nt.number theory - Generators of the Principal Unit Group in Local Fields of Characteristic 0

Let $p$ be a prime, $L$ be a finite, totally ramified-extension of $Q_p,$ and $U$ be the group of principal units in the ring of integers of $L.$ Then $U$ is a finitely generated $mathbb{Z}_p$-module under exponentiation. I'm looking for an explicit set of generators for $U$ under this action; is there a resource where I can find this information?

Thanks Robin but am I afraid I need more control on the set of generators. In particular, I'm interested if G is a set of generators of the value m = max {$v_L(g-1):gin G$}

lo.logic - Degrees many-one below $0^omega$

Sorry for answering my own question, but I'm still hoping for a characterization of some kind.

A partial answer: Let $S$ be the result of a normal jump inversion forcing for $0^{omega}$ with added requirements that the n^th column is $0^n$ modulo a finite amount. This yields a degree many-one below $0^{omega}$, even many-one above each $0^n$, whose jump is $0^{omega}$.

The reason S is many-one below $0^{omega}$ is that we know that the algorithm gives uniformly an answer for the n^th bit from $0^{n}$, since no higher requirements have been initialized by that point. So, running the turing algorithm from $0^{omega}$ is the same as running the corresponding algorithm from $0^n$. The outcome is an arithmetical fact, so can be checked in one query of $0^{omega}$.

It appears that the many-one degrees below $0^{omega}$ (many-one) above each $0^n$ is a rich enough structure, since it contains $[S,0^{omega}]_m$.

It seems possible that similarly intertwining requirements, we can get a set whose double-jump is $0^{omega}$ above each $0^n$.

ct.category theory - Equimorphisms of categories

Actually, the analogy to deformation retracts in Reid's answer can be made quite precise. In the canonical or categorical model structure on Cat (often problematically called the "folk model structure"), the acyclic cofibrations are the equivalences of categories which are injective on objects. These are evidently precisely the functors that occur as F in your equimorphisms.

So probably a more correct thing to say than my original answer would be that it depends on whether you are treating categories truly "categorically," or whether you are using "too-strict" notions for convenience in getting at the weaker "correct" notions. Model category theory, and homotopy theory more generally, is all about doing the latter. Category theorists tend to work directly with the "right" notions, because for ordinary categories doing so is pretty easy--but of course when you get up to higher categories, some model category theory frequently turns out to be useful.

Your original question, though, suggested that you were thinking about whether such equimorphisms "arise naturally" between the ordinary sort of large categories that appear in mathematical practice. I think I would still say that whenever that happens, it's an accident of the chosen set-theory rather than anything really interesting. The "too-strict" notions really only have technical, rather than fundamental, importance.

How do you switch between representations of an algebraic group and its Lie algebra?

Suppose that the algebraic group $G$ over $k$ acts on the vector space $V$, i.e. that
there is map of algebraic groups $G to GL(V).$ Passing to Lie algebras (= Zariski tangent
space to the identity) is a functor, and so we get a map on Lie algebras
$mathfrak g to End(V),$ which is the corresponding Lie algebra representation.

Another way to do this, closer to the differential point of view (and which you will
need anyway to identify the Lie algebra of $GL(V)$ with $End(V)$) is as follows:
$G(k[epsilon])$ acts on $V[epsilon]$ (take dual number-valued points of the
original morphism). In particular, the Zariski tangent space at the identity
(which on the one hand is $mathfrak g$, by definition, and on the other hand
is the subgroup of $G(k[epsilon])$ consisting of elements mapping to the identity
under the specialization $epsilon mapsto 0$) acts on $V[epsilon]$ by endomorphisms
which reduce to the identity after setting $epsilon = 0$. One checks that such
a map is of the form $v mapsto v + L(v)epsilon,$ where $L in End(V)$.
Sending it to $L$ gives the required map $mathfrak g to End(V)$.

(Note: we are taking $g in G(k[epsilon])$ lying over the identity, applying
the representation $rho$ to get $rho(g)$, then forming the difference quotient
$(rho(g) - 1)/epsilon.$ Hopefully the connection with differentiation is clear.)

One has to be slightly cautious about going back from $mathfrak{g}$ to $G$,
since there are the following subtleties which any approach has to take into account:
the field $k$ had better have char. 0; the group $G$ had better be linear algebraic,
and furthermore either nilpotent, or simply connected semi-simple; and the representation
has better be finite-dimensional.

One could try the following: take a finite dimensional representation $V$ of
$mathfrak{g}$; extend it to a rep. of the universal enveloping algebra $U(mathfrak{g})$;
use the fact that $V$ is finite-dimensional to extend the rep'n to a certain completion
of $U(mathfrak{g})$; inside this completion, look at the group-like elements under the
canonical co-multiplication on $U(mathfrak{g})$; show that these elements form a linear algebraic group $G$ with Lie algebra $mathfrak g$. (The intuition is that we can map
$mathfrak{g}$ into a well-chosen completion of $U(mathfrak{g})$ via a formal version
of the exponential map.)

[This last suggestion is based on a discussion in Serre's Lie algebras/Lie groups book, but I don't remember
if he carefully treats this algebraic group context; it may be that he is rather focussing on the Lie group setting.]

ca.analysis and odes - variational formulation: boundedness of the bilinear form

The simplest case of the problem I'm thinking about involves an elliptic differential operator, $Lu = -u'' + qu$, on the interval $(0,1)$, with homogeneous Dirichlet boundary conditions. I want to show that the bilinear form on $H_0^1 subset H_1$ defined by

$a(u,v) = int_0^1 u'v' + quv~dx$

is bounded for the $H^1$-norm, i.e., $|a(u,v)| leq M|u|_1|v|_1$ for some constant $M>0$.

My question: can I assume that the linear coefficient $q$ is $L^1$ or even $L^2$ and still guarantee boundedness?

I was thinking that this is possible, but the only books that I have lying around discussing this consider only the case when $q$ is smooth or $L^infty$. I've played around with the Cauchy-Schwartz inequality for the term $int quv$ but am not getting anywhere.

ag.algebraic geometry - Motivation for the étale topology over other possibilities

In the search for a Weil cohomology theory $H$ over a field $K$ (with $text{char}(K)=0$) for varieties in characteristic $p$, a classical argument by Serre shows that the coefficient field cannot be a subfield of $mathbb{R}$ or of $mathbb{Q}_p$; an obvious choice is to take $mathbb{Q}_ell$ for a prime $ell neq p$.

Now, we can try to make a Weil cohomology theory by taking the sheaf cohomology with constant sheaves with the Zariski topology, but this does not work as all cohomology vanishes.
Grothendieck's insight was that we can find a different topology, for example the étale topology. Then we can build a Weil cohomology theory with coefficients in $mathbb{Q}_ell$ by taking cohomology with coefficients in the constant sheaves $mathbb{Z}/ ell^nmathbb{Z}$ and then taking the inverse limit with respect to $n$ and tensor with $mathbb{Q}_ell$: this gives $ell$-adic cohomology.

But it is not so clear to me why the étale topology is best suited at this task. What happens if we repeat the above procedure on other sites? Does the cohomology theory we get fail to be a Weil cohomology theory?

P.S.: Information for fields other than $mathbb{Q}_ell$ would also be nice!

ca.analysis and odes - On the extension of a limit

We know that $lim_{prightarrowinfty}leftVert left(x_{1},cdots,x_{n}right)rightVert _{p}=maxleft{ left|x_{1}right|,cdots,left|x_{n}right|right} =:leftVert xrightVert _{infty}$
for any $left(x_{1},cdots,x_{n}right)in R^{n}$. Now
do we have $lim_{prightarrow0}leftVert left(x_{1},cdots,x_{n}right)rightVert _{p}=leftVert left(x_{1},cdots,x_{n}right)rightVert _{0}:=mbox{cardinality}left{ x_{i}:x_{i}neq0right} $?

algebraic curves - Closest point on Bezier spline

If you have a Bezier curve $(x(t),y(t))$, the closest point to the origin (say) is given by the minimum of $f(t) = x(t)^2 + y(t)^2$. By calculus, this minimum is either at the endpoints or when the derivative vanishes, $f'(t) = 0$. This latter condition is evidently a quintic polynomial. Now, there is no exact formula in radicals for solving the quintic. However, there is a really nifty new iterative algorithm based on the symmetry group of the icosahedron due to Doyle and McMullen. They make the point that you use a dynamical iteration anyway to find radicals via Newton's method; if you think of a quintic equation as a generalized radical, then it has an iteration that it just as robust numerically as finding radicals with Newton's method.

Contrary to what lhf said, Cardano's formula for the cubic polynomial is perfectly stable numerically. You just need arithmetic with complex numbers even if, indeed exactly when, all three roots are real.

There is also a more ordinary approach to finding real roots of a quintic polynomial. (Like Cardano's formula, the Doyle-McMullen solution requires complex numbers and finds the complex roots equally easily.) Namely, you can use a cutoff procedure to switch from divide-and-conquer to Newton's method. For example, if your quintic $q(x)$ on a unit interval $[0,1]$ is $40-100x+x^5$, then it is clearly close enough to linear that Newton's method will work; you don't need divide-and-conquer. So if you have cut down the solution space to any interval, you can change the interval to $[0,1]$ (or maybe better $[-1,1]$), and then in the new variable decide whether the norms of the coefficients guarantee that Newton's method will converge. This method should only make you feel "a little dirty", because for general high-degree polynomials it's a competitive numerical algorithm. (Higher than quintic, maybe; Doyle-McMullen is really pretty good.)

See also this related MO question on the multivariate situation, which you would encounter for bicubic patches in 3D. The multivariate situation is pretty much the same: You have a choice between polynomial algebra and divide-and-conquer plus Newton's method. The higher the dimension, the more justification there is for the latter over the former.

dg.differential geometry - Orthogonal complements in Hilbert bundles

I shall prefix this with my standard "rubber stamps":

1. This is not really an answer, but is a bit longer than a comment allows.


2. You should read "A Convenient Setting of Global Analysis" by Kriegl and Michor. In particular, section 45 (Manifolds of Riemannian Metrics) and section 27.11ff (Submanifolds, in particular there's a good discussion of the necessity of the splitting condition).

I should also say that, as I mentioned in the comments above, I don't have access to the two articles cited so I can only speculate on what they are trying to achieve.

Firstly, many times in infinite dimensional analysis one wants to work with a space $X$ but it's tricky, so we work instead with a space $Y$. Only there's not one particular choice for $Y$, there's lots. And sometimes if we can say something for every such $Y$ in a compatible way then we can deduce that it also holds for our original $X$. The key here is the "in a compatible way". A common example is studying some infinite dimensional Frechet manifold (such as here) by expressing it as an inverse limit of Hilbert manifolds. Now each of those Hilbert manifolds has its own Hilbertian structure, and so (for example) is diffeomorphic to an open subset of some Hilbert space. But just because these exist, doesn't mean that they exist nicely with respect to each other. So to make some general statement about all of them, sometimes you have to sacrifice the really strong structure you have and work with something in the middle.

In the case of loop spaces, to take an example I do know something a little about, then one can always consider the various Sobolev spaces of loops, say $L^s M$. Each of those is a Hilbert manifold and its tangent bundle is thus a Hilbert bundle and can be given a strong metric. However, the "natural" structure group of $T L^s M$ is $LO_n$, loops on the orthogonal group. This only acts orthogonally on the "usual" Hilbert completion of $Lmathbb{R}^n$. So any other orthogonal structure requires changing the structure group. Admittedly, the space of all the choices is contractible, but that's homotopy theory and by passing to Hilbertian manifolds one is essentially declaring that one wants to do analysis so one would have to keep track of all the homotopies involved and keep taking them into account. (To make the point a little clearer, it's the difference between knowing that a solution exists and actually going out and finding it. If you really need to know the solution, knowing that it exists gives you a little hope but doesn't really help you actually write it down.)

I would also like to say that the usual classification of orthogonal structures into just "weak" and "strong" is a little simplistic. There is almost always some addition structure in the background (usually related to the structure group) and taking it into account can give a much finer picture. I have such a finer classification in my paper How to Construct a Dirac Operator in Infinite Dimensions together with examples of the different types.

So to return to the actual question. Let me see if I can simplify it a little. We can work locally, and in the actual question you are only concerned with what happens over the submanifold. So we have some open subset of a model space, $U$, and a Hilbert space, $H$, two trivial bundles over $U$ modelled on $H$, say $E_1$ and $E_2$, and an inclusion of bundles $E_1 to E_2$ such that the image of each fibre is closed. Then we want to know if $E_2^top$ is locally trivial. We can, if we choose, impose some codimensionality conditions on the inclusions.

Adjointing, we have a smooth map $theta : U to operatorname{Incl}(H,H)$ where $operatorname{Incl}(H,H)$ is the space of closed linear embeddings of $H$ in itself.

Now we see where the crux of the matter lies. What topology do we have on $operatorname{Incl}(H,H)$? We have lots of choices. The two most popular are the strong and weak topologies. Without making further assumptions, we can only assume that we have the weak topology! Where this distinction comes into play is that the weak topology is very badly behaved. With the weak topology, $operatorname{Incl}(H,H)$ is not an ANR so there's no nice extension results. With the strong topology, it's a CW-complex and so lots of nice things follow - in particular, orthogonal complementation will be continuous and the complement will be a locally trivial bundle.

Imposing finite codimensionality doesn't help either. That's a bit like saying that you know that you end up in Fredholm operators. Again, in the strong case then that's okay since the index is well-defined and so the complement will have constant dimension and be locally trivial. With the weak topology, the index is not continuous as, with the weak topology, the space of Fredholm operators is contractible.

So there's where to find your counterexample: find such an inclusion $E_1 to E_2$ such that the adjoint map is only continuous into the weak topology, not the strong one. A good source of such examples is with the obvious representation of a Lie group $G$ on $L^2(G)$. I expect that with a bit of bundle-crunching, this could be turned into an example.

It is just possible that the bits of your question that I threw out would save you (namely that the bundles were tangent bundles) but I doubt it since I expect that you could take an example as I outlined above and consider that as the inclusion of manifolds, then the tangent bundles of that inclusion would have the same properties of the inclusion itself. That is, if you can find a counterexample, $E_1 to E_2$, to my version of your question then viewing $Y = E_1$ and $X = E_2$ then I think you get a counterexample to your original setting.

Reference request for type theory

The kind of type theory you're asking about, Russell's simple theory of types, is from about the early 1900's. Here's a reference:

• Russell, Bertrand: Mathematical Logic as Based on the Theory of Types. Amer. J. Math. 30 (1908), no. 3, 222--262.

Recent work in type theory is somewhat different, continuing the tradition of Per Martin-Löf. In addition to his work (referenced by Andrej), I would also recommend the following book by Luo:

• Luo, Zhaohui: Computation and reasoning. A type theory for computer science. International Series of Monographs on Computer Science, 11. The Clarendon Press, Oxford University Press, New York, 1994. xii+228 pp. ISBN: 0-19-853835-9.

For the relation between set theory, type theory, and category theory, you might want to have a look at this preprint by Steve Awodey.

There's also an n-lab page, and the type theory page at Stanford Encyclopedia of Philosophy has a reference section.

soft question - Famous mathematical quotes

Dirichlet allein, nicht ich, nicht Cauchy, nicht Gauß, weiß, was ein vollkommen strenger Beweis ist, sondern wir lernen es erst von ihm. Wenn Gauß sagt, er habe etwas bewiesen, so ist es mir sehr wahrscheinlich, wenn Cauchy es sagt, ist ebensoviel pro als contra zu wetten, wenn Dirichlet es sagt, ist es gewiß; ich lasse mich auf diese Delikatessen lieber gar nicht ein.

C. G. J. Jacobi, writing to von Humboldt, in 1846.

Without pretty ßs: Only Dirichlet, Not I, not Cauchy, not Gauss, knows what a perfectly rigourous proof is, but we learn it only from him. When Gauss says he has proved something, I think it is very likely; when Cauchy says it, it is a fifty-fifty bet; when Dirichlet says it, it is certain; I prefer not to go into these delicate matters.

rt.representation theory - Schemes of Representations of Groups

Let $G$ be a group, say finitely presented as $langle x_1,ldots,x_k|r_1,ldots,r_ellrangle$. Fix $ngeq 1$ a natural number. Then there exists a scheme $V_G(n)$ contained in $GL(n)^k$ given by the relations. This scheme parameterizes $n$ dimensional representations of $G$.

Now, I've known this scheme since I first started learning algebraic geometry (one of the first examples shown to me of an algebraic set was $V_{S_3}(2)$) but I've never found a good reference for this. So my first question is:

Is there a good reference for the geometry of schemes of representations?

Now, I have some much more specific questions. The main one being a point I'd been wondering about idly and tangentially since reading about some open problems related to the Calogero-Moser Integrable System:

Are there natural conditions on $G$ that will guarantee that $V_G(n)$ be smooth? Reduced? Now, this is on the affine variety, I know that the projective closure will generally be singular, but in the case of $V_{S_3}(2)$, I know that the affine variety defined above is actually smooth, of four irreducible components.

Finally, for any $G$ and $n$, we have $V_G(n)subset V_G(n+1)$ (By taking the subscheme where the extra row and column are zeros, except on the diagonal, where it is 1). We can take the limit and get an ind-scheme, $V_G$. What is the relationship between $V_G$ and the category $Rep(G)$? Can the latter be realized as a category of sheaves on the former? I know nothing here, and as I said, most of these questions are the result of idle speculation while reading about something else.

Edit: It occurs to me that as defined, $V_G(n)$ and $V_G$ may not be invariants of $G$, but really of the presentation. So two things to add: one, $V_G(n)$ is intended to really be the scheme $Hom(G,GL(n))$ (there's some issues I want to sweep under the rug with finitely generated infinite groups here, which is part of why I was thinking in presentations), and second, the situations that I'm thinking of are often the data of group with a presentation, so for that situation, $V_G(n)$ as defined should be good enough.

na.numerical analysis - Smoothing out Noisy Data

I recently launched a rocket with an altimeter that is accurate to roughly 10 ft. The recorded data is in time increments of 0.05 sec per sample and a graph of altitude vs. time looks pretty much like it should when zoomed out.

The problem is when I try to calculate other values such as velocity or acceleration from the data, the accuracy of the measurements makes the calculated values pretty much worthless. What techniques can I use to smooth out the data so that I can calculate (or approximate) reasonable values for the velocity and acceleration? It is important that major events remain in place in time, most notably the 0 for for the first entry and the highest point during flight (2707).

The altitude data follows and is measured in ft above ground level. The first time would be 0.00 and each sample is 0.05 seconds after the previous sample. The spike at the beginning of the flight is due to a technical problem that occurred during liftoff and removing the spike would be best.

All help is greatly appreciated.

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graph theory - About the Shannon Switching Game

I was playing around with the Shannon Switching Game for some planar graphs, trying to get some intuition for the strategy, when I noticed a pattern. Since I only played on planar graphs, I'll restrict the problem to those for now, but we can also ask the problem for non-planar graphs.

Call a graph feasible if $E ge 2V-2$, where $E$ is the number of edges and $V$ the number of vertices on the graph. Call a graph a minimal feasible graph if none of its proper subgraphs containing at least two vertices are feasible.

Is every minimal feasible graph a winning position for Short, regardless of which pair of vertices he has to connect? In other words, for those who don't know the solution to the Shannon Switching Game, is it true that every minimal feasible graph contains two edge-disjoint spanning trees?

If this is true, I think it provides a more "intuitive" way of looking at the Shannon Switching Game in actual play, where you can't spend your time drawing cospanning trees on your notebook - look for the smallest subgraph with more edges than it deserves, mentally collapse it to a point, and repeat.

multi-index Dirichlet series

See P. Deligne, Multizeta values, Notes d'exposes, IAS Princeton, for the deep mathematical aspects of this.

Also for a general relevance philosophy, see Kontsevich and Zagier, Periods, Mathematics Unlimited(2001). An electronic version is available here.

There are various references, including those of Zudilin, Cartier, Zagier, Terasoma, Oesterle(On polylogarithms), Manin(iterated integrals and ....). Please look into mathscinet.

There seem to be many papers by Dorian Goldfeld and collaborators, too.

co.combinatorics - Notation for the all-ones vector

What's the most common way of writing the all-ones vector, that is, the vector, when projected onto each standard basis vector of a given vector space, having length one? The zero vector is frequently written $vec{0}$, so I'm partial to writing the all-ones vector as $vec{1}$, but I don't know how popular this is, and I don't know if a reader might confuse it with the identity matrix.

I'm writing for a graph theory audience, if that helps pick a notation.

homological algebra - Chain maps of complex

I read in my book a chain map q is a kernel of p iff each q(n) is a kernel of p(n). I think there's something wrong with this, it has to do with domain and codomain. Instead of using chain complexes I will give an example of a functor category where I think there's something wrong (after all, chain maps are like natural transformations).

Take J to be the category with 2 objects and 1 non identity between them, and Ab be the abelian groups, so every functor J->Ab is like an arrow in Ab, so I'm just gonna call the functors arrows in Ab. Now let f1 be the 0 arrow Z->Z, where Z is the integers, and f2 be the 0 arrow R->R, R the real numbers. Then define the two components of a natural transformation q: f1->f2 to be the inclusion Z->R, this is clearly natural. Next let g1 be the identity R->R, and g2 be the identity R/Z->R/Z, and define the two components of a natural transformation p:g1->g2 to be the projection onto quotient. Clearly q1 is a kernel of p1 and the same for q2,p2, but the codomain of q is f2 the 0 arrow R->R not the identity R->R (the domain of p), so q can't possibly be a kernel of p. I fiddled around a bit and think that if we assume to codomain of q to equal the domain of p, then it works out. I think natural transformations are not just determined by their components; domain and range also matter.

Is my reasoning correct? I really not confident on this stuff so can someone give me corrections or assurance. Thanks

nt.number theory - Inequality with Euler's totient

As a matter of fact your sum diverges, a little manipulation shows that
$$sum_{n leq X} frac{(-1)^n}{phi(n)} = sum_{n leq X, 2|n} frac{1}{phi(n)} -
sum_{n leq X, (n,2)=1} frac{1}{phi(n)} = sum_{n leq X/2} frac{1}{phi(2n)}
-sum_{n leq X,(n,2)=1}frac{1}{phi(n)}$$
The above equala to
$$sum_{n leq X/2, (n,2)=1} frac{1}{phi(2n)} + sum_{n leq X/2, 2|n} frac{1}{phi(2n)}
- sum_{n leq X, (n,2)=1} frac{1}{phi(n)}$$ By multiplicativity of $phi(n)$ we have $phi(2n) = phi(n)$ when $(n,2)=1$. Thus the first sum above is a sum over $1/phi(n)$ and the above equation simplifies to
$$ - sum_{X/2 < n leq X, (n,2)=1} frac{1}{phi(n)} + sum_{n leq X/4} frac{1}{phi(4n)}$$ It follows that
$$sum_{n leq X} frac{(-1)^n}{phi(n)}= -sum_{X/2 < n leq X, (n,2)=1} frac{1}{phi(n)} + sum_{n leq X/4} frac{1}{phi(4n)} sim c cdot log{X}$$
because the first sum on the right converges to a constant, while the second sum on the right is asymptotically $c cdot log{X}$.

EDIT: Put details, erased mention of an earlier confusion about $(-1)^{n+1}$ not being a multiplicative function :P (it is!)

lo.logic - Encoding fuzzy logic with the topos of set-valued sheaves

First I must warn you that there is a difference between fuzzy logics and topos theory. There are some categories of fuzzy sets which are almost toposes, but not quite - they form a quasitopos, which is like a topos, but epi + mono need not imply iso. There is a construction of such a quasitopos in Johnstone's Sketches of an Elephant - Vol 1 A2.6.4(e).

Now for the three valued logic I think a good example is a time-like logic. Suppose you have a fixed point T in time. This gives you two regions of time - before T and after T. Our logic will have three truth values - always true, true after T but not before, and never true. Note that we don't have a case "true before T, but not after", since once something is true, it is always true from that time on. Like knowledge of mathematical theorems (assuming there are no mistakes!).

The topos with this logic is the arrow category of set: Set$^to$. Objects consist of Set functions $A to B$, and morphism consist of pairs of set functions forming a commutative square.

For other three valued logics look at different Heyting algebras, but pay close attention to the implication operation, as it is a vital part of topos logic. For the true, false, maybe case I am not sure on how to construct a Heyting algebra which reflects this logic.

The algebraicity of Hodge structure map

It looks false to me. Let $V=mathbb{Q}^{2}$, and let $V(mathbb{R})=V^{0}oplus V^{2}$ where $V^{0}$ is the line defined by $y=ex$ and $V^{2}$
is the line defined by $y=pi x$. Give $V^{0}$ the unique Hodge structure of
type $(0,0)$ and $V^{2}$ the unique Hodge structure of type $(1,1)$. To say
that $w$ is defined over the subfield $mathbb{Q}^{mathrm{al}}$ of
$mathbb{C}$ means that the gradation $V(mathbb{R})=V^{0}oplus V^{2}$
arises from a gradation of $V(mathbb{Q}{}^{mathrm{al}})$ by tensoring up,
but this isn't true. Perhaps the all the "resources" have additional
conditions, or perhaps they are all ...

Added: When you are defining a Shimura variety, the weight homomorphism w factors through a Q-subtorus of GL(V), and then it is true that w is defined over the algebraic closure of Q (because, for tori T,T', the group Hom(T,T') doesn't change when you pass from one algebraically closed field to a larger field).

ag.algebraic geometry - Hyperelliptic loci in Teichmueller spaces

There's a slight issue I believe with the other answers. If we consider moduli space as an orbifold (of complex dimension $3g-3$), and the hyperelliptic locus an immersed suborbifold (of complex dimension $2g-1$ or so), then we may (essentially) identify the hyperelliptic locus with the orbifold of $2g+2$ points on $S^2$, obtained by quotienting each Riemann surface by the hyperelliptic involution. However, how does one know that this space doesn't "cross" itself? Imagine by analogy an immersed geodesic curve on a hyperbolic surface, such that each complementary component is a disk: the preimage in the universal cover is connected, when taken as a union of geodesics, even though each geodesic lift is embedded.

This sort of crossing does not occur for the hyperelliptic locus. If two branches of the hyperelliptic universal cover in Teichmuller space were to intersect, then there would be a single Riemann surface fixed by two distinct hyperelliptic involutions. But a hyperelliptic involution fixes precisely the $2g+2$ Weierstrauss points of the surface, and is therefore uniquely determined, a contradiction. So in fact the hyperelliptic locus is "embedded", in the sense that each lift corresponds to a fixed set of a hyperelliptic involution, and distinct hyperelliptic involutions give distinct components in Teichmuller space.

ag.algebraic geometry - Dimension of H^0(S,O_{S}(-C))

I'm not sure about the general case, but this is at least true if the divisor associated to the curve C is ample:

In this case the line bundle L associated to the divisor is ample, so the Kodaira-Nakano vanishing theorem applies. By Serre duality we get

$$ 0 = H^{2,1}(S,L) = H^1(S, Omega^2 otimes L) = H^1(S,L*)^* = H^1(S,O(-C))^* $$

as the dual of $L$ is the line bundle associated to the divisor $-C$. Remember that $mathcal I_C = O(-C)$, then the long exact sequence associated to

$$ 0 to O_S(-C) to O_S to O_C to O $$

gives the exact sequence

$$ 0 to H^0(S,O_S(-C)) to mathbb C to mathbb C to 0 $$

as both $O_S$ and $O_C$ only have constant global sections. The vanishing of $H^0$ follows.

The general case would hold in the same way if $H^1(S,O(-C)) = H^1(S,mathcal I_C) = 0$ for any smooth curve $C$ in $S$.

ct.category theory - Pushout over a whole diagram

Suppose I have a diagram $D$ over a category $C$, where $D$ (as a graph) is a single-rooted directed acyclic graph, and all 'joins' in this DAG are actually colimits. Let the root of this diagram be the object $c$ of $C$. Suppose I also have a (new) arrow $a : crightarrow e$ where $e$ is not in diagram $D$. From this arrow, one can successively compute a number of pushouts, to obtain a new diagram $D'$ "over" $D$. [Assume that by base category $C$ has all filtered colimits, so that this all makes sense].

Question: what is the proper name for this construction? Where can I find a solid mathematical presentation of this? [I have seen this used in a number of places, but never given a proper reference]

---

Motivation: My category $C$ is a category of (presentations of) theories (with sorts, signatures and axioms), and my diagram $D$ is built by successively extending a base theory $c$ by adding new sorts/signatures/axioms and explicit pushouts. If I now extend the base theory $c$ in a 'new' direction, I should be able to 'replay' the extension given by arrow $a:crightarrow e$ 'over' all of $D$ to get a (parallel) diagram of theories $D'$ based on $e$. The aim is to maximize re-use of concepts (given by each extension).

[I realize that such a construction might well give me inconsistent theories, i.e. that some of the nodes in $D$ and $D'$ will have no models. They will, however, be perfectly fine as (presentations of) inconsistent theories.]

matrices - Linear Algebra Problems?

Is there any good reference for difficult problems in linear algebra? Because I keep running into easily stated linear algebra problems that I feel I should be able to solve, but don't see any obvious approach to get started.

Here's an example of the type of problem I am thinking of: Let $A, B$ be $ntimes n$ matrices, set $C = AB-BA$, prove that if $AC=CA$ then $C$ is nilpotent. (I saw this one posed on the KGS Go Server)

Ideally, such a reference would also contain challenging problems (and techniques to solve them) about orthogonal matrices, unitary matrices, positive definiteness... hopefully, all harder than the one I wrote above.

nt.number theory - Density of numbers having large prime divisors (formalizing heuristic probability argument)

If I recall correctly, this was an exercise in Mathematics for the Analysis of Algorithms. I don't have access to a library right now, so I can't check that.

In any case, here is a proof. Fix a positive integer $N$. We will be counting the number of $n$ in ${ 1,2, ldots, N }$ such that the largest prime divisor of $n$ is $leq sqrt{n}$.

We can break this count into two parts: (1) Those $n$ which are divisible by $p$ where $p leq sqrt{N}$ and $n leq p^2$ and (2) Those $n$ which are divisible by $p > sqrt{n}$.

Case (1) is easier. We are looking at $sum_{p leq sqrt{N}} p = int_{t=0}^{sqrt{N}} t d pi(t)$. (This is a Riemann-Stieltjes integral.) Integrating by parts, this is $int_{0}^{sqrt{N}} pi(u) du + O( sqrt{N} pi(sqrt{N}))$. Since $pi(u) = O(u / log u)$ as $u to infty$, this integral is
$O left( int^{sqrt{N}} pi(u) du right) = Oleft(sqrt{N} frac{sqrt{N}}{log N} right) = O(N/log N)$, and the second term is also $O(N/log N)$. So case 1 contributes density zero.

Case (2) is the same idea — integrate by parts and use the prime number theorem — but the details are messier because we need a better bound.

We are trying to compute
$$sum_{sqrt{N} leq p leq N} lfloor frac{N}{p} rfloor = int_{sqrt{N}}^N lfloor frac{N}{t} rfloor dpi(t) = int_{sqrt{N}}^N left( frac{N}{t} + O(1) right) dpi(t).$$

The error term is $O(pi(N)) = N/log N$ so, again, it doesn't effect the density. Integrating the main term by parts, we have
$$int_{sqrt{N}}^N left( frac{partial}{partial t} frac{N}{t} right) pi(t) dt +O(N/log N).$$
Where the error term is $left( N/t pi(t) right)|^N_{sqrt{N}}$.

Now, $pi(t) = Li(t) + O(t/(log t)^K)$ for any $K$, by the prime number theorem, where $Li(t) = int^t du/log u$. So the main term is
$$int_{sqrt{N}}^N left( frac{partial}{partial t} frac{N}{t} right) Li(t) dt + Oleft( int^N frac{N}{t^2} frac{t}{(log t)^K} dt right).$$
The error term is $O left( N/(log N)^{K-1} right)$.

In the main term, integrate by parts, "reversing" our previous integration by parts. We get
$$int_{sqrt{N}}^N frac{N}{t} frac{dt}{log t} + O(N/log N).$$

Focusing on the main term once more, we have
$$N int_{sqrt{N}}^N frac{dt}{t log t} = N log 2.$$

Putting all of our error terms together, the number of integers with large prime factors is
$$N log 2 + O(N/log N).$$

---

In summary, integration by parts, the Prime Number Theorem, and aggressive pruning of error terms.

As I recall, the follow-up exercise in Mathematics of the Analysis of Algorithms is to obtain a formula of the form
$$N log 2 + c N/log N + O(N/(log N)^2).$$
That's a hard exercise! If you want to learn a lot about asymptotic technique, I recommend it.

st.statistics - How would you compute that "average" ?

I created a DJ-ing application that allows you to mix your MP3s with a real turntable.
So I generated an audio timecode to burn on a CD, left channel is the absolute position, right channel is a synchronization sine which frequency is 2205 Hz. Left channel is same frequency except that it represents binary sequences.

Precision is an absolute pre-requisite, the pitch on my Pioneer CDJ-1000 has a precision of 0.02% which is good enough to properly beatmatch songs. I have no problem on getting the absolute position but with the right channel.
After some searching, I found the Zero Crossing Rate algorithm to try detect the pitch. When I process data in real-time however, the pitch moves constantly.

After some searching I found out that as samples are discrete values, so not continuous if I'm right, there's not enough precision with a sample rate of 44100 Hz. Period of 20 samples is 2205 Hz, period of 21 samples is 2100 Hz; so the problem needs something more high-level.

I found then harmonic average, and moving average, results are somewhat more stable. FFT is the best but costs a lot of CPU, using a size of 65536 and time to compute it, whereas with the zero-crossing rate, I can update the value very frequently. The latest candidate, I still need to test is the Standard Deviation.

There must be some math formula that helps for that particular problem, but as I know not much about maths, I am somewhat lost.

Do you have any ideas ?

Thanks a lot :-)))

Examples of noncommutative analogs outside operator algebras?

Some non-commutative analogs in lattice theory.

von Neumann's coordinatization theorem is the
non-commutative analogue of Stone's definitional equivalence between Boolean algebras
(complemented distributive lattices) and Boolean rings (associative rings with 1 where all
elements are idempotent).

The Baer - Inaba - Jonnsson - Monk coordinatization theorem gives the non-commutative
analogue of direct products of finite chains (Łukasiewicz propositional logics);
for this one uses not the usual formulation of the theorem
(which coordinatizes primary lattices with modules over a artinian ring where
one-sided ideals are two-sided and form a chain), but a reformulation that gives
a true equivalence between the lattice (of submodules of the module) and the ring
(of endomorphisms of the module); this way one has a true analogue of the (dual)
equivalence between commutative C^*-algebras and compact Hausdorff spaces
(or better, their lattice of open sets).

One has a common generalization of the two cases above: equivalence between lattices of
subobjects and rings of endomorphisms for finitely presented modules
(of geometric dimension at least 3) over a "auxiliary" ring which is WQF
(weakly quasi Frobenius, a.k.a. IF, injectives are flat).
The categories of finitely presented modules over such auxiliary
rings are exactly (up to equivalence) the abelian categories with an object which is
injective, projective and finitely generates and finitely cogenerates every object.

The ultimate generalization is G.Hutchinson's coordinatization theorem,
a correspondence between arbitrary abelian categories and
modular lattices with 0 where each element can be doubled
and intervals are projective (in lattice theory, sense i.e. the classical projective
geometry meaning) to initial intervals. When one looks at this theorem together
with the Freyd - Mitchell embedding theorem, one has that three languages are fully
adequate and equivalent ways to do linear algebra:
(1) the usual language of sums and products (modules over associative rings);
(2) the language of category theory (abelian categories);
(3) the old fashioned language of synthetic geometry of incidence (joins and meets in
suitable modular lattices).

In these equivalences, the lattices are the (pointless, noncommutative) spaces,
and the rings are the rings of coordinates or functions over the space (I am not
considering the distinction between equivalences and dual equivalences because
I am more interested in structures, with their unique concept of isomorphism attached,
rather than more general morphisms, which depend upon the particular way to define a
structure. But the complementarity between the structural and categorical views,
where neither subsumes the other, is another long theme).

Since all these ideas have their origin in von Neumann works about continuous geometries
and rings of operators, I now explain the relation with operator algebras.

First note that the usual definition of pointless topological space as complete
Heyting algebra is not a true generalization of the "topological space" concept:
they are a true generalization of sober spaces,
but to generalize topological spaces one must consider pairs: a complete boolean
algebra (which in the atomic case is the same thing as a set) with a complete
Heyting subalgebra (the lattice of open sets). To obtain the non-commutative
analogue, complete boolean algebras are generalized to meet-continuous geometries,
and algebras of measurable sets are replaced with suitable structures (projection
ortholattices of von Neumann algebras) which are embedded in the meet-continuous
geometries in the same way as a right nonsingular ring is embedded in its maximal
ring of right fractions (a regular right self-injective ring).

A meet continuous geometry is a complete lattice which is modular, complemented
and meet distubutes over increasing joins (not arbitray joins, like Heyting algebras).
These structures were introduced by von Neumann and Halperin in 1939; they are a common
generalization of (possibly reducible) continuous geometries (the subcase where
join distrubutes over decreasing meets) and (possibly reducible and infinite dimensional)
projective geometries (the atomic subcase). For them one has a dimension and decomposition
theory much like the one for continuous geometries and rings of operators (the theory of
S.Maeda, in its last version of 1961, is sufficient; one does not need the 2003 theory
by Wehrung and Goodearl). One can define the components of various types, in particular
I_1 (the boolean component, i.e. classical logic), the I_2 component (the 2-distibutive
component i.e. subdirect product of projective lines; physically these are "spin
factors" and quantum-logically it is the non-classical component which nonetheless has
non-contextual hidden variables), the I_3 nonarguesian component (subdirect product
of projective nonarguesian planes i.e. irreducible projective geometries that cannot
be embedded in larger irreducible projective geometries; quantum-logically this means
that interacion with other components is only possible classically, without superposition).
Once these bad low dimensional components are disregarded, von Neumann coordinatization
theorem gives a equivalence between the meet-continuos geometries and the right
self-injective von Neumann regular rings. So meet-continuous geometries are pointeless
quantum (i.e. non-commutative) sets in the same way as complete Boolean algebras are
(commutative) pointless sets. Regular rings are the coordinate rings of these quantum
sets in the same way as (commutative, regular) rings of step functions (with values in
a field) are the ring equivalent of a boolean algebra (classical propositional logic);
the important new fact is that in the "truly non-commutative case" the regular ring
is uniquely and canonically determined by the lattice (in the distributive case, on the
contrary, it is not: one can use step functions with values in any field, and one can
change the field with the point; commutative [resp. strongly] regular rings are the
subrings of direct products of [skew] fields which are stable for the generalized inverse
operation).

The above "propositional logics" are without the negation operator; on the other hand,
the projection ortholattices of von Neumann algebras are complete orthomodular lattices
with sufficiently many completely additive probability measures and sufficiently many
internal simmetries (von Neumann said that the strict logic of orthocomplementation
and the probability logic of the states uniquely determine each other by means of his
symmetry axioms in his characterization of finite factors as continuos geometries with
a transition probability. One should also note how much more physically meaningful are
von Neumann axioms when compared with the "modern" ones based on Soler's theorem,
but this is another large topic). Using Gleason's theorem (and as always in absence of
the bad low-dimensional components) one obtains an equivalence between von Neumann's
"rings of operators" (i.e. real von Neumann algebras, or their self-adjoint part,
real JBW-algebras) and their projection ortholattices (the normal measures on the
ortholattice give the predual of the ring of operators). One can see these logics
inside a meet-continuos geometry by equipping the geometry with a linear orthogonality
relation which has for each element a maximum orthogonal element
(pseudo-orthocomplementation in part analogous to "external" in a stonean topological
space, in part anti-analogous as it happens with Lowere closure when compared to
Kuratowski closure). The regular ring of the lattice is the ring generated by all
complementary pairs in the lattice (which are the idempotents of the ring: kernel and
image) with the
relations corresponding to the partial operation e+f-ef which is defined on idempotents
whenever fe=0 (at the lattice level this partial operation is implemented with disjoint
join of the images and co-disjoint meet of the kernels); in the case associated to
a "ring of operators", this regular ring is the ring of maximal right quotients of
the von Neumann algebra, and conversely the algebra is recovered from the lattice with
orthogonality by taking the subring generated by orthogonal projections (idempotents
whose kernel and image are orthogonal); by a theorem of Berberian the algebra is ring
generated by its self-adjoint idempotents, and there is clearly at most one involution
on the algebra which fixes such generators.

Hence, in summary, in absence of bad low dimensional components (whose exclusion
is physically meaningful, see their meanings above) one has equivalences
between the following concepts:

(0) right self-injective regular rings with a suitable additional structure
(to associate a orthogonal projection onto the closure of the image to any element)
(1) real von Neumann algebras
(2) real JBW-algebras (the Jordan algebra of self-adjoint operators i.e. observables)
(3) the effect algebra (of operators with spectrum in [0,1]) i.e. unsharp quantum logic
(4) the projection ortholattice (the sharp quantum logic)
(5) the pointeless quantum set (meet-continuous geometry) with a suitable orthogonality
(6) the convex compact set of normal states

Any of the above is a adequate starting point for quantum foundation since all
the other points of view can be canonically recovered.

All this is restricted to the level of non-commutative measure spaces; a locally compact
topological space is something more precise (like a C^* algebra when compared to a
von Neumann algebra), and a (Riemannian) metric space is something still more precise
(and in particular a differentiable structure, which can be seen as an equivalence class
of riemannian structures: note that a isometry for the geodesic metric between complete
Riemannian manifolds is automatically differentiable, so the differentiable structure
must be definable from the geodesic metric, and infact Busemann and Menger had such a
explicit definition of the tangent spaces from the global metric. The topological
structure is then another equivalence class of metrics, for another weaker equivalence).

Given the equivalence (0)--(6) above, one can note that all the concepts which are used
in Connes definition of spectral triples, and analogues structures, can be seen
equivalently from each of the above points of view. In this way, all of the above points
of view are a possible starting point for non-commutative geometry.

[Sorry for the too long post and for my bad pseudo-english language]

puzzle - Origin of Fujimura set

If we have 10 coins arranged in an equilateral triangle and we want to know the minimum number of coins we can remove so that none of the remaining coins form an equilateral triangle the remaining coins form a Fujimura set. See here for more on this problem.

We have been looking at these sets and some generalizations in Polymath1. In the paper "Density Hales-Jewett and Moser numbers" the problem has come up of finding a citation for the original problem. In Martin Gardners article "“Eccentric Chess and Other Problems” which later appeared in his book Mathematical Circus he cites a a “recent book” of Fujimura. And we are trying to find the cited book. At least one person has checked Fujimura's book The Tokyo Puzzles and did not find it there.

So the question is if anyone knows of the book by Fujimura where the problem was introduced.

gn.general topology - A question about local connectedness

Any indecomposable continuum has the property you desire.

(A continuum is indecomposable if it cannot be written as a union of two proper subcontinua.)

One way to see this is that any indecomposable continuum has uncountably many composants, all of which are mutually disjoint, and all of which are dense in $x$. (Here the composant of a point $x$ in $X$ is the union of all proper subcontinua of $X$ that contain $x$.)

Here is a more direct proof: Suppose that $C$ is a proper subcontinuum of $X$ that is a neighborhood of $x$. Then every connected component of $V := Xsetminus C$ contains a point of $C$ in its boundary. (This is known as the 'boundary bumping theorem'.)

If $overline{V}$ is connected, then $overline{V}$ and $C$ are proper subcontinua of $X$ whose union is $X$.

If $V$ is disconnected, decompose $V$ into two relatively closed disjoint subsets $A$ and $B$; then $Acup C$ and $Bcup C$ are the desired subcontinua.

A simple example of an indecomposable continuum is given by the Knaster bucket-handle, see

http://commons.wikimedia.org/wiki/File%3aThe_Knaster_%22bucket-handle%22_continuum.svg .

The solenoid, mentioned in another answer, is another indecomposable continuum. You can also get such examples from "Lakes of Wada" continua. Of course the double Cantor brush given by Victor is not indecomposable (and in fact hereditarily decomposable).

gn.general topology - What is an explicit example of a sequence converging to two different points?

Here are two relevant facts:

1) In a Hausdorff space, a sequence converges to at most one point.

2) A first-countable space in which each sequence converges to at most one point is Hausdorff.

See e.g. pages 4 to 5 of

http://math.uga.edu/~pete/convergence.pdf

for the (easy) proofs of these facts, together with the definition of first-countable. See p. 6 for an example showing that 2) does not hold with the hypothesis of first-countability dropped.

It seems like a worthwhile exercise to use 2) to find spaces that have the property you want. For instance, the cofinite topology on a countably infinite set is first-countable and not Hausdorff, so there must be non-uniquely convergent sequences.

Addendum: Here are some further simple considerations which unify some of the other examples given.

For a topological space $X$, consider the specialization relation: a point $x$ specializes to the point $y$ if $y$ lies in the closure of ${x}$. This implies that any sequence which converges to $x$ also converges to $y$. (If in the previous sentence we replace "sequence" by "net", we get a characterization of the specialization relation.) The specialization relation is always reflexive and transitive, so is a quasi-order.

Note that a topological space is T_1, or separated, iff the specalization relation is simply equality. Thus in a space which is not separated, there exist distinct points $x$ and $y$ such that every net which converges to $x$ also converges to $y$. If $X$ is first countable, we may replace "net" by "sequence".

A topological space $X$ satisfies the T_0 separation axiom, or is a Kolmogorov space, if for any distinct points $x,y in X$, there is an open set containing exactly one of $x$ and $y$. A space is Kolmogorov iff the specalization relation is anti-symmetric, i.e., is a partial ordering. Thus in a non-Kolmogorov space, there exist distinct points $x$ and $y$ such that a net converges to $x$ iff it converges to $Y$. (If $X$ is first countable...)

An example of a first countable non-Kolmogorov space is a pseudo-metric space which is not a metric space (a pseudo-metric is like a metric except $rho(x,y) = 0 iff x = y$ is weakened to $rho(x,x) = 0$). In particular, the topology defined by a semi-norm which is not a norm always gives such examples.

ag.algebraic geometry - Is the blowup of a normal scheme necessarily normal?

Is the blowup of an integral normal Noetherian scheme along a coherent sheaf of ideals necessarily normal?

I can show that there is an open cover of the blowup by schemes of the form $text{Spec } C$, where $B subset C subset B_g$ for some integrally closed domain $B$ and some $g in B$, but I don't see why this would imply that $C$ is integrally closed. Intuitively, it seems reasonable that a blowup would be at least as "nice" as the original scheme, but that intuition may have more to do with how blowups are generally used than what they are capable of.

fa.functional analysis - Dense inclusions of Banach spaces and their duals

Just to flesh out Bill's answer and comments thereon, we have the following facts. Let $X,Y$ be Banach spaces and $T : X to Y$ a bounded linear operator.

1. If $T$ has dense range then $T^*$ is injective.

Since this is a standard homework problem I'll just give a hint. Suppose $f in Y^*$ with $T^*f=0$. This means that $f(Tx)=0$ for every $x$, i.e. $f$ vanishes on the range of $T$...

2. Suppose further that $X$ is reflexive. If $T$ is injective then $T^*$ has dense range.

Hint: By Hahn-Banach, to show that $T^*$ has dense range, it suffices to show that if $u in X^{**}$ vanishes on the range of $T^*$, then $u=0$. And by reflexivity, $u in X^{**}$ is represented by some $x in X$...

Note that the proofs I have in mind don't need to discuss the weak-* topology (at least not explicitly).

We cannot drop reflexivity in 2. Consider the inclusion of $ell^1$ into $ell^2$. It is injective, but its adjoint is the inclusion of $ell^2$ into $ell^infty$, whose range is not dense.

(Fun fact: you can't prove 2 without Hahn-Banach or some other consequence of the axiom of choice. If you work in ZF + dependent choice (DC), it's consistent that $ell^1$ is reflexive, but we still have the injective map from $ell^1$ to $ell^2$ whose adjoint doesn't have dense range. So under those axioms it's consistent that 2 is false.)

reference request - Geometric interpretation of exceptional Symmetric spaces

Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are related to real forms of exceptional Lie algebra.
Most of these symmetric spaces admit at least one geometric interpretation usually in terms of complex and real Grassmannians and their generalizations to quaternions and octonions($mathbb{H}$), octonions($mathbb{mathbb{O}}$), bioctonions ($mathbb{C}otimes mathbb{O}$), quateroctonions ($mathbb{H}otimes mathbb{O}$) and octooctonions ($mathbb{O}otimes mathbb{O}$). See for example the Wikipedia's entry for Symmetric Spaces.

Two of the exceptional symmetric spaces, don't seem to have such a geometric interpretation as far as I know.
In Cartan notation, these two spaces are called $EI$ and $EV$ and correspond respectively to the exceptional symmetric spaces $frac{E_7}{SU(8) / mathbb{Z}_2 }$ and $frac{E_6}{USp(4)/ mathbb{Z}_2}$ of respective rank and dimension$(4,42)$ and $(7,70)$.

Now that the stage is set, here is my question:

What is the geometric description of
the symmetric spaces
$frac{E_{7}}{SU(8)/ mathbb{Z}_2}$ and
$frac{E_6}{Sp(4)/ mathbb{Z}_2}$?

References on the subject are also welcome.
This question is motivated by an answer to
this MO question.

In order to give a more precise idea of the kind of answer I expect, let me give some examples: the symmetric space $frac{F_4}{mathrm{Spin}(9)}$ is geometrically described as the Cayley projective plane $mathbb{O}P^2$, the space $frac{E_6}{mathrm{SO}(10) mathrm{SO}(2)}$ is geometrically the Caylay bioctonion plane $(mathbb{C}otimes mathbb{O}) mathbb{P}^2$ and the symmetric space
$frac{E_6}{F_4}$ is the space of isometrically equivalent collineations of the Cayley plane $mathbb{O}mathbb{P}^2$.

NB:These two spaces also show up as scalar manifolds in maximal supergravity theories, this is for example review in this article of Boya. But for this question, I won't consider supergravity as a geometric interpretation.

Updates

Richard Borcherds has provided an answer thanks to a reference to the book "Einstein manifolds", but the book gives the answer without any proofs or explanation. So we now have an answer but we don't understand it. So if anyone could help with explaining how "antichains" enter the story, it will be highly appreciated. I have put some extra information in the comments.

fa.functional analysis - What's a natural candidate for an analytic function that interpolates the tower function?

The question is often phrased, can tetration or iterated exponentiation be naturally extended to the real and complex numbers. Using the notation $^{1}a=a, ^{2}a=a^a, ^{3}a=a^{a^a}$, how do you compute a number like $^{.5}2$, and what are the properties of $^{x}e$ ?

The Derivatives of Iterated Functions

Consider the smooth function $f(z): mathbb{C} rightarrow mathbb{C}$ and its iterates $f^{;:t}(z), t in mathbb{N} $. The standard convention of using a coordinate translation to set a fixed point at zero is invoked, $f(0)equiv 0$, giving $f(z)=sum_{n=1}^{infty} frac{f_n}{n!} z^n$ for $0leq |z|< R$ for some positive $R$. Note that $f(z)$ is the exponential generating function of the sequence $f_0, f_1, ldots ,f_infty$, where $f_0=0$ and $f_1$ will be written as $lambda$. The expression $f_j^k$ denotes $(D^j f(z))^k |_{z=0}$ . Note: The symbol $t$ for time assumes $t in mathbb{N}$, that time is discrete. This allows the variable $n$ to be used solely in the context of differentiation. Beginning with the second derivative each component will be expressed in a general form using summations and referred to here as Schroeder summations.

The First Derivative

The first derivative of a function at its fixed point $Df(0)=f_1$ is often represented by $lambda$ and referred to as the multiplier or the Lyapunov characteristic number; its logarithm is known as the Lyapunov exponent. Let $g(z)=f^{t-1}(z)$, then

$ Df(g(z)) = f'(g(z))g'(z)$

$ = f'(f^{t-1}(z))Df^{t-1}(z) $

$ = prod^{t-1}_{k_1=0}f'(f^{t-k_1-1}(z))$

$ Df^t(0) = f'(0)^t $

$ = f_1^t = lambda^t $

The Second Derivative

$D^2f(g(z)) = f''(g(z))g'(z)^2+f'(g(z))g''(z)$

$= f''(f^{t-1}(z))(Df^{t-1}(z))^2+f'(f^{t-1}(z))D^2f^{t-1}(z) $

Setting $g(z) = f^{t-1}(z)$ results in

$ D^2f^t(0) = f_2 lambda^{2t-2}+lambda D^2f^{t-1}(0)$.

When $lambda neq 0$, a recurrence equation is formed that is solved as a summation.

$ D^2f^t(0) = f_2lambda^{2t-2}+lambda D^2f^{t-1}(0)$

$ = lambda^0f_2 lambda^{2t-2}$

$ +lambda^1f_2 lambda^{2t-4}$

$+cdots$

$+lambda^{t-2}f_2 lambda^2$

$+lambda^{t-1}f_2 lambda^0$

$ = f_2sum_{k_1=0}^{t-1}lambda^{2t-k_1-2} $

The Third Derivative

Continuing on with the third derivative,
$ D^3f(g(z)) = f'''(g(z))g'(z)^3+3f''(g(z))g'(z)g''(z)+f'(g(z))g'''(z)$

$ = f'''(f^{t-1}(z))(Df^{t-1}(z))^3 $

$ +3f''(f^{t-1}(z))Df^{t-1}(z)D^2f^{t-1}(z)$

$ +f'(f^{t-1}(z))D^3f^{t-1}(z)$

$ D^3f^t(0) = f_3lambda^{3t-3}+3 f_2^2sum_{k_1=0}^{t-1}lambda^{3t-k_1-5} +lambda D^3f^{t-1}(0) $

$ = f_3sum_{k_1=0}^{t-1}lambda^{3t-2k_1-3} +3f_2^2 sum_{k_1=0}^{t-1} sum_{k_2=0}^{t-k_1-2} lambda^{3t-2k_1-k_2-5} $

Note that the index $k_1$ from the second derivative is renamed $k_2$ in the final summation of the third derivative. A certain amount of renumbering is unavoidable in order to use a simple index scheme.

Iterated Functions

Putting the pieces together and setting the fixed point at $f_0$ gives,

$f^t(z) = sum_{j=0}^infty D^j f^t(f_0) (z-f_0)^j $

$ = f_0+lambda^t (z-f_0)+( f_2sum_{k_1=0}^{t-1}lambda^{2t-k_1-2}) (z-f_0)^2$

$+ (f_3sum_{k_1=0}^{t-1}lambda^{3t-2k_1-3} +3f_2^2 sum_{k_1=0}^{t-1} sum_{k_2=0}^{t-k_1-2} lambda^{3t-2k_1-k_2-5}) (z-f_0)^3+ ldots $

So far we have covered a decent amount of algebra, but still $t in mathbb{N}$. The equation $f^t(z)$ , $t in mathbb{N}$ is important because it is convergent when $f(z)$ is convergent.

Hyperbolic Fixed Points

When $lambda$ is neither zero nor a root of unity $lambda^t neq 1, t in mathbb{N}$, then the nested summations simplify to

$f^t(z)=f_0 + lambda ^t (z-f_0)+frac{lambda ^{-1+t} left(-1+lambda ^tright) f_2}{2 (-1+lambda )} (z-f_0)^2 $

$ + frac{1}{6} left(frac{3 lambda ^{-2+t} left(-1+lambda ^tright) left(-lambda +lambda ^tright) f_2^2}{(-1+lambda )^2 (1+lambda )}+frac{lambda ^{-1+t} left(-1+lambda ^{2 t}right) f_3}{-1+lambda ^2}right) (z-f_0)^3+ldots $

Hyperbolic Tetration

Let $a_0$ be a limit point for $f(z)=a^z$, so that $a^{a_0}=a_0$. Also $a_1=lambda$. This results in a definition for tetration of complex points for all except the set of points with rationally neutral fixed points. For the real numbers $a=e^{e^{-1}}approx 1.44467, a=e^{-e}approx 0.065988 $ have rationally neutral fixed points while $a=1$ is a superattractor. All other real values of $a$ are defined by hyperbolic tetration.

$ {}^t a = a_o + lambda ^tleft(1-a_oright)+frac{lambda ^{-1+t} left(-1+lambda ^tright) text{Log}left(a_oright){}^2}{2 (-1+lambda )}left(1-a_oright){}^2 $

$ + frac{1}{6}text{ }left(frac{3 lambda ^{-2+t} left(-1+lambda ^tright) left(-lambda +lambda ^tright)text{ }text{Log}left(a_oright){}^4}{(-1+lambda )^2 (1+lambda )}+frac{lambda ^{-1+t} left(-1+lambda ^tright) left(1+lambda ^tright)text{ }text{Log}left(a_oright){}^3}{(-1+lambda ) (1+lambda )}right)left(1-a_oright){}^3+ldots $

Summary

One issue that some researchers have with this approach is that it results in $^x e: mathbb{R} rightarrow mathbb{C} $.

Because this derivation is based on the Taylor series of $f^n(z)$, if $f(z)$ is convergent then $f^n(z)$ is convergent where $n in mathbb{N}$.

ac.commutative algebra - Are valuation rings regular?

This question is short, and to the point:

Valuation rings are certainly integrally closed, but are they regular?

The motivation is that I'm trying to understand the resolution of singularities of algebraic surfaces as was done originally, and I'm playing around with some of the ideas involved.

soft question - What's your favorite equation, formula, identity or inequality?

It's too hard to pick just one formula, so here's another: the Cauchy-Schwarz inequality:

||x|| ||y|| >= |(x.y)|, with equality iff x&y are parallel.

Simple, yet incredibly useful. It has many nice generalizations (like Holder's inequality), but here's a cute generalization to three vectors in a real inner product space:

||x||² ||y||² ||z||² + 2(x.y)(y.z)(z.x) >= ||x||²(y.z)² + ||y||²(z.x)² + ||z||²(x.y)², with equality iff one of x,y,z is in the span of the others.

There are corresponding inequalities for 4 vectors, 5 vectors, etc., but they get unwieldy after this one. All of the inequalities, including Cauchy-Schwarz, are actually just generalizations of the 1-dimensional inequality:

||x|| >= 0, with equality iff x = 0,

or rather, instantiations of it in the 2^nd, 3^rd, etc. exterior powers of the vector space.

hilbert spaces - Norm on quotient algebra of a tensor algebra

Intuitively, I would expect Q in this case to be a two dimensional vector space with basis 1,[a]. Is this correct? Projecting onto the orthogonal complement seems to give zero however.

The first part is correct. This may be easiest to see by considering the isomorphism, in the present case where $V$ is one dimensional, of $T(V)$ with $mathbb{R}[x]$. Then $Q$ is $mathbb{R}[x]/(x^2-x)$, which is 2-dimensional by the division algorithm.

The orthogonal complement of $I(S)$ in this case is not zero, but it is one dimensional and your concern is valid. Thinking again in $mathbb{R}[x]$ for convenience, the condition that $sum_k a_k x^k$ is orthogonal to $x^{m+2} - x^{m+1}$ for all $mgeq0$ says that $a_{m+2}=a_{m+1}$ for all $mgeq0$; thus, $a_k=0$ for $kgeq1$. Note that the conclusion is the same if we first complete to a Hilbert space by taking power series with square summable coefficients, so that unfortunately Andrew Stacey's desperate hope appears to be shattered. I have been assuming in this paragraph that $a$ (or $x$) has norm 1; it makes no difference for the conclusion in the non-completed case, but it simplifies the relation. (In other cases when $dim(V)>1$ the Hilbert space quotient can be far too large for the vector space isomorphism to be possible, having dimension $2^{aleph_0}$, so completion is probably not the way to go.)

This shows that there is no hope in general of using the standard inner product to define a norm on $T(V)/I(S)$. However, the problem in the above case arose because the ideal was not homogeneous; this forces relations on coefficients of different degree for elements of the orthogonal complement, thereby forcing too many coefficients to be zero. For this construction to work I recommend considering the case where $S$ is a set of homogeneous elements of $T(V)$ (i.e., each element of $S$ lies in one of the tensor powers $V^{otimes k}$). I don't know what else to tell you in the nonhomogeneous case.

Gratuitous Add-on

The Hilbert space completion of the free algebra $T(V)$ is the full (or free) Fock space $F(V)$. The latter is not an algebra, but a Banach algebra completion of $T(V)$ can be obtained by first representing $T(V)$ on $F(V)$ by choosing an orthonormal basis $v_1,ldots,v_n$ for $V$ and sending $v_jin T(V)$ to the "creation operator" $S_j$ on $F(V)$ defined by $S_j(w)=v_jotimes w$, and then closing the image in your favorite topology. These algebras were first studied by G. Popescu circa 1991 (but over $mathbb{C}$ rather than $mathbb{R}$ and with either norm or ultraweak closure), and a similar construction was earlier used by D. Evans to study the Cuntz algebra $mathcal{O}_n$ in the 1980 paper "On $mathcal{O}_n$".

qa.quantum algebra - Reference for the existence of a Shapovalov-type form on the tensor product of integrable modules

I know a couple of ways to get a Shapovalov type form on a tensor product. The details of what I say depends on the exact conventions you use for quantum groups. I will follow Chari and Pressley's book.

The first method is to alter the adjoint slightly. If you choose a * involution that is also a coalgebra automorphism, you can just take the form on a tensor product to be the product of the form on each factor, and the result is contravariant with respect to *. There is a unique such involution up to some fairly trivial modifications (like multiplying $E_i$ by $z$ and $F_i$ by $z^{-1}$). It is given by:
$$
*E_i = F_i K_i, quad *F_i=K_i^{-1}E_i, quad *K_i=K_i,
$$
The resulting forms are Hermitian if $q$ is taken to be real, and will certainly satisfy your conditions 1) ad 3). Since the $K_i$s only act on weight vectors as powers of $q$, it almost satisfies 2).

The second method is in case you really want * to interchange $E_i$ with exactly $F_i$. This is roughly contained in this http://www.ams.org/mathscinet-getitem?mr=1470857 paper by Wenzl, which I actually originally looked at when it was suggested in an answer to one of your previous questions.

It is absolutely essential that a * involution be an algebra-antiautomorphism. However, if it is a coalgebra anti-automorphism instead of a coalgebra automorphism there is a work around to get a form on a tensor product. There is again an essentially unique such involution, given by

$$
*E_i=F_i, quad *F_i=E_i, quad *K_i=K_i^{-1}, quad *q=q^{-1}.
$$

Note that $q$ is inverted, so for this form one should think of $q$ as being a complex number of the unit circle. By the same argument as you use to get the Shapovalov form, then is a unique sesquilinear *-contravariant form on each irreducible representation $V_lambda$, up to overall rescaling.

To get a form on $V_lambda otimes V_mu$, one should define
$$(v_1 otimes w_1, v_2 otimes w_2)$$
to be the product of the form on each factor applied to $v_1 otimes w_1$ and $R( v_2 otimes w_2)$, where $R$ is the universal $R$ matrix. It is then straightforward to see that the result is *-contravariant, using the fact that $R Delta(a) R^{-1} =Delta^{op}(a).$

If you want to work with a larger tensor product, I believe you replace $R$ by the unique endomorphism $E$ on $otimes_k V_{lambda_k}$ such that $w_0 circ E$ is the braid group element $T_{w_0}$ which reverses the order of the tensor factors, using the minimal possible number of positive crossings. Here $w_0$ is the symmetric group element that reverses the order of the the tensor factors.

The resulting form is *-contravariant, but is not Hermitian. In Wenzl's paper he discusses how to fix this.

Now 1) and 2) on your wish list hold. As for 3): It is clear from standard formulas for the $R$-matrix (e.g. Chari-Pressley Theorem 8.3.9) that $R$ acts on a vector of the form $b_lambda otimes c in V_lambda otimes V_mu$ as multiplication by $q^{(lambda, wt(c))}$. Thus if you embed $V_mu$ into $V_lambda otimes V_mu$ as $w rightarrow b_lambda otimes w$, the result is isometric up to an overall scaling by a power of $q$. This extends to the type of embedding you want (up to scaling by powers of $q$), only with the order reversed. I don't seem to understand what happen when you embed $V_lambda$ is $V_lambda otimes V_mu$, which confuses me, and I don't see your exact embeddings.

real analysis - Cartesian product of test function spaces

Mini introduction

Suppose $U subset mathbb R^n, V subset mathbb R^m$ are two open sets. If we take http://en.wikipedia.org/wiki/Distributions_space#Test_function_space">test functions $f_i in mathfrak D (U),~g_i in mathfrak D (V)$ for $1 leq i leq n$, then $f_1(x)g_1(y) + dots + f_n(x)g_n(y)$ is an element of $mathfrak D (U times V)$, so we have an inclusion:
$$operatorname{span}left(mathfrak D (U) times mathfrak D (U) right) subset mathfrak D (U times V)$$
where "span" means linear span.

Question

Is it true that
$$overline{operatorname{span}left(mathfrak D (U) times mathfrak D (U) right)} = mathfrak D (U times V)$$
where line means the closure in topology of $mathfrak D (U times V)$?

at.algebraic topology - Analogs of left, right, inner, and Kan fibrations in CGWH

The analogs are:

• Serre fibrations (map with lifting property with respect to $I^ntimes 0to I^{n+1}$)


• trivial Serre fibrations (map with lifting property with respect to $S^{n-1}to D^n$)


• retracts of maps built by attaching $S^{n-1}to D^n$


• retracts of maps built by attaching $I^ntimes 0to I^{n+1}$.

There are any number of references to the Quillen model structure on Top, starting with Quillen's book; you can google for the article by Dwyer and Spalinksi on "Homotopy Theories and Model Categories".

Hovey's book "Model Categories" also does this, and gives significant attention to the cases of k-spaces and CGWH; the description of the fibrations/cofibrations is the same in these categories, but some care is needed to make sure the proof that you get a model category goes through. Hirschhorn's book may also do this, though somebody seems to have my copy, so I can't check.

I can't imagine you can define left/right/inner fibrations in Top or CGWH. Topologically, an n-simplex has no "left" or "right"; you always have a homeomorphism $Delta^nto Delta^n$ which permutes the vertices however you please.

computability theory - Why relativization can't solve NP !=P?

If this problem is really stupid, please close it. But I really wanna get some answer for it. And I learnt computational complexity by reading books only.

When I learnt to the topic of relativization and oracle machines, I read the following theorem:

There exist oracles A, B such that $P^A = NP^A$ and $P^B neq NP^B$.

And then the book said because of this, we can't solve the problem of NP = P by using relativization. But I think what it implies is that $NP neq P$. The reasoning is like this:

First of all, it is quite easy to see that:

$$A = B Leftrightarrow forall text{oracle O, }A^O = B^O$$

Though I think it is obvious, I still give a proof to it:

A simple proof of NP != P ?

And the negation of it is:

$$A neq B Leftrightarrow exists text{oracle O such that } A^O neq B^O$$

Therefore since there is an oracle B such that:

$$ NP^B neq P^B$$

we can conclude that $ NP neq P $

What's the problem with the above reasoning?

gt.geometric topology - Nice proof of the Jordan curve theorem?

Carsten Thomassen's proof is relatively simple:

Carsten Thomassen, The Jordan-Schönflies theorem and the classification of surfaces. Amer. Math. Monthly 99 (1992), no. 2, 116-130.

By the way, the Jordan Curve Theorem has a formal proof (one that can be checked by a computer):
Thomas C. Hales, The Jordan curve theorem, formally and informally. Amer. Math. Monthly 114 (2007), no. 10, 882-894.

Hales bases the formal proof on Thomassen's.

The following is a survey on the older papers on the subject:

H. Guggenheimer, The Jordan curve theorem and an unpublished manuscript by Max Dehn. Archive for History of Exact Sciences 17 (1977), 193-200.