abstract algebra - Best way to teach concept of real numbers using a hands-on activity?

There is an argument to be made that the real numbers, by which I mean the completed reals, does not belong in an algebra course. In the answer above, all of the motivation comes from the real-algebraic-closure of $mathbb{Q}$ (the largest algebraic field extension not containing $sqrt{-1}$).

The reason one might want to introduce the whole continuum comes from numbers like $e$ and $pi$, which are transcendental (the fact that these numbers are transcendental is not immediate and requires a proof that I would consider past middle-school level). If you're willing to state those facts without proof, you can give a moral argument for $e$ by showing that it is the limit of the sequence $((1+1/n)^n)_{nin mathbb{N}}$, which is Cauchy, and its inclusion in the real numbers follows from the completeness of $mathbb{R}$. However, this argument may still be somewhat sophisticated for a middle-school algebra course.

Edit: On Prof. Clark's suggestion, I've copied my comments into the body text (with an additional section as well):

The notion of a sequence converging to a limiting value has a very intuitive geometric interpretation, so it wouldn't be hard to give a geometric argument (say on a graph, for example) that $e$ is a real number, since after relatively few iterations, the graph does level out. Showing that it is not the solution to a polynomial is effectively proving that it is transcendental, and I can't think of an informal argument showing this, but since this answer is community wiki, if someone has an idea, this would give a "moral" argument for the study of the "whole" continuum.

I find this approach useful because it can be introduced using the compound interest formula, which is often taught in an introductory algebra class.

We can see this as follows $$A=P(1+r/n)^{nt}.$$

Let $N:=n/r$. Then we have $$A=P(1+1/N)^{Nrt},$$
which we can rewrite as $$A=P((1+1/N)^N)^{rt}.$$

If we increase $N$ and leave $r$ fixed, this is equivalent to increasing $n$ (this is obvious because $r>0$), which amounts to increasing the number of compounding periods per unit time. Taking the limit (in some informal geometric sense), we can see that as we increase the number of compounding periods, we approach the continuously compounded interest formula $$A=Pe^{rt}.$$

oc.optimization control - Navigation solution for frictionless vehicles.

Looked around a bit and couldn't seem to find a similar question. (either that or it was worded with vocabulary above the multivariable calculus I've taken. :))

Roughly worded: I would like to develop an algorithm (either in the form of "action to take each discrete time step" or "do these actions at exactly these times") for navigating a rigid body vehicle time-optimally in a frictionless environment from point A to point B.

This specific instance happens to be a Spaceship, whom I want to get from point A to point B utilizing Forward thrusters, reverse thrusters (for braking), torque thrusters, and an omnidirectional thruster for minor position/velocity corrections. The environment is 2 dimensional, though if someone knows preexisting work in 3 dimensions I can extrapolate a simpler solution from that. I've looked around the web a bit, and was unable to find anything other than some work on steering behaviors, which always assume point particles and thus do not factor torque into the equations.

I've worked for a couple of days on this problem using standard Newtonian equations, (p(t) = at^2/2 + vt + p(0)), but the entire problem is polluted by the torque calculations, such that torque is applied to turn towards and then slow down and stop at an angle that changes based on the objects velocity and the time it would take to turn to that angle. :-S 9 pages of scribbling and several frustrated nights later, a friend Reccomend I ask here.

A generic solution (which incorporates initial velocity and angle) with a high degree of accuracy would be preferred, though in the end if I have to just "fake it" to look nice (space based RTS), that would be fine too.

nt.number theory - Existence of multi-variable p-adic L-functions

What's the "state of the art" in constructing multi-variable p-adic L-functions for number fields?

More precisely: if K is a number field, and $K_{infty} / K$ is an infinite Galois extension, unramified outside a finite set S of primes of K, containing the cyclotomic $mathbb{Z}_p$ - extension K_cyc of K and whose Galois group G is abelian with an open subgroup isomorphic to ${mathbb{Z}_p}^d$ for some $d$, then does there exist an element of the Iwasawa algebra $mathbb{Z}_p[[G]]$ (or some localisation of it) whose values at finite order characters χ of $G$ are the special values of the Hecke L-functions $L_S(chi, 1)$ (with the Euler factors at primes in S removed)?

When K is totally real Leopoldt's conjecture forces $d = 1$; but I'm interested in other cases. I know there is a 2-variable Iwasawa main conjecture for imaginary quadratic fields, which I understand has been proved by Rubin, but I'm just asking about the existence of the L-function (not about any connection to annihilators of class groups). What is known in this direction for more general $K$?

Suggestions for a good Measure Theory book

Well, I personally HATE Halmos' Measure Theory, even though an entire generation grew up on it. My favorite book on measure and integration is available in Dover paperback and is one of my all time favorite analysis texts: Angus Taylor's General Theory Of Functions And Integration. Lots of wonderful examples and GREAT exercises along with discussions of point set topology, measure theory both on $mathbb{R}$ and in abstract spaces and the Daneill approach. And all written by a master analyst with lots of references for further reading. It's one of my all time favorites and I heartily recommend it.
Folland's Real Analysis is a fine book, but it's much harder and it's really more of a general first year graduate analysis course. On the plus side, it does have many applications, including probability and harmonic analysis. It's definitely worth having, but it's going to take a lot more effort then Taylor. The IDEAL thing to do would be to work through both books simultaneously for a fantastic course in first year graduate analysis.
And please don't torture yourself with Rudin's Real And Complex Analysis. It's
sole purpose seems to be to see how much analysis can be crammed incomprehensibly into a single text. Folland is the same level and is much more accessible.
That should get you started. Good luck!

reference request - A Riemannian metric on S^2 times S^2 of nonnegative curvature that is not a product

One has to be careful with perturbations of metrics of nonnegative curvature, because that may introduce negative curvature.

Here's another approach which gives you a nonnegatively curved metric.

Start with $S^3times S^2$ with the product of round metrics. (Note that the round metric on S^3 is its biinvariant metric).

Consider the $S^1$ action on $S^3 times S^2$ where it acts as the Hopf action on $S^3$ and simultaneously rotates the $S^2$ factor $2k$ times around for some integer $k$.

To make it explicit, thinking of $S^2$ as the unit sphere in the imaginary quaternions, the action can be described as $z*(p,q) = (zp, z^k q overline{z}^k)$.

The action is clearly free and the quotient is diffeomorphic to $S^2 times S^2$. Since the circle is acting isometrically, there is an induced submersion metric on $S^2times S^2$. By the O'Neill formulas for a submersion, this metric has nonnegative curvature. When k = 0, one gets the usual product of round metrics, but when $kneq 0$ the metric is, in general, not a product.

Edit I'm now no longer certain that for $kneq 0$, the metric is not a product. I am confident that for $kneq 0$, the metric is not a product of round metrics, but I don't see any reason they can't be a product of two nonnegatively curved metrics.

However, here is an example (sorry for doubling the length of my post!): Let $G = S^3times S^3$. Let $g_0$ denote a biinvariant metric on $G$. Writing $mathfrak{g}$ for the Lie algebra of $G$, set $mathfrak{p}$ to be the Lie algebra of the diagonal $S^3$ and choose $mathfrak{q}$ to be $g_0$-orthogonal to $mathfrak{p}$.

Fix a positive real number $t$ and define a new inner product $g_1 = g_0|_{mathfrak{q}} + frac{t}{t+1}g_0|_{mathfrak{p}}$ and left translate it around $G$ to give a left invariant, right $Delta S^3$ invariant metric. Such a metric is called a Cheeger deformation of $g_0$ and it is known that $g_1$ has nonnegative sectional curvature.

Give $Gtimes G$ the product metric $g_0+g_1$ and consider the space $Delta S^3 backslash Gtimes G/ T^2$ where the $T^2$ acts on $Gtimes G$ as $(z,w)*(p,q,r,s) = (pz^{-1}, q, rw^{-1},sw^{-1})$.

(The map $Gtimes Grightarrow G$ sending $(p,q,r,s)$ to $(r^{-1}p, s^{-1}q)$, or something like it if I've made a mistake, induces a diffeomorphism between $Delta S^3backslash Gtimes G/T^2$ and $G/T^2 = S^2times S^2$, where the $G/T^2$ is referring to the action of $T^2$ on $G$ spelled out before the edit with k=1).

As above, there is an induced submersion metric of nonnegative sectional curvature by the O'Neill formulas. Finally, to prove that this is NOT a product metric, one observes that at generic points, there is a unique plane with 0 sectional curvature, while for a product metric, there should be infinitely many planes of 0 curvature.

The observation comes from

P.Müter, Krümmungserhöhende Deformationen mittels Gruppenaktionen, Ph.D. thesis, University of Münster, 1987.

modular tensor categories - Do all 3D TQFTs come from Reshetikhin-Turaev?

If you have a 3d TQFT, with no anomaly, and which goes down to points, and where things are sufficiently finite and semisimple, then I think you can show that it comes from a Turaev-Viro type construction on the 2-category Z(pt).

If you have a 3d TQFT, possibly with anomaly, which goes down to circles, and where things are sufficiently finite and semisimple, then I agree with Noah: Z(S^1) is a MTC and the RT construction on the MTC reproduces the TQFT.

Relating these two statements, TV(C) = RT(double(C)), where C is a 2-cat, double(C) is the Drinfeld double (or maybe center), TV is the Turaev-Viro construction, and RT is the Reshetikhin-Turaev construction.

it.information theory - Question regarding divergence

The inequality $D(P'|Q') ge D(P^star| Q^star)$ does not need to hold.

Here is an example.

Let $A$ be the set ${1,2,3,...,n}$. Let $E$ be the set of measures $P$ on $A$ such that $P({1}) = 0$. Projecting a measure $P$ on $E$ using $D$ is equivalent to conditioning $P$ on $ A- {1}$. Choose $P'$ and $Q'$ such that they both put equal and nonzero mass on ${1}$. By direct computation one sees: $D(P^star| Q^star) = frac{1}{1-P'({1})} D(P'|Q') > D(P' | Q')$.

The details of the above computation are as follows.

For ease of notation set $n=3$. Let $E$ be the set of measures $P$ with $P({1}) =epsilon$; to obtain the example above, one sets $epsilon = 0$. Let us parametrize the measures on ${1,2,3}$ as follows: $P({1}) = p_1$, $P({2}) =p_2$ and $P({3}) = 1-p_1 -p_2$. Our problem is:
$$
inf_{ Q in E}left[ p_1 log frac{p_1}{q_1} + p_2 log frac{p_2}{q_2} + (1-p_1 -p_2) logfrac{ 1- p_1 - p_2}{ 1- q_1 - q_2 } right].
$$
Let $F$ denote the expression after the $inf$.
$F$ is strictly convex in $Q$ and therefore will have a unique optimizer. In the above coordinates, the normal to $E$ is the vector $(1,0)$. Then
$$
frac{partial F} {partial q_1} = -frac{p_1}{q_1} + frac{1-p_1-p_2}{1-q_1-q_2} = lambda
$$
and
$$
frac{partial F} {partial q_2} = -frac{p_2}{q_2} + frac{1-p_1-p_2}{1-q_1-q_2} = 0.
$$
We have the constraint that $Qin E$, i.e., $q_1 =epsilon$.
From the last two equalities one infers:
$$
q_2 = frac{(1-epsilon) p_2}{ 1-p_1}.
$$

Going back to the coordinates $(p_1,p_2,p_3)$ to denote a measure on ${1,2,3}$,
projecting a measure on $E$ using $D$ corresponds to the following map:
$$
(p_1,p_2,p_3) rightarrow left(epsilon, (1-epsilon)frac{p_2}{p_2+p_3}, (1-epsilon)frac{p_3}{p_2 + p_3}right).
$$
For $epsilon =0$, this is the same as conditioning $P$ on ${2,3}$.

One obtains the expression for the relative entropy given above by directly computing it using this formula for the projections.