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https://mathoverflow.net/questions/11077 | 21 | Can someone please explain what the symbol $\stackrel{!}{=}$, consisting of an exclamation mark (!) above an equals sign (=) means? Below is the example I'm trying to decipher:
---
The normalization factor is chosen such that in average, Dynamic Θ Time passes as fast as physical time. In practice it is determiend... | https://mathoverflow.net/users/3062 | What does ! above = mean | In my classes I use it to indicate anxiety. So an equals with ! over it means "we want to show this equality is true". An equals without ! means "I am asserting this is true".
I don't know how universal this convention is, though. I do know I'm not the only person to use this convention.
| 12 | https://mathoverflow.net/users/1465 | 11078 | 7,530 |
https://mathoverflow.net/questions/11073 | 1 | Let {a1,a2,...,an} and {b1,b2,...,bn} be two bases for a vector space E. Fix p, 1 ≤ p ≤n. Is there a permutation σ such that
{a1,a2,...,ap,bσ(p+1),...,bσ(n)} and {bσ(1),bσ(2),...,,bσ(p),ap+1,...,an} are both bases of E?
This question is the last exercise of the first chapter in the book Linear Algebra by Greub. I can... | https://mathoverflow.net/users/3061 | On permutation of elements of two bases of a vector space (Greub´s book) | It's also Theorem 7.2 in [Prasolov's Problems in Linear Algebra](http://www2.math.su.se/~mleites/books/prasolov-1994-problems.pdf), which gives a proof and attributes it to Green 1973.
| 4 | https://mathoverflow.net/users/2530 | 11080 | 7,532 |
https://mathoverflow.net/questions/10907 | 2 | Let X be a separable reflexive Banach space and f:X\to\mathbb{R} be a
Lipschitz function. Say that a point x in X is a *local supporting point
of* f if there exist x^\* in X^\* and an open neighborhood U of x
such that either x^\* (y-x)\leq f(y)-f(x) for all y in U or
x^\* (y-x)\geq f(y)-f(x) for all y in U.
Question... | https://mathoverflow.net/users/3038 | Local supporting points of Lipschitz functions | My guess is that you did not formulate question correctly --- in the present form the answer is NO.
One can take strictly saddle $f$ on $\mathbb R^2$, say $f(x,y)=\sqrt{1+x^2}-\sqrt{1+y^2}$.
| 2 | https://mathoverflow.net/users/1441 | 11083 | 7,533 |
https://mathoverflow.net/questions/11069 | 6 | While reading Demazure-Gabriel's construction of $\mathcal{S}ch$ as a full subcategory of $\mathcal{P}sh(CRing^{op})$, I've been trying to translate their exposition into the language of covering sieves and Grothendieck topologies. The requirement that a scheme be a sheaf in the Zariski topology on $CRing^{op}$ is esse... | https://mathoverflow.net/users/1353 | Induced Grothendieck topology on a presheaf or sheaf category of a site? | The answer to your first question is yes. Suppose $C$ a site.
The category $C^{\sim}$ of sheaves on $C$ simply has the canonical topology (SGA 4, Vol. 1, Exp. II, 2.5). The sheaves here are precisely the representable sheaves. (I'm ignoring questions about universes.)
The category $C^{\wedge}$ of presheaves on $C$ ... | 8 | https://mathoverflow.net/users/3049 | 11085 | 7,534 |
https://mathoverflow.net/questions/11081 | 19 | There is a conjecture of Kontsevich which states that Hochschild (co)homology of the Fukaya category of a compact symplectic manifold $X$ is the (co)homology of the manifold. (See page 18 of Kontsevich's "Homological algebra of mirror symmetry" paper and page 16 of Costello's paper "TCFTs and CY categories".)
Moreov... | https://mathoverflow.net/users/83 | Hochschild (co)homology of Fukaya categories and (quantum) (co)homology | The statement that $HF^{\ast}(X,X)$ is isomorphic to $QH^\ast(X)$ is a version of the Piunikhin-Salamon-Schwarz (PSS) isomorphism (proved, under certain assumptions, in McDuff-Salamon's book "J-holomorphic curves in symplectic topology"). PSS is a canonical ring isomorphism from $QH^{\ast}(X)$ to the Hamiltonian Floer ... | 23 | https://mathoverflow.net/users/2356 | 11098 | 7,544 |
https://mathoverflow.net/questions/11105 | 11 | Is there an easy example of an integral domain and two elements on it which do not have a greatest common divisor? It will have to be a non-UFD, obviously.
"Easy" means that I can explain it to my undergrad students, although I will be happy with any example.
| https://mathoverflow.net/users/3065 | An example of two elements without a greatest common divisor | [Here's](http://en.wikipedia.org/wiki/Greatest_common_divisor) an example stolen blatantly from wikipedia.
Let $R=\mathbb{Z}[\sqrt{-3}]$, let $a=4=2\*2=(1+\sqrt{-3})(1-\sqrt{-3})$ and $b=2(1+\sqrt{-3})$. Now, $2$ and $1+\sqrt{-3}$ are both maximal among divisors, but are not associates, thus, there is not GCD.
| 14 | https://mathoverflow.net/users/622 | 11107 | 7,549 |
https://mathoverflow.net/questions/11106 | 11 | I want to start out by making this clear: I'm NOT looking for the modern proofs and rigorous statements of things.
What I am looking for are references for classical enumerative geometry, back before Hilbert's 15th Problem asked people to actually make it work as rigorous mathematics. Are there good references for th... | https://mathoverflow.net/users/622 | Classical Enumerative Geometry References | And actually, as a partial answer to my own question, I just stumbled across Schubert's "[Kalkul](http://books.google.com/books?id=1zQAAAAAQAAJ&printsec=frontcover&dq=hermann+schubert&lr=&ei=_dJGS8SYGoaszATUh-XNAg&cd=11#v=onepage&q=&f=false)" on Google Books, and it looks complete, which makes me rather happy, though o... | 3 | https://mathoverflow.net/users/622 | 11116 | 7,556 |
https://mathoverflow.net/questions/11117 | 15 | Any Eilenberg-MacLane space $K(A,n)$ for abelian $A$ can be given the structure of an $H$-space by lifting the addition on $A$ to a continuous map $K(A\times A,n)=K(A,n)\times K(A,n)\to K(A,n)$.
Does somebody know an explicit way to describe this structure in the cases $K({\mathbb Z}/2{\mathbb Z},1)={\mathbb R}P^\inf... | https://mathoverflow.net/users/3067 | H-space structure on infinite projective spaces | Look at $\mathbb R^\infty\setminus 0$ as the space of non-zero polynomials, which you can multiply. Pass to the quotient to construct the projective space and, from the multiplication, its $H$-space product.
The complex case is quite the same.
**NB:** Jason asks in a comment below if this is the same $H$-space stru... | 20 | https://mathoverflow.net/users/1409 | 11121 | 7,558 |
https://mathoverflow.net/questions/11053 | 23 | Let $p$ be an odd prime and $\left( \frac{a}{p} \right)$ the Legendre symbol. The **Gauss sum**
$\displaystyle g\_p(a) = \sum\_{k=0}^{p-1} \left( \frac{k}{p} \right) \zeta^{ak},$
where $\zeta\_p = e^{ \frac{2\pi i}{p} }$, is a periodic function of period $p$ which is sometimes invoked in proofs of quadratic recipro... | https://mathoverflow.net/users/290 | What's the relationship between Gauss sums and the normal distribution? | It's true (as the answer below and some of the commenters note) that it's easy to interpret this question in a way that makes it seem trivial and uninteresting. I'm quite sure, however, that pursuing typographical similarity between $e^{x^2}$ and $\zeta^{m^2}$ leads to interesting mathematics, and so here's a more seri... | 12 | https://mathoverflow.net/users/nan | 11138 | 7,572 |
https://mathoverflow.net/questions/11133 | 5 | Can anyone name a undecidable problem that is genuinely graph-related? (Genuine means: not a standard one in graph's disguise.)
| https://mathoverflow.net/users/2672 | Undecidable graph problems? | From a MathSciNet search:
---
Földes, Stéphane; Steinberg, Richard
A topological space for which graph embeddability is undecidable.
J. Combin. Theory Ser. B 29 (1980), no. 3, 342--344.
From the introduction: ``From Edmonds' permutation theorem and a generalization due to Stahl, it follows that graph embeddabil... | 12 | https://mathoverflow.net/users/1149 | 11142 | 7,575 |
https://mathoverflow.net/questions/11130 | 1 | Prove that the [Normal (Gaussian) Distribution](https://en.wikipedia.org/wiki/Normal_distribution) with a given Variance $ {\sigma}^{2} $ maximizes the [Differential Entropy](https://en.wikipedia.org/wiki/Differential_entropy) among all distributions with defined and finite 1st Moment and Variance which equals $ {\sigm... | https://mathoverflow.net/users/2285 | Differential Entropy of Random Signal | Cover and Thomas's book is indeed the right place to learn about this.
The statement basically follows by convexity, in the form of Jensen's inequality. Here is the way it is usually presented:
Let $f$ be the probability density of a real random variable. Then the Shannon entropy is given by
$-\int f\log f dx$
... | 5 | https://mathoverflow.net/users/613 | 11144 | 7,577 |
https://mathoverflow.net/questions/11109 | 7 | I understand the definition of a Killing Form $B$ as $B(X,Y)=Tr(ad(X)ad(Y))$
And when the Lie group is semi-simple the negative of the Killing Form can serve as a Riemannian metric.
{Wonder if thats why some people say that a semi-simple Lie Group has only one metric!}
I would like to know what is the difference ... | https://mathoverflow.net/users/2678 | A terminology issue with the Killing Form | Let me add a few comments to the answers by Mariano and Theo.
There is a one-to-one correspondence between bi-invariant metrics (of any signature) in a Lie group and ad-invariant nondegenerate symmetric bilinear forms on its Lie algebra.
In a simple Lie algebra every nondegenerate symmetric bilinear form is propor... | 15 | https://mathoverflow.net/users/394 | 11152 | 7,581 |
https://mathoverflow.net/questions/11149 | 5 | I could sample a set of m elements from the uniform distribution over a universe $U$ of n >> m elements. Alternately, I could select a random probability distribution $\mathcal{D}$, and sample $m$ elements from $\mathcal{D}$.
EDIT (per Michael Lugo): When I say "select a random probability distribution", I mean sele... | https://mathoverflow.net/users/3028 | Sample from uniform distribution vs. Sample from random distribution | You're right that things change for m>1; I was thinking sloppily.
Assume $U=\{1,\ldots,n\}$ for concreteness. If $Y\_1,\ldots,Y\_m$ are chosen independently and uniformly from $U$, then for any $k\_1,\ldots,k\_m\in U$, we of course have
$$
\Pr[Y\_1=k\_1,\ldots,Y\_m=k\_m] = \frac{1}{n^m}.
$$
On the other hand, if $x... | 3 | https://mathoverflow.net/users/1044 | 11160 | 7,587 |
https://mathoverflow.net/questions/11164 | 16 | Notation
--------
Let $\mathfrak g$ be a the Lie algebra of an algebraic group $G\subseteq GL(V)$ over a(n algebraically closed) field $k$ (I'm actually thinking $G=GL\_n$, so $\mathfrak g=\mathfrak{gl}\_n$). Then any element $X$ of $\mathfrak g$ can be uniquely written as the sum of a semi-simple (diagonalizable) el... | https://mathoverflow.net/users/1 | What does the nilpotent cone represent? | The ideal which defines the nilpotent cone is generated by the homogeneous elements of positive degree in $\mathbb{C}[\mathfrak g]^G$. In the case of $GL\_n$, this ideal is generated by the functions $\mathrm{tr}\;X^k$ for $k$ up to the dimension, and also by the coefficients of the characteristic polynomial. See, for ... | 14 | https://mathoverflow.net/users/1409 | 11167 | 7,591 |
https://mathoverflow.net/questions/11087 | 5 | In an [opinion piece](http://www.ams.org/notices/201001/rtx100100005p.pdf) which appeared in the AMS Notices of January 2010, John Wermer tells us that he once heard about a seminar given by Grothendieck which was described as "a telegram by Grothendieck to Serre". Is this anecdote recorded somewhere?
| https://mathoverflow.net/users/1409 | A telegram by Grothendieck to Serre | I think I heard Grothendieck's talk at the 1958 ICM described as a "a telegram by Grothendieck to Serre". But this would have been in a conversation (with neither Serre nor Grothendieck). I don't know whether anyone said it in a lecture, much less wrote it down.
MR0130879 (24 #A733) Grothendieck, Alexander The cohomo... | 15 | https://mathoverflow.net/users/930 | 11172 | 7,595 |
https://mathoverflow.net/questions/11178 | 5 | As far as I know, one way to take a homotopy colimit in a model category is to replace (up to acyclic fibration) all arrows in the diagram with cofibrations, and take the strict colimit of the resulting diagram.
In Top with the model structure given by Serre fibrations, cofibrations, and weak equivalences, if one wan... | https://mathoverflow.net/users/2536 | Homotopy Pushouts via Model Structure in Top | Question 1: The model category $\mathcal{C}$ should be *left proper*, i.e. the pushout of a weak equivalence along a cofibration is again a weak equivalence. (Dually, there is a notion of right proper.) Top is left proper, as is any model category in which every object is cofibrant, such as SSet. There is some informat... | 9 | https://mathoverflow.net/users/126667 | 11180 | 7,601 |
https://mathoverflow.net/questions/11177 | 8 | I have heard that for a locally ringed space $X$ whose topology is second countable and Hausdorff, $X$ is a smooth manifold if and only if it is locally ringed space which is locally isomorphic to the sheaf of differentiable functions of some open sets in $\mathbb{R}^n$.
Question: What about the maps between smooth m... | https://mathoverflow.net/users/nan | Smooth maps considered as locally ringed space morphisms? | For a morphism $f: (X,{\mathcal O}\_X)\to (Y,{\mathcal O}\_Y)$ of locally ringed spaces the locality of the maps $f^{\sharp}\_x: {\mathcal O} \_{Y,f(x)}\to {\mathcal O} \_{X,x}$ implies that for each function $\lambda\in{\mathcal O}\_Y(U)$ we have $v(f^{\sharp}\_U(\lambda)) = f^{-1}(v(\lambda))$ (here $v(\lambda)$ deno... | 9 | https://mathoverflow.net/users/3067 | 11184 | 7,603 |
https://mathoverflow.net/questions/11182 | 7 | Remove the closure of simply connected region from the interior of a simply connected region. Is it true that the resulting domain can be mapped conformally to some annulus?
| https://mathoverflow.net/users/2938 | Riemann mapping for doubly connected regions | The answer is yes. This is a special case of theorem 10 in Ahlfors' *Complex Analysis*, section 5, chapter 6. (Special in that the theorem more generally says that if the complement of the domain has $n$ connected components not reduced to points in the extended plane, then the domain is equivalent to an annulus from w... | 7 | https://mathoverflow.net/users/1409 | 11185 | 7,604 |
https://mathoverflow.net/questions/11192 | 34 | Let $\mathcal{X}$ be a real or complex Banach space.
It is a well known fact that $\mathcal{X}$ is a Hilbert space (i.e. the norm comes from an inner product) if the parallelogram identity holds.
Question: Are there other (simple) characterizations for a Banach space to be a Hilbert space?
| https://mathoverflow.net/users/2989 | When is a Banach space a Hilbert space? | From [this article](https://doi.org/10.3103/S0027132208050070) by O. N. Kosukhin:
>
> A real Banach space $(X, \|\cdot\|)$ is a Hilbert space if and only if for any three points $A$, $B$, $C$ of this space not belonging to a line there are three altitudes in the triangle $ABC$ intersecting at one point.
>
>
>
... | 24 | https://mathoverflow.net/users/1847 | 11194 | 7,610 |
https://mathoverflow.net/questions/11188 | 3 | let $(X\_i)\_{i \in I}$ be an infinite family of sets with $|X\_i| \geq 2$. we define an equivalence relation on $X = \prod\_{i \in I} X\_i$ by $x \sim y \Leftrightarrow \{i : x\_i \neq y\_i\}$ is finite. what is the cardinality of $X/\sim$? we may endow the $X\_i$ with group structures and write this set as $\prod\_{i... | https://mathoverflow.net/users/2841 | cardinality of product modulo direct sum | The cardinality of the reduced product is always the same as that of the product, modulo omitting finitely many unusually large $X\_i$'s. (Even when $I$ is finite, in which case all $X\_i$'s should be omitted.)
Arrange the sets $X\_i$ in nondecreasing order of size in a wellordered sequence $(X\_\alpha)\_{\alpha<\tau... | 5 | https://mathoverflow.net/users/2000 | 11204 | 7,616 |
https://mathoverflow.net/questions/11203 | 1 | In my ODE class, we proved that if $\exp(L) = \exp(L')$ then the eigenvalues are congruent mod $2 \pi i$. Here, $L$ and $L'$ are two $n \times n$ matrices. I wanted to know if something more precise was true.
In a way, we should expect that matrix logs are multiple valued since this is the case in $\mathbb{C}$.
$$\... | https://mathoverflow.net/users/1358 | Matrix logarithms are not unique | No, they cannot. Note that $\exp(PAP^{-1})=P\exp(A)P^{-1}$, so wlog, both are in Jordan form. Then, we can compute by exponentiating Jordan blocks, and the first will have a two by two block (or two one by one) depending on whether it is diagonal or not $\delta=0,1$, and three $\exp(\lambda\_2)$ eigenvalues. The second... | 8 | https://mathoverflow.net/users/622 | 11205 | 7,617 |
https://mathoverflow.net/questions/11208 | 1 | How do you recenter a spherical coordinate system. For example, if the center were at $\left (0, 0, 0 \right )$ and I wanted to move the center of the spherical coordinate system to $\left (\rho\_{1}, \Theta\_{1}, \Phi\_{1} \right )$, then what transformation would I apply to $\left (\rho\_{2}, \Theta\_{2}, \Phi\_{2} \... | https://mathoverflow.net/users/3084 | Recentering a Spherical Coordinate Sytem | This is going to be unsightly...
The following Mathematica code:
```
Needs["VectorAnalysis`"]
Simplify@ CoordinatesFromCartesian[
CoordinatesToCartesian[{r, theta, phi}, Spherical]
+ CoordinatesToCartesian[{r0, theta0, phi0}, Spherical],
Spherical
]
```
gives the following output (doct... | 5 | https://mathoverflow.net/users/1409 | 11214 | 7,621 |
https://mathoverflow.net/questions/11207 | 9 | Here, I'm primarily concerced about zeta functions of hypersurfaces over fields of finite characteristic.
Assume $F\_q$ to be a finite field with q elements. Consider the zeta function of the hypersurface defined by $-y\_0^2+y\_1^2+y\_2^2+y\_3^2=0$ in $\mathbb{P}^3$.
If $-1$ is a square in $F\_q$, the zeta functi... | https://mathoverflow.net/users/2902 | How does the order of a pole of a zeta function indicate any geometric information? | For a smooth projective surface, the order of the pole at 1/q is conjectured to be the rank of the Neron-Severi group of the surface. That's a conjecture of Tate and is an analog of the Birch and Swinnerton-Dyer conjecture. Tate has formulated a more general conjecture for higher dimensional varieties too. For the case... | 15 | https://mathoverflow.net/users/2290 | 11215 | 7,622 |
https://mathoverflow.net/questions/11209 | 11 | I can (barely) understand the definition of the higher algebraic K-groups a la the plus construction right now (I have some past familiarity with K-theory for C\*-algebras and can recall the rudiments of the situation for vector bundles). So by "simple", I mean to a mathematical layman. If you have a complicated answer... | https://mathoverflow.net/users/1847 | Is there a simple relationship between K-theory and Galois theory? | Perhaps the other Bloch-Kato conjecture is more relevant; it relates *Milnor's* higher $K$-groups and Galois cohomology.
The following text is lifted from the expository account [on the arXiv.](http://arxiv.org/abs/math/0311099)
Let $F$ be a field, $n>0$ an integer which is invertible in $F$, $\bar F$ a
separable c... | 12 | https://mathoverflow.net/users/2821 | 11218 | 7,625 |
https://mathoverflow.net/questions/11219 | 59 | I am interested in learning about Shimura curves. Unlike most of the people who post reference requests however (see this [question](https://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry) for example), my problem is not sorting through an abundance of books but rather dealing with what appea... | https://mathoverflow.net/users/nan | What is a good roadmap for learning Shimura curves? | First of all, Kevin is being quite modest in his comment above: his paper
---
Buzzard, Kevin. Integral models of certain Shimura curves. Duke Math. J. 87 (1997), no. 3, 591--612.
---
contains many basic results on integral models of Shimura curves over totally real fields, and is widely cited by workers in ... | 42 | https://mathoverflow.net/users/1149 | 11229 | 7,631 |
https://mathoverflow.net/questions/11226 | 41 | Typically, in the functor of points approach, one constructs the category of algebraic spaces by first constructing the category of locally representable sheaves for the global Zariski topology (Schemes) on $CRing^{op}$. That is, taking the full subcategory of $Psh(CRing^{op})$ which consists of objects $S$ such that $... | https://mathoverflow.net/users/1353 | Commutative rings to algebraic spaces in one jump? | Yes. The category of algebraic spaces is the smallest subcategory of the category of sheaves of sets on Aff, the opposite of the category of rings, under the etale topology which (1) contains Aff, (2) is closed under formation of quotients by etale equivalence relations, and (3) is closed under disjoint unions (indexed... | 43 | https://mathoverflow.net/users/1114 | 11234 | 7,636 |
https://mathoverflow.net/questions/11238 | 17 | Before I ask the question, I need to recall what Bernoulli numbers
$(B\_k)\_{k\in\mathbb{N}}$ are, and what von Staudt and Clausen discovered about
them in 1840. The numbers $B\_k\in\mathbb{Q}$ are the coefficients in the formal power series
$$
{T\over e^T-1}=\sum\_{k\in\mathbb{N}}B\_k{T^k\over k!}
$$
so that $B\_0=1$,... | https://mathoverflow.net/users/2821 | von Staudt-Clausen over a totally real field | I also can't answer the question, but I'll say some things that could help. One thing von Staudt-Clausen tells you is the denominator of the Bernoulli number $B\_k$: it is precisely, the product of primes p for which $p-1\mid k$ (when $p-1\nmid k$, a result of Kummer says that $B\_k/k$ is p-integral). As Buzzard commen... | 8 | https://mathoverflow.net/users/1021 | 11242 | 7,641 |
https://mathoverflow.net/questions/11239 | 11 | In another [question](https://mathoverflow.net/questions/11182/riemann-mapping-for-doubly-connected-regions) here on MO, Anweshi asks if any doubly connected region in the complex plane can be conformally mapped to some annulus. The answer to this is yes. But the fact is that two annuli are conformally equivalent iff t... | https://mathoverflow.net/users/135 | Conformal maps of doubly connected regions to annuli. | By "see" I will assume you mean in a geometric sense. Then your question falls within a standard topic in geometric complex analysis. First some terminology: A doubly connected domain $R$ on the Riemann sphere is called a *ring domain*, and if you map it onto $r < |z| < s$ as a canonical domain, then $\mathrm{mod}(R) =... | 16 | https://mathoverflow.net/users/3304 | 11243 | 7,642 |
https://mathoverflow.net/questions/11248 | 3 |
>
> What can be said about the polynomials $f\in\mathbb Q[x, y]$ which are nonnegative on $\mathbb R\times \mathbb R$?
>
>
>
**Motivation:** this may lead to progress in the question about [polynomial onto map $\mathbb Z\times \mathbb Z\to\mathbb N$](https://mathoverflow.net/questions/9731/polynomial-representin... | https://mathoverflow.net/users/65 | Nonnegative polynomial in two variables | The following theorem of Artin -- his solution of Hilbert's 17th problem, but in a stronger form than Hilbert himself asked for -- answers the question.
Theorem (Artin, 1927): Let $F$ be a subfield of $\mathbb{R}$ that has a unique ordering, and let $f(t) = f(t\_1,\ldots,t\_n) \in F(t\_1,\ldots,t\_n)$ be a rational f... | 5 | https://mathoverflow.net/users/1149 | 11256 | 7,650 |
https://mathoverflow.net/questions/11240 | 31 | I'm trying to understand the notion of an *accessible category*. This isn't the first time I've tried to do this; but every time I try to make sense of the definition, I become perturbed by the following issue: *not every small category is accessible*.
You can find a definition of accessible category at [nLab](http:/... | https://mathoverflow.net/users/437 | Why aren't all small categories accessible? | To me, the "obvious" guess at (2) would be a category whose idempotent-splitting-completion (aka "Cauchy completion" or "Karoubi envelope") is accessible. While I don't have an explicit counterexample, I doubt that these have all the same good properties. The two properties you mention are special cases of closure unde... | 17 | https://mathoverflow.net/users/49 | 11264 | 7,657 |
https://mathoverflow.net/questions/4463 | 6 | It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs."
I have two closely related questions.
1. Is my understanding of the usage of "elementary" correct?
2. It appears that advanced techniques from other areas (e.g. algeb... | https://mathoverflow.net/users/136 | Definition of elementary number theory | Your usage of "elementary" is correct; your definition is the one that most number theorists would use. You don't have to take my word for it however; just consider the first sentence of [Selberg's Elementary Proof of the Prime Number Theorem](http://www.jstor.org/pss/1969455):
*In this paper will be given a new proo... | 4 | https://mathoverflow.net/users/nan | 11265 | 7,658 |
https://mathoverflow.net/questions/11246 | 2 | Let $a\_n$ be a bounded sequence of positive real numbers. Is it the case that the Dirichlet series $\sum \frac{a\_n}{n^s}$ can be meromorphically continued up to the right of zero, with at the most a pole at $1$?
This question occurred to me while trying to extend the Dedekind zeta function up to zero. If the above ... | https://mathoverflow.net/users/2938 | Continuation up to zero of a Dirichlet series with bounded coefficients | The following paper seems to (among other things) give a detailed construction roughly along the lines of my comment above:
---
Bhowmik, Gautami, Schlage-Puchta, Jan-Christoph
Natural boundaries of Dirichlet series. (English summary)
Funct. Approx. Comment. Math. 37 (2007), part 1, 17--29.
In this paper, the au... | 3 | https://mathoverflow.net/users/1149 | 11266 | 7,659 |
https://mathoverflow.net/questions/11268 | 2 | Let ${\mathcal D}$ be a triangulated category, ${\mathcal C}$ a triangulated subcategory and $Q: {\mathcal D}\to {\mathcal D}/{\mathcal C}$ the corresponding Verdier-localization. Now suppose we have a triangulated functor ${\mathbb F}: {\mathcal D}\to {\mathcal T}$ to some other triangulated category ${\mathcal T}$.
... | https://mathoverflow.net/users/3108 | Derived Functors in arbitrary triangulated categories | Yes, there exists such a treatment by Deligne, see "Cohomologie a supports propres", SGA4, Tome 3, Lect. Notes Math. 305, subsections 1.2.1-1.2.2. Basically, what one needs is that for any object X in D there exists a morphism X→Y in D with a cone in C such that for any morphism Y→Z in D with a cone in C there exists a... | 7 | https://mathoverflow.net/users/2106 | 11272 | 7,663 |
https://mathoverflow.net/questions/10919 | 7 | Two questions:
1. Suppose *a* and *b* are two uncountable cardinals. Consider the symmetric groups on sets of sizes *a* and *b* respectively (the symmetric group on a set is the group of all bijections from the set to itself, under composition). Consider the first-order theories of these as "pure groups" (i.e., just ... | https://mathoverflow.net/users/3040 | elementary equivalence of infinitary symmetric groups | For each ordinal $\alpha < \omega^\omega$ the symmetric group on $\aleph\_{\alpha}$ is first order definable in the class of all symmetric groups; i.e. there is a sentence in the first order language of groups that is true in $Sym(A)$ iff $|A| = \aleph\_\alpha$. See Mckenzie - On elementary types of symmetric groups, A... | 12 | https://mathoverflow.net/users/2689 | 11276 | 7,665 |
https://mathoverflow.net/questions/11283 | 11 | This question is partly inspired by a problem in Stewart's Calculus: "Find the area of the region enclosed by $y=x^2$ and $x=y^2$."
Suppose $f\colon [0,1]\to [0,1]$ is a convex increasing function that fixes 0 and 1. The graphs $y=f(x)$ and $x=f(y)$ enclose a convex region. To find its area, students will likely inte... | https://mathoverflow.net/users/2912 | Determination of a symmetric convex region by parallel sections | Yes.
Assume we have two distinct functions $f$ and $g$ such that $f^{-1}-f\equiv g^{-1}-g$.
Take a sequence $x\_n=f(x\_{n-1})$.
Clearly $f(x\_n)-g(x\_n)=0$ or $(-1)^n[f(x\_n)-g(x\_n)]$ has the same sign for all $n$.
Sinse $\int\_0^1f=\int\_0^1g$, there are two sequences $x\_n$ and $y\_n$ as above such that $f(x\_n... | 9 | https://mathoverflow.net/users/1441 | 11288 | 7,672 |
https://mathoverflow.net/questions/11295 | 1 | (I am a very, very new to mathematics, so I apologise in advance for posing a question so basic, but am out of ideas).
In [*Idoneal Numbers and some Generalizations*](http://www.mast.queensu.ca/~kani/papers/idoneal.pdf), pp. 15, Ernst Kani quotes Euler's criterion for idoneal numbers:
>
> An integer n ≥ 1 is id... | https://mathoverflow.net/users/3114 | Interpreting Euler's Criterion for Idoneal Numbers | In the remark below the statement of Euler's criterion in the paper you linked to, notice that Grube, who tried to correct Euler's original "proof" of this criterion, actually only provided a correct version in one direction -
>
> Frei [17], p. 57, points out that Grube only proved one direction of this criterion ... | 0 | https://mathoverflow.net/users/1916 | 11299 | 7,676 |
https://mathoverflow.net/questions/10463 | 8 | Is there a simple example of a smooth proper variety $X$ over $K=\mathbb{Q}\_p$ ($p$ prime) such that
--- $X(K)\neq\emptyset$,
--- the $l$-adic étale cohomology $H^i(X\times\_K\bar K,\mathbb{Q}\_l)$ is unramified for every prime $l\neq p$ and every $i\in\mathbb{N}$,
--- the $p$-adic étale cohomology $H^i(X\times\... | https://mathoverflow.net/users/2821 | A nice variety without a smooth model | As Minhyong suggests, a curve $C$ (with $C(\mathbb{Q}\_p)\neq0$) which has bad reduction but whose jacobian $J$ has good reduction would do the affair. This works because the cohomology of $C$ is essentially the same as that of $J$, and because an abelian $\mathbb{Q}\_p$-variety has good reduction if and only if its $l... | 3 | https://mathoverflow.net/users/2821 | 11302 | 7,678 |
https://mathoverflow.net/questions/11306 | 44 | From the example $D\_4$, $Q$, we see that the character table of a group doesn't determine the group up to isomorphism. On the other hand, Tannaka duality says that a group $G$ is determined by its representation ring $R(G)$.
What is the additional information contained in $R(G)$ as opposed to the character table?
... | https://mathoverflow.net/users/nan | Character table does not determine group Vs Tannaka duality | If I'm not mistaken, the extra information is not contained in the representation ring, you have to look at the category of representations. In particular, you want to look at the representation category equipped with its forgetful functor to vector spaces. Then the group can be recovered as the automorphisms of this f... | 23 | https://mathoverflow.net/users/321 | 11308 | 7,680 |
https://mathoverflow.net/questions/11307 | 3 | What would you call the product of an annulus and $S^1$ (a 'thickened' torus like 3-manifold)?
More generally, is there an archive or list online of names assigned to various (non-standard) manifolds by people? Or a set convention by which to name them?
| https://mathoverflow.net/users/3121 | What do you call the product of a circle and an annulus? | I would call it a thickened torus. I don't know how standard that is, but it is quite normal to speak of thickened manifolds, where one means that manifold times a closed interval.
I have long felt that there should be a mathematical dictionary- not an encyclopaedia, by a dictionary- in order to fix and record stand... | 6 | https://mathoverflow.net/users/2051 | 11315 | 7,684 |
https://mathoverflow.net/questions/11296 | 39 | A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent among von Neumann, Zermelo, etc. ordinals, an absolutist if she makes an argument for a priviledged reduction, such as ide... | https://mathoverflow.net/users/2833 | Were Bourbaki committed to set-theoretical reductionism? | First, most mathematicians don't really care whether all sets are "pure" -- i.e., only contain sets as elements -- or not. The theoretical justification for this is that, assuming the Axiom of Choice, every set can be put in bijection with a pure set -- namely a von Neumann ordinal.
I would describe Bourbaki's approa... | 29 | https://mathoverflow.net/users/1149 | 11319 | 7,686 |
https://mathoverflow.net/questions/11322 | 10 | Somewhere in Mumford's GIT, he seems to imply that any linear algebraic group is rational? This seems strange to me. Is it true?
| https://mathoverflow.net/users/nan | any linear algebraic group rational? | A (reduced, irreducible) linear algebraic group over an algebraically closed field is rational -- i.e., birational to projective space. This is a nontrivial result.
Briefest sketch of proof [EDIT: in characteristic 0 only; see the comment below]: use the Levi decomposition to reduce to the case of reductive groups, t... | 18 | https://mathoverflow.net/users/1149 | 11324 | 7,690 |
https://mathoverflow.net/questions/11297 | 6 | Hironaka's theorem guarantees an existence of resolution of singularities in characteristic 0. If I am not wrong, it also guarantees (or at least some other result does), that if the resolution is a singular point, one can get the "Exceptional Fiber" to be a simple normal crossing divisor. Very likely, if the singular ... | https://mathoverflow.net/users/2870 | Resolution of singularities, nature of | Hironaka in fact says that you can resolve singularities by a sequence of blow ups, and the universal property of blowing up is that the exceptional locus is a Cartier divisor. So in fact, the exceptional locus of the whole thing will be a Cartier divisor. Making sure that the exceptional locus is a snc divisor is call... | 8 | https://mathoverflow.net/users/622 | 11326 | 7,692 |
https://mathoverflow.net/questions/11327 | 13 | This is a very minor point, but one which had been grating me for a while. I apologize for asking a relatively trivial question, but nevertheless hope that it is suitable for MO since it should have a definite answer.
In Mumford's books, for instance Curves on Surfaces or Red Book, there is thing called "prescheme" w... | https://mathoverflow.net/users/2938 | Preschemes and schemes | The prescheme usage is outdated. As indicated in [nLab](https://ncatlab.org/nlab/show/EGA),
>
> our schemes are in EGA called preschemes; EGA’s schemes are what we call separated schemes
>
>
>
| 18 | https://mathoverflow.net/users/nan | 11328 | 7,693 |
https://mathoverflow.net/questions/11301 | 29 | The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x\_1,x\_2,\dots] \to H^\ast(M\_g ; \mathbb{Q})$ sending $x\_i$ to the kappa class $\kappa\_i$, is an isomorphism in degrees less than $n$, for sufficiently large $g$. Here $M\_g$ denotes the moduli of genus $g$ curves, and the degree ... | https://mathoverflow.net/users/83 | Mumford conjecture: Heuristic reasons? Generalizations? ... Algebraic geometry approaches? | All current proofs of Mumford's conjecture in fact prove a far stronger result, the "Strong Mumford conjecture", first formulated by Ib Madsen. This says the following (where by "moduli space" in the following we mean a homotopy type classifying concordance classes of surface bundles with perhaps some extra structure):... | 26 | https://mathoverflow.net/users/318 | 11339 | 7,698 |
https://mathoverflow.net/questions/11318 | 4 | I read that the Brauer-Manin obstruction $A(\mathbb{A}\_K)^{\mathbf{Br}}$ of an Abelian variety $A$ over a number field $K$ equals (naturally?) its Tate-Shafarevich group $\mathrm{III}(A)$.
Is this true? And if so, where can I find a proof?
| https://mathoverflow.net/users/nan | Brauer-Manin obstruction and Tate-Shafarevich group of an Abelian variety | The quotient of what you called the Brauer-Manin obstruction by the closure of $A(K)$ within it is related to the divisible part of Sha. In particular, if Sha has no divisible part (e.g. if it is finite) then the Brauer-Manin obstruction
is the closure of $A(K)$. See L. Wang, Brauer-Manin obstruction to weak approximat... | 4 | https://mathoverflow.net/users/2290 | 11340 | 7,699 |
https://mathoverflow.net/questions/11294 | 3 | The Kendall tau distance was originally defined as a correlation coefficient. It seems clear to me that every metric function $d$ that is bounded by $b$, induces a correlation coefficient. That is:
Let $d$ be a metric. Then $-d(x,y)/b+1$ is a correlation coefficient.
I wonder if the converse is true, too:
Let $c(... | https://mathoverflow.net/users/3116 | Do all correlation coefficients induce a pseudometric? | The statement is false. Consider the Pearson sample correlation coefficient:
c(x,y) = (x-mean(x)).(y-mean(y))/sqrt(|x-mean(x)|^2\*|y-mean(y)|^2)
Here is an example, where the triangle inequality for the distance defined in
the question is not satisfied
x = [0.5847 -0.3048 -0.4431 0.5032 -0.3401],
y = [0.2018 ... | 3 | https://mathoverflow.net/users/1059 | 11341 | 7,700 |
https://mathoverflow.net/questions/11338 | 3 | Iasked me the question what the interpretation of the irreducibility of a moduli space is for the functor it represents. For proper, there is the valuative criterion and for (formally) smooth, there is the infinitesimal lifting property, but I don't know something similar for irreducible. Can someone shed some light on... | https://mathoverflow.net/users/nan | functorial meaning of irreducibility of a moduli space | Your question is a bit vague, but let me try to say something. Being irreducible is a global property so there can be no local characterization like being smooth, regular, etc. If $\mathcal{M}$ is the "space" representing your functor, we say that it is irreducible if it admits a surjective map from an irreducible vari... | 8 | https://mathoverflow.net/users/397 | 11347 | 7,705 |
https://mathoverflow.net/questions/10569 | 14 | [**Edit:** Question 1 has been moved [elsewhere](https://mathoverflow.net/questions/11316/smooth-proper-schemes-over-z-with-points-everywhere-locally) so that an answer to Question 2 can be *accept*ed.]
**Question 2.** *Is there a number field* $K$, *and a smooth proper scheme* $X\to\operatorname{Spec}(\mathfrak{o})$... | https://mathoverflow.net/users/2821 | Smooth proper schemes over rings of integers with points everywhere locally | Chandan asked Vladimir and me for an example of an elliptic curve over a real quadratic field that has everywhere good reduction and non-trivial sha, with an explicit genus $1$ curve representing some element of sha. Here's one we found:
The elliptic curve $y^2+xy+y = x^3+x^2-23x-44$ over $\mathbb Q$ (Cremona's refer... | 12 | https://mathoverflow.net/users/3132 | 11356 | 7,710 |
https://mathoverflow.net/questions/11354 | 11 | Let $G$ be a finite group. If $L$ is a finite free $\mathbf{Z}$-module with an action of $G$, say $L$ is *induced* if it's isomorphic as a $G$-module to $Ind\_H^G(\mathbf{Z})$ with $H$ a subgroup of $G$ and $H$ acting trivially on $\mathbf{Z}$. And, for want of better terminology, let's say $L$ is a *sum-of-induceds* i... | https://mathoverflow.net/users/1384 | Extension of induced reps over Z: is it a sum of induced reps? | In my interpretation of your description, a sum-of-induceds is just the permutation representation of some (not necessarily connected) G-set. The representation $\mathrm{Hom}(V\_1,V\_2)$ for two permutation representations is itself permutation: it's the permutation action on product of the G-sets.
So, $\mathrm{Ext}^... | 12 | https://mathoverflow.net/users/66 | 11357 | 7,711 |
https://mathoverflow.net/questions/11366 | 21 | I ran into the following problem. I have some lengthy paper in which I develop some theory to attack some problem. While I was working on this paper, I got some nice result which might interest a bigger audience and can stand by itself (that is, it is interesting out of context of the big work, and has a fairly short p... | https://mathoverflow.net/users/2042 | When to split/merge papers? | According to [Gian-Carlo Rota](http://www.math.vt.edu/people/day/advice/YMN4_25.html), one of the secrets of mathematical success is to publish the same result many times.
| 26 | https://mathoverflow.net/users/297 | 11369 | 7,720 |
https://mathoverflow.net/questions/10468 | 9 | I know that many graph problems can be solved very quickly on graphs of bounded degeneracy or arboricity. (It doesn't matter which one is bounded, since they're at most a factor of 2 apart.)
From Wikipedia's article on the clique problem I learnt that finding cliques of any constant size k takes linear time on graph... | https://mathoverflow.net/users/1042 | Algorithms on graphs of bounded degeneracy/arboricity | Bounded degeneracy or arboricity just means that the graph is sparse (number of edges is proportional to number of vertices in all subgraphs).
Some ideas that have been used for fast algorithms on these graphs:
* Order the vertices so that each vertex has only d neighbors that are later in the ordering, where d is ... | 11 | https://mathoverflow.net/users/440 | 11374 | 7,725 |
https://mathoverflow.net/questions/11377 | 2 | The standard definition is that $a\in\mathbb{Z}$ is a primitive root modulo $p$ if the order of $a$ modulo $p$ is $p-1$.
Let me rephrase, to motivate my generalization: $a\in\mathbb{Z}$ is a primitive root modulo $p$ if the linear recurrence defined by $x\_0=1$, $x\_n=ax\_{n-1}$ for $n\geq1$ has maximum possible peri... | https://mathoverflow.net/users/1916 | Generalization of primitive roots | If $(a\_1,\ldots,a\_r)$ is an $r$-primitive root in your definition, then the polynomial $x^r-a\_1x^{r-1}-\cdots-a\_r$ is irreducible in $Z/pZ[x]$ and any of
its roots is a generator (i.e. a primitive root) of the multiplicative group of the field of $p^r$ elements and conversely. This maximal period is $p^r-1$. See, e... | 7 | https://mathoverflow.net/users/2290 | 11381 | 7,729 |
https://mathoverflow.net/questions/11335 | 18 | This is a question I have since longer time, but I have absolutely no idea how to proceed on it.
Let $p>3$ be a prime. Prove that $\displaystyle\sum\limits\_{k=1}^{p-1}\frac{1}{k}\binom{2k}{k}\equiv 0\mod p^2$. Here, we work in $\mathbb{Z}\_{\mathbb{Z}\setminus p\mathbb{Z}}$ (that is, $\mathbb{Z}$ localized at all nu... | https://mathoverflow.net/users/2530 | A binomial sum is divisible by p^2 | Your binomial sum is divisible by $p^2$ as is shown in a recent paper of Sun and Tauraso:
<https://arxiv.org/abs/0805.0563>
Even more remarkably, they compute that it's equivalent to $\frac{8}{9}p^2B\_{p-3} \bmod p^3$, as well as various other congruences for the corresponding alternating sum.
| 18 | https://mathoverflow.net/users/3143 | 11401 | 7,742 |
https://mathoverflow.net/questions/11404 | 22 | I tried to read the definition of it on Rubin's book "Euler systems" but it looks highly technical. Can anyone shed some light on it? In particular, is there some starting examples?
The wiki entry is too short and does not contain much useful information.
And what is the motivation of such objects? In particular, why... | https://mathoverflow.net/users/1238 | what is an Euler system and the motivation for it? | My understanding is that they're named "Euler systems" because that "Frobenius acting on T" in the definition (line 4, p. 22 of Rubin's book) is an "Euler factor" as in Euler's product decomposition of the Riemann zeta function.
The two easiest examples of Euler systems are the so-called cyclotomic units (not the ro... | 19 | https://mathoverflow.net/users/1018 | 11405 | 7,743 |
https://mathoverflow.net/questions/11393 | 16 | I'm aware to varying extents of the existence of certain decompositions of the space of $k$-forms on a compact complex or compact Riemannian manifold that split into closed, co-closed, and harmonic forms, and that the space of harmonic forms becomes isomorphic to the de Rham cohomology groups. However, these are define... | https://mathoverflow.net/users/344 | Why the similarity between Hodge theory for compact Riemannian and complex manifolds? | For any Riemannian (in particular, for any Hermitean) manifold there is the decomposition: all forms are the direct sum of the harmonic ones, exact ones and those in the image of the conjugate operator of the de Rham differential. Every de Rham cohomology class is represented by a unique $d$-harmonic form.
For any co... | 14 | https://mathoverflow.net/users/2349 | 11407 | 7,744 |
https://mathoverflow.net/questions/11398 | 3 | Hi,
I know that the answer is no, yet I dont know how to prove it wrong. Finding a counterexample is not a good solution because it is a past written exam question with no calculators allowed. The first counterexample is about 30000... Is there a, simple preferred, solution to this problem? (This is asked in a CS dis... | https://mathoverflow.net/users/3142 | Is the product of first $n$ prime numbers $+1$ another prime number? | Here's a possible intended solution to show that $30031 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 + 1$ is composite without factoring it. Recall the Fermat primality test: if $a^{n-1} \not \equiv 1 \bmod n$, then $n$ cannot be prime. It turns out that $2^{30030} \equiv 21335 \bmod 30031$, so $30031$ must indeed be ... | 10 | https://mathoverflow.net/users/290 | 11412 | 7,747 |
https://mathoverflow.net/questions/11427 | 11 | Can anyone give me references where I would see a detailed exposition of how to translate gauge field theory as known to physicists into the language of connections. I am looking for a detailed exposition on the mathematical formulation of Yang-Mills field theory. Something which might also give an exposition about Che... | https://mathoverflow.net/users/2678 | Looking for reference on gauge fields as connections. | *[Geometry, Topology, and Physics](http://rads.stackoverflow.com/amzn/click/0750306068)* by Nakahara
*[Classical Theory of Gauge Fields](http://rads.stackoverflow.com/amzn/click/0691059276)* by Rubakov
*[Modern Geometry](http://rads.stackoverflow.com/amzn/click/0387961623)*, Part 2 by Dubrovin, Fomenko, and Novikov... | 5 | https://mathoverflow.net/users/1847 | 11433 | 7,760 |
https://mathoverflow.net/questions/11422 | 28 | If we have a finite dimensional Lie algebra g, then the flag variety of g is a projective scheme.
My question is what is Hirzebruch-Riemann-Roch formula for this projective scheme? Are there any interesting results?
I wonder know whether Riemann-Roch in this setting have some beautiful representation theory explan... | https://mathoverflow.net/users/3156 | What is the Hirzebruch-Riemann-Roch formula for the flag variety of a Lie algebra? | Riemann-Roch for the flag variety is the Weyl Character formula!
More specifically, let $L$ be an ample line bundle on $G/B$, corresponding to the weight $\lambda$. According to [Borel-Weil-Bott](http://en.wikipedia.org/wiki/Borel%25E2%2580%2593Weil%25E2%2580%2593Bott_theorem), $H^0(G/B,L)$ is $V\_{\lambda}$, the irr... | 53 | https://mathoverflow.net/users/297 | 11440 | 7,764 |
https://mathoverflow.net/questions/11091 | 20 | I was sure it is known, but it appears to be an open problem (see the answer of Terry Tao).
>
> Assume for a group $G$ there is a
> polynomial $P$ such that given
> $n\in\mathbb N$ there is set of
> generators $S=S^{-1}$ such that
> $$|S^n|\leqslant P(n)\cdot|S|\ \ (\text{or stronger condition}\ |S^k|\leqslant P(k)... | https://mathoverflow.net/users/1441 | А generalization of Gromov's theorem on polynomial growth | My [paper with Shalom](http://terrytao.wordpress.com/2009/10/23/a-finitary-version-of-gromovs-polynomial-growth-theorem/) does settle the question when $S = S\_n$ is known to have size polynomial in n (and maybe is allowed to grow just a little bit faster than this, something like $n^{(\log \log n)^c}$ or so), but I do... | 17 | https://mathoverflow.net/users/766 | 11447 | 7,770 |
https://mathoverflow.net/questions/11435 | 12 | Suppose *A* is a set and *S* is a collection of subsets closed under arbitrary unions and intersections. Can we find a collection *F* of functions from *A* to itself such that a subset *B* of *A* is in *S* if and only if $f(B) \subseteq B$ for all $f \in F$ (in other words, is *S* precisely the collection of invariant ... | https://mathoverflow.net/users/3040 | Collection of subsets closed under union and intersection | The answer is **Yes**. Furthermore, such a family can be found of size at most the cardinality of A, even when S is much larger.
The key to the solution is to realize that every such family S arises as the collection of downward-closed sets for a certain partial pre-order on A, which I shall define. (Conversely, ever... | 15 | https://mathoverflow.net/users/1946 | 11451 | 7,772 |
https://mathoverflow.net/questions/11436 | 1 | Suppose we have a monoidal category $(\mathcal{C},\otimes)$. I am interested in the conditions or situations where the following do and do not hold:
1. For any objects *A*, *B* of $\mathcal{C}$, the induced map $\operatorname{End}(A) \times \operatorname{End}(B) \to \operatorname{End}(A \otimes B)$ is injective (by i... | https://mathoverflow.net/users/3040 | Monoidal operations on categories where the maps on Aut, End are injective | Your conditions don't seem to obtain very often, unfortunately.
Let's begin with the one-object case (where the one object is the unit, as it must be). This is the same thing as a monoid object in monoids, or equivalently, a commutative monoid $A$, by Eckmann-Hilton. Your requirement (1) appears to be that the multip... | 3 | https://mathoverflow.net/users/3049 | 11452 | 7,773 |
https://mathoverflow.net/questions/11457 | 9 | In their paper *[Computing Systems of Hecke Eigenvalues Associated to Hilbert Modular Forms](http://arxiv.org/abs/0904.3908)*, Greenberg and Voight remark that
...it is a folklore conjecture that if one orders totally real fields by their discriminant, then a (substantial) positive proportion of fields will have stri... | https://mathoverflow.net/users/nan | Strict Class Numbers of Totally Real Fields | One heuristic is the following: if one imagines that the residue at $s = 1$ of the $\zeta$-function doesn't grow too rapidly, then the value is a combination of the regulator and the class number. I don't know any reason for the regulator not to also grow (there
are a lot of units, after all!), and hence one can imagin... | 6 | https://mathoverflow.net/users/2874 | 11463 | 7,778 |
https://mathoverflow.net/questions/11450 | 19 | I wonder why the Artin conjecture is so important. The only reason I could figure out is that one could use the holomorphy of Artin L-series and Weil's converse theorem to show modularity of two-dimensional Galois representations.
Are there other reasons?
| https://mathoverflow.net/users/nan | Applications of Artin's holomorphy conjecture | From a modern view-point, its importance stems from the relationship with modularity/automorphy. Namely, a stronger conjecture, due to Langlands, should be true:
if $\rho:G\_K \to GL\_n({\mathbb C})$ is a continuous irreducible representation, for some number field $K$, then there is a cuspidal automorphic representati... | 26 | https://mathoverflow.net/users/2874 | 11466 | 7,780 |
https://mathoverflow.net/questions/11488 | 26 | Is there a characterization of the class of varieties which can be described as an intersection of quadrics, apart from the taulogical one?
Lots of varieties arise in this way (my favorite examples are the Grassmanianns and Schubert varieties and some toric varieties) and I wonder how far can one go.
| https://mathoverflow.net/users/1409 | Varieties cut by quadrics | In fact the answer *is* in some sense tautological: every projective variety can be realized as a scheme-theoretic intersection of quadrics! See e.g.
D. Mumford, "Varieties defined by quadratic equations", Questions on algebraic varieties, C.I.M.E. Varenna, 1969 , Cremonese (1970) pp. 29–100,
for quantitative refin... | 32 | https://mathoverflow.net/users/1149 | 11490 | 7,800 |
https://mathoverflow.net/questions/11480 | 18 | Let $V=L$ denote the axiom of constructibility. Are there any ***interesting*** examples of set theoretic statements which are independent of $ZFC + V=L$? And how do we construct such independence proofs? The (apparent) difficulty is as follows: Let $\phi$ be independent of $ZFC + V=L$. We want models of $ZFC + V=L + \... | https://mathoverflow.net/users/2689 | On statements independent of ZFC + V=L | There are numerous examples of such statements. Let me organize some of them into several categories.
First, there is the hierarchy of [large cardinal](http://en.wikipedia.org/wiki/Large_cardinal) axioms that are relatively consistent with V=L. See the [list of large cardinals](http://en.wikipedia.org/wiki/List_of_l... | 26 | https://mathoverflow.net/users/1946 | 11491 | 7,801 |
https://mathoverflow.net/questions/11495 | 3 | It's sometimes convenient to have different notations for "$A$ is a subset of $B$" depending on what the inclusion map does:
1. If it's non-surjective, $A\subsetneq B$ or $A\subset B$, depending on your religion
2. If it's surjective, $A=B$ :)
3. If the image is a precompact set, $A\Subset B$
Does there exist notat... | https://mathoverflow.net/users/2912 | Notation for "the inclusion map is a homotopy equivalence" | $A\stackrel{\sim}{\hookrightarrow}B$? Alternatively, using Oberdiek's stackrel.sty you could say something like
```
A \mathrel{\raisebox{2pt}{$\stackrel[\raisebox{1pt}{$\sim$}]{}\subset$}} B
```
and play a little with the raiseboxes so that this aligns more or less correctly (this depends on your final font, and... | 6 | https://mathoverflow.net/users/1409 | 11496 | 7,804 |
https://mathoverflow.net/questions/11502 | 35 | [Une traduction française suit la version anglaise.]
The question is only about elliptic curves $E$ over $\mathbb{Q}$ and concerns only the aspect
(order of vanishing of $L(E,s)$ at $s=1$)$\ =\ $(rank of $E(\mathbb{Q})$).
Let $r$ be the LHS and $d$ the RHS, so that (a special case of ) the Birch and Swinnerton-D... | https://mathoverflow.net/users/2821 | The current status of the Birch & Swinnerton-Dyer Conjecture | The parity conjecture is known, i.e. it is known that if the order of vanishing of the $L$-function is even/odd, then the corank of $p$-Selmer is of the same parity (and I think this is known for every $p$ at this point; Nekovar handled the good ordinary or multiplicative case,
and B.D. Kim the good supersingular case.... | 27 | https://mathoverflow.net/users/2874 | 11507 | 7,811 |
https://mathoverflow.net/questions/11508 | 18 | Given points $u, v\_1, \dots,v\_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v\_1, \dots, v\_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in the obvious way. I have two questions:
1. Is there a different (or more efficient algorithm) for this? If not, t... | https://mathoverflow.net/users/825 | Deciding membership in a convex hull | It isn't just that you can do the problem with linear programming. The reduction to linear programming actually shows that the convex hull problem is equivalent by dualization to the feasible-point problem in linear programming. In other words, you are asking whether there exist coefficients $\alpha\_1,\ldots,\alpha\_n... | 15 | https://mathoverflow.net/users/1450 | 11510 | 7,813 |
https://mathoverflow.net/questions/11464 | 18 | There was a problem in an Olympiad selection test, which went as follows: Consider the set $\{1,2,\dots,3n \}$ and partition it into three sets *A*, *B* and *C* of size *n* each. Then, show that there exist *x*, *y* and *z*, one in each of the three sets, such that *x + y = z*.
This has a tricky-to-get but otherwise ... | https://mathoverflow.net/users/3040 | Balancing problem | [This](http://www.emis.de/journals/INTEGERS/papers/a9int2003/a9int2003.pdf) article is a nice survey of "Rainbow Ramsey theory". In this jargon what you are trying to prove is that the vector $(1,1,\dots,1,-1,-1,\dots,-1)$ is rainbow partition $m$-regular.
The case of $(1,1,-1,-1)$ being rainbow partition 4-regular, ... | 16 | https://mathoverflow.net/users/2384 | 11512 | 7,815 |
https://mathoverflow.net/questions/11537 | 1 | I know that if we have some data representing some wave, for example image line values, we can use fourier transform to get frequency function of that wave. But we have N values at points x=0...N-1 And we get only N frequencies at the output. So I want to analyze the wave everywhere in the range [0, N-1] For example at... | https://mathoverflow.net/users/3195 | Calculating fourier transform at any frequency | You need to have some a priori knowledge about your wave between and beyond the sampling points to get a meaningful guess about the full Fourier transform. The $N$ values that you get doing the discrete Fourier transform have anything to do with the continuous Fourier transform only for indices much less than $N$. Note... | 7 | https://mathoverflow.net/users/1131 | 11551 | 7,838 |
https://mathoverflow.net/questions/11549 | 8 | In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets?
**Vitali's Lemma:**
Let $E$ be a set of finite outer measure and $G$ a collection of intervals that cover $E$ in the sense of Vitali. ... | https://mathoverflow.net/users/2907 | Why is this generality in Vitali's Lemma useful? | The short answer (to the question in the title) is so that you don't need to worry about whether E is measurable. If you happen to know that E is measurable then you can drop "outer" everywhere.
There may be a longer answer (better addressing the question as stated in the page) involving specific applications where E... | 8 | https://mathoverflow.net/users/1044 | 11552 | 7,839 |
https://mathoverflow.net/questions/11536 | 9 | Theorem 3.2 of the [paper](http://arxiv.org/abs/1001.0056) "Quantum cohomology of the Springer resolution" by Braverman, Maulik and Okounkov relates equivariant quantum cohomology of the cotangent bundle of $G/B$ to the trigonometric Dunkl operators. A naive guess (or maybe [WAG](https://mathoverflow.net/questions/9764... | https://mathoverflow.net/users/nan | Quantum equivariant $K$-theory and DAHA. | My best guess is that either
1. this is true for $\mathbb{CP}^1$, and it's pretty easy to generalize that given what's already in that paper, or
2. this is false for $\mathbb{CP}^1$, and you're hosed.
The bit one has to understand is the map from the 2 point genus 0 moduli space to the Steinberg variety. BMO get aw... | 2 | https://mathoverflow.net/users/66 | 11556 | 7,842 |
https://mathoverflow.net/questions/11558 | -2 | I want to create Taylor series of a complex function that has complex conjugate in it. Obviously I cannot do a total derivative but derivations over real and imag parts exist.
Bonus question: Can I produce a Taylor series using only derivations over real part?
| https://mathoverflow.net/users/269 | Taylor series of a complex function that is not holomorphic | Another option is $\sum c\_{mn}z^m \bar z^n$, which still keeps track of the complex structure. For instance, harmonic functions will have $c\_{mn}=0$ unless $mn=0$.
| 2 | https://mathoverflow.net/users/2912 | 11568 | 7,850 |
https://mathoverflow.net/questions/11580 | 6 | There are a few algorithms around for finding the minimal bounding rectangle (OBB) containing a given (convex) polygon.
Does anybody know about an algorithm for finding a minimal-area bounding quadrilateral (any quadrilateral, not just rectangles)?
I've been refered to this site from stackoverflow.com ([original po... | https://mathoverflow.net/users/3204 | Minimum-area bounding quadrilateral algorithm | I think what you want is "Geometric applications of a matrix searching algorithm", Aggarwal et al, Algorithmic 1987, [doi:10.1007/BF01840359](http://dx.doi.org/10.1007/BF01840359). It builds on previous work of Aggarwal, Chang, and Yap (their reference [2]) to show that the minimum area enclosing k-gon of a geometric f... | 6 | https://mathoverflow.net/users/440 | 11584 | 7,862 |
https://mathoverflow.net/questions/11576 | 6 | This may be totally trivial or wrong. I am just posting this because I am sick and tired of trying to understand this myself and I am sure someone out here can just answer it out of his head in 2 minutes.
Let $G$ be a finite group, and $k$ an algebraically closed field whose characteristic is not a divisor of $\left|... | https://mathoverflow.net/users/2530 | Apocryphal Maschke theorem? | The result about bimodules is true, and standard. Here is one way to see it.
By Frobenius reciprocity, $\operatorname{Hom}\_G(V,k[G]) = \operatorname{Hom}\_k(V,k)$, since $k[G]$ is the induction (or coinduction, depending on your terminology) of the trivial representation of the trivial subgroup of $G$ to $G$.
Sinc... | 12 | https://mathoverflow.net/users/2874 | 11590 | 7,867 |
https://mathoverflow.net/questions/11567 | 32 | There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}\_X$ modules obeying some nice condition. Then $H^i(X, E)=0$ for $i>0$."
What is the best way to formulate this result? I ask because I'm sur... | https://mathoverflow.net/users/297 | What is the right version of "partitions of unity implies vanishing sheaf cohomology" | Although we clearly all have more or less the same answers, here is how I like to organize things.
I) Let $\mathcal F$ be a sheaf of abelian groups on the topological space $X$. It is said to be soft if every section $s \in \Gamma (A,\mathcal F)$ over a closed subset $A\subset X$ can be extended to $X$.
Notice caref... | 13 | https://mathoverflow.net/users/450 | 11594 | 7,870 |
https://mathoverflow.net/questions/11589 | 6 | Let ${\mathcal C}$ be the class of topological spaces which carry a CW-structure (note that I do not want to fix some particular CW-structure).
Is it true that for a covering map $E\stackrel{f}{\to} B$ with $E\in{\mathcal C}$ we have $B\in{\mathcal C}$, too?
It *is* true that the total space of a covering lies in ... | https://mathoverflow.net/users/3108 | Properties of the class of topological spaces possessing a CW-structure | Ok, here is a simple example when this fails: let $X$ be $\mathbf{R}^2$ minus the origin and consider the automorphism of $X$ given by $(x,y)\mapsto (2x,y/2)$. This automorphism generates a group $G$. The quotient map $X\to X/G$ is a covering, but $X/G$ is not Hausdorff.
This answers the question as it is stated, but... | 10 | https://mathoverflow.net/users/2349 | 11602 | 7,874 |
https://mathoverflow.net/questions/10237 | 24 | The [Convolution Theorem](http://mathworld.wolfram.com/ConvolutionTheorem.html%20%22Convolution%20Theorem%22) is often exploited to compute the convolution of two sequences efficiently: take the (discrete) Fourier transform of each sequence, multiply them, and then perform the inverse transform on the result.
The sam... | https://mathoverflow.net/users/2361 | does the "convolution theorem" apply to weaker algebraic structures? | In general, it is a major open question in discrete algorithms as to which algebraic structures admit fast convolution algorithms and which do not. (To be concrete, I define the *$(\oplus,\otimes)$ convolution* of two $n$-vectors $[x\_0,\ldots,x\_{n-1}]$ and $[y\_0,\ldots,y\_{n-1}]$, to be the vector $[z\_0,\ldots,z\_{... | 24 | https://mathoverflow.net/users/2618 | 11606 | 7,877 |
https://mathoverflow.net/questions/11610 | 10 | A Poisson manifold is a real manifold $M$ along with a Lie bracket $[\cdot,\cdot]$ on $C^\infty(M)$ which is a derivation in each variable. Poisson manifolds are interesting for a few reasons, among them:
1. You can define the notion of an integrable system structure on a Poisson manifold, which allows them to be app... | https://mathoverflow.net/users/622 | Examples of Poisson schemes | Of course, thinking of Poisson things in a relative way isn't going to give anything new, since a family of Poisson things is itself Poisson.
But to answer whether people think about these things in an abstract way, the answer is yes. There's now a huge and absolutely beautiful theory of symplectic singularities; her... | 8 | https://mathoverflow.net/users/66 | 11615 | 7,883 |
https://mathoverflow.net/questions/11531 | 7 | ### Preliminaries
Let $n \in \mathbb{N}$ and $v$ be a vertex of a graph $G$. Let the *$n$-neighbourhood of $v$*, $N\_n(v)$, be the induced subgraph of $G$ containing $v$ and all vertices at most $n$ edges away from $v$.
With $\epsilon(v)$ the [eccentricity](http://mathworld.wolfram.com/GraphEccentricity.html) of $v... | https://mathoverflow.net/users/2672 | Yet another graph invariant: the similarity matrix | This is nice. What is the motivation for dividing by $\epsilon(u)+\epsilon(v)$?
To partly answer your first question, I suspect the notion of $n$-neighborhood is relatively common, but one place where it is very common, if not a staple, in Finite Model Theory (read also Database Theory). It is a key component of vari... | 3 | https://mathoverflow.net/users/2000 | 11619 | 7,887 |
https://mathoverflow.net/questions/11622 | 24 | I'm a little embarrassed to ask this one, but it could help for a class I'm teaching, so here goes:
Let $X$ be a metric space. We all know that $X$ admits a **completion**, which is a complete metric space $\hat{X}$ together with an isometric embedding $\iota: X \hookrightarrow \hat{X}$ with dense image. Moreover, on... | https://mathoverflow.net/users/1149 | What is the "right" universal property of the completion of a metric space? | This doesn't quite answer your questions about (2), but one could say the following:
given a metric, there is an underlying uniform structure. We can then form the completion
with respect to this uniform structure, which is universal for uniformly continuous maps into complete uniform spaces. That this map is injective... | 15 | https://mathoverflow.net/users/2874 | 11625 | 7,891 |
https://mathoverflow.net/questions/11614 | 9 | Let's consider a vector bundle $E$ of rank $n$ over a compact manifold $X$. Consider the associated Grassmannian bundle $G$ for some $k < n$, obtained by replacing each fiber $E\_x$ by $Gr(k,E\_x)$.
Let's suppose that there is a full flag of subbunldes $F\_1 \subset F\_2 \dots \subset F\_n \subset E$. I think that in... | https://mathoverflow.net/users/2260 | Grassmannian bundle theorem | Re 1.: the reasoning is correct additively, but not multiplicatively (did you check the case $k=1$?).
Re 2.: given any complex bundle $E$ over $X$ of rank $n$, the cohomology of the associated Grassmannian bundle of $k$-planes is $$H^{\bullet}(X,\mathbf{Z})\otimes\mathbf{Z}[c\_1,\ldots,c\_k,c\_1',\ldots,c\_{n-k}']/(1... | 17 | https://mathoverflow.net/users/2349 | 11627 | 7,893 |
https://mathoverflow.net/questions/11628 | 4 | As is well known, one can view $\mathbb{CP}^n$ as a quotient of the unit $(2n + 1)$-sphere in $\mathbb{C}^{n+1}$ under the action of $U(1)$, since every line in $\mathbb{C}^{n+1}$ intersects the unit sphere in a circle.
Moreover, we have $S^{2n + 1} = SU(n+1)/SU(n)$, where $SU(n)$ embeds into the bottom right-hand c... | https://mathoverflow.net/users/1977 | Complex Projective Space as a $U(1)$ quotient | The U(1) group is the torus in SU(N+1) which commutes with SU(N). In the example
given in the question where SU(N) is chosen as the bottom N-dimensional block. U(1) consists
of the diagonal matrices
diag{exp(N\*i\*theta), exp(-i\*theta), . . . . (N-times) exp(-i\*theta)}
Please observe that the restriction to the ... | 4 | https://mathoverflow.net/users/1059 | 11640 | 7,901 |
https://mathoverflow.net/questions/11641 | 6 | There are subsets of the real line that has infinite counting measure, but Lebegue measure 0, so the Lebegue measure is used for measuring larger sets than the counting measure. My question is: Is there a translation invariant measure m such that for some sets with Lebegue measure 0 the m-measure is infinite and for so... | https://mathoverflow.net/users/2097 | Measure between the counting measure and the Lebegue measure | Hausdorff measures of dimensions between 0 and 1 are a continuous spectrum of examples.
| 10 | https://mathoverflow.net/users/2368 | 11642 | 7,902 |
https://mathoverflow.net/questions/11629 | 9 | The basic logic course in school gives the impression that logic has both the syntax and the semantics aspects. Recently, I wonder whether the syntax part still plays an essential role in the current studies. Below are some of my observations, I hope the idea from the community can make them more complete.
Model theo... | https://mathoverflow.net/users/2701 | How much of the current logic is about syntax? | I think your observation is a very good one, but this phenomenon is limited to classical logic and does not continue to hold when we move to intuitionistic or substructural logics.
One way of understanding the role of syntax is to take the connectives of logic as explaining what counts as a legitimate proof of that p... | 7 | https://mathoverflow.net/users/1610 | 11646 | 7,905 |
https://mathoverflow.net/questions/11583 | 1 | The following question is related to "Remark 2.2" in Christophe Cazanave's paper "[Algebraic homotopy classes of algebraic functions](http://arxiv.org/abs/0912.2227v1)". I decided to add the arxiv article-id to the questions title to invite other people who like to study this article to do the same. My hope is that thi... | https://mathoverflow.net/users/2146 | A rational point in the scheme of pointed degree n rational functions [0912.2227] | I'm not so good on the scheme-theoretic language, so let me embed $F\_n$ as the affine variety $\text{res}\\_{n,n}(X^n + ..., b\_{n-1} X^{n-1} + ...) y = 1$ one dimension up. Then a morphism $k[a\_0, ... a\_{n-1}, b\_0, ... b\_{n-1}, y]/(\text{stuff}) \to R$ is precisely (assuming that Cazanava means either $k = \mathb... | 3 | https://mathoverflow.net/users/290 | 11656 | 7,911 |
https://mathoverflow.net/questions/11664 | 13 | Using the chern character, it can be shown that there is no complex structure on $S^n$ if $n > 6$: See May's book: if $S^{2n}$ has a complex structure, let $\tau$ be the tangent bundle. $c\_n(\tau) = \chi(S^{2n}) = 2$ must be divisible by $(n-1)!$ by Husemöller, Fibre bundles, chapter 20, Theorem 9.8. So the only case ... | https://mathoverflow.net/users/nan | complex structure on S^n | It is known that $S^4$ doesn't even have an almost complex structure. (The sphere $S^6$ does have some, but whether any of them is the underlying almost complex structure of a complex manifold is open.) The lack of almost complex structure can be proved a number of ways, one way is by showing that an almost complex, co... | 27 | https://mathoverflow.net/users/622 | 11667 | 7,918 |
https://mathoverflow.net/questions/11660 | 3 | This question was motivated by the answers in [D-module as quasi coherent sheaves on deRham stack](https://mathoverflow.net/questions/11200/geometry-of-triangulated-category-and-d-modules-theory). What I am interested in is the case of D-module on flag variety of Lie algebra. So,in this case, if we realize the D-module... | https://mathoverflow.net/users/1851 | Is D-module on flag variety of Lie algebra a scheme? | The de Rham space of a scheme is essentially never a scheme or algebraic space (unless I guess you're Spec of an Artin ring, in which case you'll get a discrete set of points). In particular this applies to the flag variety. I'm not sure which perspective of NC AG you're taking, but certainly if you define the field as... | 10 | https://mathoverflow.net/users/582 | 11671 | 7,920 |
https://mathoverflow.net/questions/11669 | 13 | Hi,
Currently, I'm taking matrix theory, and our textbook is Strang's Linear Algebra. Besides matrix theory, which all engineers must take, there exists linear algebra I and II for math majors. What is the difference,if any, between matrix theory and linear algebra?
Thanks!
| https://mathoverflow.net/users/3142 | What is the difference between matrix theory and linear algebra? | Let me elaborate a little on what Steve Huntsman is talking about. A matrix is just a list of numbers, and you're allowed to add and multiply matrices by combining those numbers in a certain way. When you talk about matrices, you're allowed to talk about things like the entry in the 3rd row and 4th column, and so forth... | 55 | https://mathoverflow.net/users/290 | 11679 | 7,924 |
https://mathoverflow.net/questions/11677 | 21 | Are there infinitely many (linearly independent) cuspidal eigenforms for $\Gamma(1)$ with integer coefficients?
Someone told me that the Hecke algebra is conjectured to act irreducibly on the space of modular forms of level 1, so there would be no eigenforms if $\mathrm{dim} S\_k > 1$, i.e. for $k \geq 12$.
| https://mathoverflow.net/users/nan | modular eigenforms with integral coefficients [Maeda's Conjecture] | I will replace *modular form* by *cuspform* in your question, just to avoid the trivial answer of *yes*, because of the Eisenstein series. Given this, numerical evidence suggests that the eigenforms of weight $k$ are all conjugate (in the sense that their $q$-expansions are all algebraically conjugate under the action... | 17 | https://mathoverflow.net/users/2874 | 11681 | 7,926 |
https://mathoverflow.net/questions/11674 | 28 | I am not by any means an expert in category theory. Anyway whenever I have studied a concept in category theory I have always had the feeling that most of the subtleties introduced are artificial.
For a few examples:
-one does not usually consider isomorphic, but rather equivalent categories
-universal objects ar... | https://mathoverflow.net/users/828 | Can skeleta simplify category theory? | The first, and perhaps most important, point is that *hardly any categories that occur in nature are skeletal*. The axiom of choice implies that every category is equivalent to a skeletal one, but such a skeleton is usually artifical and non-canonical. Thus, even if using skeletal categories simplified category theory,... | 66 | https://mathoverflow.net/users/49 | 11687 | 7,931 |
https://mathoverflow.net/questions/11688 | 25 | How does one prove that the splitting of primes in a non-abelian extension of number fields is not determined by congruence conditions?
| https://mathoverflow.net/users/nan | Why do congruence conditions not suffice to determine which primes split in non-abelian extensions? | Here's an answer for the special case when the base field is $\mathbb{Q}$. It involves a large bit of class field theory over $\mathbb{Q}$, so I'll be terse.
We start with the lemma which Buzzard mentioned.
**Lemma** - Let $K$, $L$ be finite Galois extensions of $\mathbb{Q}$. Then $K$ is contained in $L$ if and onl... | 17 | https://mathoverflow.net/users/nan | 11692 | 7,933 |
https://mathoverflow.net/questions/11690 | 13 | Let $R$ be a complete discrete valuation ring and let $K$ be its field of fractions. Suppose $X$ is a smooth rigid-anaytic space over $K$. Often it is convenient to have a model of $X$ whose reduction has singularities which are as mild as possible--a semistable model. This amounts to having an admissible covering of $... | https://mathoverflow.net/users/271 | Cohomology of rigid-analytic spaces | Here's a first pass at your question; hopefully it will suggest something more definitive.
Let's imagine we were in the simplest case, where $X$ is a disk, with its smooth model
being the formal affine line over $R$, and that $Z$ was the sub-disk of elements of
absolute value less than or equal the absolute value of... | 6 | https://mathoverflow.net/users/2874 | 11696 | 7,937 |
https://mathoverflow.net/questions/11704 | 8 | Let k be a perfect field. By a k-variety, I shall mean a geometrically reduced separated scheme of finite type over k. I think that is enough conditions that the following data determine an affine k-variety:
1. A subset $X(\bar{k})$ of $\bar{k}^n$ which is defined by polynomials
2. A continuous action of $\mathop{\ma... | https://mathoverflow.net/users/1046 | Points of a variety defined by Galois descent | Despite what you learn in logic classes, there's truth to the conventional wisdom that you can't prove a negative. That is, it's a lot harder to explain why something necessarily can't work than why it can. (One almost invariably settles for: "it cannot work under the following explicit conditions, plus possibly others... | 6 | https://mathoverflow.net/users/1149 | 11706 | 7,943 |
https://mathoverflow.net/questions/11691 | 5 | Let the sequence $u\_1, u\_2, \ldots$ satisfy $u\_{n+1} = u\_n - u\_n^2 + O(u\_n^3)$. Then it can be shown that if $u\_n \to 0$ as $n \to \infty$, then $u\_n = n^{-1} + O(n^{-2} \log n)$. (See N. G. de Bruijn, *Asymptotic methods in analysis*, Section 8.5.)
This can be used to obtain asymptotics for $v\_{n+1} = Av\_n... | https://mathoverflow.net/users/143 | Asymptotics of iterated polynomials | Have you tried the method of section 8.7, i.e., solving Abel's equation $\phi(P(x))-\phi(x)=1$? Here we expect $\phi(t)=t^{-1}+\sum\_{n=1}^{\infty}c\_n t^n$, and you can find the coefficients of $\phi$ one by one. For example, I took $P(x)=x-x^2+x^3+x^4$ and immediately found $c\_1=-2$, $c\_2=-5/2$, $c\_3=-7/2$, $c\_4=... | 4 | https://mathoverflow.net/users/2912 | 11707 | 7,944 |
https://mathoverflow.net/questions/11709 | 3 | I'm wondering if there's an algorithm, or a program I can use, to compute integral closures. Specifically, what I have in mind are variants of questions of the sort: what is the integral closure of ℤ[x] in Quot(ℚ[x,y]/fℚ[x,y]), for some specific f(x,y).
| https://mathoverflow.net/users/3238 | Computing Integral Closures | If you are looking for prepackaged software, [Macaulay II](http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.3.1/share/doc/Macaulay2/IntegralClosure/html/index.html) can do the job.
| 2 | https://mathoverflow.net/users/297 | 11712 | 7,948 |
https://mathoverflow.net/questions/11631 | 24 | Obviously, graph invariants are wonderful things, but the usual ones (the Tutte polynomial, the spectrum, whatever) can't always distinguish between nonisomorphic graphs. Actually, I think that even a combination of the two I listed will fail to distinguish between two random trees of the same size with high probabilit... | https://mathoverflow.net/users/382 | Complete graph invariants? | A complete graph invariant is computationally equivalent to a canonical labeling of a graph. A canonical labeling is by definition an enumeration of the vertices of every finite graph, with the property that if two graphs are isomorphic as unlabeled graphs, then they are still isomorphic as labeled graphs. If you have ... | 28 | https://mathoverflow.net/users/1450 | 11715 | 7,950 |
https://mathoverflow.net/questions/11718 | 1 | Let $R$ be the ring of Dirichlet series with integer coefficients. I'd often wondered about whether $R$ was a UFD; [this post](https://mathoverflow.net/questions/5522/dirichlet-series-with-integer-coefficients-as-a-ufd) cleared that up, because it turns out that $R\simeq\mathbb{Z}[[x\_1,x\_2,\cdots]]$ (the $x\_i$ corre... | https://mathoverflow.net/users/1916 | Fractional powers of Dirichlet series? | For your first question: let $R$ be any commutative ring, and let $D(R)$ be the ring of formal Dirichlet series over $R$, i.e., the set of all functions $f: \mathbb{Z}^+ \rightarrow R$ under pointwise addition and convolution product.
Then the unit group of $R$ is precisely the set of formal Dirichlet series $f$ suc... | 3 | https://mathoverflow.net/users/1149 | 11724 | 7,953 |
https://mathoverflow.net/questions/6918 | 10 | I'm looking for natural conditions on $a\_{ij}$ to guarantee that the null space of the $n\times m$ matrix $A=(a\_{ij})$ has a nice basis.
The null space of { {1,-2,1,0,0}, {0,1,-2,1,0}, {0,0,1,-2,1} } is the set vectors $\langle x\_1,x\_2,x\_3,x\_4,x\_5\rangle^T$ with $x\_1,\dots,x\_5$ in arithmetic progression or c... | https://mathoverflow.net/users/935 | Matrices whose nullspace is nicely shaped | After quite a bit of tinkering, I decided that the example and a more fully realized generalization merited separate answers, not least because my initial answer entered community wiki due to the number of edits I made.
Let $N$ be a null space matrix for $A$, i.e., the columns of $N$ are annihilated by $A$. We want v... | 1 | https://mathoverflow.net/users/1847 | 11725 | 7,954 |
https://mathoverflow.net/questions/11723 | 6 | Let X\_0(p) be the modular curve of level p where p is prime. The Jacobian variety J\_0(p) has a natural family of quotients defined over Q with dimensions summing to dim(J\_0(p)), each quotient corresponding to a Galois conjugacy class of normalized Hecke eigenforms on X\_0(p), the dimension of a quotient being equal ... | https://mathoverflow.net/users/683 | Distribution of dimensions of factors of the Jacobian of X_0(p) | Let me begin with what was formerly a comment above: the feeling among most experts is probably\* that for each fixed $d$, the number of isogeny factors of $J\_0(p)$ of dimension $d$ should be small compared to the dimension of $J\_0(p)$, i.e. $o\_d(p)$.
In what follows, I will cite some results which point in this ... | 4 | https://mathoverflow.net/users/1149 | 11727 | 7,956 |
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