238 | Convergence of Gaussian measures | Somehow I didn't register how strong the assumptions Tom was making were, hence the fact that my other answer missed the point.
Unless I'm still missing something, this is very easy. Say $Z$ is a Gaussian random vector in $X$ with covariance $K$ and mean $0$. You want to show that $Z+x\_n \to Z+x$ weakly, i.e. $\math... | 5 | https://mathoverflow.net/users/1044 | 16518 | 11,070 |
https://mathoverflow.net/questions/16416 | 30 | I'm looking for a good book in commutative algebra, so I ask here for some advice. My ideal book should be:
-More comprehensive than Atiyah-MacDonald
-More readable than Matsumura (maybe better organized?)
-Less thick than Eisenbud, and more to the point
To put this in context, I'm an algebraic geometer, so I k... | https://mathoverflow.net/users/828 | Reference book for commutative algebra | For a reference on Cohen-Macaulay and Gorenstein rings, you can try "Cohen-Macaulay rings" by Bruns-Herzog.
Also, Huneke's [lecture note](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.7033) "Hyman Bass and Ubiquity: Gorenstein Rings" is a great introduction to Gorenstein rings, very easy to read and to ... | 31 | https://mathoverflow.net/users/2083 | 16519 | 11,071 |
https://mathoverflow.net/questions/16177 | 3 | Let us assume two samples, A and B, where A are the results obtained with some standard method, and B are the results obtained with a new method, which is not necessarily more accurate, but has additional advantages (eg: lower cost).
So I'm interested in testing if B is "as good as" A, using non-inferiority hypothes... | https://mathoverflow.net/users/2735 | Can t-test be used for non-inferiority hypothesis testing? | Follow-up on my question, after some more research on the question.
The method of using two one-sided two-sample t-tests for clinical equivalence testing is shown in [Schuirmann 1987](https://doi.org/10.1007/BF01068419 "Schuirmann, D.J. A comparison of the Two One-Sided Tests Procedure and the Power Approach for asse... | 1 | https://mathoverflow.net/users/2735 | 16520 | 11,072 |
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