orbit - Derivation of the formula for longitude of ascending node for a satellite

I've been looking into the document IS-GPS-200H to understand how to calculate satellite location in the ECEF coordinate.

I am having problem understanding the formula to derive $Omega$, the longitude of the ascending node (LAN) relative to Greenwich at given time $t$:

$$
Omega = Omega_0 + left( dot{Omega - w} right)times t_k - w times t_{oe}
$$

where:
$$
Omega_0: text{LAN relative to vernal equinox, at the beginning of the week}\
dot{Omega}: text{angular velocity for LAN, relative to vernal equinox.}\
w: text{angular velocity of earth, relative to vernal equinox.}\
t_k: t - t_{oe}\
t_{oe}: text{ephemeris reference epoch}\
$$
(and let us denote the beginning of the week as $t_0$ for brevity).

But if I try to work out this from scratch:

1. At $t = t_0$, LAN was $Omega_0$. But since what we really need is the difference of LAN and longitude of Greenwich, we also need to know $w_0$, the initial longitude of Greenwich at $t = t_0$.
   $$
   Omega(t = t_0) = Omega_0 - w_0
   $$


2. At the ephemeris reference epoch time $t = t_{oe}$, LAN and the earth both rotate with their respective angular momentum and hence:
   $$
   Omega(t = toe) = Omega_0 + w_0 + (dot{Omega} - w) times t_{oe}
   $$


3. As time varies from $toe$ to $t$, again LAN and the earth both rotate with their own respective angular momentum and hence
   $$
   Omega(t) = Omega_0 + w_0 + (dot{Omega} - w) times t_{oe} + (dot{Omega} - w) times t_k
   $$
   which obviously differs from the right formula by $w_0 + dot{Omega} times t_{oe}$.

My question is where am I making mistakes/misunderstanding the eqution?
Explain also why we don't need to know $w_0$ or equivalent input, that would be greatly appreciated.

real analysis - How can I measure the Morse index in infinite dimensions?

It seems that simply one can't measure it. Here below is briefly described
an example of a nondegenerate indefinite inner product space having
no cardinal-valued Morse index.

Consider $(ell^{2},<.,.>)$ as
naturally embedded (via Riesz) into its (huge) algebraic dual, say
$mathcal{A}$, let $mathcal{F}$ be the real vector space of all
finitely supported functions from $mathbb{R}$ to $mathbb{ell^{textrm{2}}}$,
and put $V$:= $mathcal{Atimes F}$. Next, write $mathcal{A}$
as a direct sum
$mathcal{A}=ell^2oplus{E}$ (hence
$dim E=2^{c}$),
let $pi:mathcal{A}$ $to$ ${E}$
be the attached algebraic projection, and let $[.,.]$
be a scalar product on $Etimes E$. If $u=$($varphi,f)$, and $v=$($psi,g)$
are in $V$, then define the bilinear symmetric pairing

$a(u,v)$:= - $[pi varphi,pi psi]$ + $varphi(sum_{tinmathbb{R}}g(t))+psi(sum_{tinmathbb{R}}f(t))$ + < $ sum_{tinmathbb{R}} f(t),sum_{tinmathbb{R}}g(t))> $
- $sum_{tinmathbb{R}}$ <
$f(t),g(t)$> .

Define also the subspace $W$ of $V$ by $W$ := {$ {{ (varphi,f)|:varphi=-sum_{tinmathbb{R}}f(t)} }$}.
Then is not hard to see that:

1) $W$ is negative definite (w.r.t. $a$), and $W^{bot}$ = { 0 } , hence $a$ is non-degenerate.

2) [Using the C-B-S inequality and Riesz] Any maximal negative
definite subspace $mathcal{N}$ of $V$ containing $W$ is a linear
subspace of $ell^{2}timesmathcal{F}$, hence $dim$ $mathcal{N}$
= $c$.

3) Any maximal negative definite subspace $mathcal{M}$ of $V$ containing
$Etimes ${ 0 }$ $ has $dimmathcal{M}$ $>$ $c$.

Consequently, $mathcal{M}$ and $mathcal{N}$ are not isomorphic
as real vector spaces.

the sun - Cooling of stars

In our current universe white dwarfs are the first ones that should cool, because they are already "cold" (not producing anything anymore, just radiating heat) remnants of a former star. The time for this to happen is disputed (10^15 or 10^37 years), but is far bigger than the age of universe, so nobody expects to find one "cooled star" yet. See this article for details on "black dwarfs" as they are called.

In your sudden stop hypothetical universe probably the first one to cool would be the smallest and coolest stars, a M9V red dwarf (with 7.5% solar mass, 8% solar radius, 2,300K temperature). Note that brown dwarfs are categorized as substellar objects, so they shouldn't be considered. I don't think the poorly understood "Mpemba effect" can be yet applied to stars.

nt.number theory - Forms over finite fields and Chevalley's theorem

Perhaps it is obvious to most readers, but about a year ago I spent several days trying to determine for which pairs (d,n) there existed an anisotropic degree d form in n variables over a finite field $mathbb{F}_q$. The question was motivated by Exercise 10.16 in Ireland and Rosen's classic number theory text: "Show by explicit calculation that every cubic form in two variables over $mathbb{F}_2$ has a nontrivial zero."

As many students have discovered over the years, this is false: e.g. take

$f(x_1,x_2) = x_1^3 + x_1^2 x_2 + x_2^3$.

I knew about the existence and anisotropy of norm hypersurfaces for all $n = d$. But what about $n < d$? I confess that I spent some time proving this result in several special cases and even dragged a postdoc into it. Here is a copy of the sheepish email I sent out (in particular to Michael Rosen) later on:

If K is a field, and f(x_1,...,x_n) is an anisotropic form of degree d in n variables, then f(x_1,...,x_{n-1},0) is an anisotropic
form of degree d in n-1 variables.

So let K be any field which admits field extensions of every positive degree d. Then for all d there is an anisotropic norm form
N in d variables of degree d. For any n < d, setting (d-n) of the variables equal to 0 gives an anisotropic form of degree d in
n variables. In particular, this proves "the converse of Chevalley-Warning".

So, not so fascinating after all, then.

I think it is still nontrivial to ask what happens if the hypersurface f is required to be geometrically irreducible. For instance, despite the fact that (q,3,3) is anisotropic, every geometrically irreducible cubic curve over a finite field has a rational point.

AS's question about classifying anisotropic hypersurfaces with $d = n$ is interesting. It may also be interesting to look at the case $d < n$. It is certainly not clear to me that all such anistropic hypersurfaces come from intersecting a norm hypersurface of larger dimension with a linear subspace.

I also want to add that the following generalization seemed less trivial to me (and I still don't know the answer): Chevalley-Warning is also true for sytems of polynomial equations $f_1(x_1,ldots,x_n) = ldots = f_r(x_1,ldots,x_n)$ so long as the sum of the degrees of the $f_i$'s is strictly less than $n$. What kind of counterexamples can we construct here when $d = d_1 + ldots + d_r geq n$?

gt.geometric topology - Does a triangulation without fixed simplex property always exist?

EDITED. The arugment related to Mostov rigidity is completed according to a nice suggestion of Tom Church

The answer to the first question is no. There exsit manifolds of dimension 3 such that every simlicial map of the manfiold to itself (for any simplicial decomposition) has a fixed point (and hence a fixed simplex). At the same time every 3-mafiold adimits a smooth self-map without fixed points.

Namely, take $M^3$ with vanishing first and second homology ($H_1(M^3,R)=H_2(M^3,R)=0$) and that is a hyperbolic 3-manifold. Moreover take such $M^3$ that does not have isometries. Existence of such manifolds is a standard result of 3-dimensional hyperbolic geometry. Let us prove that every such manifold gives us an example.

Proof. All compact 3-manifolds have zero Euler characteristics, so on $M^3$ there is a non-vanishing vector field $v$. Take the flow $F_t$ generated by $v$ in small time $t$. This will give us a family of diffeo $F_t$ of $M^3$ that don't have fixed points for small $t$. So $M^3$ in not FPP.

Now, let us show that $M^3$ has FSP. Take any simplicial decomposition of $M^3$. First we state a simple lemma (without a a proof)

Lemma. Consider a simplicial decomposition of a compact orienable manifold. Suppose we have a simpicial map from it to itself, that send simplexes of highest dimensions to simplexes of higher dimentions (i.e. don't collapse them) and don't indentify them. Then this is an automorphism of finite order.

Corollary. Every non-identical simplicial map $phi$ from $M^3$ to itself either collapses a simplex of dimension 3 or identifies two such simplexes. In paricular the generator of $H^3(M^3,mathbb Z)$ is sent to zero by this map.

This corollary together with Lefshetz fixed point theorem implies immediately that $phi$
has a fixed point, and so it proves FSP for $M^3$ (we use that $H_1(M^3)=H_2(M^3)=0$).

Proof of corollary. If $phi$ does not collapse 3 simplexes of $M^3$ or identify them, then it is a homeomorphism of $M^3$ of finite order (Lemma above). From Mostov rigidity it follows that this automorphism is homotopic to the identity. In order to show that it IS in fact the identity we need to use a more involved statement suggseted by Tom Church below. Namely, a paritial case of Proposition 1.1 in http://www.math.uchicago.edu/~farb/papers/hidden.pdf
says that for a hyperbolic 3-manifold the group of isometries of any Riemanninan metric on it is isomorphic to a subgroup of the group of hyperbolic isometries. By our choice the group of hyperbolic isometries of $M^3$ is trivial. It is clear that $phi$ preserves a Riemannian metric on $M^3$. So by Prop 1.1 it is the identity.

From this it immediately follows that $phi$ sends $H^3(M^3, Z)$ to zero (since the volume is contacted). End of proof.

Complex orientations on homotopy

I am wondering if there is a more "geometric" formulation of complex orientations for cohomology theories than just a computation of $E^*mathbb{C}$P$^{infty}$ or a statement about Thom classes. It seems that later in Hopkins notes he says that the complex orientations of E are in one to one correspondence with multiplicative maps $MU rightarrow E$, is there a treatment that starts with this perspective? How do the complex orientations of a spectrum E help one compute the homotopy of $E$, or the $E$-(co)homology of MU? Further, what other kinds of orientations could we think about, are there interesting $ko$ or $KO$ orienations? how much of these $E$-orientations of X is detected by E-cohomology of X?

I do have some of the key references already in my library, for example the notes of Hopkins from '99, Rezk's 512 notes, Ravenel, and Lurie's recent course notes. If there are other references that would be great. I am secretly hoping to get some insight from some of the experts. (I guess I should really also go through Tyler's abelian varieties paper)

(sorry for the on and off texing but the preview is giving me weird feedback.)

EDIT: I eventually found the type of answer i was looking for in some notes of Mark Behrens on a course he taught. This answer is that a ring spectrum $R$ is complex orientable is there is a map of ring spectra $MU to R$. This also appears in COCTALOS by Hopkins but neither source takes this as the more fundamental concept. Anyway, the below answer is more interesting geometrically.

ac.commutative algebra - reduced ⊗ reduced = reduced; what about connected?

Several questions actually.

All rings and algebras are supposed to be commutative and with $1$ here.

(1) Let $k$ be a field, and let $A$ and $B$ be two $k$-algebras. I need a proof that if $A$ and $B$ are reduced (i. e., the only nilpotents are $0$) and $mathrm{char}k=0$, then $Aotimes_k B$ is reduced as well.

The condition $mathrm{char}k=0$ can be replaced by "$k$ is perfect", but I already know a proof for the $mathrm{char}k>0$ case (the main idea is that every nilpotent $x$ satisfies $x^{p^n}=0$ for some $n$, where $p=mathrm{char}k$), so I am only interested in the $mathrm{char}k=0$ case.

Please don't use too much algebraic geometry - what I am looking for is a constructive proof, and while most ZFC proofs can be made constructive using Coquand's dynamic techniques, the more complicated and geometric the proof, the more work this will mean.

BTW the reason why I am so sure the above holds is that some algebraist I have spoken with has told me that he has a proof using minimal prime ideals, but I haven't ever seen him afterwards.

Ah, and I know that this is proven in Milne's Algebraic Geometry for the case $k$ algebraically closed.

(2) What if $k$ is not a field anymore, but a ring with certain properties? $mathbb{Z}$, for instance? Can we still say something? (Probably only to be thought about once (1) is solved.)

(3) Now assume that $k$ is algebraically closed. Can we replace reduced by connected (which means that the only idempotents are $0$ and $1$, or, equivalently, that the spectre of the ring is connected)? In fact, this even seems easier due to the geometric definition of connectedness, but I don't know the relation between $mathrm{Spec}left(Aotimes_k Bright)$ and $mathrm{Spec}A$ and $mathrm{Spec}B$. (I know that $mathrm{Spm}left(Aotimes_k Bright)=mathrm{Spm}Atimesmathrm{Spm}B$ however, but this doesn't help me.)

PS. All algebras are finitely generated if necessary.