User ketil tveiten - MathOverflow

Is there a standard name for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial?

2013-01-29T16:43:53Z

Is there any existing standard terminology for functions of the form $x^\alpha p(x)$, where $p(x)$ is a polynomial and $\alpha$ is e.g. a complex number? I haven't been able to come up with any good name (e.g. generalised polynomial or near-polynomial don't sound so good), and no one I've talked to knows any standard terminology, so I thought I'd ask here.

Edit: the motivation for this is the following: if you have a polynomial solution $f(z)$ to a hypergeometric differential equation (i.e. some $_pF_q$ that is a polynomial), then $x^\alpha f(x)$ (for a certain given multiindex $\alpha$) is a solution to the associated GKZ $A$-hypergeometric system in $p+q$ variables. I'm currently working on generalising some results on polynomial solutions to the hypergeometric equation, and I'd like a good name for this kind of power-times-polynomial thing.

Answer by Ketil Tveiten for D-module that is coherent as O-module

2013-01-10T10:47:11Z

No, you can generally only get locally freeness on an open dense set. This is Lemma VII.9.3 in Algebraic D-modules by Armand Borel et al.

EDIT: just realised that Lemma talks about where $\mathcal{D}_X$-coherent modules are $\mathcal{O}_X$-locally free, not whether $\mathcal{O}_X$-coherent ones are. A $\mathcal{D}_X$-coherent module needs a good filtration by $\mathcal{O}_X$-coherent modules to be $\mathcal{O}_X$-coherent, so you may have to check that. Section II of Borel et al. should cover this, have a look there.

Monodromy of covering map related to symmetric group

2012-07-05T12:38:58Z

Let $X:=\mathbb{C}^n$, and let the symmetric group $S_n$ act by permutation of coordinates in the obvious way; let $X_n:=X/S_n$ be the quotient by the group action. Now, $X_n\simeq \mathbb{C}^n$, so we get a map $\pi: X\to X_n$ with finite fibers. Away from the discriminant $\Delta:=\prod_{i\lt j}(x_i-x_j)$, $\pi$ is a covering map $X\setminus\Delta\to X_n\setminus\pi(\Delta)$, and this map has monodromy group $S_n$. In similar fashion, we can let the subgroup $S_k$ act on the first $k$ coordinates, and let $X_k$ be the resulting quotient; again away from the discriminant the map $X\to X_k$ is a covering map and has monodromy group $S_k$. (We could of course generalise to $S_\lambda$ for a partition $\lambda$ of $n$, but let's keep it simple.)

Now, $\pi:X\to X_n$ factors through any of the $X_k$, so we have maps $X_k\to X_n$, and in particular $X_n\to X_{n+1}$, and away from the branching locus they are also covering maps.

Question: What are the monodromy groups of these covering maps?

This seems like the sort of thing someone might have worked out long ago, is anything known about this?

Answer by Ketil Tveiten for Reference wanted - preservation of constructible sheaves (in classical topology) by all functors

2012-01-26T09:06:44Z

You could try Sheaves in Topology by Alexandru Dimca. There are no prerequisites other than basic sheaf theory, so you don't have to worry about microlocal troubles or anything else.

What conditions are needed for $-\otimes_A B$ to be faithful?

2010-12-02T13:51:15Z

For $A$ a (commutative) ring, $f:A\to B$ an $A$-algebra, what conditions do we need on $A$ and $B$ (and $f$) for the functor $-\otimes_A B:A-mod\to B-mod\quad$ to be faithful (i.e. injective on $Hom$-sets)?

I can't seem to come up with anything other than the rather obvious condition that tensoring with $B$ shouldn't kill anything (or at least not too much), but this is hardly satisfying. It seems like it would be faithful in general, but I fail to come up with an argument as to why this should be true, and precisely when it is (if at all). Dare I beg the aid of the MO?

Note: this is basically the same as $f^*$ being faithful for a morphism $f:X\to Y$ of schemes, which reduces to the above. Hence the algebraic geometry tag.

Answer by Ketil Tveiten for introductory book on spectral sequences

2011-10-31T09:31:18Z

I would recommend that everyone's very first (zeroth?) introduction would be Timothy Chow's excellent short article You Could Have Invented Spectral Sequences. It doesn't give a lot of technical details, but it will definitely remove your fear before you start on a more advanced exposition.

Link: http://www.ams.org/notices/200601/fea-chow.pdf

Answer by Ketil Tveiten for undergraduate logic textbook

2011-06-15T08:52:11Z

I would recommend A Friendly Introduction to Mathematical Logic by Christopher Leary. It covers all the important things up to the Incompleteness Theorem, and really is Friendly.

(I have heard rumours that it's out of print, though.)

What is known about the number of permissible simplicial complexes given the number of k-cells?

2011-05-27T11:50:00Z

Motivation: I am working on a problem that reduces to finding simplicial complexes given some data (details are unnecessary), but all I have managed to wrangle from my input is the number of cells of each dimension. What I want is to find the possible complexes with these numbers of cells. This really is not my field of expertise (if anything is), so I turn to your collective wisdom.

So suppose I am given a vector of positive integers, e.g. $(8,12,6,1)$. What simplicial complexes can I build with number of $k$-cells given by the vector? (i.e. in the example, 8 0-cells, 12 1-cells, etc.)

I expect that for most such vectors there would be several possible complexes one could build out of them, but I have no idea if (a) this is true, or (b) how to find them, or even (c) to find the number of possibilities. Has anyone done anything similar before? Is anything known about this? I'd appreciate any suggestions or references.

(I was not really sure which tags to apply. Feel free to retag.)

Answer by Ketil Tveiten for Why is "abelian" infrequently capitalized?

2010-11-05T14:46:54Z

I was told by a liguistics professor that de-capitalisation happened when the word was sufficiently common to become 'everyday', so to speak. So, perhaps we have 'abelian' rather than 'Abelian' because we use the word a lot. Certainly, we use 'abelian' and 'noetherian' all the time, whereas say, 'Cohen-Macaulay' doesn't appear quite so often.

It seems like this happens more often to 'algebra words' (abelian, noetherian, artinian) than, say 'geometry words' (Euclidian, Riemannian). Could it be that algebraists use the name-words more in 'everyday' speech than geometers? Maybe they even have sloppier attitudes to grammar-pedantry, or like to save their shift keys, who knows?

(Of course, this is a very tentative hypothesis and I expect to see it demolished by counterexamples.)

Are there any good computer programs for drawing (algebraic) curves?

2010-10-15T11:22:26Z

I realise that I lack some intuition into how a curve (or surface, or whatever) looks geometrically, from just looking at the equation. Thus, I sometimes resort to some computer program (such as Mathematica) to draw me a picture. The problem is, all these programs require input of the form $y=f(x)$, whereas my curve might be something like $y^3+x^3-6x^2 y=0$, and transforming this into the former form is not always easy, and always misses some information. So, my question:

Are there any programs that can take an equation ($p(x,y)=0$, say) as input and return a graph of its zero-set?

Update: So, lots of good answers, I wish I could accept them all. I'll accept Jack Huizenga's answer, for the reason of personal bias that I already have Mathematica available.

Quick ways to calculate cohomology of vector bundle/local system from transition functions?

2010-05-14T16:29:35Z

Suppose I have a vector bundle (or local system, or something else given by transition functions) on a Riemann surface (or generally a (complex) manifold), and I want to compute its cohomology. The trouble is, I only know the transition functions, not anything about the sections.

All I can think of is Cech cohomology, i.e. giving a cover and doing lots of calculation. To make things work, it seems to me that the cover needs to have very many opens (effectively a triangulation), which complicates calculation.

Does anyone know if there is a better way? Alternately, if anyone knows of ways to make the above approach more efficient, I'd be glad to know.

(Perhaps this is a stupid (trivial) question, but I can't find any references about this.)

Answer by Ketil Tveiten for What are some examples of narrowly missed discoveries in the history of mathematics?

2010-03-25T18:35:08Z

When I took a course in set theory, I was told that in the early 1920's, Thoralf Skolem essentially proved what is now Gödel's Completeness Theorem, but was unaware of its significance because the concept of completeness was not understood fully at the time.

Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?

2009-11-04T19:31:38Z

Let X be a complex analytic space. It is a 'well known fact' that the categories of local systems on X (i.e. locally constant sheaves with stalk C^n), and of (holomorphic) vector bundles on X with flat connection, are equivalent. I've been looking for a proof of this, but every reference I can find merely says something like 'this is well known' without further argument. Does anyone know of a proof?

Is there a name for this property of a topology?

2010-03-19T16:34:22Z

This property seems like it should have a nice name, but I can't find one anywhere. Does anyone know a name for this?

For each non-empty open set $U$, there exist proper open subsets $\{U_i\}_{i\in I}$ such that $U=\cup_i U_i$.

I suppose this could also be formulated as each nonempty open set having an open cover of proper subsets, or being the colimit of its open subsets.

(Also, apologies if this is something obvious I should have thought of.)

How does one get the short exact sequence in a two-column spectral sequence?

2010-01-20T17:31:01Z

In a two-column double complex, one gets from the associated spectral sequence short exact sequences $0\to E_2^{1,n-1}\to H^n\to E_2^{0,n}\to 0$, where $H^n$ is the cohomology of the total complex, but I have never seen the construction of this sequence. Any text I've seen merely states it as a fact, or leaves it as an exercise which I have had no luck trying to solve. Can anyone give a construction or good reference?

Comment by Ketil Tveiten

2013-05-03T10:02:47Z

Why don't you post these answers as comments to the relevant posts?

Comment by Ketil Tveiten

2013-04-23T16:56:23Z

I have sometimes seen semicolon used instead of colon or vertical bar, but I very much prefer the bar anyway, even if it gets ugly.

Comment by Ketil Tveiten

2013-04-12T16:07:42Z

Alexandre: I suppose the thing I object to is the word "omnipresent". Observing lots of examples of $\mathbb{Z}/2$-symmetry does not mean that it is somehow a fundamental thing, it only means that $\mathbb{Z}/2$-symmetry is a thing we are good at recognising. Most objects (or living things) in nature don't have any symmetry at all, and in a similar way, most objects in mathematics have no symmetry, it's just that we tend to work with those objects that are nice enough that we can do something with them, and having some kind of low-order symmetry is an easy way to be nice.

Comment by Ketil Tveiten

2013-04-09T08:30:27Z

An example of a "useful" composition of left and right derived functors is the direct image of D-modules, which is a $R\pi_*$ applied to a derived tensor product.

Comment by Ketil Tveiten

2013-04-05T09:02:30Z

I don't like this answer. If we understand well how to use hammers, we are going to notice a lot of nails, but that doesn't mean nails are somehow truly ubiquitous or favoured by the gods, it just means that we recognise them when we see them. Bonus points to anyone who makes good use of Jellyfish Algebras, btw.

Comment by Ketil Tveiten

2013-03-07T12:29:37Z

Perhaps the OP is looking for some other characterisation of algebraic numbers than "is the root of a monic rational polynomial"?

Comment by Ketil Tveiten

2013-02-26T08:29:11Z

The nlab page for derived functors nlab.mathforge.org/nlab/show/derived+functor talks about Kan extensions, though I'm not competent enough to decide if that helps you.

Comment by Ketil Tveiten

2013-02-12T11:11:37Z

Is there any reason why you don't define it in terms of the derived direct image $f_+$? That would seem to be the natural thing to do from a $D$-module perspective...

Comment by Ketil Tveiten

2013-01-31T14:44:55Z

I can't see how the question wasn't answered in the Stackexchange thread. Try reading it again, and consulting your Galois theory textbook?

Comment by Ketil Tveiten

2013-01-30T12:39:18Z

The first link is broken, should be en.wikipedia.org/wiki/Amoeba_(mathematics).

Comment by Ketil Tveiten

2013-01-10T16:37:50Z

You could also check J.E. Björk, Analytic D-modules and their applications, he spells out the coherence stuff in a little more detail. Might be hard to find a copy, though.

Comment by Ketil Tveiten

2012-12-25T13:11:04Z

How is $max(x,-x)$ not convex?

Comment by Ketil Tveiten

2012-12-06T13:14:47Z

I don't understand, what's the purpose of a method that only works if you know the answer beforehand?

Comment by Ketil Tveiten

2012-12-06T13:02:55Z

@Brian Rushton: Think of cutting a paper Möbius strip (aka. paper-strip-glued-with-a-half-twist) along the midline. You get the paper-strip-glued-with-a-twist, which is the required nonstandard cylinder embedding.

Comment by Ketil Tveiten

2012-10-30T09:45:23Z

+1 for "just well forgotten".