Question:
I'm currently looking into parallel methods for ODE integration. There is a lot of new and old literature out there describing a wide range of approaches, but I haven't found any recent surveys or overview articles describing the topic in general.
There's the book by Burrage [1], but it's almost 20 years old and hence does not cover many of the more modern ideas like the parareal algorithm.
[1] K. Burrage, Parallel and Sequential Methods for Ordinary Differential Equations, Clarendon Press, Oxford, 1995

I'm not aware of any recent overview articles, but I am actively involved in the development of the PFASST algorithm so can share some thoughts.
There are three broad classes of time-parallel techniques that I am aware of:

across the method — independent stages of RK or extrapolation integrators can be evaluated in parallel; see also the RIDC (revisionist integral deferred correction algorithm)
across the problem — waveform relaxation
across the time-domain — Parareal; PITA (parallel in time algorithm); and PFASST (parallel full approximation scheme in space and time).

Methods that parallelize across the method usually perform very close to spec but don't scale beyond a handful of (time) processors.  Typically they are relatively easier to implement than other methods and are a good if you have a few extra cores lying around and are looking for predictable and modest speedups.
Methods that parallelize across the time domain include Parareal, PITA, PFASST.  These methods are all iterative and are comprised of inexpensive (but inaccurate) "coarse" propagators and expensive (but accurate) "fine" propagators.  They achieve parallel efficiency by iteratively evaluating the fine propagator in parallel to improve a serial solution obtained using the coarse propagator.
The Parareal and PITA algorithms suffer from a rather unfortunate upper bound on their parallel efficiency $E$: $E < 1/K$ where $K$ is the number of iterations required to obtain convergence throughout the domain.  For example, if your Parareal implementation required 10 iterations to converge and you are using 100 (time) processors, the largest speedup you could hope for would be 10x.  The PFASST algorithm relaxes this upper bound by hybridizing the time-parallel iterations with the iterations of the Spectral Deferred Correction time-stepping method and incorporating Full Approximation Scheme corrections to a hierarchy of space/time discretizations.
Lots of games can be played with all of these methods to try and speed them up, and it seems as though the performance of these across-the-domain techniques depends on what problem you are solving and which techniques are available for speeding up the coarse propagator (coarsened grids, coarsened operators, coarsened physics etc.).
Some references (see also references listed in the papers):

This paper demonstrates how various methods can be parallelised across the method:  A theoretical comparison of high order explicit Runge-Kutta, extrapolation, and deferred correction methods; Ketcheson and Waheed.
This paper also shows a nice way of parallelizing across the method, and introduces the RIDC algorithm: Parallel high-order integrators; Christlieb, MacDonald, Ong.
This paper introduces the PITA algorithm: A Time-Parallel Implicit Method for Accelerating the Solution of Nonlinear Structural Dynamics Problems; Cortial and Farhat.
There are lots of papers on Parareal (just Google it).
Here is a paper on the Nievergelt method: A minimal communication approach to parallel time integration; Barker.
This paper introduces PFASST: Toward an efficient parallel in time method for partial differential equations; Emmett and Minion;
This papers describes a neat application of PFASST: A massively space-time parallel N-body solver; Speck, Ruprecht, Krause, Emmett, Minion, Windel, Gibbon.

I have written two implementations of PFASST that are available on the 'net: PyPFASST and libpfasst.


Question:
I've heard that the wonderful smell of a fresh rain is actually chemicals  released from the trees and grass and other plants.

What is the process that allows these chemicals to be released?
What are the chemicals that create that smell?
How is it advantageous for the plant to release the chemicals rather than hold onto them?

That molecule is called Geosmin. It is mainly produced 1 by Actinomycetes such as Streptomyces which are filamentous bacteria that live in soil. Other organisms also produce geosmin:

Cyanobacteria
Certain fungi
An amoeba called Vanella
A liverwort

It is an intracellular metabolite and cell damage is the primary reason attributed to its release. However oxidant exposure and transmembrane pressure also causes geosmin release in cyanobacteria. It seems that the release is triggered by some kind of stress.
I am not quite sure about their advantage to the host species.

1 or perhaps the most well-studied in


Question:
The 2023 Nobel Prize in Physics was announced today, and it was awarded to Pierre Agostini, Ferenc Krausz and Anne L’Huillier, for

“experimental methods that generate attosecond pulses of light for the study of electron dynamics in matter”.

The documents released by the Nobel Foundation along with the announcement (the popular science background  and the more detailed scientific background do a good job of explaining the basics, but:
why are attosecond pulses exciting, and what can you do with them that you cannot do in any other way?

What's the big deal?
When quantum mechanics was being discovered and formalized, in the 1920s and 1930s, our view of physics was deeply rooted in the macroscopic world. We understood that microscopic entities like atoms and molecules existed, and we arrived reasonably quickly at a good understanding of their basic structure, but for a very long time they were very remote objects, whose behaviour was so abstract and disconnected from our everyday experience that it was even kind of pointless to really interrogate it.
So, as an example, if you heated up a vial with sodium, then the gas sample in the vial might emit or absorb light at a particular wavelength, and if you worked out the quantum-mechanical maths then you could predict what those wavelengths should be, in terms of quantum jumps between energy levels $-$ but, could you really say what each individual atom in the gas was doing? How could you be sure that those "quantum jumps" were even real, if you only ever had access to the macroscopic gas sample, and never to any individual atom?
Moreover, that same quantum-mechanical maths predicts that the dynamics in an atom will be blazingly fast, and indeed many orders of magnitude faster than any experimental techniques available at the time. So, could you really talk about the electrons "moving"? This was aggravated by the fact that the particular choices of quantum-mechanical maths that made sense for this type of experiment talked much more about "orbitals" and "energy levels", with those mysterious quantum jumps to link them $-$ so maybe it makes more sense to treat those orbitals and energy levels as the "real" objects, and disregard the notion that there is any movement in the micro-world?
However, we live in a very different world now. Not only do we have tools like scanning electron microscopy that allow us to observe the atoms that make up a metal surface, we are also now able to hold and control a single atom with delicate electrical "tweezers", which then allows us to interrogate it directly. And when we look, much to our chagrin, that individual atom is indeed performing the fabled quantum jumps. More generally, since the turn of the millenium the name of the game (and indeed the routine) has been the observation and control of individual quantum systems.
A similar story holds for the dynamics of microscopic systems, and for our ability to observe them directly. The discoveries of the laser, and then Q-switching and mode locking allowed laser pulses to get pretty fast, first faster than a microsecond ($10^{-6}\:\rm s$) and then faster than a nanosecond ($10^{-9}\:\rm s$), respectively, and work in the 1970s and 1980s allowed us to create pulses as short as a picosecond ($10^{-12}\:\rm s$) and shorter. If you really push a laser system, using technology known as Chirped Pulse Amplification (which I wrote about previously here when it won its Nobel Prize), you can get down to a few femtoseconds ($10^{-15}\:\rm s$). This is very fast for a pulse of light, and it is actually so fast that the pulse of light is no longer a periodic electric-field oscillation, and instead it lasts only for a few cycles. But it is still not fast enough.
Why? Because atoms are even faster.
To understand how fast atoms are, it is enough to do some basic dimensional analysis. The dynamics of the electrons inside an atom are governed by the Schrödinger equation,
$$
i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m_e}\nabla^2\psi -\frac{e^2}{r}\psi,
$$
and this has only three core constants involved: the reduced Planck constant, $\hbar$, the electron's mass, $m_e$, and the electron charge $e$. (Or, if you work in SI units, the Coulomb constant $e^2/4\pi\epsilon_0$.) And, as it turns out, those constants can be combined into a unique timescale, known as the atomic unit of time,
$$
t_\mathrm{a.u.} = \frac{\hbar^3}{m_ee^4} = 24\:\rm as,
$$
which is measured in attoseconds: $1\:\rm as = 10^{-18}\:\rm s$. As a rule of thumb, the dynamics might be somewhat faster, or somewhat slower, depending on the atom and the conditions, but it will generally stick to that rough order of magnitude.
And that means, in turn, that those dynamics might seem completely out of reach, because the period of oscillation of optical light is still rather slower than this. (For light of wavelength $550\:\rm nm$, the period is of about $2\:\rm fs$.) So that might make you think that a direct observation of something as fast as atomic dynamics must be out of reach.
So how do you make an attosecond pulse?
This is the real breakthrough that is being rewarded with today's announcement. Our workhorse is a process known as high-harmonic generation, which uses a highly nonlinear interaction between a gas and a pulse of laser light to generate sharp bursts of radiation $-$ the famed attosecond pulses $-$ which can be much shorter than the period of the pulse that drives the process, and can be as short as a few dozen attoseconds.
From an experimental perspective, what you have to do is simply start with a laser pulse with a fairly long wavelength and slow period (usually in the near-infrared), shine it into a gas cell, and make sure that the pulse is intense. How intense? very intense. Intense enough to directly yank electrons out of the gas atoms and shake them about once they're free. (And, indeed, intense enough that the pulse will burn out the laser amplifier if you let it, as explained in the thread about Chirped Pulse Amplification.)
This was done in 1987 by a team led by Anne L'Huillier, and the surprising observation was that the gas emitted harmonics, i.e., additional wavelengths of light at sub-multiples of the original driving wavelength. This was known to occur (second-harmonic generation is almost as old as the laser itself), but L'Huillier and colleagues discovered that if the driving pulse is intense enough, it can generate all sorts of harmonics at crazy high orders, with a very slow decline in emission as the order increases. (Up until the signal reaches a cutoff and decays exponentially, of course.)
What's going on? the basic physics was worked out by Paul Corkum (who was very high in the shortlist for getting the Nobel Prize if it ever did get awarded to attosecond science), and it is known as the three-step model.

Image taken from D. Villeneuve, Contemp. Phys. 59, 47 (2018)
In essence, the laser can be thought of as a constant force (and therefore a linear ramp in potential energy) which slowly oscillates and tilts around the potential well that the atomic electron sits in. At the maximum of field intensity, this is enough to yank the electron away (though more on this later), at which point the electron will freely oscillate in the field, gaining energy from the electric field of the light ... up until it crashes into the potential well that it just left, at which point it can recombine back with the ion it left behind, and emit its (now considerable) kinetic energy as a sharp burst of radiation.
The coolest things about this collision are that it is very energetic (so the burst of radiation has a high photon energy, and therefore very high frequencies), and that it is very short (it is over in a flash), and it is this short duration that means that the pulses of radiation emitted will be extremely short.
The other parts of the Nobel Prize are being awarded for the explicit creation and detection of these sharp bursts of light.

One thing that happens quite often is that (because the driving pulse is long, and has many periods where the three-step model can happen), the emission is often in the shape of an attosecond-pulse train, sometimes with several dozen sharp bursts following each other in quick succession. Pierre Agostini was the first to directly observe the duration of the bursts within such a train, using a technique known as RABBITT (attoscience has since acquired an "animal theme" for our acronyms), and his group was able to show that they were indeed very short, down to as little as $250\:\rm as$.

Alternatively, you might want to invest some (considerable) time and energy into finding a way to "gate" the emission, so that there is only one burst in the train. (For a fresh-off-the-press review of different ways to "gate" the emission see e.g. this preprint.) This gating was achieved by Ferenc Krausz's group, who were able to isolate a single pulse with a duration of $650\:\rm as$.

Of course, the field has continued to innovate, making things more reliable and robust, but also pushing down the shortest duration achievable. If I understand correctly, the current record is $43\:\rm as$, which is very, very short.
(Another cool record is how high you can push the order of nonlinearity in the process, for which, if I understand correctly, a 2012 classic still holds the prize with a minimal order of nonlinearity of 4,500.)
What can you use these pulses for?
We're now down to the most interesting part. Say that you have made one of these attosecond pulses. What can you do with it?
Directly observing the wave oscillations of light
For me, the most exciting application from the "classic" experiments in attoscience is a setup known as "attosecond streaking".
The basic idea is to take a short attosecond pulse, and overlap it, inside a gas sample, with a slower pulse of infrared light.

The short pulse has enough photon energy to ionize the gas, and we know that this must happen within the duration of the short pulse. After this ionization, the slower infrared pulse has an electric field which oscillates, and this will impact the final energy and momentum of the electron, but the extent of this effect will depend on when the electron is released, so by changing the time delay between the two, we can scan against this electric field.
$\qquad$

The end result, shown above, is a direct observation of the oscillations of the electric field (raw data on the left, and reconstructed electric field on the right), which is a task that was considered somewhere between impossible and unthinkable for many, many decades after we understood that light was a wave (but only had indirect ways to prove it).
I've discussed this experiment previously here. For more details (and the source of the figures), see the landmark publication:

Direct measurement of light waves. E. Goulielmakis et al. Science 305, 1267 (2004); author eprint.

Directly observing electron motion in real time
Similarly to observing the motion of the electric field of light, we can also observe the motion of electrons inside an atom. I have discussed this in detail in Is there oscillating charge in a hydrogen atom?, but the short story is that if you prepare an electron in a quantum superposition of two different energy levels, such as the combination
$$
\psi = \psi_{1s} + \psi_{2p}
$$
of the hydrogen $1s$ and $2p$ levels, the charge density in the atom will oscillate over time:

Mathematica source through Import["
This is not a hypothetical or purely theoretical construct, and we can directly observe it in experiment. The first landmark test, reported in

Real-time observation of valence electron motion. E. Goulielmakis et al. Nature 466, 739 (2010).

was able to show a clear oscillation in how much a short pulse was absorbed by an oscillating charge distribution caused by spin-orbit interactions (where different parts of the oscillations correspond to different orientations of the charge density, and therefore to different absorption profiles), showing a clear corresponding oscillation in the absorbance:

Similarly, a much-beloved example is the observation of charge oscillation dynamics in a bio-relevant molecule, phenylalanine, which was reported in

Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses. F. Calegari et al., Science 346, 336 (2014),

and where the ionization of the molecule by a (relatively) short laser pulse (in the near-infrared) is then probed by a (very) short attosecond burst. The resulting dynamics inside the molecule are fairly complicated,

but they lead to clear oscillations in the signal (with the graph below showing the overall decay, and the oscillations on top of an exponential background) at a very short timescale that is only observable thanks to the availability of attosecond pulses.

Watching quantum interference build up in real time
I will do one more direct-timing-of-observation, because I think they're really cool. This one is again about a quantum superposition, but one that happens with a free electron. When you ionize an atom, the electron gets released, and one photon gets absorbed. And, more importantly, the details of the energy states that the electron gets released into will be imprinted into the absorbance spectrum of the light.
In particular, it is possible to tune things so that you are ionizing close to a resonance: the electron can either ionize directly, or it can spend some time in a highly-excited autoionizing state (also explained here and here) that will fall apart after some time. The end result is that the electron will go into a superposition of both pathways, which will interfere in its spectrum and cause a wonky, nontrivial shape in the absorption spectrum.
However, if we have short pulses of radiation, we are able to control how long we let the electron to sit in that autoionizing state, before we come in with a second pulse of light to disrupt it, and kill the interference:

And indeed, when we do this, the build-up of the line and the development of the interference features (and particularly that sharp dip on the right-hand side of the line) is very clearly seen in experiment:

And, just to add some more pretty pictures, here it is all stacked together, on the left-hand figure, and on the right a similar experiment showing very clearly the destructive interference building up over time:

For more details, and the sources of the figures, see

Observing the ultrafast buildup of a Fano resonance in the time domain. A. Kaldun et al. Science 354, 738 (2016)

and

Attosecond dynamics through a Fano resonance: Monitoring the birth of a photoelectron. V Gruson et al. Science 354, 734 (2016)

Moreover, it is also possible to use these types of resonances to enhance high-harmonic generation itself, in a process known as resonant HHG. For a nice review written by a colleague (in a paper I coauthored) see Eur. Phys. J D 75, 209 (2021) (arXiv:2101.09335).
Further reading
Long as this post is, I have only just scratched the surface. Here are some additional places to read more about the field:

Attosecond science. D. Villeneuve, Contemp. Phys. 59, 47 (2018) (author eprint)

Attosecond science. P.B. Corkum & F. Krausz. Nature Physics 3, 381 (2007) (author eprint)

The physics of attosecond light pulses. P. Agostini & L.F. DiMauro. Reports on Progress in Physics 67, 813 (2004) (author eprint)

Attosecond electromagnetic pulses: generation, measurement, and application. Attosecond metrology and spectroscopy. M.Yu. Ryabikin et al. *Physics-Uspekhi 66, 360 (2023)

Shining the shortest flashes of light on the secret life of electrons. M. Khokhlova, E. Pisanty & A. Zair. Advanced Photonics 5, 060501 (2023)


Question:
I've noticed lately that a lot of people are developing tensor equivalents of many methods (tensor factorization, tensor kernels, tensors for topic modeling, etc) I'm wondering, why is the world suddenly fascinated with tensors? Are there recent papers/ standard results that are particularly surprising, that brought about this? Is it computationally a lot cheaper than previously suspected?
I'm not being glib, I sincerely am interested, and if there are any pointers to papers about this, I'd love to read them.

This is not an answer to your question, but an extended comment on the issue that has been raised here in comments by different people, namely: are machine learning "tensors" the same thing as tensors in mathematics?
Now, according to the Cichoki 2014, Era of Big Data Processing: A New Approach via Tensor Networks and Tensor Decompositions, and Cichoki et al. 2014, Tensor Decompositions for Signal Processing Applications,

A higher-order tensor can be interpreted as a multiway
  array, [...]
A tensor can be thought of as a multi-index numerical array, [...]
Tensors (i.e., multi-way arrays) [...]

So in machine learning / data processing a tensor appears to be simply defined as a multidimensional numerical array. An example of such a 3D tensor would be $1000$ video frames of $640\times 480$ size. A usual $n\times p$ data matrix is an example of a 2D tensor according to this definition.
This is not how tensors are defined in mathematics and physics!
A tensor can be defined as a multidimensional array obeying certain transformation laws under the change of coordinates (see Wikipedia or the first sentence in MathWorld article). A better but equivalent definition (see Wikipedia) says that a tensor on vector space $V$ is an element of $V\otimes\ldots\otimes V^*$. Note that this means that, when represented as multidimensional arrays, tensors are of size $p\times p$ or $p\times p\times p$ etc., where $p$ is the dimensionality of $V$.
All tensors well-known in physics are like that: inertia tensor in mechanics is $3\times 3$, electromagnetic tensor in special relativity is $4\times 4$, Riemann curvature tensor in general relativity  is $4\times 4\times 4\times 4$. Curvature and electromagnetic tensors are actually tensor fields, which are sections of tensor bundles (see e.g. here but it gets technical), but all of that is defined over a vector space $V$.
Of course one can construct a tensor product $V\otimes W$ of an $p$-dimensional $V$ and $q$-dimensional $W$ but its elements are usually not called "tensors", as stated e.g. here on Wikipedia:

In principle, one could define a "tensor" simply to be an element of any tensor product. However, the mathematics literature usually reserves the term tensor for an element of a tensor product of a single vector space $V$ and its dual, as above.

One example of a real tensor in statistics would be a covariance matrix. It is $p\times p$ and transforms in a particular way when the coordinate system in the $p$-dimensional feature space $V$ is changed. It is a tensor. But a $n\times p$ data matrix $X$ is not.
But can we at least think of $X$ as an element of tensor product $W\otimes V$, where $W$ is $n$-dimensional and $V$ is $p$-dimensional? For concreteness, let rows in $X$ correspond to people (subjects) and columns to some measurements (features). A change of coordinates in $V$ corresponds to linear transformation of features, and this is done in statistics all the time (think of PCA). But a change of coordinates in $W$ does not seem to correspond to anything meaningful (and I urge anybody who has a counter-example to let me know in the comments). So it does not seem that there is anything gained by considering $X$ as an element of $W\otimes V$.
And indeed, the common notation is to write $X\in\mathbb R^{n\times p}$, where $R^{n\times p}$ is a set of all $n\times p$ matrices (which, by the way, are defined as rectangular arrays of numbers, without any assumed transformation properties).
My conclusion is: (a) machine learning tensors are not math/physics tensors, and (b) it is mostly not useful to see them as elements of tensor products either.
Instead, they are multidimensional generalizations of matrices. Unfortunately, there is no established mathematical term for that, so it seems that this new meaning of "tensor" is now here to stay.


Question:
My biology teachers never explained why animals need to breathe oxygen, just that we organisms die if we don't get oxygen for too long. Maybe one of them happened to mention that its used to make ATP. Now in my AP Biology class we finally learned the specifics of how oxygen is used in the electron transport chain due to its high electronegativity. But I assume this probably isn't the only reason we need oxygen.
What other purposes does the oxygen we take in through respiration serve? Does oxygen deprivation result in death just due to the halting of ATP production, or is there some other reason as well? What percentage of the oxygen we take in through respiration is expelled later through the breath as carbon dioxide?

Superoxide, O2− is created by the immune system in phagocytes (including neutrophils, monocytes, macrophages, dendritic cells, and mast cells) which use NADPH oxidase to produce it from O2 for use against invading microorganisms. However, under normal conditions, the mitochondrial electron transport chain is a major source of O2−, converting up to perhaps 5% of O2 to superoxide. [1]
As a side note, there are two sides to this coin.  While this is a useful tool against microorganisms, the formation of the reactive oxygen species has been incriminated in autoimmune reactions and diabetes (type 1). [2]

[1] Packer L, Ed. Methods in Enzymology, Volume 349. San Diego, Calif: Academic Press; 2002
[2] Thayer TC, Delano M, et al. (2011) Superoxide production by macrophages and T cells is critical for the induction of autoreactivity and type 1 diabetes,60(8), 2144-51.


Question:
In the highly-rated TV series, Breaking Bad, Walter White, a high school chemistry teacher recently diagnosed with cancer, takes to making the illicit drug, crystal meth (methamphetamine), by two main routes.
First, along with his RV-driving accomplice, Jessie Pinkman, Mr. White uses the common small-scale route starting with (1S,2S)-pseudoephedrine (the active ingredient in Sudafed®).  This method features the use of an optically active starting material to make an optically active end product, (S)-methamphetamine.  However, making (S)-methamphetamine on a large scale is limited because it is hard to get sufficient quantities of (1S,2S)-pseudoephedrine.
In the second route, Mr. White uses his knowledge of chemistry to move to an alternative synthesis starting with phenylacetone (also know as P2P or phenyl-2-propanone):

Racemic methamphetamine was obtained by the Winnebago-based chemists by reductive amination of P2P using methylamine and hydrogen over activated aluminum.
While his blue-colored product is considered by his customers to be exceptionally pure, Mr. White clearly knows about the issue of producing the correct enantiomer.  In fact, he raises this topic more than once in the series.
Since the show might not want to tell us the answer, I am wondering what what other possible methods Mr. White could have used to obtain an enantiomerically pure product?

Intriguing question.
First, the best yield would be achieved by selectively producing one enantiomer instead of the other. In this case, White wants D-methamphetamine (powerful psychoactive drug), not L-methamphetamine (Vicks Vapor Inhaler). Reaction processes designed to do this are known as "asymmetric synthesis" reactions, because they favor production of one enantiomer over the other.
The pseudoephedrine method for methamphetamine employs one of the more common methods of asymmetric synthesis, called "chiral pool resolution". As you state, starting with an enantiomerically-pure sample of a chiral reagent (pseudoephedrine) as the starting point allows you to preserve the chirality of the finished product, provided the chiral point is not part of any "leaving group" during the reaction. However, again as you show, phenylacetone is achiral, and so the P2P process cannot take advantage of this method.
There are other methods of asymmetric synthesis, however none of them seem applicable to the chemistry shown or described on TV either; none of the reagents or catalysts mentioned would work as chiral catalysts, nor are they bio- or organocatalysts. Metal complexes with chiral ligands can be used to selectively catalyze production of one enantiomer, however the aluminum-mercury amalgam is again achiral. I don't remember any mention of using organocatalysis or biocatalysis, but these are possible.
The remaining route, then, is chiral resolution; let the reaction produce the 50-50 split, then separate the two enantiomers by some means of reactionary and/or physical chemistry. This seems to be the way it works in the real world. The advantage is that most of the methods are pretty cheap and easy; the disadvantage is that your maximum possible yield is 50% (unless you can then run a racemization reaction on the undesireable half to "reshuffle" the chirality of that half; then your yield increases by 50% of the last increase each time you run this step on the undesirable product).
In the case of methamphetamine, this resolution is among the easiest, because methamphetamine forms a "racemic conglomerate" when crystallized. This means, for the non-chemists, that each enantiomer molecule prefers to crystallize with others of the same chiral species, so as the solution cools and the solvent is evaporated off, the D-methamphetamine will form one set of homogeneous crystals and the L-methamphetamine will form another set. This means that all White has to do is slow the evaporation of solvent and subsequent cooling of the pan, letting the largest possible crystals form. Then, the only remaining trick is identifying which crystals have which enantiomer (and as these crystals are translucent and "optically active", observing the polarization pattern of light shone through the crystals will identify which are which).


Question:
Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of number of edges on a complete graph starts $0,1,3,6,10$, so the next term must be 15 etc."
Granted, this second statement is less logically unsound than the first since it's not difficult to see the reason why the sequence must continue as such; nevertheless, the statement was made on a premise that boils down to "interesting patterns must always continue".
I try to counter this logic by creating a ridiculous argument like "the numbers $1,2,3,4,5$ are less than $100$, so surely all numbers are", but this usually fails to be convincing.
So, are there any examples of non-trivial patterns that appear to be true for a large number of small cases, but then fail for some larger case? A good answer to this question should:

be one which could be explained to the layman without having to subject them to a 24 lecture course of background material, and
have as a minimal counterexample a case which cannot (feasibly) be checked without the use of a computer.

I believe conditions 1. and 2. make my question specific enough to have in some sense a "right" (or at least a "not wrong") answer; but I'd be happy to clarify if this is not the case. I suppose I'm expecting an answer to come from number theory, but can see that areas like graph theory, combinatorics more generally and set theory could potentially offer suitable answers.

I'll translate an entry in the blog Gaussianos ("Gaussians") about Polya's conjecture, titled:
A BELIEF IS NOT A PROOF.

We'll say a number is of even kind if in its prime factorization, an even number of primes appear. For example $6 = 2\cdot 3$ is a number of even kind. And we'll say a number is of odd kind if the number of primes in its factorization is odd. For example, $18 = 2·3·3$ is of odd kind. ($1$ is considered of even kind).
Let $n$ be any natural number. We'll consider the following numbers:

$E(n) =$ number of positive integers less or equal to $n$ that are of even kind.
$O(n) =$  number of positive integers less or equal to $n$ that are of odd kind.

Let's consider $n=7$. In this case $O(7) = 4$ (number 2, 3, 5 and 7 itself) and $E(7) = 3$ ( 1,  4 and 6). So $O(7) >E(7)$.
For $n = 6$: $O(6) = 3$ and $E(6) = 3$. Thus $O(6) = E(6)$.
In 1919 George Polya proposed the following result, know as Polya's Conjecture:
For all $n > 2$, $O(n)$ is greater than or equal to $E(n)$.
Polya had checked this for $n < 1500$. In the following years this was tested up to $n=1000000$, which is a reason why the conjecture might be thought to be true. But that is wrong.
In 1962, Lehman found an explicit counterexample: for $n = 906180359$, we have $O(n) = E(n) – 1$, so:
$$O(906180359) < E(906180359).$$
By an exhaustive search, the smallest counterexample is $n = 906150257$, found by Tanaka in 1980.
Thus Polya's Conjecture is false.
What do we learn from this? Well, it is simple: unfortunately in mathematics we cannot trust intuition or what happens for a finite number of cases, no matter how large the number is. Until the result is proved for the general case, we have no certainty that it is true.


Question:
A common bioinformatics task is to decompose a DNA sequence into its constituent k-mers and compute a hash value for each k-mer. Rolling hash functions are an appealing solution for this task, since they can be computed very quickly. A rolling hash does not compute each the hash value from scratch with each k-mer: rather it updates a running hash value using an update strategy and a sliding window over the data.
It's also very useful for many applications to have a k-mer hash to the same value as its reverse complement. Unless the data were generated using a strand-specific sample prep, it's impossible to distinguish a k-mer from its reverse complement, and they should be treated as the same sequence.
Are there any rolling hashes that will map reverse complements to the same value? If not, how would we develop such an algorithm?
UPDATE: Ideally the hash function would be able to support k > 32, which would be lossy unless using something larger than a 64-bit integer.
ANOTHER UPDATE: I don't think it's necessary to store both the running k-mer and its reverse complement in a single value. If storing two k-mer strings and/or two hash values makes this easier, I'm totally cool with that.

A rolling hash function for DNA sequences called ntHash has recently been published in Bioinformatics and the authors dealt with reverse complements:

Using this table, we can easily compute the hash value for the reverse-complement (as well as the canonical form) of a sequence efficiently, without actually reverse- complementing the input sequence, as follows:
...

EDIT (by @user172818): I will add more details about how ntHash works. The notations used in its paper are somewhat uncommon. The source code is more informative.
Let's first define rotation functions for 64-bit integers:
rol(x,k) := x << k | x >> (64-k)
ror(x,k) := x >> k | x << (64-k)

We then define a hash function h() for each base. In the implementation, the authors are using:
h(A) = 0x3c8bfbb395c60474
h(C) = 0x3193c18562a02b4c
h(G) = 0x20323ed082572324
h(T) = 0x295549f54be24456
h(N) = 0

The rolling hash function of a forward k-mer s[i,i+k-1] is:
f(s[i,i+k-1]) := rol(h(s[i]),k-1) ^ rol(h(s[i+1]),k-2) ^ ... ^ h(s[i+k-1])

where ^ is the XOR operator. The hash function of its reverse complement is:
r(s[i,i+k-1]) := f(~s[i,i+k-1])
               = rol(h(~s[i+k-1]),k-1) ^ rol(h(~s[i+k-2]),k-2) ^ ... ^ h(~s[i])

where ~ gives the reverse complement of a DNA sequence. Knowing f(s[i,i+k-1]) and r(s[i,i+k-1]), we can compute their values for the next k-mer:
f(s[i+1,i+k]) = rol(f(s[i,i+k-1]),1) ^ rol(h(s[i]),k)  ^ h(s[i+k])
r(s[i+1,i+k]) = ror(r(s[i,i+k-1]),1) ^ ror(h(~s[i]),1) ^ rol(h(~s[i+k]),k-1)

In other words, for the forward kmer for each additional base, XOR the following three values together:

a single left rotation of the previous hash, f(s[i,i+k-1])
a $k$-times left rotation of the base hash of s[i]
the base hash of s[i+k]

Similarly for the reverse kmer, XOR the following three values together:

a single right rotation of the previous reverse hash, r(s[i,i+k-1])
a single right rotation of the base hash of the reverse complement of s[i]
a $k-1$-times left rotation of the base hash of the reverse complement of s[i+k]

This works because rol, ror and ^ can all be switched in order. Finally, for a k-mer s, the hash function considering both strands is the smaller between f(s) and r(s):
h(s) = min(f(s),r(s))

This is a linear algorithm regardless of the k-mer length. It only uses simple arithmetic operations, so should be fairly fast. I have briefly tested its randomness. It seems comparable to murmur. ntHash is probably the best algorithm so far if you want to hash an arbitrarily long k-mer into 64 bits.


Question:
HardOCP has an image with an equation which apparently draws the Batman logo.  Is this for real?

Batman Equation in text form:
\begin{align}
&\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-3}}+\left(\frac y3\right)^2\sqrt{\frac{\left|y+\frac{3\sqrt{33}}7\right|}{y+\frac{3\sqrt{33}}7}}-1 \right) \\
&\qquad \qquad \left(\left|\frac x2\right|-\left(\frac{3\sqrt{33}-7}{112}\right)x^2-3+\sqrt{1-(||x|-2|-1)^2}-y \right) \\
&\qquad \qquad \left(3\sqrt{\frac{|(|x|-1)(|x|-.75)|}{(1-|x|)(|x|-.75)}}-8|x|-y\right)\left(3|x|+.75\sqrt{\frac{|(|x|-.75)(|x|-.5)|}{(.75-|x|)(|x|-.5)}}-y \right) \\
&\qquad \qquad \left(2.25\sqrt{\frac{(x-.5)(x+.5)}{(.5-x)(.5+x)}}-y \right) \\
&\qquad \qquad \left(\frac{6\sqrt{10}}7+(1.5-.5|x|)\sqrt{\frac{||x|-1|}{|x|-1}} -\frac{6\sqrt{10}}{14}\sqrt{4-(|x|-1)^2}-y\right)=0
\end{align}

As Willie Wong observed, including an expression of the form $\displaystyle \frac{|\alpha|}{\alpha}$ is a way of ensuring that $\alpha > 0$. (As $\sqrt{|\alpha|/\alpha}$ is $1$ if $\alpha > 0$ and non-real if $\alpha < 0$.)

The ellipse $\displaystyle \left( \frac{x}{7} \right)^{2} + \left( \frac{y}{3} \right)^{2} - 1 = 0$ looks like this:

So the curve $\left( \frac{x}{7} \right)^{2}\sqrt{\frac{\left| \left| x \right|-3 \right|}{\left| x \right|-3}} + \left( \frac{y}{3} \right)^{2}\sqrt{\frac{\left| y+3\frac{\sqrt{33}}{7} \right|}{y+3\frac{\sqrt{33}}{7}}} - 1 = 0$ is the above ellipse, in the region where $|x|>3$ and $y > -3\sqrt{33}/7$:

That's the first factor.

The second factor is quite ingeniously done. The curve $\left| \frac{x}{2} \right|\; -\; \frac{\left( 3\sqrt{33}-7 \right)}{112}x^{2}\; -\; 3\; +\; \sqrt{1-\left( \left| \left| x \right|-2 \right|-1 \right)^{2}}-y=0$ looks like:

This is got by adding $y = \left| \frac{x}{2} \right| - \frac{\left( 3\sqrt{33}-7 \right)}{112}x^{2} - 3$, a parabola on the positive-x side, reflected:

and $y = \sqrt{1-\left( \left| \left| x \right|-2 \right|-1 \right)^{2}}$, the upper halves of the four circles $\left( \left| \left| x \right|-2 \right|-1 \right)^2 + y^2 = 1$:

The third factor $9\sqrt{\frac{\left( \left| \left( 1-\left| x \right| \right)\left( \left| x \right|-.75 \right) \right| \right)}{\left( 1-\left| x \right| \right)\left( \left| x \right|-.75 \right)}}\; -\; 8\left| x \right|\; -\; y\; =\; 0$ is just the pair of lines y = 9 - 8|x|:

truncated to the region $0.75 < |x| < 1$.

Similarly, the fourth factor $3\left| x \right|\; +\; .75\sqrt{\left( \frac{\left| \left( .75-\left| x \right| \right)\left( \left| x \right|-.5 \right) \right|}{\left( .75-\left| x \right| \right)\left( \left| x \right|-.5 \right)} \right)}\; -\; y\; =\; 0$ is the pair of lines $y = 3|x| + 0.75$:

truncated to the region $0.5 < |x| < 0.75$.

The fifth factor $2.25\sqrt{\frac{\left| \left( .5-x \right)\left( x+.5 \right) \right|}{\left( .5-x \right)\left( x+.5 \right)}}\; -\; y\; =\; 0$ is the line $y = 2.25$ truncated to $-0.5 < x < 0.5$.

Finally, $\frac{6\sqrt{10}}{7}\; +\; \left( 1.5\; -\; .5\left| x \right| \right)\; -\; \frac{\left( 6\sqrt{10} \right)}{14}\sqrt{4-\left( \left| x \right|-1 \right)^{2}}\; -\; y\; =\; 0$ looks like:

so the sixth factor $\frac{6\sqrt{10}}{7}\; +\; \left( 1.5\; -\; .5\left| x \right| \right)\sqrt{\frac{\left| \left| x \right|-1 \right|}{\left| x \right|-1}}\; -\; \frac{\left( 6\sqrt{10} \right)}{14}\sqrt{4-\left( \left| x \right|-1 \right)^{2}}\; -\; y\; =\; 0$ looks like

As a product of factors is $0$ iff any one of them is $0$, multiplying these six factors puts the curves together, giving: (the software, Grapher.app, chokes a bit on the third factor, and entirely on the fourth)


Question:
Diborane has the interesting property of having two 3-centered bonds that are each held together by only 2 electrons (see the diagram below, from Wikipedia).  These are known as "banana bonds."
I'm assuming there is some sort of bond hybridization transpiring, but the geometry doesn't seem like it is similar to anything I'm familiar with Carbon doing.  What sort of hybridization is it, and why don't we see many (any?) other molecules with this bond structure?

Look carefully, it's (distorted) tetrahedral--four groups at nearly symmetrically positions in 3D space{*}. So the hybridization is $sp^3$.

As you can see, the shape is distorted, but it's tetrahedral. Technically, the banana bonds can be said to be made up of orbitals similar to $sp^3$ but not exactly (like two $sp^{3.1}$ and two $sp^{2.9}$ orbitals--since hybridization is just addition of wavefunctions, we can always change the coefficients to give proper geometry). I'm not too sure of this though.
$\ce{B}$ has an $2s^22p^1$ valence shell, so three covalent bonds gives it an incomplete octet. $\ce{BH3}$ has an empty $2p$ orbital. This orbital overlaps the existing  $\ce{B-H}$ $\sigma$ bond cloud (in a nearby $\ce{BH3}$), and forms a 3c2e bond.
It seems that there are a lot more compounds with 3c2e geometry. I'd completely forgotten that there were entire homologous series' under 'boranes' which all have 3c2e bonds (though not the same structure)
And there are Indium and Gallium compounds as well. Still group IIIA, though these are metals. I guess they, like $\ce{Al}$, still form covalent bonds.
So the basic reason for this happening is due to an incomplete octet wanting to fill itself.
Note that "banana" is not necessarily only for 3c2e bonds. Any bent bond is called a "banana" bond.
Regarding similar structures, $\ce{BeCl2}$ and $\ce{AlCl3}$ come to mind, but both of them have the structure via dative(coordinate) bonds. Additionally, $\ce{BeCl2}$ is planar.
Sneaks off and checks Wikipedia. Wikipedia says $\ce{Al2(CH3)6}$ is similar in structure and bond type.
I guess we have less such compounds because there are comparatively few elements ($\ce{B}$ group pretty much) with $\leq3$ valence electrons which form covalent bonds(criteria for the empty orbital). Additionally, $\ce{Al}$ is an iffy case--it like both covalent and ionic bonds. Also, for this geometry (either by banana bonds or by dative bonds), I suppose the relative sizes matter as well--since $\ce{BCl3}$ is a monomer even though $\ce{Cl}$ has a lone pair and can form a dative bond.
*Maybe you're used to the view of tetrahedral structure with an atom at the top? Mentally tilt the boron atom till a hydrogen is up top. You should realize that this is tetrahedral as well.


Question:
I am the resident Bioinfo Geek in a hospital academic lab that routinely employs NGS as well as CyTOF and other large volume data producing technologies. I am sick of our current "protocol" for metadata collection and association with the final products (miriad excel sheets and a couple poorly designed RedCap DBs).
I want to implement a central structured, controlled datastore that will take care of this. I know that the interface to the technicians how will be inputing the data is crucial to its adoption, but this is not the focus of THIS particular question: Does there exist a schema or schema guidelines for this type of database?
I would rather use a model that has been developed by people who know how to do this well. I know of BioSQL but it seems more geared towards full protein/nucleotide records like those found in uniprot or genbank. That is not what we have here. What I want is something similar to the system touched on in this preprint:
Alternatively, can anyone provide links to where I might find relevant guidelines or supply personal advice?

The Global Alliance for Genomics and Health has been working on the issue of representing sequencing data and metadata for storage and sharing for quite some time, though with mixed results.  They do offer a model and API for storing NGS data in their GitHub repository, but it can be a bit of a pain to get a high-level view.  I am not sure if any better representation of this exists elsewhere.
I can say from personal experience (having built over a dozen genomic databases), there is no ideal data model and storage best practices.  Genomic data comes in many shapes and sizes, and your needs are going to vary from every other organization, so what works for one bioinformatics group won't necessarily work for you.  The best thing to do is design and implement a model that will cover all of the data types in your workflow and downstream analyses you might do with the data and metadata.


Question:
There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property.  To my knowledge, not many "highly" discontinuous Darboux functions are known--the only one I am aware of being the Conway base 13 function--and few (none?) of these are derivatives of differentiable functions.  In fact they generally cannot be since an application of Baire's theorem gives that the set of continuity points of the derivative is dense $G_\delta$.
Is it known how sharp that last result is?  Are there known Darboux functions which are derivatives and are discontinuous on "large" sets in some appropriate sense?

What follows is taken (mostly) from more extensive discussions in the following sci.math posts:
 [23 January 2000]
 [6 November 2006]
 [20 December 2006]
Note: The term interval is restricted to nondegenerate intervals (i.e. intervals containing more than one point).
The continuity set of a derivative on an open interval $J$ is dense in $J.$ In fact, the continuity set has cardinality $c$ in every subinterval of $J.$ On the other hand, the discontinuity set $D$ of a derivative can have the following properties:

$D$ can be dense in $\mathbb R$.
$D$ can have cardinality $c$ in every interval.
$D$ can have positive measure. (Hence, the function can fail to be Riemann integrable.)
$D$ can have positive measure in every interval.
$D$ can have full measure in every interval (i.e. measure zero complement).
$D$ can have a Hausdorff dimension zero complement.
$D$ can have an $h$-Hausdorff measure zero complement for any specified Hausdorff measure function $h.$

More precisely, a subset $D$ of $\mathbb R$ can be the discontinuity set for some derivative if and only if $D$ is an $F_{\sigma}$ first category (i.e. an $F_{\sigma}$ meager) subset of $\mathbb R.$
This characterization of the discontinuity set of a derivative can be found in the following references: Benedetto [1] (Chapter 1.3.2, Proposition, 1.10, p. 30); Bruckner [2] (Chapter 3, Section 2, Theorem 2.1, p. 34); Bruckner/Leonard [3] (Theorem at bottom of p. 27); Goffman [5] (Chapter 9, Exercise 2.3, p. 120 states the result); Klippert/Williams [7].
Regarding this characterization of the discontinuity set of a derivative, Bruckner and Leonard [3] (bottom of p. 27) wrote the following in 1966: Although we imagine that this theorem is known, we have been unable to find a reference. I have found the result stated in Goffman's 1953 text [5], but nowhere else prior to 1966 (including Goffman's Ph.D. Dissertation).
Interestingly, in a certain sense most derivatives have the property that $D$ is large in all of the ways listed above (#1 through #7).
In 1977 Cliff Weil [8] published a proof that, in the space of derivatives with the sup norm, all but a first category set of such functions are discontinuous almost everywhere (in the sense of Lebesgue measure). When Weil's result is paired with the fact that derivatives (being Baire $1$ functions) are continuous almost everywhere in the sense of Baire category, we get the following:
(A) Every derivative is continuous at the Baire-typical point.
(B) The Baire-typical derivative is not continuous at the Lebesgue-typical point.
Note that Weil's result is stronger than simply saying that the Baire-typical derivative fails to be Riemann integrable (i.e. $D$ has positive Lebesgue measure), or even stronger than saying that the Baire-typical derivative fails to be Riemann integrable on every interval. Note also that, for each of these Baire-typical derivatives, $\{D, \; {\mathbb R} - D\}$ gives a partition of $\mathbb R$ into a first category set and a Lebesgue measure zero set.
In 1984 Bruckner/Petruska [4] (Theorem 2.4) strengthened Weil's result by proving the following: Given any finite Borel measure $\mu,$ the Baire-typical derivative is such that the set $D$ is the complement of a set that has $\mu$-measure zero.
In 1993 Kirchheim [5] strengthened Weil's result by proving the following: Given any Hausdorff measure function $h,$ the Baire-typical derivative is such that the set $D$ is the complement of a set that has Hausdorff $h$-measure zero.
[1] John J. Benedetto, Real Variable and Integration With Historical Notes, Mathematische Leitfäden. Stuttgart: B. G. Teubne, 1976, 278 pages. [MR 58 #28328; Zbl 336.26001]
[2] Andrew M. Bruckner, Differentiation of Real Functions, 2nd edition, CRM Monograph Series #5, American Mathematical Society, 1994, xii + 195 pages. [The first edition was published in 1978 as Springer-Verlag's Lecture Notes in Mathematics #659. The second edition is essentially unchanged from the first edition with the exception of a new chapter on recent developments (23 pages) and 94 additional bibliographic items.] [MR 94m:26001; Zbl 796.26001]
[3] Andrew M. Bruckner and John L. Leonard, Derivatives, American Mathematical Monthly 73 #4 (April 1966) [Part II: Papers in Analysis, Herbert Ellsworth Slaught Memorial Papers #11], 24-56. [MR 33 #5797; Zbl 138.27805]
[4] Andrew M. Bruckner and György Petruska, Some typical results on bounded Baire $1$ functions, Acta Mathematica Hungarica 43 (1984), 325-333. [MR 85h:26004; Zbl 542.26004]
[5] Casper Goffman, Real Functions, Prindle, Weber & Schmidt, 1953/1967, x + 261 pages. [MR 14,855e; Zbl 53.22502]
[6] Bernd Kirchheim, Some further typical results on bounded Baire one functions, Acta Mathematica Hungarica 62 (1993), 119-129. [94k:26008; Zbl 786.26002]
[7] John Clayton Klippert and Geoffrey Williams, On the existence of a derivative continuous on a $G_{\delta}$, International Journal of Mathematical Education in Science and Technology 35 (2004), 91-99.
[8] Clifford Weil, The space of bounded derivatives, Real Analysis Exchange 3 (1977-78), 38-41. [Zbl 377.26005]


Question:
The help pages in R assume I know what those numbers mean, but I don't.
I'm trying to really intuitively understand every number here. I will just post the output and comment on what I found out. There might (will) be mistakes, as I'll just write what I assume. Mainly I'd like to know what the t-value in the coefficients mean, and why they print the residual standard error.
Call:
lm(formula = iris$Sepal.Width ~ iris$Petal.Width)

Residuals:
     Min       1Q   Median       3Q      Max
-1.09907 -0.23626 -0.01064  0.23345  1.17532

This is a 5-point-summary of the residuals (their mean is always 0, right?). The numbers can be used (I'm guessing here) to quickly see if there are any big outliers. Also you can already see it here if the residuals are far from normally distributed (they should be normally distributed).
Coefficients:
                 Estimate Std. Error t value Pr(>|t|)
(Intercept)       3.30843    0.06210  53.278  < 2e-16 ***
iris$Petal.Width -0.20936    0.04374  -4.786 4.07e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Estimates $\hat{\beta_i}$, computed by least squares regression. Also, the standard error is $\sigma_{\beta_i}$. I'd like to know how this is calculated.  I have no idea where the t-value and the corresponding p-value come from. I know $\hat{\beta}$ should be normal distributed, but how is the t-value calculated?
Residual standard error: 0.407 on 148 degrees of freedom

$\sqrt{ \frac{1}{n-p} \epsilon^T\epsilon }$, I guess. But why do we calculate that, and what does it tell us?
Multiple R-squared: 0.134,  Adjusted R-squared: 0.1282

$ R^2 = \frac{s_\hat{y}^2}{s_y^2} $, which is $ \frac{\sum_{i=1}^n (\hat{y_i}-\bar{y})^2}{\sum_{i=1}^n (y_i-\bar{y})^2} $. The ratio is close to 1 if the points lie on a straight line, and 0 if they are random.  What is the adjusted R-squared?
F-statistic: 22.91 on 1 and 148 DF,  p-value: 4.073e-06

F and p for the whole model, not only for single $\beta_i$s as previous. The F value is $ \frac{s^2_{\hat{y}}}{\sum\epsilon_i} $. The bigger it grows, the more unlikely it is that the $\beta$'s do not have any effect at all.

Five point summary
yes, the idea is to give a quick summary of the distribution. It should be roughly symmetrical about mean, the median should be close to 0, the 1Q and 3Q values should ideally be roughly similar values.
Coefficients and $\hat{\beta_i}s$
Each coefficient in the model is a Gaussian (Normal) random variable. The $\hat{\beta_i}$ is the estimate of the mean of the distribution of that random variable, and the standard error is the square root of the variance of that distribution. It is a measure of the uncertainty in the estimate of the $\hat{\beta_i}$.
You can look at how these are computed (well the mathematical formulae used) on Wikipedia. Note that any self-respecting stats programme will not use the standard mathematical equations to compute the $\hat{\beta_i}$ because doing them on a computer can lead to a large loss of precision in the computations.
$t$-statistics
The $t$ statistics are the estimates ($\hat{\beta_i}$) divided by their standard errors ($\hat{\sigma_i}$), e.g. $t_i = \frac{\hat{\beta_i}}{\hat{\sigma_i}}$. Assuming you have the same model in object modas your Q:
> mod <- lm(Sepal.Width ~ Petal.Width, data = iris)

then the $t$ values R reports are computed as:
> tstats <- coef(mod) / sqrt(diag(vcov(mod)))
(Intercept) Petal.Width
  53.277950   -4.786461

Where coef(mod) are the $\hat{\beta_i}$, and sqrt(diag(vcov(mod))) gives the square roots of the diagonal elements of the covariance matrix of the model parameters, which are the standard errors of the parameters ($\hat{\sigma_i}$).
The p-value is the probability of achieving a $|t|$ as large as or larger than the observed absolute t value if the null hypothesis ($H_0$) was true, where $H_0$ is $\beta_i = 0$. They are computed as (using tstats from above):
> 2 * pt(abs(tstats), df = df.residual(mod), lower.tail = FALSE)
 (Intercept)  Petal.Width
1.835999e-98 4.073229e-06

So we compute the upper tail probability of achieving the $t$ values we did from a $t$ distribution with degrees of freedom equal to the residual degrees of freedom of the model. This represents the probability of achieving a $t$ value greater than the absolute values of the observed $t$s. It is multiplied by 2, because of course $t$ can be large in the negative direction too.
Residual standard error
The residual standard error is an estimate of the parameter $\sigma$. The assumption in ordinary least squares is that the residuals are individually described by a Gaussian (normal) distribution with mean 0 and standard deviation $\sigma$. The $\sigma$ relates to the constant variance assumption; each residual has the same variance and that variance is equal to $\sigma^2$.
Adjusted $R^2$
Adjusted $R^2$ is computed as:
$$1 - (1 - R^2) \frac{n - 1}{n - p - 1}$$
The adjusted $R^2$ is the same thing as $R^2$, but adjusted for the complexity (i.e. the number of parameters) of the model. Given a model with a single parameter, with a certain $R^2$, if we add another parameter to this model, the $R^2$ of the new model has to increase, even if the added parameter has no statistical power. The adjusted $R^2$ accounts for this by including the number of parameters in the model.
$F$-statistic
The $F$ is the ratio of two variances ($SSR/SSE$), the variance explained by the parameters in the model (sum of squares of regression, SSR) and the residual or unexplained variance (sum of squares of error, SSE). You can see this better if we get the ANOVA table for the model via anova():
> anova(mod)
Analysis of Variance Table

Response: Sepal.Width
             Df  Sum Sq Mean Sq F value    Pr(>F)
Petal.Width   1  3.7945  3.7945   22.91 4.073e-06 ***
Residuals   148 24.5124  0.1656
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The $F$s are the same in the ANOVA output and the summary(mod) output. The Mean Sq column contains the two variances and $3.7945 / 0.1656 = 22.91$. We can compute the probability of achieving an $F$ that large under the null hypothesis of no effect, from an $F$-distribution with 1 and 148 degrees of freedom. This is what is reported in the final column of the ANOVA table. In the simple case of a single, continuous predictor (as per your example), $F = t_{\mathrm{Petal.Width}}^2$, which is why the p-values are the same. This equivalence only holds in this simple case.


Question:
It's one of my real analysis professor's favourite sayings that "being obvious does not imply that it's true".
Now, I know a fair few examples of things that are obviously true and that can be proved to be true (like the Jordan curve theorem).
But what are some theorems (preferably short ones) which, when put into layman's terms, the average person would claim to be true, but, which, actually, are false
(i.e. counter-intuitively-false theorems)?
The only ones that spring to my mind are the Monty Hall problem and the divergence of $\sum\limits_{n=1}^{\infty}\frac{1}{n}$ (counter-intuitive for me, at least, since $\frac{1}{n} \to 0$
).
I suppose, also, that $$\lim\limits_{n \to \infty}\left(1+\frac{1}{n}\right)^n = e=\sum\limits_{n=0}^{\infty}\frac{1}{n!}$$ is not obvious, since one 'expects' that $\left(1+\frac{1}{n}\right)^n \to (1+0)^n=1$.
I'm looking just for theorems and not their (dis)proof -- I'm happy to research that myself.
Thanks!

Theorem (false):

One can arbitrarily rearrange the terms in a convergent series without changing its value.


Question:
I am looking for a tool, preferably written in C or C++, that can quickly and efficiently count the number of reads and the number of bases in a compressed fastq file. I am currently doing this using zgrep and awk:
zgrep . foo.fastq.gz |
     awk 'NR%4==2{c++; l+=length($0)}
          END{
                print "Number of reads: "c;
                print "Number of bases in reads: "l
              }'

The files I need to process are regular ASCII text files (fastq) compressed using gzip (usually GNU gzip, possibly BSD gzip sometimes if clients are using macs).
zgrep . will print non-blank lines from the input file and the awk 'NR%4==2 will process every 4th line starting with the second (the sequence).
This works fine, but can take a very long time when dealing with large files such as WGS data. Is there a tool I can use (on Linux) that will give me these values? Or, if not, I'm also open to suggestions for speeding up the above command.
I know that the FASTQ specification (such as it is) allows for line breaks in the sequence and qual strings, so simply taking the second of every group of 4 lines is not guaranteed to work (see here). That's another reason why I'd rather use a more sophisticated, dedicated tool. However, I have yet to actually encounter a file with >4 lines per record in the wild, so I am not too worried about that.

It's difficult to get this to go massively quicker I think - as with this question working with large gzipped FASTQ files is mostly IO-bound. We could instead focus on making sure we are getting the right answer.
People deride them too often, but this is where a well-written parser is worth it's weight in gold. Heng Li gives us this FASTQ Parser in C.
I downloaded the example tarball and modified the example code (excuse my C...):
#include <zlib.h>
#include <stdio.h>
#include "kseq.h"
KSEQ_INIT(gzFile, gzread)

int main(int argc, char *argv[])
{
    gzFile fp;
    kseq_t *seq;
    int l;
    if (argc == 1) {
        fprintf(stderr, "Usage: %s <in.seq>\n", argv[0]);
        return 1;
    }
    fp = gzopen(argv[1], "r");
    seq = kseq_init(fp);
    int seqcount = 0;
    long seqlen = 0;
    while ((l = kseq_read(seq)) >= 0) {
        seqcount = seqcount + 1;
        seqlen = seqlen + (long)strlen(seq->seq.s);
    }
    kseq_destroy(seq);
    gzclose(fp);
    printf("Number of sequences: %d\n", seqcount);
    printf("Number of bases in sequences: %ld\n", seqlen);
    return 0;
}

Then make and kseq_test foo.fastq.gz.
For my example file (~35m reads of ~75bp) this took:
real    0m49.670s
user    0m49.364s
sys     0m0.304s

Compared with your example:
real    0m43.616s
user    1m35.060s
sys     0m5.240s

Konrad's solution (in my hands):
real    0m39.682s
user    1m11.900s
sys     0m5.112s

(By the way, just zcat-ing the data file to /dev/null):
real    0m38.736s
user    0m38.356s
sys     0m0.308s

So, I get pretty close in speed, but am likely to be more standards compliant. Also this solution gives you more flexibility with what you can do with the data.
And my horrible C can almost certainly be optimised.

Same test, with kseq.h from Github, as suggested in the comments:
My machine is under different load this morning, so I've retested. Wall clock times:
OP: 0m44.813s
Konrad: 0m40.061s
zcat > /dev/null: 0m34.508s
kseq.h (Github): 0m32.909s
So most recent version of kseq.h is faster than simply zcat-ing the file (consistently in my tests...).


Question:
I've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity.
Thanks.

The name "ring" is derived from Hilbert's term "Zahlring" (number ring), introduced in his Zahlbericht for certain rings of algebraic integers. As for why Hilbert chose the name "ring", I recall reading speculations that it may have to do with cyclical (ring-shaped) behavior of powers of algebraic integers. Namely, if $\:\alpha\:$ is an algebraic integer of degree $\rm\:n\:$  then $\:\alpha^n\:$ is a $\rm\:\mathbb Z$-linear combination of lower powers of $\rm\:\alpha\:,\:$ thus so too are all higher powers of $\rm\:\alpha\:.\:$ Hence all powers cycle back onto  $\rm\:1,\:\alpha,\:,\ldots,\alpha^{n-1}\:,\:$ i.e. $\rm\:\mathbb Z[\alpha]\:$ is a finitely generated $\:\mathbb Z$-module. Possibly also the motivation for the name had to do more specifically with rings of cyclotomic integers. However, as plausible as that may seem, I don't recall the existence of any historical documents that provide solid evidence in support of such speculations.
Beware that one has to be very careful when reading such older literature. Some authors mistakenly read modern notions into terms which have no such denotation in their original usage. To provide some context I recommend reading Lemmermeyer and Schappacher's Introduction to the English Edition of Hilbert’s Zahlbericht. Below is a pertinent excerpt.

Below is an excerpt from Leo Corry's Modern algebra and the rise of mathematical structures, p. 149.

Below are a couple typical examples of said speculative etymology of the term "ring" via the "circling  back" nature of integral dependence, from Harvey Cohn's Advanced Number Theory, p. 49.

$\quad$The designation of the letter $\mathfrak D$ for the integral domain has some historical importance going back to Gauss's work on quadratic forms. Gauss $\left(1800\right)$ noted that for certain quadratic forms $Ax^2+Bxy+Cy^2$ the discriminant need not be square-free, although $A$, $B$, $C$ are relatively prime. For example, $x^2-45y^2$ has $D=4\cdot45$. The $4$ was ignored for the reason that $4|D$ necessarily by virtue of Gauss's requirement that $B$ be even, but the factor of $3^2$ in $D$ caused Gauss to refer to the form as one of "order $3$." Eventually, the forms corresponding to a value of $D$ were called an "order" (Ordnung). Dedekind retained this word for what is here called an "integral domain."
$\quad$The term "ring" is a contraction of "Zahlring" introduced by Hilbert $\left(1892\right)$ to denote (in our present context) the ring generated by the rational integers and a quadratic integer $\eta$ defined by $$\eta^2+B\eta+C=0.$$ It would seem that module $\left[1,\eta\right]$ is called a Zahlring because $\eta^2$ equals $-B\eta-C$ "circling directly back" to an element of $\left[1,\eta\right]$ . This word has been maintained today. Incidentally, every Zahlring is an integral domain and the converse is true for quadratic fields.

and from Rotman's Advanced Modern Algebra, p. 81.


Question:
I need to determine the KL-divergence between two Gaussians. I am comparing my results to these, but I can't reproduce their result. My result is obviously wrong, because the KL is not 0 for KL(p, p).
I wonder where I am doing a mistake and ask if anyone can spot it.
Let $p(x) = N(\mu_1, \sigma_1)$ and $q(x) = N(\mu_2, \sigma_2)$. From Bishop's
PRML I know that
$$KL(p, q) = - \int p(x) \log q(x) dx + \int p(x) \log p(x) dx$$
where integration is done over all real line, and that
$$\int p(x) \log p(x) dx = -\frac{1}{2} (1 + \log 2 \pi \sigma_1^2),$$
so I restrict myself to $\int p(x) \log q(x) dx$, which I can write out as
$$-\int p(x) \log \frac{1}{(2 \pi \sigma_2^2)^{(1/2)}} e^{-\frac{(x-\mu_2)^2}{2 \sigma_2^2}} dx,$$
which can be separated into
$$\frac{1}{2} \log (2 \pi \sigma_2^2) - \int p(x) \log e^{-\frac{(x-\mu_2)^2}{2 \sigma_2^2}} dx.$$
Taking the log I get
$$\frac{1}{2} \log (2 \pi \sigma_2^2) - \int p(x) \bigg(-\frac{(x-\mu_2)^2}{2 \sigma_2^2} \bigg) dx,$$
where I separate the sums and get $\sigma_2^2$ out of the integral.
$$\frac{1}{2} \log (2 \pi \sigma^2_2) + \frac{\int p(x) x^2 dx - \int p(x) 2x\mu_2 dx + \int p(x) \mu_2^2 dx}{2 \sigma_2^2}$$
Letting $\langle \rangle$ denote the expectation operator under $p$, I can rewrite this as
$$\frac{1}{2} \log (2 \pi \sigma_2^2) + \frac{\langle x^2 \rangle - 2 \langle x \rangle \mu_2 + \mu_2^2}{2 \sigma_2^2}.$$
We know that $var(x) = \langle x^2 \rangle - \langle x \rangle ^2$. Thus
$$\langle x^2 \rangle = \sigma_1^2 + \mu_1^2$$
and therefore
$$\frac{1}{2} \log (2 \pi \sigma_2^2) + \frac{\sigma_1^2 + \mu_1^2 - 2 \mu_1 \mu_2 + \mu_2^2}{2 \sigma_2^2},$$
which I can put as
$$\frac{1}{2} \log (2 \pi \sigma_2^2) + \frac{\sigma_1^2 + (\mu_1 - \mu_2)^2}{2 \sigma_2^2}.$$
Putting everything together, I get to
\begin{align*}
KL(p, q) &= - \int p(x) \log q(x) dx + \int p(x) \log p(x) dx\\\\
&= \frac{1}{2} \log (2 \pi \sigma_2^2) + \frac{\sigma_1^2 + (\mu_1 - \mu_2)^2}{2 \sigma_2^2} - \frac{1}{2} (1 + \log 2 \pi \sigma_1^2)\\\\
&= \log \frac{\sigma_2}{\sigma_1} + \frac{\sigma_1^2 + (\mu_1 - \mu_2)^2}{2 \sigma_2^2}.
\end{align*}
Which is wrong since it equals $1$ for two identical Gaussians.
Can anyone spot my error?
Update
Thanks to mpiktas for clearing things up. The correct answer is:
$KL(p, q) = \log \frac{\sigma_2}{\sigma_1} + \frac{\sigma_1^2 + (\mu_1 - \mu_2)^2}{2 \sigma_2^2} - \frac{1}{2}$

OK, my bad. The error is in the last equation:
\begin{align}
KL(p, q) &= - \int p(x) \log q(x) dx + \int p(x) \log p(x) dx\\\\
&=\frac{1}{2} \log (2 \pi \sigma_2^2) + \frac{\sigma_1^2 + (\mu_1 - \mu_2)^2}{2 \sigma_2^2} - \frac{1}{2} (1 + \log 2 \pi \sigma_1^2)\\\\
&= \log \frac{\sigma_2}{\sigma_1} + \frac{\sigma_1^2 + (\mu_1 - \mu_2)^2}{2 \sigma_2^2} - \frac{1}{2}
\end{align}
Note the missing $-\frac{1}{2}$. The last line becomes zero when $\mu_1=\mu_2$ and $\sigma_1=\sigma_2$.


Question: Many numerical approaches to CFD can be extended to arbitrarily high order (for instance, discontinuous Galerkin methods, WENO methods, spectral differencing, etc.).  How should I choose an appropriate order of accuracy for a given problem?

In practice, most people stick to relatively low orders, usually first or second order.  This view is often challenged by more theoretical researchers that believe in more accurate answers .  The rate of convergence for simple smooth problems is well documented, for example see Bill Mitchell's comparison of hp adaptivity.
While for theoretical works it is nice to see what the convergence rate are, for more application oriented among us this concern is balanced with constitutive laws, necessary precision, and code complexity.  It doesn't make much since in many porous media problems that solve over a highly discontinuous media to have high order methods,  the numerical error will dominate the discretization errors.  The same concern applies for problems that include a large number of degrees of freedom.  Since low-order implicit methods have a smaller bandwidth and often a better conditioning, the high order method becomes too costly to solve.  Finally the code complexity of switching orders and types of polynomials is usually too much for the graduate students running the application codes.


Question:
Reading this question, Why are there no wheeled animals?, I wondered why no organisms seem to make use of the tensile and other strengths of metal, as we do in metal tools and constructions. I am obviously not talking about the microscopic uses of metal, as in human blood etc.
Why are there no plants with metal thorns? No trees with "reinforced" wood? No metal-plated sloths? No beetles with metal-tipped drills? Or are there?
I can think of some potential factors why there are none (or few), but I do not know whether they are true:

Is metal too scarce near the surface?
Are there certain chemical properties that make metal hard to extract and accumulate in larger quantities?
Is metal too heavy to carry around, even in a thin layer or mesh or tip?
Can metal of high (tensile etc.) strength only be forged under temperatures too high to sustain inside (or touching) organic tissue, and is crystallised metal too weak?
Are functionally comparable organic materials like horn, bone, wood, etc. in fact better at their tasks than metal, and do we humans only use metal because we are not good enough at using e.g. horn to make armour or chitin to make drills?

As a predator, I would like to eat a lot of vertebrates and save up the metal from their blood to reinforce my fangs...

A bonus question: are there any organisms that use the high electric conductivity of metal? Animals depend upon electric signals for their nervous system, but I do not think nerves contain much metal. The same applies to the few animals that use electricity as a weapon.

There are some cases of bio-metallic materials, as hinted at by the comments. But these are relatively small amount of metal.
It's not that there is a lack of metal available. Iron in particular is the fourth most common element in the earth's crust. Most soil that has a reddish color has iron in it.  There are several reasons you don't see iron exoskeletons on animals all the time.
Firstly, metallic iron (in chemistry terms, fully reduced, oxidation state 0) has a high energetic cost to create.
Iron is the second most common metal after aluminum on the earth's crust but it's almost entirely present in oxidized states - that's to say: as rust.  Most biological iron functions in the +2/+3 oxidation state, which is more similar to rust than metal.  Cytochromes and haemoglobin are examples of how iron is more valuable as a chemically active biological agent than a structural agent, using oxidized iron ions as they do. Aluminium, the most common metal on Earth, has relatively little biological activity - one might assume because its redox costs are even higher than iron.
As to why reduced biometal doesn't show up very often, inability of biological systems to deposit reduced (metallic) metals is not one of them.  There are cases of admittedly small pieces of reduced metal being produced by biological systems. The Magnetosomes in magnetotactic bacteria are mentioned, but there are also cases of reduced gold being accumulated by microorganisms.
Bone and shell are examples of biomineralization where the proteins depositing the calcium carbonate or other minerals in the material are structured by the proteins to be stronger than they would be as a simple crystal.  most of the examples here have very little or no metal, but rather minerals like the Chrysomallon squamiferum cited by @navyguymarko and @loki'sbane here.  The Iron Sulfide looks metallic but it is a mineral, akin to a bone.
While iron skeletons might seem to be an advantage, they are electrochemically unstable - oxygen and water will tend to oxidize (rust) them quickly and the organism would have to spend a lot of energy keeping it in working form.  Electrical conductivity sounds useful, but the nervous system favors exquisite levels of control over bulk current flow, even in cases like electric eels, whose current is produced by gradients from acetylcholine.
What's more, biological materials actually perform as well as or better than metal when they need to.  Spider silk has a greater tensile strength than steel (along the direction of the thread). Mollusk shells are models for tank armor - they are remarkably resistant to puncture and breakage.  Bone is durable for most purposes and flexible in addition.
The time it would take for metallized structures to evolve biologically are likely too long. By the time the metalized version of an organ or skeleton got started, the bones, shells and fibers we know probably have a big lead and selective advantage.


Question:
I am currently using an SVM with a linear kernel to classify my data.  There is
no error on the training set.  I tried several values for the parameter $C$
($10^{-5}, \dots, 10^2$).  This did not change the error on the test set.
Now I
wonder: is this an error caused by the ruby bindings for libsvm I am using
(rb-libsvm) or is this theoretically explainable?
Should the parameter $C$ always change the performance of the classifier?

In a SVM you are searching for two things: a hyperplane with the largest minimum margin, and a hyperplane that correctly separates as many instances as possible. The problem is that you will not always be able to get both things. The c parameter determines how great your desire is for the latter.
I have drawn a small example below to illustrate this. To the left you have a low c which gives you a pretty large minimum margin (purple). However, this requires that we neglect the blue circle outlier that we have failed to classify correct. On the right you have a high c. Now you will not neglect the outlier and thus end up with a much smaller margin.

So which of these classifiers are the best? That depends on what the future data you will predict looks like, and most often you don't know that of course.
If the future data looks like this:

then the classifier learned using a large c value is best.
On the other hand, if the future data looks like this:

then the classifier learned using a low c value is best.
Depending on your data set, changing c may or may not produce a different hyperplane. If it does produce a different hyperplane, that does not imply that your classifier will output different classes for the particular data you have used it to classify. Weka is a good tool for visualizing data and playing around with different settings for an SVM. It may help you get a better idea of how your data look and why changing the c value does not change the classification error.
In general, having few training instances and many attributes make it easier to make a linear separation of the data. Also that fact that you are evaluating on your training data and not new unseen data makes separation easier.
What kind of data are you trying to learn a model from? How much data? Can we see it?


Question: LEDs are an old technology, why did the industries take so long to put them into light bulbs? Was there any technological gap missing?

It is not possible to produce white light without an efficient blue LED, either using RGB LEDs or a blue LED + yellow phosphor.
The breakthrough was the invention of the high-brightness Gallium-Nitride blue LED by Shuji Nakamura at Nichia
in the early 1990s.
It still took a while to get the overall efficiency up to the level of fluorescent bulbs, and it's only in the last decade that LEDs finally came out on top.


Question:
We use electromagnetic communication everywhere these days. Cell phones, wifi, old-school radio transmissions, television, deep space communication, etc.
I'm curious about some of the possible reasons we have never seen biological systems having evolved to use electromagnetic, i.e. radio, for communication. The one obvious exception to this are organisms that generate their own light, i.e. bioluminescence. Cuttlefish are masters of this, and many other species as well.
It seems like bio-radio could have offered all kinds of evolutionary advantages for animals capable of using it.
Are their basic physical limits in chemistry, or excess energy requirements or something that would basically have made this impossible? Or was this perhaps just something that life never evolved to use, but would otherwise be possible in evolution?

There is a very different mechanism for generation (and detection) of ultraviolet, visible and infrared light vs radio waves.
For the first, it is possible to generate it using chemical reactions (that is, chemiluminescence, bioluminescence) with a typical energy of order of 2 eV (electronovolts). Also, it is easy to detect with similar means - coupling to a bond (e.g. using opsins).
For much longer electromagnetic waves, and much lower energies per photon, such mechanism does not work. There are two reasons:

typical energy levels for molecules (but it can be worked around),
thermal noise has energies (0.025 eV) which are higher than radio wave photon energies (<0.001 eV) (it rules out both controlled creation and detection using molecules).

In other words - radiation which is less energetic than thermal radiation (far infrared) is not suitable for communication using molecular mechanisms, as thermal noise jams transmission (making the sender firing at random and making the receiver being blind by noise way stronger than the signal).
However, one can both transmit, and detect it, using wires. In principle it is possible; however, without good conductors (like metals, not - salt solutions) it is not an easy task (not impossible though).


Question: Are there any free open source software tools available for simulating Oxford Nanopore reads?

Simulators designed specifically for Oxford Nanopore:

NanoSim
NanoSim-H
SiLiCO
ReadSim
DeepSimulator

General long read simulators:

Loresim
Loresim 2
FASTQsim
LongISLND

For an exhaustive list of existing read simulators, see page 15 of my thesis, Novel computational techniques for mapping and
classifying Next-Generation Sequencing data.


Question:
From my understanding of light, you are always looking into the past based on how much time it takes the light to reach you from what you are observing.
For example when you see a star burn out, if the star was 5 light years away then the star actually burnt out 5 years ago.
So I am 27 years old, if I was 27 light years away from Earth and had a telescope strong enough to view Earth, could I theoretically view myself being born?

Yes, you can.  And you do not even need to leave the Earth to do it.
You are always viewing things in the past, just as you are always hearing things in the past.  If you see someone do something, who is 30 meters away, you are seeing what happened $(30\;\mathrm{m})/(3\times10^8\;\mathrm{m}/\mathrm{s}) = 0.1\;\mu\mathrm{s}$
in the past.
If you had a mirror on the moon (about 238K miles away), you could see about 2.5 seconds into earth's past.  If that mirror was on Pluto, you could see about 13.4 hours into Earth's past.
If you are relying on hearing, you hear an event at 30 m away about 0.1 s after it occurs.  That is why runners often watch the starting pistol at an event, because they can see a more recent picture of the past than they can hear.
To more directly answer the intent of your question:  Yes, if you could magically be transported 27 lightyears away, or had a mirror strategically placed 13.5 lightyears away, you could see yourself being born.


Question:
I have come across many sorting algorithms during my high school studies. However, I never know which is the fastest (for a random array of integers). So my questions are:

Which is the fastest currently known sorting algorithm?
Theoretically, is it possible that there are even faster ones? So, what's the least complexity for sorting?

In general terms, there are the $O(n^2)$ sorting algorithms, such as insertion sort, bubble sort, and selection sort, which you should typically use only in special circumstances; Quicksort, which is worst-case $O(n^2)$ but quite often $O(n\log n)$ with good constants and properties and which can be used as a general-purpose sorting procedure; the $O(n\log n)$ algorithms, like merge-sort and heap-sort, which are also good general-purpose sorting algorithms; and the $O(n)$, or linear, sorting algorithms for lists of integers, such as radix, bucket and counting sorts, which may be suitable depending on the nature of the integers in your lists.
If the elements in your list are such that all you know about them is the total order relationship between them, then optimal sorting algorithms will have complexity $\Omega(n\log n)$. This is a fairly cool result and one for which you should be able to easily find details online. The linear sorting algorithms exploit further information about the structure of elements to be sorted, rather than just the total order relationship among elements.
Even more generally, optimality of a sorting algorithm depends intimately upon the assumptions you can make about the kind of lists you're going to be sorting (as well as the machine model on which the algorithm will run, which can make even otherwise poor sorting algorithms the best choice; consider bubble sort on machines with a tape for storage). The stronger your assumptions, the more corners your algorithm can cut. Under very weak assumptions about how efficiently you can determine "sortedness" of a list, the optimal worst-case complexity can even be $\Omega(n!)$.
This answer deals only with complexities. Actual running times of implementations of algorithms will depend on a large number of factors which are hard to account for in a single answer.


Question: There is probably some reason for this, but I can't figure out what it is.  I agree that it probably doesn't happen 100% of the time, but most all of the time, the cream is clinging to just one of the cookie sides.

The "stuff" sticks to itself better than it sticks to the cookie. Now if you pull the cookies apart, you create a region of local stress, and one of the two interfaces will begin to unstick. At that point, you get something called "stress concentration" at the tip of the crack (red arrow) - where the tensile force concentrates:

To get the stuff to start separating at a different part of the cookie, you need to tear the stuffing (which is quite good at sticking to itself) and initiate a delamination at a new point (where there is no stress concentration).
Those two things together explain your observation.
Cookie picture credit (also explanation about manufacturing process introducing a bias)
Update
A plausible explanation was given in this article describing work by Cannarella et al:

Nabisco won’t divulge its Oreo secrets, but in 2010, Newman’s Own—which makes a very similar “Newman-O”—let the Discovery Channel into its factory to see how their version of cookies are made. The key aspect for twist-off purposes: A pump applies the cream onto one wafer, which is then sent along the line until a robotic arm places a second wafer on top of the cream shortly after. The cream always adheres better to one of these wafers—and all of the cookies in a single box end up oriented in the same direction.
  Which side is the stronger wafer-to-cream interface? “We think we know,” says Spechler. The key is that fluids flow better at high temperatures. So the hot cream flows easily over the first wafer, filling in the tiny cracks of the cookie and sticking to it like hot glue, whereas the cooler cream just kind of sits on the edges of those crevices.


Question:
What is the difference between minimum spanning tree algorithm and a shortest path algorithm?
In my data structures class we covered two minimum spanning tree algorithms (Prim's and Kruskal's) and one shortest path algorithm (Dijkstra's).
Minimum spanning tree is a tree in a graph that spans all the vertices and total weight of a tree is minimal. Shortest path is quite obvious, it is a shortest path from one vertex to another.
What I don't understand is since minimum spanning tree has a minimal total weight, wouldn't the paths in the tree be the shortest paths? Can anybody explain what I'm missing?
Any help is appreciated.

You are right that the two algorithms of Dijkstra (shortest paths from a single start node) and Prim (minimal weight spanning tree starting from a given node) have a very similar structure. They are both greedy (take the best edge from the present point of view) and build a tree spanning the graph.
The value they minimize however is different. Dijkstra selects as next edge the one that leads out from the tree to a node not yet chosen closest to the starting node. (Then with this choice, distances are recalculated.) Prim choses as edge the shortest one leading out of the tree constructed so far. So, both algorithms chose a "minimal edge". The main difference is the value chosen to be minimal. For Dijkstra it is the length of the complete path from start node to the candidate node, for Prim it is just the weight of that single edge.
To see the difference you should try to construct a few examples to see what happens, That is really instructive. The simplest example that shows different behaviour is a triangle $x,y,z$ with edges $\{x,y\}$ and  $\{x,z\}$ of length 2, while  $\{y,z\}$ has length 1. Starting in $x$ Dijkstra will choose  $\{x,y\}$ and  $\{x,z\}$ (giving two paths of length 2) while Prim chooses  $\{x,y\}$ and $\{y,z\}$ (giving spanning tree of weight 3).

As for Kruskal, that is slightly different. It solves the minimal spanning tree, but during execution it chooses edge that may not form a tree, they just avoid cycles. So the partial solutions may be disconnected. In the end you get a tree.


Question:
The power spectrum of a signal can be calculated by taking the magnitude squared of its Fourier transform. Being an audio person, the signal of interest for me would be a time series.
How does this representation differ from a PSD (power spectral density), and importantly, in what practical situations should one use a PSD instead of the power spectrum described above?

The power spectral density describes the density of power in a stationary random process $X(t)$ per unit of frequency. By the Wiener-Khinchin theorem, it can be calculated as follows for a wide-sense stationary random process:
$$
S_{xx}(f) = \int_{-\infty}^{\infty} r_{xx}(\tau) e^{-j2\pi f \tau} d\tau
$$
where $r_{xx}(\tau)$ is the autocorrelation function of the process $X(t)$:
$$
r_{xx}(\tau) = \mathbb{E}\left(X(t)X(t - \tau)\right)
$$
This is only valid for a wide-sense stationary process because its autocorrelation function is only a function of the time lag $\tau$ and not the absolute time $t$; stated differently, this means that its second-order statistics don't change as a function of time.
With that said, if you have a sufficiently-detailed and accurate statistical model for your signal, then you can calculate its power spectral density using the relationship above. As an example, this can be used to calculate the power spectral density of communications signals, given the statistics of the information symbols carried by the signal and any pulse shaping employed during transmission.
In most practical situations, this level of information is not available, however, and one must resort to estimating a given signal's power spectral density. One very straightforward approach is to take the squared magnitude of its Fourier transform (or, perhaps, the squared magnitude of several short-time Fourier transforms and average them) as the estimate of the PSD. However, assuming that the signal you're observing contains some stochastic component (which is often the case), this is again just an estimate of what the true underlying PSD is based upon a single realization (i.e. a single observation) of the random process. Whether the power spectrum that you calculate bears any meaningful resemblance to the actual PSD of the process is situation-dependent.
As this previous post notes, there are many methods for PSD estimation; which is most suitable depends upon the character of the random process, any a priori information that you might have, and what features of the signal you're most interested in.


Question:
Air is 1% argon.  Argon is heavier than air.
Why doesn't the argon concentrate in low-lying areas, choking out life there?

It does.  You would find the average percentage of the atmosphere that is argon is very slightly higher at the floor of valleys.  However, bear in mind first of all it wouldn't be anywhere near a complete stratification -- a layer of pure argon, then another of pure N2, and so on.  A mixture of nearly ideal gases doesn't do that, at least at equilibrium, because it would eliminate the considerable entropy of mixing.  (It can happen in liquids because liquids have strong intermolecular forces that normally favor separation and oppose the entropy of mixing.)  Another way to think about it is that since the atoms and molecules in gases don't (much) interact, there's nothing stopping an individual argon atom going slightly faster than nearby nitrogen and oxygen molecules from bouncing up higher than they do.
What you would get in a theoretical ideal (uniform gravitational field, complete stillness -- no wind -- and uniform temperature) would be an exponential fall of pressure with altitude, and the exponential for heavier gases would be steeper than for lighter gases.  That would result in enrichment of the heavier gases at lower altitudes.  A little work starting from the Boltzmann distribution of gravitational potential energies of each type of atom and molecule would get you an ideal estimate of the argon excess as a function of altitude.
In practice the lower atmosphere has so much mixing due to wind and big thermal gradients that I doubt you could even measure the mild excess of argon and other heavy gases.
There is one fascinating short-term exception, which bears directly on your question.  Sometimes volcanoes will belch out a considerable quantity of CO2, which is significantly denser than air, and this CO2 can accumulate briefly in a thick layer at the bottom of a valley or over a lake, if there isn't much wind.  It can persist for some hours, perhaps days, before it diffuses away and is mixed with the rest of the atmosphere.
Then indeed the valley bottom becomes an invisible death trap for humans and animals: walk into the valley, or be unable to exit fast enough when it happens, and you will suffocate for no reason you can see.  The most famous example of this is the Lake Nyos disaster in 1986 which killed thousands of humans and animals.  I think the government now has mixing devices installed in that lake to prevent any future sudden release of CO2.


Question: Why is it that when you look in the mirror left and right directions appear flipped, but not the up and down?

Here's a video of physicist Richard Feynman discussing this question.
Imagine a blue dot and a red dot.  They are in front of you, and the blue dot is on the right.  Behind them is a mirror, and you can see their image in the mirror.  The image of the blue dot is still on the right in the mirror.
What's different is that in the mirror, there's also a reflection of you.  From that reflection's point of view, the blue dot is on the left.
What the mirror really does is flip the order of things in the direction perpendicular to its surface.  Going on a line from behind you to in front of you, the order in real space is

Your back
Your front
Dots
Mirror

The order in the image space is

Mirror
Dots
Your front
Your back

Although left and right are not reversed, the blue dot, which in reality is lined up with your right eye, is lined up with your left eye in the image.
The key is that you are roughly left/right symmetric.  The eye the blue dot is lined up with is still your right eye, even in the image.  Imagine instead that Two-Face was looking in the mirror. (This is a fictional character whose left and right side of his face look different.  His image on Wikipedia looks like this:)

If two-face looked in the mirror, he would instantly see that it was not himself looking back!  If he had an identical twin and looked right at the identical twin, the "normal" sides of their face would be opposite each other.  Two-face's good side is the right.  When he looked at his twin, the twin's good side would be to the original two-face's left.
Instead, the mirror Two-face's good side is also to the right.  Here is an illustration:

So two-face would not be confused by the dots.  If the blue dot is lined up with Two-Face's good side, it is still lined up with his good side in the mirror.  Here it is with the dots:

Two-face would recognize that left and right haven't been flipped so much as forward and backward, creating a different version of himself that cannot be rotated around to fit on top the original.


Question:
In most audio processing tasks, one of the most used transformations is MFCC (Mel-frequency cepstral coefficients).
I mostly know the math that's behind the MFCC: I understand both the filterbank step and the Mel frequency scaling.
What I don't get is the DCT (Discrete Cosine Transform) step: What kind of information do I get in this step? What is a good visual representation of this step?

You can think of the DCT as a compression step. Typically with MFCCs, you will take the DCT and then keep only the first few coefficients. This is basically the same reason that the DCT is used in JPEG compression. DCTs are chosen because their boundary conditions work better on these types of signals.
Let's contrast the DCT with the Fourier transform. The Fourier transform is made up of sinusoids that have an integer number of cycles. This means, all of the Fourier basis functions start and end at the same value -- they do not do a good job of representing signals that start and end at different values. Remember that the Fourier transform assumes a periodic extension: If you imagine your signal on a sheet of paper, the Fourier transform wants to roll that sheet into a cylinder so that the left and right sides meet.
Think of a spectrum that is shaped roughly like a line with negative slope (which is pretty typical). The Fourier transform will have to use a lot of different coefficients to fit this shape. On the other hand, the DCT has cosines with half-integer numbers of cycles. There is, for example, a DCT basis function that looks vaguely like that line with negative slope. It does not assume a period extension (instead, an even extension), so it will do a better job of fitting that shape.
So, let's put this together. Once you've computed the Mel-frequency spectrum, you have a representation of the spectrum that is sensitive in a way similar to how human hearing works. Some aspects of this shape are more relevant than others. Usually, the larger more overarching spectral shape is more important than the noisy fine details in the spectrum. You can imagine drawing a smooth line to follow the spectral shape, and that the smooth line you draw might tell you just about as much about the signal.
When you take the DCT and discard the higher coefficients, you are taking this spectral shape, and only keeping the parts that are more important for representing this smooth shape. If you used the Fourier transform, it wouldn't do such a good job of keeping the important information in the low coefficients.
If you think about feeding the MFCCs as features to a machine learning algorithm, these lower-order coefficients will make good features, since they represent some simple aspects of the spectral shape, while the higher-order coefficients that you discard are more noise-like and are not important to train on. Additionally, training on the Mel spectrum magnitudes themselves would probably not be as good because the particular amplitude at different frequencies are less important than the general shape of the spectrum.


Question:
There are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge, etc.
So I'd like to know what mathematical proofs you've come across that you think other mathematicians should know, and why.

Here is my favourite "wow" proof .
Theorem
There exist two  positive irrational numbers $s,t$  such that $s^t$ is rational.
Proof
If $\sqrt2^\sqrt 2$ is rational, we may take $s=t=\sqrt 2$ .
 If $\sqrt 2^\sqrt 2$ is irrational , we may take $s=\sqrt 2^\sqrt 2$ and  $t=\sqrt 2$ since  $(\sqrt 2^\sqrt 2)^\sqrt 2=(\sqrt 2)^ 2=2$.