TITLE: Nuclear Salt Water Rockets: viability and follow-up
QUESTION [4 upvotes]: This is the original paper by R. Zubrin proposing the Nuclear Salt Water Rocket design.
Basically the design is that a capillar set of pipes store a uranium salt-water solution, inside a cadmium matrix, which helps to keep it below criticality. The fluid is simply flown at high speed into a combustion chamber where it reaches criticality, heats quickly and is expelled from some unspecified nozzle.
The paper is from 1991, and i'm sure that the design has been revisited many times, so there must be good critiques and discussion of the problems that would have to be addressed, but i could not find any serious follow-up literature about the viability of the design, just, you know, the usual blog and mailing list informal ranting that you can find from a simple google search.
Have there been any simulations or attempts to evaluate the design with more detail in the literature that i'm not aware? The original paper doesn't go into much discussion about the temperatures, which i presume are quite high, and most of the informal viability analysis i've read about focus on that aspect.

REPLY [2 votes]: Although the paper was published in the early 1990s, I could identify two citations of it only:

McNutt Jr., R.L., Andrews, G.B., McAdams, J., Gold, R.E., Santo, A., Oursler, D., Heeres, K., Fraeman, M. & Williams, B. 2003, Low-cost interstellar probe, Acta Astronautica, vol. 52, no. 2-6, pp. 267-279.
R.L. McNutt Jr., G.B. Andrews, J.V. McAdams, R.E. Gold, A.G. Santo, Douglas A. Ousler, K.J. Heeres, M.E. Fraeman, B.D. Williams, A Realistic Interstellar Probe, In: Klaus Scherer, Horst Fichtner, Hans Jörg Fahr and Eckart Marsch, Editor(s), COSPAR Colloquia Series, Pergamon, 2001, vol. 11, pp. 431-434.

There are issues with citation databases and search engines, their are not conclusive, I know. But just two follow-up papers/abstracts/proceedings in 20 years is usually a sign, that there has not been any further investigation - at least in civilian public research.


TITLE: Decomposing a symmetric matrix as a sum of nilpotent matrices
QUESTION [3 upvotes]: Assume that a real-valued symmetric matrix $M$ with trace zero can be written as
$$
M = A + A^T,
$$
with $A^2=0$.

Given that $M$ is known, how (if possible) can $A$ be found?

The diagonal elements of $A$ are just half those of $M$, that is
$$
A_{ii}=M_{ii}/2.
$$
But for the non diagonal ones, considering only the decomposition above, the number of unknowns is the double of the number of equations:
$$
M_{ij}=A_{ij}+A_{ji},
$$
with $i\neq j$, implying that is not possible to get an unique solution.
Using the nilpotency property of $A$, nonlinear equations such as $M^2=AA^T+A^TA$ or $A^TMA=0$ can be generated, but I can't see how using these nonlinear relations can lead to an unique solution.

REPLY [1 votes]: I just read the user1551's solution (that is perfect as usual). We can also say
since $S,K^2$ commute, we may assume (up to an orthogonal change of basis) that $S=diag((\lambda_i)),K^2=diag((-\lambda_i^2))$.
Since $K,S^2$ commute, we may assume (up to a permutation of the new basis) that $S^2=diag(\lambda_1^2 I_{i_1},\cdots,\lambda_k^2 I_{i_k})$, where the $(\lambda_i)$ are distinct  and $K=diag(K_1,\cdots,K_k)$, where $K_i$ is skew with the correct dimension. After, it's not difficult to conclude (as user1551) that $S=Pdiag(S_1,\cdots,S_k,0_{n-2k})P^T$ where $P=[q_1,q'_1\cdots,q_k,q'_k,\cdots]$ is orthogonal and $S_i=\begin{pmatrix}\lambda_i&0\\0&-\lambda_i\end{pmatrix}$ (where the $(\lambda_i)$ are not necessarily distinct).
Then, a particular solution is $K=Pdiag(L_1,\cdots,L_k,0_{n-2k})P^T$ where $L_i=\begin{pmatrix}0&\lambda_i\\-\lambda_i&0\end{pmatrix}$.
In other words, if $S=\sum_{i=1}^k\lambda_i(q_i{q_i}^T-q'_i{q'_i}^T)$, then a particular solution is $K=\sum_{i=1}^k \lambda_i (q_i{q'_i}^T-q'_i{q_i}^T)$.


TITLE: Which is bigger, the number of neurons in the brain or the all the stars in the observable universe?
QUESTION [0 upvotes]: In other words, is 100 billion larger than $10^{22}$ or ....?
Are there also other interesting comparisons of systems with large number of members?  i.e. the sand on the beach, the atoms in the air, etc.
Update: If we have $10^{11}$ per human brain, is the sum of all neurons on the planet larger than the number of stars in the observable universe?

REPLY [1 votes]: 100 billion is the number $1$ followed by eleven zeroes: $100,000,000,000$, so it's $10^{11}$, which means $10^{22}$ is bigger because it has a bigger exponent with the same base.


TITLE: Embedding fields into the complex numbers $\mathbb{C}$.
QUESTION [7 upvotes]: Let $k$ be a field of characteristic $0$ with $\mathrm{trdeg}_\mathbb{Q}(k)$ at most the cardinality of the continuum. I want to prove the existence of a field homomorphism $k\rightarrow\mathbb{C}$. (I hope this statement is even true, I made it up on my own.)
Let $S$ be a transcedence base for $k/\mathbb{Q}$, $S'$ one for $\mathbb{C}/\mathbb{Q}$. Let $S\rightarrow S'$ be an injection. The induced map $\mathbb{Q}[S]\rightarrow\mathbb{Q}[S']\rightarrow\mathbb{Q}(S')\rightarrow\mathbb{C}$ is injective, and hence (by the mapping property of the fraction field) induces a map $\mathbb{Q}(S)\rightarrow\mathbb{C}$. But as $k/\mathbb{Q}(S)$ is algebraic, we get an induced map $k\rightarrow\mathbb{C}$.
Is this proof correct?

REPLY [2 votes]: Your proof seems all right to me.
As an aside, this method can be used to prove that $\mathbb{C}$ admits infinitely many proper subfields isomorphic to itself: simply choose any non-surjective injection $S'\to S'$, which will induce a non-surjective morphism $\mathbb{Q}(S')\to \mathbb{Q}(S')$ and thus a non-surjective morphism $\mathbb{C}\to \mathbb{C}$.


TITLE: Photon statistics of an incandescent light source
QUESTION [7 upvotes]: We usually calibrate the cameras on our microscopes by capturing 20 images of a blurry (not sharp) fluorescent particle. For each pixel in this stack of 20 images we calculate the intensity variance. By plotting the variance against the intensity we obtain the scale factor that allows us to convert the ADU (arbitrary data unit) into detected photoelectrons.
However, this approach relies on a light source with Poisson statistics.
This is applicable for fluorescence light.
But can we also use light of a halogen lamp instead of fluorescent light?
I heard that photons leaving a heated metal surface have a particular statistic. I think the argument was that they are coming out in bunches and it has something to do with the workfunction. It would be nice if someone could give a good reference on this.

The picture above shows the calibration curves for an EMCCD camera.
The left graph corresponds to the readout mode you would use for bright images and has a readnoise of 15 electrons per pixel.
The graph on the right shows the result for the low noise mode. In this mode the charge that has been accumulated under the pixels is transferred through an electron multiplication register. Impact ionization increases the number of electrons. The gain is usually up to 300x.
The inset on the lower right is how a calibration image looks like. Note that this one is fake.

REPLY [1 votes]: Incandescent (i.e. thermal) light sources exhibit noise which is beyond the standard Poissonian noise limit.  The statistics of any particular mode of a thermal light source are given by the Bose-Einstein distribution and therefore fall into the category of super-Poissonnian noise.  For light with multiple modes $N_m$, the variance in the number of photons in any given window of time is given by
$$
(\Delta n)^2=\bar{n}+\frac{\bar{n}^2}{N_m}
$$
where $\bar{n}$ is the average number of photons within the window.
However, macroscopic thermal light sources have a very large number of modes. In this case, the statistics of thermal light sources reduce to Poissonian statistics with
$$
(\Delta n)^2=\bar{n}.
$$
So, for most practical experiments, an incandescent light source can be treated as a shot noise limited source.
Source: Fox, M. (2006). Quantum Optics: An Introduction (Vol. 6). Oxford University Press. ISBN: 9780198566731 (Specifically Section 5.5.1; Relevant chapters as pdf)


TITLE: Finding the smallest $\sigma$-algebra generated by $\mathcal{U}=\{\{1,3,5\},\{1,5,7,9\}\}$
QUESTION [2 upvotes]: Let $\Omega=\{1,3,5,7,9\}$ and $\mathcal{U}=\{\{1,3,5\},\{1,5,7,9\}\}$. I am trying to find the smallest $\sigma$-algebra, $\sigma_\mathcal{U}$, generated by $\mathcal{U}$.
By considering the partition $\{3\}, \{1,5\}, \{7,9\}$, I found that
$$\sigma_\mathcal{U}=\{\emptyset,\{1,3,5,7,9\},\{3\},\{1,5\},\{7,9\},\{1,3,5\},\{1,5,7,9\},\{3,7,9\}\}.$$
I know that the $\sigma_\mathcal{U}$ generated by $\mathcal{U}$ is the power set of my partition and should therefore contain eight elements.
Is my logic correct? I'm somewhat of a novice at measure theory.

REPLY [1 votes]: Your logic is indeed correct. A $\sigma$-algebra containing $\mathcal U$ would have to contain $\mathcal V = \{\{3\}, \{1,5\}, \{7,9\}\}$ and hence the power set $\mathcal P(\mathcal V)$.
As the smallest $\sigma$-algebra containing $\mathcal U$ is the intersect of all $\sigma$-algebras containing $\mathcal U$, it is equal to $ \sigma_{\mathcal U}$.


TITLE: A height in isosceles triangle
QUESTION [2 upvotes]: isosceles triangle with height and distances
$$M \in  AB$$
$$MM_1 \bot  AC, M_1 \in AC$$
$$MM_2 \bot  BC, M_2 \in BC$$
$$AH \bot  BC, H \in BC$$
I have to show that: $$MM_1 + MM_2 = AH.$$
I've forgotten how this happens and I hope you will help me. :)

REPLY [2 votes]: Because $$\frac{AC\cdot MM_1}{2}+\frac{BC\cdot MM_2}{2}=\frac{BC\cdot AH}{2}$$ and since $AC=BC,$ we are done!