You might consider the "QR algorithm": given A, factor A as QR (Q orthogonal and R triangular), then let A' = RQ. Repeat with A' as the new A, ad infinitum.
In a way, though, all implementable diagonalization algorithms are approximate, since it's impossible to diagonalize a general matrix in a finite number of elementary operations.
This question is based on the following phrase:
"In a sense, $textrm{Spec} mathbf{Z}$ looks topologically like a 3-dimensional sphere viewed as the Hopf fibration over $mathbf{S}^2$."
See page 88 of Algebraic Geometry II by Shafarevich.
I find this remark very interesting but I can't seem to parse it.
I always just viewed $textrm{Spec} mathbf{Z}$ as an arithmetic analogue of $mathbf{P}^1(mathbf{C}) = mathbf{S}^2$. This remark would add "something" to that in a sense.
Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus solvable radical) and that all such semisimple subalgebras (Levi factors) are conjugate in a strong sense: see Jacobson, Lie Algebras, III.9, for example. This carries over to connected linear algebraic
groups, but in prime characteristic there are counterexamples going back perhaps
to Chevalley that involve familiar group schemes like $SL_2$ over rings of Witt
vectors. Recent posts here have somewhat ignored that difficulty, having just characteristic 0 in mind. Borel and Tits redefined "Levi factor" to be a reductive complement to the unipotent radical, which is makes no real difference in characteristic 0 but allows them to concentrate on positive answers for parabolic subgroups of reductive groups in general. Other familiar subgroups of reductive groups like the identity component of the centralizer of a unipotent element require much more subtle treatment, as in work of George McNinch.
Whether or not the characteristic $p$ question is important, it has remained
open for many decades (say over an algebraically closed field). I gave up after one forgettable paper (Pacific J. Math. 23, 1967). The problem is still
easy to state:
Are there effective necessary or sufficient conditions for existence or uniqueness of Levi factors in a connected linear algebraic group over an algebraically closed field of prime characteristic?
It's clear that a scheme-theoretic viewpoint may be needed. Possibly the known
counterexamples using Witt vectors suggest in some way all possible counterexamples? (Or is the question hopeless to resolve completely?)
EDIT: For online access to my 1967 paper, via Project Euclid, see
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pjm/1102991730. Here
Chevalley's counterexample is mentioned only in the abstract, but in remarks
later on it is noted that Borel-Tits (III.15) gave an example involving two
Levi subgroups which fail to be conjugate; see NUMDAM link to PDF version of
Publ. Math. IHES 27 (1965) at http://www.numdam.org:80/?lang=en
In April
1967 Tits responded to my inquiry with a letter outlining the behavior of the
group scheme $SL_2$ over the ring of Witt vectors of length 2, which gives a
6-dimensional algebraic group over the underlying field with unipotent radical of dimension 3 but no Levi factor. He remarked that he got this counterexample from P. Roquette but had also been told about Chevalley's counterexample.
ADDED: The question as formulated probably doesn't have a neat answer, but meanwhile George McNinch has delved much deeper (over more general fields) in his new arXiv preprint
1007.2777. Some technical steps rely on the forthcoming book Pseudo-reductive groups (Cambridge, 2010) by Conrad-Gabber-Prasad.
The basis for the deterministic polynomial-time algorithm for primality of Agrawal, Kayal and Saxena is (the degree one version of) the following generalization of Fermat's theorem.
Suppose that P is a polynomial with integer coefficients, and that p is a prime number. Then
$(P(X))^pequiv P(X^p) (mod p)$.
Surely this result was known previously, but I have not been able to find a reference in the literature on the AKS algorithm (which means that the authors also did not know of a reference). Does anyone here know of one?
Furthermore, there is a converse to the lemma in the AKS paper:
If n is a composite number, then $(X+a)^nnot equiv X^n+a (mod n)$ whenever a is coprime to n.
Again, it is easy to generalize this statement. For example, if P is a polynomial which has at least two nonzero coefficients and such that all nonzero coefficients are coprime to n, then $P(X)^nnotequiv P(X^n) (mod n)$ for composite n.
On the other hand, clearly some conditions are necessary; for example $(3X+4)^6equiv 3X^6+4 (mod 6)$.
Is there a best possible statement? And, again, is there a reference?
I get that the answer is "no" for an abelian variety over the algebraic closure of Fp with complex multiplication by a ring with a unit of infinite order. Since you say you have already thought through the abelian variety case, I wonder whether I am missing something.
More generally, let X be any variety over the algebraic closure of Fp with an automorphism f of infinite order. A concrete example is to take X an abelian variety with CM by a number ring that contains units other than roots of unity. Any finite collection of closed points of X will lie in X(Fq) for some q=p^n. Since X(Fq) is finite, some power of f will act trivially on X(Fq). Thus, any finite set of closed points is fixed by some power of f.
As I understand the applications to descent theory, this is still uninteresting. For that purpose, we really only need to kill all automorphisms of finite order, right?
Just yesterday I heard of the notion of a fundamental group of a topos, so I looked it up on the nLab, where the following nice definition is given:
If $T$ is a Grothendieck topos arising as category of sheaves on a site $X$, then there is the notion of locally constant, locally finite objects in $T$ (which I presume just means that there is a cover $(U_i)$ in $X$ such that each restriction to $U_i$ is constant and finite). If $C$ is the subcategory of $T$ consisting of all the locally constant, locally finite objects of $T$, and if $F:Crightarrow FinSets$ is a functor ("fiber functor"), satisfying certain unnamed properties which should imply prorepresentability, then one defines $pi_1(T,F)=Aut(F)$.
Now, if $X_{et}$ is the small étale site of a connected scheme $X$, then it is well known the category of locally constant, locally finite sheaves on $X$ is equivalent to the category of finite étale coverings of $X$, and with the appropriate notion of fiber functor it surely follows that the étale fundamental group and the fundamental group of the topos on $X_{et}$ coincide.
Similarly, as the nlab entry mentions, if $X$ is a nice topological space, locally finite, locally constant sheaves correspond to finite covering spaces (via the "éspace étalé"), and we should recover the profinite completion of the usual topological fundamental group.
Before I come to my main question: Did I manage to summarize this correctly, or is there something wrong with the above?
My question:
Has the fundamental group of other topoi been studied, and in what context or disguise might we already know them? For example, what is known about the fundamental group of the category of fppf sheaves over a scheme $X$?
Too long for a comment:
Yes. One family of examples is the singular K3 surfaces - a recent paper generalizing this is
http://arxiv.org/pdf/0904.1922
This is a consequence of a result of Livné about the modularity of 2-dimensional orthogonal Galois representations.
Rigid Calabi-Yau 3-folds also give examples, after Serre's conjecture, cf. the following paper:
http://arxiv.org/pdf/0902.1466
(Although this implication was already in Serre's original paper: you can deduce a similar result for any motive with the right Hodge numbers).
These examples are however "close" to abelian varieties in some sense, so you might not find them very satisfying. I don't know of any others though.
Edit: I want also to mention information that potential automorphy theorems can give you. For example, in his thesis Barnet-Lamb showed that the zeta function of the Dwork hypersurface in $mathbb{P}^4$ has meromorphic continuation, by showing that the cohomology is automorphic after possibly restricting to a totally real field extension of $mathbb{Q}$.
Actually, I'm not quite sure where this is explained completely basically in the literature. So maybe here's a hint, at least for Q1. The definition of $C^*(G)$ is that it's the completion of L^1(G) with respect to the biggest C*-norm. So if $phi:Grightarrow H$ is a continuous group homo, we immediately get a *-homomorphism $ell^1(G) rightarrow ell^1(H)$, and so by inclusion, a *-homo $ell^1(G) rightarrow C^*(H)$. But this defines some C*-norm on $ell^1(G)$, so the norm on $C^*(G)$ must dominate this, and hence we get a continuous extension to $C^*(G) rightarrow C^*(H)$.
For Q2, find a proof in the literature (I think this goes back to Godemont?) that $C^*_r(G) = C^*(G)$ if and only if the left-regular representation contains the trivial representation, if and only if G is amenable. Put another way, if G is not amenable, then the trivial homomorphism $Grightarrow{1}$ doesn't induce a map $C^*_r(G)rightarrow C^*({1}) = mathbb C$.
This is a powerful technique that can prove consistency, conservativity (that a statement about a small system
which is a theorem in a more expressive one was already a theorem of the smaller one) etc. Applied to programming languages, it can show that if the result of a program in its denotational semantics is a number (as opposed to undefined) then when you run the program it is guaranteed to terminate (maybe after the Sun has gone supernova) and return that number.
It works by tying the syntax and the semantics together in lock-step, so that (maybe easy) observations about the semantic structure have direct consequences for the existence of a proof.
See
Section 7.7 in my book "Practical Foundations of Mathematics" for one categorical treatment, although there is a vast literature in theoretical computer science about this.
Of course, the construction uses structural recursion over the syntax. Amongst its consequences are consistency results. For anyone aware of Godel's incompleteness theorem, this should set some alarm bells ringing.
The solution is that the semantic structure (often a Grothendieck topos) is logically much stronger than the syntactic one. If, for example, the latter is the logic of an elementary topos then the former must enjoy (some fragment of) the axiom-scheme of replacement.
PS The actual categorical construction is extremely simple. The "lock-step" property has to be proved as a theorem for each type constructor (eg function-spaces)
and is valid in many cases, although not higher order logic.
Edited answer (see below for my original, less useful reply): Since your polynomial p(z) of degree 2d is palindromic, rewrite it as z^d q(z+1/z) for some polynomial q(x) of degree d. Then p(z) has 2d roots on the circle if and only if q(x) has d real roots in the interval [-2,2]. (Equivalently, q(x-2) should have d nonnegative real roots, while q(x+2) should have no positive roots.) Now you can try to extract information using e.g. Sturm sequences to try to count real roots in that interval.
I had previously posted: Since the real line parameterizes the circle (minus a point), you can transform your problem into counting the real zeros of an associated real polynomial. (See p. 182 of Rodriguez Villegas's book "Experimental Number Theory" for details; this is viewable in Google Books.)
There certainly have to exist necessary & sufficient criteria for all the zeros of a real polynomial to be real, involving rather complicated inequalities in the coefficients of the polynomial, generalizing the condition b^2-4ac >= 0 for quadratics; or for a specific polynomial you can try a real-root-counting algorithm (see Sturm's theorem on Wikipedia).
I'm curious about Faltings' "A p-adic Simpson correspondence ". Do you know more detailed, introductory, expositions, surveys, texts of seminars on that?
Edit: Annette Werner's survey "Vector Bundles on Curves over C_p" seems to be related.
Edit: The first part of a "new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0". An other related article.
Edit: today new in arxiv - "Non-abelian Hodge theory for algebraic curves over characteristic p"
There are all these examples surrounding fixed point theorems.
The following is somehow a cliche. Take a sheet of paper, crunch it, and put it on top of its original position. Then there is a point that lies on the vertical of its previous position. This illustrates the fixed point theorem for contractions in Banach spaces.
There are also a lot of examples in probability theory.
Here is one related to the harmonic series. In your youth, you may have collected cards depicting soccer players, martians, whatever. There is a finite number of cards to collect, say n. Each packet of corn flakes comes with one of them, at random. And of course you want your mom to buy this precise brand of flakes so as to get the whole collection. May be you have wondered what is the average number of packets she should buy so that you can complete the collection. The answer is
$$
n sumlimits_{k=1}^n {1 over k}
$$
This is asymptotic to $nlog n + ngamma+ 1/2$, where $gamma$ is the Euler constant. This is the simplest mundane example I know involving that constant. So for example, if there are $n=150$ cards to collect, you need to buy an average of 519 cards.
I haven't checked all the details, but I think the story could go like this. (I have to apologize: it's a bit long.)
(1) Let $F:mathsf Arightarrow mathsf B$ be an additive left exact functor between two abelian categories. Take an injective resolution of an object $A$ in $mathsf A$:
$$0rightarrow A rightarrow I^0 rightarrow I^1 rightarrow cdots $$
Let us call $i: A rightarrow I^0$ the first morphism. Apply $F$ to this exact sequence:
$$0rightarrow FA rightarrow FI^0 rightarrow FI^1 rightarrow cdots $$
Now, the total right derived functor of $F$ applied to $A$ (thought as a complex concentrated in degree zero) is the complex
$$mathbb RF(A) = [ FI^0 rightarrow FI^1 rightarrow FI^2 rightarrow cdots ]$$
and the classical right derived functors of $F$ are its cohomology:
$R^nF(A) = H^n(mathbb RF(A)) = H^n(FI^)$.
These ${R^nF}_n$ are a universal cohomological delta-functor and we have a natural transformation of functors
$$qF Rightarrow (mathbb RF)q$$
which is essentially
$$Fi: FA rightarrow mathbb RF(A)$$
(here we have extended $F$ degree-wise to the category of complexes, and this is the degree zero of the natural transformation, because $mathbb RF(A)^0 = FI^0$ ).
(2) Now, let $T^n : mathsf A rightarrow mathsf B$ be a cohomological delta-functor and $f^0 : F Rightarrow T^0$ a natural transformation. We have to extend this $f^0$ to a unique morphism of delta-functors ${ f^n : R^nF Rightarrow T^n }$.
To do this, observe that, in general, given two right-derivable functors between two, say, model categories $$F,G: mathsf C rightarrow mathsf D$$, and a natural transformation between them $t: F Rightarrow G $, we have a natural transformation between the total right derived functors $mathbb Rt : mathbb RF Rightarrow mathbb RG$ because of the universal property of the derived functors:
Indeed, if $f : qF Rightarrow (mathbb RF)q$ and $g : qG Rightarrow (mathbb RG)q$
are the universal morphisms of the derived functors, then we have a natural transformation
$$gt : F Rightarrow (mathbb R G)q$$
and, so, because of the universal property of derived functors, a unique natural transformation $mathbb R t : mathbb R F rightarrow mathbb R G$ such that $(mathbb R t)qf = g$.
(3) So, take our $f^0 : F Rightarrow T^0$ , extend it to a natural transformation between the degree-wise induced functors between complexes. Passing to the derived functors, we obtain
$$mathbb R f^0 : mathbb R F Rightarrow mathbb R T^0.$$
Taking cohomology, for each $n$, we get
$$H^n(mathbb R f^0) : H^n (mathbb R F) Rightarrow H^n (mathbb R T^0).$$
But these are the classical right derived functors, so we have natural transformations
$$R^nf : R^n F Rightarrow R^nT^0$$
and because the classical right derived functors are universal delta-functors, we have unique natural transformations
$$i^n : R^nT^0 Rightarrow T^n$$
which extend the identity
$$i^0 : R^0T^0 = T^0.$$
The composition
$$i^n circ R^f : R^F Rightarrow T^n$$
is, I think, the required morphisms of delta-functors that we need.
Consider a compact differentiable manifold $M$. We say that $f:Mto M$ and $g: M to M$ are topologically conjugated if there exists $h:Mto M$ a homeomorphism such that $fcirc h= h circ g$. The conjugacy class of a homeomorphism $f$ is the set of all $g$ such that $g$ is topologically conjugated to $f$.
If a homeomorphism $f: Mto M$ has infinite topological entropy (which is an invariant under topological conjugacy), then the conjugacy class of $f$ has no diffeomorphisms.
Is there any other known obstruction for a homeomorphism not to have diffeomorphisms in its conjugacy class? I would guess that yes, but I could not find one.
Is there a restriction on the dimension of the manifold?
(Remark: In dimension one, that is, in the circle, every homeomorphism has a diffeomorphism-of class $C^1$ in its conjugacy class. However, Denjoy counterexamples have no $C^2$ diffeomorphisms in theirs).
The 1970 paper of Halmos entitled "how to write mathematics" is a bit old, preposterous to the TeX era, but I think that many of his advices still make sense today. To summarize what he says, "use well chosen words instead of plethora of symbols.".
"Think about the alphabet". And don't hesitate to associate a meaningful symbol to a piece of formula that makes sense by itself, so as to keep your calculation as compact as possible. In particular, when there is some constant at the end of the computation, that comes from an agregation of many constants appearing during the computation, just call it C through the whole calculation, and at the end, give its actual value. "The value of the constant C is..." (I think I read that trick in some paper by Krantz).
"Use words correctly". Explain what is going on. Irrelevancy should be avoided. Writing "Now applying the Cauchy-Schwarz inequality leads to...", and just giving the result, may be better than actually applying it in the middle of the computation without mentioning it.
Beware of too heavy use of formulae notations. Something like "Now applying (32), we get ..." is less helpful than "Let us apply the upper bound on the curvature that was obtained with the help of the Gauss-Bonnet theorem." Give meaningful names to important formulas in your paper instead of refering to them through numbers. The section entitled "resist symbols" in the paper of Halmos gives a few other tricks to replace heavy formalism by well worded sentences.
If $fcolon X to S$ a proper flat of schemes map with $n$-dimensionnal fibers over a noetherian scheme $S$, the relative canonical sheaf $omega_{X/S}:=H^n(f^!{mathcal{O}_S})$ is a dualizing sheaf. I guess that this should imply what you want by a GAGA-type argument.
Indeed, being $f$ flat we have that $f^!G = f^*G otimes f^!{mathcal{O}_S}$ [Lipman, SLN 1960, Theorem 4.9.4] for $G in D^b_c(X)$ (or, more generally, for $G in D^+_{qc}(X)$). Now, $f^!{mathcal{O}_S}$ is concentrated between degrees 0 and $-n$ by looking at the fibers of $f$ and its description via residual complexes. Then, taking $-n$-th cohomology on both sides and one obtains the desired result.
Unfortunately, I am not familiar enough with the analytic version of the story as in Ramis-Ruget-Verdier "Dualité relative en géométrie analytique complexe", but I guess the algebraic version may give you a clue for how to transpose the result to your setting. I would bet you do not need $S$ reduced as long as $f$ is flat.
Addendum
There was a confusion on my part. I was speaking about the dualizing complex while the original question was about the dualizing sheaf. If we agree that $omega_X = H^{-n}(f^!O_B)$ for a equidimensional family (with $n$ the dimension of the fibers), then the base change isomorphism in the derived category for a square with base $u colon P to B$, ($B$ for "base scheme") a base change of the map $f colon X to B$ completing the square with $v colon F to X$ and $g colon F to P$, respectively.
Then, by the formula in the derived category,
$Lv^* f^! G cong g^! Lu^* G$
we get an isomorphism of sheaves
$H^{-n}(Lv^* f^!O_B) cong H^{-n}(g^! Lu^*O_B) = H^{-n}(g'^!O_P)$
so we see that the dualizing sheaf over $F$ is the abutment of a spectral sequence involving higher $Tor$ sheaves related to the embedding of the fiber into the space $X$. But, of course one needs some information on this embedding to get the collapse of the spectral sequence.
I'm looking for some examples of actions on Sylow p-groups, and often those actions appear in the case of finite almost simple groups. Given a finite almost simple group, I understand in principle how to calculate its Sylow p-subgroups (here p is usually 2 or 3), but perhaps I am just too slow at doing it. In particular, I am familiar with the papers of Weir and Carter-Fong.
I am not sure how to do the reverse calculation: given a p-group P (possibly described as "the Sylow p-subgroup of the almost simple group X", and X is something explicit like "PGL(3,19)" or "M11"), find all of the almost simple groups that have a Sylow p-subgroup isomorphic to P.
I am pretty sure some people know how to do this, but it's not really clear to me how to go about it. For instance, it would have never occurred to me that PSU(3,8), PSL(3,19), and 3D4(2) have isomorphic Sylow 3-subgroups.
Is there a description of how this is done?
I think there is likely to be a finite answer to the question: Obviously only finitely many (and probably O(p)) alternating groups could work for a given P. We take for granted that only finitely many sporadic groups could work. It seems that, similarly to the alternating case, there are only finitely many ranks (again probably O(p)) of Lie groups that could work, and hopefully for each Lie type (and rank), there are just some congruences on "q" that indicate which ones work and which don't.
However, I've not had much luck doing this calculation in examples, and so I am looking for papers or textbooks where this has been done. I have found some that state the result of doing something like this (post CFSG), and I have found several that do this in quite some detail, but pre-CFSG so they spend hundreds of pages eliminating impossible groups obscuring what should now be an easy calculation. I'm looking for something with the pedagogical style of the pre-CFSG papers, but that doesn't mind using the standard 21st century tools.
Alternatively: does anyone know of a vaguely feasible approach to construct all groups with given Sylow subgroups? Blackburn et al.'s Enumerating book has some upper bounds, but they are pretty outrageous and don't seem adaptable to a feasible algorithm for my problem.
I don't know if this is a "deep" or "shallow" explanation, but if anyone is still reading this thread, here is a different explanation. I'll start with the preliminary comment that the cohomology of a group $K$ is a special case of cohomology of topological spaces. In topology in general, you get the same phenomenon that $H^k(X,G)$ is well-defined either when $G$ is abelian or $k=1$.
Consider the definition of simplicial cohomology for locally finite simplicial complexes. Or, more generally, CW cohomology for locally finite, regular complexes — regular means mainly that each attaching map is embedded. You can define a $k$-cochain with coefficients in a group $G$ (or even in any set) as a function from the $k$-cells to $G$. In attempting to define the coboundary of a cochain $c$ on a $k+1$-cell $e$, you should multiply together the values of $c$ on the facets of $e$. The obvious problem is that if $G$ is non-abelian, the product is order-dependent. However, if $k=1$, geometry gives you a gift: The facets are cyclically ordered, and what you mainly wanted to know is whether the product is trivial. The criterion of whether a cyclic word is trivial is well-defined in any group, not just abelian groups. A similar but simpler phenomenon occurs for the notion of a coboundary: If $e$ is an oriented edge and $c$ is a 0-cochain, there is a non-abelian version of modifying a 1-cochain by $c$ because the vertices of $e$ are an ordered pair.
So far this is just a more geometric version of Eric Wofsey's answer. It is very close to the fact that $pi_1$ is non-abelian while higher homotopy groups are abelian — and therefore non-commutative classifying spaces exist only for $K(G,1)$. However, in this version of the explanation, something extra appears when $X=M$ is a 3-manifold.
If $M$ is a 3-manifold, then not only are the edges of a face cyclically ordered, the faces incident to an edge are also cyclically ordered. It turns out that, at least at the level of computing the cardinality of $H^1(M,G)$, you can let $G$ be both non-commutative and non-cocommutative. In other words, $G$ can be replaced by a finite-dimensional Hopf algebra $H$ which does not need to be commutative or cocommutative. Finiteness is necessary because it is a counting invariant. The resulting invariant $\#(M,H)$ was a topic of my PhD thesis and is explained here and here. Although the motivation is original, the invariant is a special case of more standard quantum invariants defined by other people. (The same construction was also later found by three physicists, but I can't remember their names at all right now.)
Many 3-manifolds are also classifying spaces of groups, so for these groups there is the same notion of noncommutative, non-cocommutative group cohomology.
If one uses the functions $f_d(t) = (cos (d t))^{frac 1d}$ (which were implicitly introduced by Colin Maclaurin) one can compute the catenaries for central gravitational fields which are proportional to a power of the distance to the Sun. These are curves of the form $rf(theta)=1$ where $f$ is a function of the above type (or a dilation and rotation thereof).
Interestingly, one can also get catenaries for parallel forces from the same functions. In fact, parametrised curves of the form $(F(t),f(t))$ where $f$ is one of the above functions and $F$ is a primitive, give the catenaries for gravitational fields which are proportional to powers of the distance to the $X$-axis. These curves, the Maclaurin catenaries, are new.
Not relevant to the question but worth mentioning is that if one rotates them about the $X$-axis, they provide examples of surfaces for which all six parameters in the fundamental forms ($E$, $F$, $G$, $L$, $M$, and $N$) are proportional to powers of the above distance.
Details in arXiv 1102.1579
Here's an informal example of a proof by coinduction.
The extended natural numbers E = {0, 1, 2, ..., ∞} are the final coalgebra for the functor F(X) = 1 + X. (If we have an F-coalgebra X, i.e. a set X with a map f : X -> 1 + X, and an element of X, what we can do is repeatedly apply f until we get the element of 1 rather than a new element of X. Counting the number of times we got new elements of X gives us an element of E. This is the unique F-coalgebra map X -> E.) In Haskell we would write
data Nat = Z | S Nat -- possible values are Z, S Z, S (S Z), ..., S (S (S ...))
We can define addition:
add :: Nat -> Nat -> Nat
add Z b = b
add (S a1) b = S (add a1 b)
I'll prove that add a Z = a by coinduction. We perform case analysis on a. If a = Z then it follows from the first equation. If a = S a1 then add a Z = add (S a1) Z = S (add a1 Z) = S a1 = a where the next-to-last step used the coinductive hypothesis.
Why isn't that circular reasoning? We were allowed to apply the coinductive hypothesis because we did so inside an application of the "constructor" S. In other words, even if you didn't believe the coinductive hypothesis, you would still conclude that add a Z and a were both of the form Z or both of the form S x--in other words, add a Z and a agree for to one observation. Using that statement where we used the coinductive hypothesis, you find that add a Z and a agree to two observations, etc. Since an element of E is determined by all the finite sequences of observations we can make, it follows that add a Z = a.
(Note that this argument works even when a = S (S (S ...))!)
Exercise: Give a formal version of this argument, using the definition of E as the final F-coalgebra.
The Suzuki and Ree groups are usually treated at the level of points. For example, if $F$ is a perfect field of characteristic $3$, then the Chevalley group $G_2(F)$ has an unusual automorphism of order $2$, which switches long root subgroups with short root subgroups. The fixed points of this automorphism, form a subgroup of $G_2(F)$, which I think is called a Ree group.
A similar construction is possible, when $F$ is a perfect field of characteristic $2$, using Chevalley groups of type $B$, $C$, and $F$, leading to Suzuki groups. I apologize if my naming is not quite on-target. I'm not sure which groups are attributable to Suzuki, to Ree, to Tits, etc..
Unfortunately (for me), most treatments of these Suzuki-Ree groups use abstract group theory (generators and relations). Is there a treatment of these groups, as algebraic groups over a base field? Or am I being dumb and these are not obtainable as $F$-points of algebraic groups.
I'm trying to wrap my head around the following two ideas: first, that there might be algebraic groups obtained as fixed points of an algebraic automorphism that swaps long and short root spaces. Second, that the outer automorphism group of a simple simply-connected split group like $G_2$ is trivial (automorphisms of Dynkin diagrams mean automorphisms that preserve root lengths).
So I guess that these Suzuki-Ree groups are inner forms... so there must be some unusual Cayley algebra popping up in characteristic 3 to explain an unusual form of $G_2$. Or maybe these groups don't arise from algebraic groups at all.
Can someone identify and settle my confusion?
Lastly, can someone identify precisely which fields of characteristic $3$ or $2$ are required for the constructions of Suzuki-Ree groups to work?
I looked at Jiang's monograph for a little while last night. Here is what I could get from it (I am now quoting from memory, so my terminology and notation may not be exactly the same). If $F$ is a field (of "numbers"), then the field $overline{F}$ of "isodual numbers" has the same underlying set and addition operation, but multiplication is replaced by the operation $x overline{bullet} y := - (xy)$. The new multiplicative identity is $-1$.
This is mathematically valid, of course: i.e., $overline{F}$ really is a field. Moreover it is isomorphic to $F$ via the map $x mapsto -x$, although I couldn't find a clear statement of that. (But somewhat later on I saw references to the isotopy $F rightarrow overline{F}$.) Physically speaking, the isodual numbers are supposed to bear the same relation to the ordinary numbers as antimatter does to matter. (I don't know what that means, but I am not a physicist and so am not even going to worry about it.)
Jiang defines a new function $J_2(omega)$, which is supposed to be some sort of repaired version of the Riemann zeta function. In one of his published works, he claims that the Riemann hypothesis is false -- in fact, he says, the zeta function has no zeros in the critical strip. [Logically speaking, wouldn't that make the Riemann Hypothesis true? Never mind.] From this definition, he immediately deduces proofs of Goldbach, twin primes, primes of the form $n^2+1$, and several other outstanding number theoretic conjectures -- literally immediately, in that I could find no argumentation for them. First these results are stated for "isonumbers" but later on they are stated for the usual integers.
That's about as far as I got. I also noticed, though, that many of the results described in this monograph were first published as papers by the journal Algebra, Groups and Geometries (founding editor: R.M. Santilli). These papers appear on MathSciNet but are not (going to be) reviewed.
The previous version of this question was rather badly broken, and I hope this version makes some sense.
There have been a few questions on MathOverflow about how much representation-theoretic information is lost when passing from a Lie group to its Lie algebra, e.g., away from the semisimple case, Lie algebras have many more representations. In the algebraic setting, there is an intermediate construction between an algebraic group and its Lie algebra, given by the formal group. One completes the algebraic group along the identity to get a formal scheme equipped with a group law, and one can pass from there to the tangent space to get the Lie algebra. In characteristic zero, the tangent space functor is an equivalence of categories from formal groups to Lie algebras, but in positive characteristic, formal groups form an honest intermediate category since the tangent space can lose a lot of information. For example, there is only one isomorphism class of one-dimensional Lie algebra, but one-dimensional formal groups have a rich arithmetic theory, with a moduli space stratified by positive integer heights. The completions at the identity of the additive group and the multiplicative group have very distinct formal group structures, and one way to explain the lack of isomorphism is by the presence of denominators in the usual logarithm and exponential power series.
It seems to me that in positive characteristic, there could be an intermediate construction between formal groups and Lie algebras, given by passing to PD rings and replacing the coordinate ring of the formal group with the divided power envelope of the identity section. If I'm not mistaken, this construction yields a group object in PD formal schemes.
Here is a bit of explanation for the uninitiated (see Berthelot-Ogus for more): PD rings are triples $(A,I,gamma)$, where $A$ is a commutative ring, $I$ is an ideal, and $gamma = { gamma_n: I to A }_{n geq 0}$ is a system of divided power operations. I think they arose when Grothendieck tried to get De Rham cohomology to give the expected answers for proper varieties in characteristic $p$, since the naïve definition tended to yield infinite dimensional spaces. There is a forgetful functor $(A,I,gamma) mapsto (A,I)$ from PD rings to ring-ideal pairs, and it has a left adjoint, called the divided power envelope. In characteristic zero, $gamma$ is canonically given as $gamma_n(x) = x^n/n!$, so both functors are equivalences in that case. The notion of PD ring can be sheafified and localizations have canonical PD structures, so one has notions of PD scheme and PD formal scheme.
Question: Do PD formal groups contain any more information than the underlying Lie algebra?
I have a suspicion that the answer is "no" and the answer to the title question is "yes". Vague word-association suggests that the divided power structure is exactly what one needs to get a formal logarithm, but maybe there is a more fundamental obstruction.
I was originally motivated by the question of how Gelfand-Kazhdan formal geometry would differ in charateristic $p$ if I switched between ordinary and PD structures (cf. David Jordan's question). Unfortunately, I was laboring under some misconceptions about formal completions, and I'm still a bit confused about the precise structure of the automorphism group of the completion (PD or ordinary) of a smooth variety at a point in characteristic $p$.
Let A be a non-negative integer square matrix with eigenvalues x1, x2, ... xn. Any symmetric function of these eigenvalues with integer matrices is an integer. I'm aware of the following results regarding the combinatorial interpretation of these integers:
If A is the adjacency matrix of a finite directed graph G, the power symmetric functions of the eigenvalues count closed walks on A with a distinguished starting point.
Similarly, the complete homogeneous symmetric functions of the eigenvalues count non-negative integer linear combinations of aperiodic closed walks on A.
(Gessel-Viennot-Lindstrom) If Aij is the number of paths from source i to sink j on, say, a 2-D lattice where the only permissible moves are to the right and up, then the elementary symmetric functions of the eigenvalues count the number non-intersecting k-tuples of paths from the sources to the sinks. In particular det A is the number of non-intersecting n-tuples of paths.
Do these results generalize to give a nice combinatorial interpretation of the value of the Schur function associated to an arbitrary partition evaluated at x1, x2, ... xn in terms of some combinatorial object attached to A? What conditions need to be placed on A so that the Schur functions are always non-negative?
Feel free to either talk about the GL(n) perspective or to frame your discussion entirely in terms of tableaux.
I solved a problem in analysis and i was thinking of generalizing this question which i couldn't succeed.
If $f:mathbb{R} to mathbb{R}$ is a continuous function which satisfies $f(x)=f(2x+1)$, for all $x in mathbb{R}$ then prove that $f$ is constant. I was able to prove it considering $g(x)=f(x-1)$ and showing that $g(x) to g(0)$.
Now my question is suppose $f: mathbb{R} to mathbb{R}$ is a continuous function and satisfies $f(p(x))=f(x)$ for every polynomial $p(x) in mathbb{R}$, then what should be the condition on $p(x)$ such that $f$ remains constant.
This is not an exact answer, but it helps to quickly determine in many cases that a given function is primitive recursive. The idea is to use a reasonable programming language in which your function can be expressed more easily than with "raw" arithmetic and primitive recursion. Of course, the programming language must be limited enough to prevent unbounded searches and other constructs that lead to general recursion. I mention here just one easy possibility.
Suppose a function $f : mathbb{N} to mathbb{N}$ is computed by a simple imperative program in polynomial running time (actually, as Grigory points out in his answer, any primitive recursive bound will do). More precisely, the program is allowed to use basic arithmetic operations, boolean values, variables, arrays, loops, and conditional statements. Then the function $f$ is primitive recursive.
The reason that such a program can only define a primitive recursive function is that the entire execution of the program may be encoded by a number $N$ whose bit size is bounded by a polynomial $p(n)$, where $n$ is the input. Because verifying that a number encodes the execution of a simple imperative program is primitive recursive, we can perform a bounded search up to $2^{p(n)}$ to find the number $N$ (such a bounded search is primitive recursive) and extract the answer from it.
Let us apply this to your question. The following Python program computes the function $pi(n)$ and uses just a couple of loops and basic arithmetic (we could replace the remainder function % with another loop). Its running time is quadratic in $n$, assuming all basic operations are constant time:
def pi(n):
'''Computes the number of primes which are less than or equal n.'''
p = 0 # the number of primes found
k = 2 # for each k we test whether it is prime
while k <= n:
j = 1 # possible divisors of k
d = 0 # number of divisors of k found
while j <= n:
if k % j == 0: d = d + 1
j = j + 1
if d == 2: p = p + 1
k = k + 1
return p
Therefore your function is primitive recursive.
In chapter 3 of What Is Mathematics, Really? (pages 43-45), Prof. Hersh writes:
How is it possible that mistakes occur in mathematics?
René Descartes's Method was so clear, he said, a mistake could only happen by inadvertence. Yet, ... his Géométrie contains conceptual mistakes about three-dimensional space.
Henri Poincaré said it was strange that mistakes happen in mathematics, since mathematics is just sound reasoning, such as anyone in his right mind follows. His explanation was memory lapse—there are only so many things we can keep in mind at once.
Wittgenstein said that mathematics could be characterized as the subject where it's possible to make mistakes. (Actually, it's not just possible, it's inevitable.) The very notion of a mistake presupposes that there is right and wrong independent of what we think, which is what makes mathematics mathematics. We mathematicians make mistakes, even important ones, even in famous papers that have been around for years.
Philip J. Davis displays an imposing collection of errors, with some famous names. His article shows that mistakes aren't uncommon. It shows that mathematical knowledge is fallible, like other knowledge.
...
Some mistakes come from keeping old assumptions in a new context.
Infinite dimensionl space is just like finite dimensional space—except for one or two properties, which are entirely different.
...
Riemann stated and used what he called "Dirichlet's principle" incorrectly [when trying to prove his mapping theorem].
Julius König and David Hilbert each thought he had proven the continuum hypothesis. (Decades later, it was proved undecidable by Kurt Gödel and Paul Cohen.)
Sometimes mathematicians try to give a complete classification of an object of interest. It's a mistake to claim a complete classification while leaving out several cases. That's what happened, first to Descartes, then to Newton, in their attempts to classify cubic curves (Boyer). [cf. this annotation by Peter Shor.]
Is a gap in a proof a mistake? Newton found the speed of a falling stone by dividing 0/0. Berkeley called him to account for bad algebra, but admitted Newton had the right answer... Mistake or not?
...
"The mistakes of a great mathematician are worth more than the correctness of a mediocrity." I've heard those words more than once. Explicating this thought would tell something about the nature of mathematics. For most academic philosopher of mathematics, this remark has nothing to do with mathematics or the philosophy of mathematics. Mathematics for them is indubitable—rigorous deductions from premises. If you made a mistake, your deduction wasn't rigorous, By definition, then, it wasn't mathematics!
So the brilliant, fruitful mistakes of Newton, Euler, and Riemann, weren't mathematics, and needn't be considered by the philosopher of mathematics.
Riemann's incorrect statement of Dirichlet's principle was corrected, implemented, and flowered into the calculus of variations. On the other hand, thousands of correct theorems are published every week. Most lead nowhere.
A famous oversight of Euclid and his students (don't call it a mistake) was neglecting the relation of "between-ness" of points on a line. This relation was used implicitly by Euclid in 300 B.C. It was recognized explicitly by Moritz Pasch over 2,000 years later, in 1882. For two millennia, mathematicians and philosophers accepted reasoning that they later rejected.
Can we be sure that we, unlike our predecessors, are not overlooking big gaps? We can't. Our mathematics can't be certain.
The reference to the said article by Philip J. Davis is:
Fidelity in mathematical discourse: Is one and one really two? Amer. Math. Monthly 79 (1972), 252–263.
From that article, I find particularly amusing the following couple of paragraphs from page 262:
There is a book entitled Erreurs de Mathématiciens, published by Maurice Lecat in 1935 in Brussels. This book contains more than 130 pages of errors committed by mathematicians of the first and second rank from antiquity to about 1900.There are parallel columns listing the mathematician, the place where his error occurs, the man who discovers the error, and the place where the error is discussed. For example, J. J. Sylvester committed an error in "On the Relation between the Minor Determinant of Linearly Equivalent Quadratic Factors", Philos. Mag., (1851) pp. 295-305. This error was corrected by H. E. Baker in the Collected Papers of Sylvester, Vol. I, pp. 647-650.
...
A mathematical error of international significance may occur every twenty years or so. By this I mean the conjunction a mathematician of great reputation and a problem of great notoriety. Such a conjunction occurred around 1945 when H. Rademacher thought he had solved the Riemann Hypothesis. There was a report in Time magazine.
I think I know how to answer this now. The main point is that OE(D) is the dual of ID. Namely: OE(D)=sheafHom(ID, OE). Thus, H^0(E,OE(D))=Hom(ID, OE).
This can be computed explicitly in any computer algebra package. Or you can see how to compute it as follows. Take a free presentation of ID as an OE-module. In the case I asked about, this yields:
OE3(-4)-->OE3(-3)-->ID.
Label the first map F. Then Hom(ID, OE) is just the kernel of the map of free modules:
Hom(OE3(-3), OE)--> Hom(OE3(-4), OE)
induced by composition with F. Thus, computing a free presentation of the ideal sheaf ID yields a presentation of H^0(E,OE(D)) as the kernel of a map of free modules.
I am interested in the following question on products of finite groups. Let $Gamma$ be a subgroup of $U_1times U_2$ such that the compositions with the canonical projections $Gamma subset U_1times U_2 rightarrow U_1$ and $Gamma subset U_1times U_2 rightarrow U_2$ are both surjective.
Does it follow that there is a group $G$ such that $Gamma$ is isomorphic to the fiber product $U_1 times_G U_2$? This means that there are surjections $pi_1:U_1rightarrow G$ and $pi_2:U_2rightarrow G$ such that $Gamma$ is the set of pairs $(u_1,u_2)$ with $pi_1(u_1)=pi_2(u_2)$.
Goursat's Lemma mentioned in this question proves the statement in the case $Gamma$ is a normal subgroup of $U_1times U_2$.
If the statement is not true without the normality assumption, then what would be a general characterization of these subgroups $Gamma$?
I came onto mathoverflow for a reference on an unrelated topic, but since I noticed this question I thought I would chime in: the image of the J homomorphism in the stable homotopy groups of spheres, by the work of Adams, is dictated by the denominators of Bernoulli numbers $-B_n/n$ (i.e. $zeta(1-2n)$) and it is these orders that are what are fundamentally related to the vector fields on spheres problem.
The image of $J$ is the $n = 1$ instance of some general $v_n$-periodic families in the stable stems called Greek letter elements (See Miller-Ravenel-Wilson's Annals paper "Periodic phemonena in the Adams-Novikov spectral sequence"). The n=2 analog is called the divided beta family.
Adams's identification of the Image of J with denomenators of Bernoulli numbers was through his "e"-invariant, which took values in Q/Z equal to the images of the Bernoulli numbers.
Gerd Laures, in his paper "The topological q-expansion paper" introduced a higher form of the "e" invariant called the "f"-invariant, which takes values in the quotient of the Katz's ring of divided congruences of modular forms by the sum of the integral q-expansions, and the classical modular forms over Q. (A kind of higher analog of Q/Z) He showed that his f invariant gave an injection from the divided beta family to this quotient, but did not characterize the image.
In my paper "congruences of modular forms and the divided beta family in homotopy theory" I showed that the divided beta family was actually in bijective correspondence with a set of congurence conditions between modular forms. Gerd and I later showed in another paper that these congruences I wrote down were precisely the ones coming from his f invariant.
But what I do not know, and would be curious if any user had an idea concerning this, is if there is a natural family of modular forms, kind of like a higher form of Eisenstein series, which would realize these beta elements in its image in the quotient of the ring of divided congruences.
The phenomenon I describe is not unique to n = 2, abstractly there is some kind of "congruence condition" amongst holomophic automorphic forms on U(1,n-1) that describes the nth Greek letter family in the stable homotopy groups of spheres - I just don't know how to express this congruence condition in clasical terms.
You are describing what is known as a
supertask, or
task involving infinitely many steps, and there are
numerous interesting examples. In a previous MO answer, for
example, I described an entertaining example about the
deal with the
Devil, which is similar to your example. Let me
mention a few additional examples here.
In the article "A beautiful supertask" (Mind,
105(417):81-84, 1996), the author Laraudogoitia considers
the situation with Newtonian physics in which there are
infinitely many billiard balls, getting progressively
smaller, with the $n^{th}$ ball positioned at $frac1n$, converging to $0$.
Now, set ball $1$ in motion, which hits ball 2 in such a
way that all energy is transferred to ball 2,
which hits ball 3 and so on. All collisions take place in
finite time, because of the positions of the balls, and so the motion disappers into the
origin; in finite time after the collisions are completed,
all the balls are stationary. Thus:
The general conclusion is that one cannot expect to prove
the principle of conservation of energy throughout time
without completeness assumptions about the nature of
time, space and spacetime.
A similar example has the balls spaced out to infinity, and
this time the collisions are arranged so that the balls
move faster and faster out to infinity (using Newtonian
physics), completing their progressively rapid interactions in finite total time. In
this case, once again, a physical system that is
energy-preserving at each step does not seem to be
energy-preserving throughout time, and the energy seems to
have leaked away out to infinity. The interesting thing
about this example is that one can imagine running it in
reverse, in effect gaining energy from infinity, where the
balls suddenly start moving towards us from infinity,
without any apparent violation of energy-conservation in
any one interaction.
Another example uses relativistic physics. Suppose that you
want to solve an existential number-theoretic question, of
the form $exists nvarphi(n)$. In general, such statements
are verified by a single numerical example, and there is in
principle no way of getting a yes-no answer to such
questions in finite time. The thing to do is to get into a
rocket ship and fly around the earth, while your graduate
student---and her graduate students, and so on in
perpetuity---search for an additional example, with the
agreement that if an example is ever found, then a signal
will be sent up to your rocket. Meanwhile, you should
accelerate unboundedly close to the speed of light, in such
a way that because of relativistic time contraction, the
eternity on earth corresponds to only a finite time on the
rocket. In this way, one will know the answer is finite
time. With rockets flying around rockets, one can in
principle learn the answer to any arithmetic statement in
finite time. There are, of course, numerous issues with
this story, beginning with the fact that unbounded energy
is required for the required time foreshortening, but
nevertheless Malament-Hogarth spacetimes can be constructed
to avoid these issues, and allow a single observer to have
access to an infinite time history of another individual.
These examples speak to an intriguing possible argument
against the Church-Turing thesis, based on the idea that
there may be unrealized computational power arising from
the fact that we live in a quantum-mechanical relativistic
world.
How about some tomography? This should work if $X$ is open. Assume $Xsubset mathbb R^3$ is nonempty and for every plane $H$ the intersection $Hcap X$ is either contractible or empty. (Note that an open subset of the plane is contractible if it is ((nonempty,) connected, and) simply connected.)
Claim 1: $X$ is contractible.
Proof: Consider the space of all pairs $(x,H)$, $xin X$ and $H$ a plane containing $x$.
Fiber this by $(x,H)mapsto x$. It's a trivial bundle over $X$ with fiber $P^2$.
Fiber the same thing by $(x,H)mapsto H$. The nonempty fibers are contractible, so the domain is homotopy equivalent to the image, which in turn fibers over $P^2$ with contractible fibers.
So $Xtimes P^2$ has the same homotopy type as $P^2$, and upon further inspection $X$ is contractible.
Claim 2: $X$ is convex.
Proof: Let $L$ be any line whose intersection with $X$ is nonempty, and play the same game again with pairs $(x,H)$, but now the plane $H$ is constrained to contain $L$. Call the space of all such pairs $Y$. On the one hand, $Y$ is equivalent to the circle $P^1$ by the same kind of argument as before. On the other hand, $Y$ is the blowup of the $3$-manifold $X$ along the $1$-manifold $Xcap L$. The complement $Y'$ of $(Xcap L)times P^1$ in $Y$ is the same as the complement of $Xcap L$ in $X$, so it is connected. Therefore all of the group $H_1(Y,Y')$ comes from $H_1(Y)$. But the latter group is of rank $1$ while the former is isomorphic to $H_0( (Xcap L)times P^1)$. (I am using mod $2$ coefficients.) It follows that $Xcap L$ is connected.
Maybe this generalizes to $mathbb R^n$. The hypothesis I am thinking of is that every nonempty hyperplane section is contractible, or equivalently $(n-2)$-connected. I don't believe that every hyperplane section simply connected is enough if $n>3$
A quick comment on the idea of "a very slick solution using the hairy ball theorem". Any such very slick proof will surely only use the fact that the Euler characteristic is non-zero, and so should apply just as well to surfaces of higher genus (at least two). For instance, if I understand it correctly, then Reid's answer above would work just as well on a higher-genus surface, by the Poincare--Hopf Theorem.
But the theorem does not hold on surfaces of higher genus.
What follows would probably be better with pictures, but I'll try to describe it without and hope for the best.
For instance, it's easy to divide a torus into two rectangles such that there are "traffic schedules" with no crashes. (OK, the Euler characteristic of a torus is zero, but bear with me.) (Also, what is to ants as a traffic schedule is to cars?)
Now consider the surface of genus two as an octagon with sides identified, as usual. You can divide this into two rectangles and two pentagons in a way that mimics two copies of the torus picture in the previous paragraph: the surface is the union of two tori with boundary, and each torus is divided into a rectangle and a pentagon. Schedule each torus as before (stretching one of the ants' route over the fifth side of the pentagon). As only one ant from each torus traverses the fifth side of the pentagon, it is easy to arrange that the ants from different tori do so at different times.
Does that make sense, or does anyone want a picture?
EDIT:
Oh, and of course it follows that the theorem is also false on any orientable surface of even higher genus, as they all cover the surface of genus two.
What carries over?
As Peter pointed out, a submodule of a free $mathbb{Z}$-module though
free need not have a complement. Indeed each submodule of a free
$mathbb{Z}$-module is free, but a quotient module need not be, for instance
$mathbb{Z}/2mathbb{Z}$. Also a $mathbb{Z}$-module is free if and oly if
it is projective; this entails that a kernel of a map of free modules
does have a complement.
The set $mathrm{Hom}(F,G)$ for free $mathbb{Z}$-modules need not be free.
If $F$ is free of countably infinite rank and $G=mathbb{Z}$, then
$mathrm{Hom}(F,G)congprod_{j=1}^inftymathbb{Z}$ which remarkably
is not free over $mathbb{Z}$. But $Fotimes G$ is free for free $F$ and $G$.
My favorite exercise is: prove that a comodule (over a coalgebra over a field) is coflat if and only if it is injective. This presumes that you already know that any coalgebra is the union of its finite-dimensional subcoalgebras, and any comodule is the union of its finite-dimensional subcomodules (if you don't know that, prove that first).
Also: construct a counterexample showing that infinite products in the category of comodules are not exact in general. Describe as explicitly as possible the cofree coalgebra with one cogenerator, i.e. the coalgebra $F$ such that coalgebra morphisms from any coalgebra $C$ into $F$ correspond bijectively to linear functions on $C$. Prove existence of the cofree coalgebra with any vector space of cogenerators.
Define a cosimple coalgebra as a coalgebra having no nonzero proper subcoalgebras, and a cosemisimple coalgebra as a direct sum of cosimple coalgebras. Prove that a coalgebra is cosemisimple if and only if the abelian category of comodules over it is semisimple. Prove that any coalgebra $C$ contains a maximal cosemisimple subcoalgebra which contains any other cosemisimple subcoalgebra of this coalgebra. Consider the quotient coalgebra of $C$ by this maximal cosemisimple subcoalgebra; it will be a coalgebra without counit. Prove that any element of this quotient coalgebra $D$ is annihilated by the iterated comultiplication map $Dto D^{otimes n}$ for a large enough $n$ (depending on the element).
Define Cotor between an unbounded complex of right comodules and an unbounded complex of left comodules as the cohomology of the total complex of the cobar complex of the coalgebra with coefficients in these two complexes of comodules, the total complex being constructed by taking infinite direct sums along the diagonal planes. Construct a counterexample showing that the Cotor between two acyclic complexes can be nonzero.
Hi, there is one such routine in SAGE (http://www.sagemath.org/)
def trace_faces(self, comb_emb):
"""
A helper function for finding the genus of a graph. Given a graph
and a combinatorial embedding (rot_sys), this function will
compute the faces (returned as a list of lists of edges (tuples) of
the particular embedding.
Note - rot_sys is an ordered list based on the hash order of the
vertices of graph. To avoid confusion, it might be best to set the
rot_sys based on a 'nice_copy' of the graph.
INPUT:
- ``comb_emb`` - a combinatorial embedding
dictionary. Format: v1:[v2,v3], v2:[v1], v3:[v1] (clockwise
ordering of neighbors at each vertex.)
EXAMPLES::
sage: T = graphs.TetrahedralGraph()
sage: T.trace_faces({0: [1, 3, 2], 1: [0, 2, 3], 2: [0, 3, 1], 3: [0, 1, 2]})
[[(0, 1), (1, 2), (2, 0)],
[(3, 2), (2, 1), (1, 3)],
[(2, 3), (3, 0), (0, 2)],
[(0, 3), (3, 1), (1, 0)]]
"""
from sage.sets.set import Set
# Establish set of possible edges
edgeset = Set([])
for edge in self.to_undirected().edges():
edgeset = edgeset.union( Set([(edge[0],edge[1]),(edge[1],edge[0])]))
# Storage for face paths
faces = []
path = []
for edge in edgeset:
path.append(edge)
edgeset -= Set([edge])
break # (Only one iteration)
# Trace faces
while (len(edgeset) > 0):
neighbors = comb_emb[path[-1][-1]]
next_node = neighbors[(neighbors.index(path[-1][-2])+1)%(len(neighbors))]
tup = (path[-1][-1],next_node)
if tup == path[0]:
faces.append(path)
path = []
for edge in edgeset:
path.append(edge)
edgeset -= Set([edge])
break # (Only one iteration)
else:
path.append(tup)
edgeset -= Set([tup])
if (len(path) != 0): faces.append(path)
return faces
I also have my own implementation which is an adaptation from the SAGE lib:
def Faces(edges,embedding)
"""
edges: is an undirected graph as a set of undirected edges
embedding: is a combinatorial embedding dictionary. Format: v1:[v2,v3], v2:[v1], v3:[v1] clockwise ordering of neighbors at each vertex.)
"""
# Establish set of possible edges
edgeset = set()
for edge in edges: # edges is an undirected graph as a set of undirected edges
edge = list(edge)
edgeset |= set([(edge[0],edge[1]),(edge[1],edge[0])])
# Storage for face paths
faces = []
path = []
for edge in edgeset:
path.append(edge)
edgeset -= set([edge])
break # (Only one iteration)
# Trace faces
while (len(edgeset) > 0):
neighbors = self.embedding[path[-1][-1]]
next_node = neighbors[(neighbors.index(path[-1][-2])+1)%(len(neighbors))]
tup = (path[-1][-1],next_node)
if tup == path[0]:
faces.append(path)
path = []
for edge in edgeset:
path.append(edge)
edgeset -= set([edge])
break # (Only one iteration)
else:
path.append(tup)
edgeset -= set([tup])
if (len(path) != 0): faces.append(path)
return iter(faces)
Given two normally distributed variables x_1, x_2, is there a non-simulation method of calculating the probability that x_1 > x_2?
Generalizing a bit, what is the probability that given a list of normally distributed variables x_i, the probability that x_a = max x_i?
As far as I know, the only significant result to speed up these calculations is that $E_2(frac{p}{q}) = frac{1}{2}|lbrace d: d | q^2, d equiv -q (mod p) rbrace|$, where $E_2(p/q)$ represents the number of decompositions into 2 unit fractions, and each matching $d$ represents the decomposition $frac{p}{q} = frac{qp}{q(q+d)} + frac{dp}{q(q+d)}$. (Take floor() or ceil() depending on whether you want to allow repeats.)
When I've coded this in the past, I called one of 4 different functions depending on a) whether $p=1$ or not, and b) whether $q/p ge min$ or not, where $min$ is the greatest denominator I'm already using. When $p=1$ and $q ge min$, in particular, we can just calculate $tau(q^2)/2$ from the factorisation of $q$; in the other cases I actually walked the factors from $q/p$ to $sqrt{q}$.
So: yes, you can count the number of matching sets without generating the 7 elements of each set, but computationally the elements are just a whisker away.
If the original poster is satisfied, that everything should be ok. However, I find this approach of giving a 'physical interpretation' of a purely mathematical idea slightly misleading. You can give a physical meaning to complex numbers, sure, but their mathematical meaning is far more interesting and compelling; I would rather speak of an application to physics.
As to fractional derivatives, they become quite easy to understand if you think that the Fourier transform takes the derivative of a function into multiplication by the variable: $widehat f'=ixicdot hat f$. So higher order derivatives can be defined as multiplication of $hat f$ by powers of $xi$, and it is no wonder that you can use this idea to define fractional derivatives, or actually generic 'functions of $d/dx$'. This leads to pseudodifferential operators etc.etc.
The main reason why this idea is not just a game but on the contrary is enormously useful, also in physics, is that using this kind of calculus you can give explicit (well, almost) expressions to fundamental things such as solutions to differential equations, and manipulate or estimate them in a very effective way.
Let me try to add a different point of view on B-fields and mirror symmetry. Ideally in mirror symmetry, given a Calabi-Yau manifold X, you would like to "construct" its mirror X', where the symplectic form on X should give you the complex structure on X'. As already mentioned, classes of symplectic forms have moduli of real dimension $h^{1,1}(X)$ and complex structures on X' have moduli of complex dimension $h^{2,1}(X') = h^{1,1}(X)$. So the kahler class is not enough to determine all complex structures on X'. In the context of the Strominger-Yau-Zaslow conjecture there is a nice interpretation of the B-field. Suppose X = $T^*B / Lambda$, where B is a smooth manifold and $Lambda$ is locally the span over the integers of 1-forms $dy_1$, ..., $dy_n$ (here $y_1$, ..., $y_n$ are coordinates which
change with affine transformations from one chart to the other). Then $X$ has a standard symplectic form. We can consider $X'= TB / Lambda'$, where $Lambda'$ is the dual lattice.
Then X' has a natural complex structure defined as follows. In standard coordinates on TB, given by $(y,x)$ --> $x partial_y$, the complex coordinates on X' are $z_k = e^{2pi i(x_k + i y_k)}$, which are well defined due to the nature of the coordinates x and y. But the above complex coordinates can be twisted locally (on a coordinate patch) by $z_k (b) = e^{2pi i(x_k + b_k + i y_k)}$, where $b = (b_1, ldots, b_n)$ is some local data. But since on overlaps $U_i cap U_j$ the coordinates have to match, we must have $b(i) - b(j) in Lambda$. It turns out that by putting $b_{ij} = b(i) - b(j)$ on overlaps, we get a cohomology class in $H^{1}(B, Lambda)$, this is the B-field. The cohomology group $H^{1}(B, Lambda)$ shoud coincide (in some cases at least) with $H^2(X, R/Z)$, which is what Kevin Lin mentioned. The elliptic curve case (mentioned by Kevin) can be seen from this point of view.
This point of view is also called "mirror symmetry without corrections" and it only approximates what happens in compact Calabi-Yaus. I have learned this in papers by Mark Gross (such as "Special lagrangian fibrations II: geometry") or the book "Calabi-Yau manifolds and related geometries" by Gross, Huybrechts and Joyce.
I would be interested to know how this interpretation connects to the other ones which have been described.
My question concerns the existence of a nice (deterministic?) method/algorithm for calculating the distribution of surface areas for two-dimensional projections of an arbitrary polytope (or convex approximation of a polytope). Less optimistically, a method of finding the minimum, maximum, and perhaps, mean surface area of the polytope's projections.
It is a relatively straightforward procedure to calculate a given two-dimensional surface projection along some orientational vector, and then calculate the approximate surface area of the projection (or its convex hull). But, beyond statistical sampling or methods related to simulated annealing, I'm having trouble imagining how to go about characterizing the full set of projections along all arbitrary vectors... and I haven't had any luck with a literature search (so far).
Note - This question is directly related to computations one might like to perform for - Characterizing a tumbling convex polytope from the surface areas of its two-dimensional projections. I hope this follow-up post is appropriate...
Ok, I thought a bit about the problem, and here is another idea. It does not provide an answer, but might give a new idea. Maybe even the sketched algorithm turns out to work well in practice.
Assume we want to find some $n in mathbb{Z}$ satisfying $C le n le D$ (for some constants $C$ and $D$, which can be assumed to be integers as well) such that $ncdot y pmod{x}$ is minimal under this condition.
For that, first use the Extended Euclidean Algorithm to compute the GCD $d$ of $x$ and $y$, as well as integers $A, B$ with $d = A x + B y$. Then we can write $d' = A' x + B' y$ with $d', A', B'$ if, and only if, $d' = d t$ for some $t in mathbb{Z}$, and $A' = A t + s y/d$, $B' = B t - s x/d$ with $s in mathbb{Z}$.
Hence, we want to make $t in mathbb{N}_{ge 0}$ as small as possible, while keeping $C le B t - s x/d le D$ for some $s in mathbb{Z}$. Such an $s$ exists if, and only if,
the closed interval $[(B t - D) frac{d}{x}, (B t - C) frac{d}{x}]$ contains an integer, or equivalently, if $lceil(B t - D) frac{d}{x}rceil le (B t - C) frac{d}{x}$.
Now $lceilfrac{a}{b}rceil = frac{a + (-a pmod{b})}{b}$, whence $lceil(B t - D) frac{d}{x}rceil = frac{(B t - D) d + (-(B t - D) d pmod{x})}{x}$. This is $le (B t - C) frac{d}{x}$ if, and only if, $(D - B t) d pmod{x} le (D - C) d$.
Therefore, an equivalent problem is finding the smallest $t ge 0$ such that $$D - B t pmod{tfrac{x}{d}} le D - C.$$
Note that without loss of generality, we can assume that $0 le B le frac{x}{d}$; in fact, in almost every case, we have $B < frac{x}{d}$ (the only exception is $d = x$ and $B = 1$, $A = 0$, in which $n cdot y pmod{x}$ is zero for all $n$). Hence, we can assume that $B d < x$. Moreover, since $1 = A frac{x}{d} + B frac{y}{d}$, we see that $B$ and $frac{x}{d}$ are coprime. In particular, $-B t pmod{frac{x}{d}}$, $t in mathbb{N}_{ge 0}$ iterates over every integer the interval $[0, frac{x}{d})$, including $0$ itself; therefore, we can always find a solution $t$ satisfying $0 le t < frac{x}{d}$, which is not surprising when considering the original problem.
One could now proceed as follows, which might lead to an algorithm which is fast in practice (in case $x > (D - C) d$): compute several solutions $t$ by choosing some random $T le B - C$ and computing $t$ such that $D - B t equiv T pmod{frac{x}{d}}$ (i.e. choose $t equiv (-T + D) frac{y}{d} pmod{frac{x}{d}}$, since $frac{y}{d}$ is the modular inverse of $B$ modulo $frac{x}{d}$) and take the minimum $t'$ over all such $t$. Hoping that at least one of these solutions is small, we are left only with a small interval $[0, t']$ to check for smaller solutions.
[Note that this is a similar problem to the one we started with: we want to find $T in [0, D - C]$ such that $(-T + D) frac{y}{d} pmod{frac{x}{d}}$ is minimal, instead of finding $n in [C, D]$ such that $n frac{y}{d} pmod{frac{x}{d}}$ is minimal.]
When we assume that $T mapsto (-T + D) B^{-1} pmod{frac{x}{d}}$ is "random", we can assume that the $t$'s we obtain are randomly distributed in the interval $[0, frac{x}{d})$, whence $t'$ can be expected to be small. Hence, this algorithm is only faster than just tying all values for $n$ if $t'$ is less than $D - C$, but this can be determined by a simple comparism.
I like your concept a lot, and have been able to find a characterization.
Suppose that $f:Nto N$ is effectively closed in your sense.
First, as you mentioned, it is easy to see that $text{ran}(f)$ is
computable, since by taking $W_e$ to be empty your equation shows that
$text{ran}(f)$ is both c.e. and co-c.e.
Second, I claim that $f$ is finite-to-one. To see this,
suppose that $f^{-1}(k)$ is infinite for some $k$. Define
a c.e. set $W_e$ as follows: At stage $s$, if we see that
$k$ is still not in $W_{rho(e),s}$, the state-$s$ approximation
to $W_{rho(e)}$, then enumerate the next element of
$f^{-1}(k)$ into $W_e$. (Although this definition may look
circular, since I am defining $W_e$ by reference to
$W_{rho(e)}$, the definition is legitimate by an
application of the Recursion Theorem. That is, I really
define $W_{r(e)}$, and then find $e$ such that $W_e=W_{r(e)}$.)
Note that if $k$ is never enumerated into $W_{rho(e)}$,
then I will eventually put all of $f^{-1}(k)$ into $W_e$,
which will result in $knotin f[N-W_e]$, but $kin
N-W_{rho(e)}$, a contradiction. Alternatively, if $kin
W_{rho(e),s}$, then $f^{-1}(k)cap W_e$ has at most
$s$ members, and so there are $ain N-W_e$ with $f(a)=k$,
placing $k$ into $f[N-W_e]$ but not in $N-W_{rho(e)}$,
again a contradiction.
A similar argument shows actually that the
function $kmapsto f^{-1}(k)$ is computable. Namely,
define the set $W_e$ by the following procedure. At stage
$s$, look at every $kleq s$, and if $knotin
W_{rho(e),s}$, then enumerate all of $f^{-1}(k)cap s$ into
$W_e$. (Again, appeal to Recursion Theorem to get such an
$e$.) In other words, as long as $k$ is not in
$W_{rho(e),s}$, then we put all elements of $f^{-1}(k)$
below $s$ into $W_e$.
If $knotin W_{rho(e)}$, then $f^{-1}(k)subset W_e$, and
so $knotin f[N-W_e]$, contradicting $kin N-W_{rho(e)}$.
Thus, $W_{rho(e)}=N$. From this, it follows that $W_e=N$.
Now, note that $kin W_{rho(e)}$ implies $kin W_{rho(e),s_k}$
for some stage $s_k$, and so $f^{-1}(k)$ is a subset of $s_k$.
By applying $f$ to each value below $s_k$, we see that the
map $kmapsto f^{-1}(k)$ is a computable function.
This means that $f$ has a particularly simple form.
Namely, there is a computable partition $N=bigsqcup_k
B_k$, with each $B_k$ finite, such that $f$ maps elements of
$B_k$ to $k$. (Note that some $B_k$ may be empty.)
Conversely, every function with such a form is
computably closed in your sense. Suppose that $f$ arises
from such a computable partition of $N$ into finite sets
$B_k$. Given any program $e$, enumerate $k$ into $W_{rho(e)}$ when
all of $B_k$ gets enumerated into $W_e$. It follows that
$f[N-W_e]=N-W_{rho(e)}$, as desired.
This provides a characterization of the effectively closed
computable functions:
Theorem. A computable function $f:Nto N$ is
effectively closed if and only if $f$ is finite-to-one and
the map $kmapsto f^{-1}(k)$ is computable.
Fractal curvature might be the answer. In differential or convex geometry, you need curvature to classify sets up to isometry. So it seems natural to introduce curvature for "fractals" in an attempt to get a finer geometric description. This has been done for mostly self-similar fractals by Winter, Zähle, Rataj, Kombrink, and me (Bohl, formerly Rothe). The full generalization to self-conformal sets is my upcoming PhD thesis.
Philosophically, and literally in differential geometry, curvature takes the second derivative of "coordinates" into account. In contrast, the Hausdorff and packing measures, most other dimensions, Minkowski content (=lacunarity), and surface content are only sensitive to the first derivative. Topological entropy is related to Gibbs / equilibrum measures if you have some kind of iterated function system, and these measures also belong to first order geometry.
Suppose $sum_{nge 1} a_n$ is a conditionally convergent series of real numbers, then by rearranging the terms, you can make "the same series" converge to any real number $x$. To do this, let $P={nge 1|a_nge 0}$ and $N={nge 1|a_n<0}$. Since $sum_{nge 1} a_n$ converges conditionally, each of $sum_{nin P}a_n$ and $sum_{nin N}a_n$ diverge and $lim a_n=0$.
Starting with the empty sum (namely zero), build the rearrangement inductively. Suppose $sum_{i=1}^m a_{n_i}=x_m$ is the (inductively constructed) $m$-th partial sum of the rearrangement. If $x_mle x$, take $n_{m+1}$ to be the smallest element of $P$ which hasn't already been used. If $x_m> x$, take $n_{m+1}$ to be the smallest element of $N$ which hasn't already been used.
Since $sum_{nin P}a_n$ diverges, there will be infinitely many $m$ for which $x_mge x$, so $n_{m+1}$ will be in $N$ infinitely often. Similarly, $n_{m+1}$ will be in $P$ infinitely often, so we've really constructed a rearrangement of the original series. Note that $|x-x_m|le max{|a_n|bigm| nnotin{n_1,dots, n_m}}$, so $lim x_m=x$ because $lim a_n=0$.
Suppose $sum_{nge 1}v_n$ is a conditionally convergent series with $v_nin mathbb R^k$. Can the sum be rearranged to converge to any given $win mathbb R^k$?
Obviously not! If $lambda$ is a linear functional on $mathbb R^k$ such that $sum lambda(v_n)$ converges absolutely, then $lambda$ applied to any rearrangement will be equal to $sum lambda(v_n)$. So let's also suppose that $sum lambda(v_n)$ is conditionally convergent for every non-zero linear functional $lambda$. Under this additional hypothesis, I'm pretty sure the answer should be "yes".
I figured that's the question you wanted to ask. The relation
$$sum_{k=0}^{n} (-1)^k {n choose k} a_k = b_n$$
for all $n$ (you did not specify this; it was very unclear) is equivalent to the relation
$$e^{x} A(-x) = B(x)$$
where $A(x) = sum_{k ge 0} frac{a_k}{k!} x^k, B(x) = sum_{k ge 0} frac{b_k}{k!} x^k$. This gives $A(x) = e^x B(-x)$, or
$$sum_{k=0}^{n} (-1)^k {n choose k} b_k = a_n.$$
So the $a_i$ are all integers if and only if the $b_i$ are all integers, and each uniquely determines the other. I don't know what else to say; you can choose either the $a_i$ or the $b_i$ arbitrarily. What exactly do you want to know?
Let $A$ be an abelian variety defined over $mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $pi_A$ of $GL(2d)/mathbb{Q}$ whose L-function agrees with the L-function attached to the Tate module of $A$. In fact, $pi_A$ should arise from an automorphic form on an orthogonal group. My question is, which (real) form of $O(2d+1)$ should $pi_A$ live on? For $d=1$ the group is $O(2,1)$, which is isogenous to $SL_2$, and for $d=2$ the group should be is $O(2,3)$, which is isogenous to $Sp_4$. But what should happen in general?
An answer to this question was given to me by Pierre Schapira. This is known as the microlocal Bertini-Sard theorem (cf. Sheaves on manifolds cor. 8.3.12).
Consider a map $f:Xto A^1$. It induces $f_pi : Xtimes_{A^1} T^*A^1 to T^*A^1$ and $f_d : Xtimes_{A^1} T^*A^1 to T^*X$. Set $Lambda = SS(M)$ the characteristic variety of $M$. This is a closed conic isotropic subset of $T^*X$. Now
$$
supp(phi_{f-t}(M)) subset [ x ~|~ f(x) = t,~(x,df(x)) in Lambda ]
$$
so
$$
[ tin A^1 ~|~ phi_{f-lambda}(M) neq 0 ] subset
[ tin A^1 ~|~ (t,dt)in f_pi f_d^{-1}(Lambda) ]
$$
Now assume that $f$ is compactifiable as $Xoverset{j}{to} bar{X} overset{bar{f}}{to} A^1$, $j$ an open immersion and $bar{f}$ proper. The closure $bar{Lambda}$ of $Lambda$ is $T^*bar{X}$ is a closed conic isotropic subset and since $bar{f}$ is proper, $bar{f}_pi bar{f}_d^{-1}(bar{Lambda})$ is a closed conic isotropic subset of $T^*A^1$. So its intersection with the nowhere vanishing section
$$
[t in A^1 ~|~ (t,dt) in bar{f}_pi bar{f}_d^{-1}(bar{Lambda})]
$$
has dimension 0. Since $f_pi f_d^{-1}(Lambda) subset bar{f}_pi bar{f}_d^{-1}(bar{Lambda})$ the same is true for
$$
[ tin A^1 ~|~ phi_{f-lambda}(M) neq 0 ] subset
[ tin A^1 ~|~ (t,dt)in f_pi f_d^{-1}(Lambda) ]
$$
and the theorem is proved.
If $f$ is algebraic it is always compactifiable. If $f$ is analytic, I don't know if the theorem still holds in general.
PS: If $f(x) = lambda $, the condition $(x,df(x)) in T^*_Z X$ just says that the fiber ${f = lambda}$ is tranverse to $Z$ at $x$. So when $SS(M) subset bigcup T^*_{S_alpha} X$ this gives the geometric interpretation that the vanishing cycles are 0 whenever the fibers are transverse to the strata.
At (about) the time Mumford was giving his lectures at Harvard, Ax was lecturing on his work with Kochen in which they proved a conjecture of Artin for almost all p by using ultrafilters. This is clearly what Mumford was thinking of. The reference for the Ax-Kochen work is:
MR0184930 Ax, James; Kochen, Simon Diophantine problems over local fields. I. Amer. J. Math. 87 1965 605--630. Ib. 87 1965 631--648.
You are right. An idependent filtering of the two signals will introduce errors because it is not contrained to fix the correlation to p. One possible approach is to perform a unified maximum likelihood estimation of both the missing samples and the correlation p. This can be done as follows: Assuming that the processes m_n and e_n have the same variance, hence we may write:
m_n = p * e_n + q * f_n, p^2 + q^2 = 1,
where f_n is a normal white noise uncorrelated to e_n and has the same variance as e_n.
The log likelihood function is proportional to:
sum_n((x_n - sum_i=1 to n(e_n))^2) + sum_n((y_n - sum_i=1 to n(p * e_n + q * f_n))^2) + lambda (p^2 + q^2 -1)
where lambda is a Lagrange multiplier, and the outer sums are of course over the known samples only.
This one will be very easy for the experts.
Let $F$ be a nonarch local field, let $ngeq1$ be an integer, choose $0leq d<n$ and let $D$ be the central simple algebra over $F$ with invariant $d/n$ in the Brauer group [EDIT: and with $F$-dimension $n^2$, so e.g. if $gcd(d,n)>1$ then $D$ will not be a division algebra but rather a matrix algebra over the division algebra with invariant $d/n$]. Let $G$ be the algebraic group $D^times$ and let $pi_0$ denote the trivial 1-dimensional representation of $G(F)$.
The local Jacquet-Langlands theorem guarantees us the existence of a smooth irreducible
representation $pi=JL(pi_0)$ of the group $GL(n,F)$ canonically associated to $pi_0$. This construction gives a completely canonical map from the set ${0,1/n,2/n,ldots,(n-1)/n}$ to the set of smooth irreducible representations of $GL(n,F)$, sending $d/n$ to $pi$.
If you put a gun to my head and asked me to guess what $pi=pi(d/n)$ was in this situation, I would probably go for the following construction: set $h=gcd(d,n)$, Let $P$ be the standard parabolic in $GL(n)$ whose Levi is $GL(h)^{n/h}$, and (non-normally) induce the trivial 1-dimensional representation of $P$ up to $G$; such a representation will, I suspect, have a canonical "biggest" irreducible subquotient, corresponding on the Galois side to an $n$-dimensional representation of the Weil-Deligne group of $F$ which is a direct sum of $h$ representations of degree $n/h$ each of which is Steinberg (in the sense that $N$ is maximally unipotent). I only envisage this because I can't imagine any other such map which agrees with what I know in the $GL(2)$ case!
Here is a consequence of my guess: if $d$ is coprime to $n$ then the trivial 1-dimensional representation of $D^times$ corresponds to the Steinberg representation of $GL(n)$, whatever $d$ is. This makes me wonder whether the following is true: say $d_1$ and $d_2$ are both coprime to $n$ and let $G_i$ be the group of units of the central simple algebra over $F$ with invariant $d_i/n$. Are the smooth irreducible representations of $G_i$ canonically in bijection with one another? Does this remain true if I relax the condition that the $d_i$ are coprime to $n$ but instead only demand that $gcd(d_1,n)=gcd(d_2,n)$?
More generally is it true that if $gcd(d_1,n)$ divides $gcd(d_2,n)$ then there's a canonical injection from the irreps of $G_1$ to the irreps of $G_2$, which is a bijection iff the gcd's coincide?
I don't know where in the literature to look for such statements :-/ so I ask here.
For fixed $k$ and large $n$ this is pretty doable. You want to find solutions to
$$x_1 + x_2 + ... + x_k = n$$
where $x_1 ge x_2 ge ... ge x_k$. Letting $y_i = x_i - x_{i+1}$ and $y_k = x_k$, this is equivalent to finding solutions to
$$y_1 + 2y_2 + ... + ky_k = n$$
where $y_i ge 0$. If $p_k(n)$ denotes the number of ways to do this, it follows by a standard generating function trick that
$$sum_{k ge 0} p_k(n) x^n = frac{1}{(1 - x)(1 - x^2)...(1 - x^k)}.$$
In principle one can find the partial fraction decomposition of the RHS, allowing us to write $p_k(n)$ as a quasi-polynomial.
In prime characteristic (or for algebraic groups rather than Lie algebras in general), the comments already posted indicate a need for caution. Jantzen's
Part I covers a lot of the ground, but he refers back at a few delicate points to Demazure-Gabriel.
Duality for general Hopf algebras is discussed in section 3.5 of Cartier's 2006 notes
http://inc.web.ihes.fr/prepub/PREPRINTS/2006/M/M-06-40.pdf
Starting with a Hopf algebra $A$, its reduced dual Hopf algebra (denoted by him $R(A)$) lives in the linear dual (a coalgebra in its own right) but is often smaller. So it's complicated to go back and forth. On the other hand, dealing with algebraic groups rather than just Lie algebras makes life a little more complicated, as suggested in his earlier discussion of the algebra of representative functions on a group; see also:
"Remark 3.7.3. Let $k$ be an algebraically closed field of arbitrary characteristic.
As in subsection 3.2, we can define an algebraic group over $k$ as a pair
$(G,O(G))$ where $O(G)$ is an algebra of representative functions on $G$ with
values in $k$ satisfying the conditions stated in Lemma 3.2.1. Let
$H(G)$ be the
reduced dual Hopf algebra of $O(G)$. It can be shown that $H(G)$ is a twisted
tensor product $G times U(G)$ where $U(G)$ consists of the linear forms on $O(G)$ vanishing on some power $mathfrak{m}^N$ of the maximal ideal $mathfrak{m}$ corresponding to the unit element of $G$; here $mathfrak{m}$ is the kernel of the counit
$epsilon: O(G) rightarrow k$. If $k$ is of characteristic 0, $U(G)$ is again the enveloping algebra of the Lie algebra $mathfrak{g}$ of
$G$. For the case of characteristic $p >0$, we refer the reader to Cartier [18] or
Demazure-Gabriel [32]."
[Note that the times symbol should be the LaTeX symbol ltimes, which apparently won't print here.]
ADDED. Given $G$, the hyperalgebra of $G$ is what Cartier denotes by $U(G)$; so it is not quite the reduced dual Hopf algebra of the algebra of regular functions on $G$ in general. In case $G$ is a connected reductive algebraic group over an algebraically closed field of characteristic $p>0$, the hyperalgebra has an explicit "divided power" construction starting with Kostant's integral form of the universal enveloping algebra of the complex analogue of the Lie algebra of $G$ (in the crucial case $G$ semisimple and simply connected), then reducing mod $p$. This is used heavily by Jantzen to investigate the rational representations of $G$ and relevant closed subgroups such as Borel subgroups, etc. (A similar construction is used by Lusztig for the quantum enveloping algebra at a root of unity.)
Jacob Lurie gave a talk last week at Peter May's birthday conference on noncommutative Poincare duality. The idea is to take an n-manifold M and a (n-1)-connected space X. Then, he showed that the compact mapping space Map_c(M,X) is isomorphic to a certain homotopy colimit over a certain category of open subsets of M. This is equivalent to the usual commutative Poincare duality. However, it is not clear (to me) what the natural generalization of the statement is to non-manifolds. So, I am not sure how to use it as a test. However, if you could use it as a test of being a manifold, it seems feasible that if the noncommutative statement held for your test space M and all (n-1)-connected spaces X for some X, then it would seem reasonable ask whether your test space is the homotopy type of an n-manifold.
The category of open sets over which Lurie takes the colimit is the category of disjoint balls (homeomorphic to R^n) in M. Thus, a guess might be something like: if M is a space, and if U is a category of open sets of M that cover M, and if Map_c(M,X) is equivalent to the homotopy colimit of Map_c(U_i,X) for all U_i in U for all (n-1)-connected X for some n, then M has the homotopy type of an n-manifold.
I have no idea if this is true, and even if it is true, it is not clear if it would be useful.