Question:
These formulae are used if the molecule has a possible plane of symmetry. One such example would be:

Here the carbons marked with an asterisk are stereogenic centres (the asterisk is not used to mark isotopes). We can clearly see that if carbon number 2 (in the entire longest chain) and the carbon number 4 have opposite stereogenic configuration, then the molecule will be achiral because it will have a plane of symmetry. For example, (2R,3S,4S)-pentanetriol will be a meso compound with a plane of symmetry. This compound clearly satisfies the criteria we have set for the type of molecules being discussed.
Let us take another example.

Again, we see that if the molecule has the same geometrical configuration at the first and the third double bond, then the molecule has a plane of symmetry. For example, (2,7)-diphenylocta-(2Z,4E,6Z)-triene clearly has a plane of symmetry.
Our goal is to find the total number of stereoisomers such compounds can have in total. We can assume that a molecule does not have both, a double bond and a chiral centre.
In our class notes, we have written these formulae:

For geometrical isomers (i.e. in case of polyenes),

If 'n' is even (here n is the number of double bonds):$$\text{Number of stereoisomers} = 2^{n-1}+2^{n/2-1}$$
If 'n' is odd, then: $$\text{Number of stereoisomers} = 2^{n-1}+2^{(n-1)/2}$$

And for optical isomers (molecules with chiral centres):

If 'n' is even (here n is the number of chiral centres): $$\text{Number of enantiomers} = 2^{n-1}$$ $$\text{Number of meso compounds} = 2^{n/2-1}$$ $$\text{Total number of optical isomers} = 2^{n-1}+2^{n/2-1}$$
If 'n' is odd: $$\text{Number of enantiomers} = 2^{n-1}-2^{(n-1)/2}$$ $$\text{Number of meso compounds} = 2^{(n-1)/2}$$ $$\text{Total number of optical isomers} = 2^{n-1}$$

How to derive these formulae?

I managed to crack the formula for optical isomers with odd chiral centers, so I'll share my attempt here. Hopefully others may innovate on it and post solutions for other formulae.

Pseudo-chiral carbon atoms - an introduction
The Gold Book defines pseudo-chiral/pseudo-asymmetric carbon atom as:

a tetrahedrally coordinated carbon atom bonded to four different entities, two and only two of which have the same constitution but opposite chirality sense.

This implies that, in your case:

If chiral carbons 2 and 4 both have configuration R (or both S), then the central carbon 3 will be achiral/symmetric, because now "two and only two of its groups which have the same constitution" will have the same chirality sense instead. (Your approach by "plane of symmetry" is wrong. Find more details on this question)
Hence, there can be two stereoisomers (r and s) possible on the 3rd carbon due to its pesudochirality. But, there will be only one if both substituents on left and right have the same optical configurations.

Building up an intuition by manual counting
For optical isomers with odd number of chiral centers and similar ends, you can guess that, if there are $n$ chiral centers, then the middle ($\frac{n+1}2$-th) carbon atom will be pseudo-chiral. To build up an intuition, we'll manually count optical isomers for $n=3$ and $n=5$:
Case $n=3$
Take the example of pentane-2,3,4-triol itself. We find four (=$2^{n-1}$) isomers:
$$
\begin{array}{|c|c|c|}\hline
\text{C2}&\text{C3}&\text{C4}\\\hline
R&-&R\\\hline
S&-&S\\\hline
R&S&R\\\hline
R&S&S\\\hline
\end{array}
$$
As expected from the relevant formula, we find that the first two ($=2^\frac{n-1}2$) are meso compounds, and the remaining two ($=2^{n-1}-2^\frac{n-1}2$) are enantiomers.
Case $n=5$
Take the example of heptane-2,3,4,5,6-pentol:

We expect $16~(=2^{n-1})$ isomers, with the C4 carbon being pseudo-chiral. To avoid a really large table, we observe that the number of meso isomers is easily countable (<< number of enantiomers). Here is a table of those four (=$2^\frac{n-1}2$) meso isomers:
$$
\begin{array}{|c|c|c|c|c|c|}\hline
\text{C2}&\text{C3}&\text{C4}&\text{C5}&\text{C6}\\\hline
R&R&-&R&R\\\hline
R&S&-&S&R\\\hline
S&R&-&R&S\\\hline
S&S&-&S&S\\\hline
\end{array}
$$
Note that the total optical isomers are given by $2^{n-1}$ isomers (more on that below). Hence, the number of enantiomers is easily $12(=2^{n-1}-2^\frac{n-1}2)$.

A formula for the number of meso isomers
As you must have observed from the table, the sequence of optical configurations, when read from the fourth carbon atom, is exactly the same towards both left and right. In other words, if we fix an arbitrary permutation for the optical configurations of the carbon atoms on the left (say RSS), then we will get only one unique permutation of the optical configurations on the right (SSR).
We know that each carbon on the left has two choices (R or S), and there are $\frac{n-1}{2}$ carbon atoms on the left. Hence, the total number of permutations will be $2\times2\times2\cdots\frac{n-1}{2}\text{ times}=2^\frac{n-1}{2}$.
Since, our description ("the sequence of optical configurations, when read from the fourth carbon atom, is exactly the same on both left and right") describes meso isomers, we have hence counted the number of meso isomers, which is $2^\frac{n-1}{2}$.

A formula for the number of total isomers
We note that there are $n$ chiral carbons (including that pseudo chiral carbon). Again, each chiral carbon has $2$ choices. Hence, the maximum possible number of optical isomers is $2\times2\times2\cdots n\text{ times}=2^n$. This is the maximum possible, not the actual total number of isomers, which is much lower.
The reduction in number of isomers is because the string of optical configurations reads exactly the same from either terminal carbon. Example: RSsRS is the same as SRsSR. This happens because the compound has "similar ends"
Hence, each permutation has been over counted exactly twice. Thus, the actual total number of isomers is half of the maximum possible, and is  $=\frac{2^n}2=2^{n-1}$.

Conclusion
Hence, we have derived that, if 'n' (number of chiral centers) is odd for a compound with similar ends, then:

$\text{Number of meso isomers} = 2^{(n-1)/2}$
$\text{Total number of optical isomers} = 2^{n-1}$
$\text{Number of enantiomers} = 2^{n-1}-2^{(n-1)/2}$


Question:
What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)?

This question is based on the question of Kevin Lin, which didn't quite fit in Mathoverflow. Answers at any level of sophistication are welcome.

The ancient Greeks had a theory that the sun, the moon, and the planets move around the Earth in circles.  This was soon shown to be wrong.  The problem was that if you watch the planets carefully, sometimes they move backwards in the sky.  So Ptolemy came up with a new idea - the planets move around in one big circle, but then move around a little circle at the same time.  Think of holding out a long stick and spinning around, and at the same time on the end of the stick there's a wheel that's spinning.  The planet moves like a point on the edge of the wheel.
Well, once they started watching really closely, they realized that even this didn't work, so they put circles on circles on circles...
Eventually, they had a map of the solar system that looked like this:

This "epicycles" idea turns out to be a bad theory.  One reason it's bad is that we know now that planets orbit in ellipses around the sun.  (The ellipses are not perfect because they're perturbed by the influence of other gravitating bodies, and by relativistic effects.)
But it's wrong for an even worse reason that that, as illustrated in this  wonderful youtube video.
In the video, by adding up enough circles, they made a planet trace out Homer Simpson's face.  It turns out we can make any orbit at all by adding up enough circles, as long as we get to vary their size and speeds.
So the epicycle theory of planetary orbits is a bad one not because it's wrong, but because it doesn't say anything at all about orbits.  Claiming "planets move around in epicycles" is mathematically equivalent to saying "planets move around in two dimensions".  Well, that's not saying nothing, but it's not saying much, either!
A simple mathematical way to represent "moving around in a circle" is to say that positions in a plane are represented by complex numbers, so a point moving in the plane is represented by a complex function of time.  In that case, moving on a circle with radius $R$ and angular frequency $\omega$ is represented by the position
$$z(t) = Re^{i\omega t}$$
If you move around on two circles, one at the end of the other, your position is
$$z(t) = R_1e^{i\omega_1 t} + R_2 e^{i\omega_2 t}$$
We can then imagine three, four, or infinitely-many such circles being added.  If we allow the circles to have every possible angular frequency, we can now write
$$z(t) = \int_{-\infty}^{\infty}R(\omega) e^{i\omega t} \mathrm{d}\omega.$$
The function $R(\omega)$ is the Fourier transform of $z(t)$.  If you start by tracing any time-dependent path you want through two-dimensions, your path can be perfectly-emulated by infinitely many circles of different frequencies, all added up, and the radii of those circles is the Fourier transform of your path.  Caveat: we must allow the circles to have complex radii.  This isn't weird, though.  It's the same thing as saying the circles have real radii, but they do not all have to start at the same place.  At time zero, you can start however far you want around each circle.
If your path closes on itself, as it does in the video, the Fourier transform turns out to simplify to a Fourier series.  Most frequencies are no longer necessary, and we can write
$$z(t) = \sum_{k=-\infty}^\infty c_k e^{ik \omega_0 t}$$
where $\omega_0$ is the angular frequency associated with the entire thing repeating - the frequency of the slowest circle.  The only circles we need are the slowest circle, then one twice as fast as that, then one three times as fast as the slowest one, etc.  There are still infinitely-many circles if you want to reproduce a repeating path perfectly, but they are countably-infinite now.  If you take the first twenty or so and drop the rest, you should get close to your desired answer.  In this way, you can use Fourier analysis to create your own epicycle video of your favorite cartoon character.
That's what Fourier analysis says.  The questions that remain are how to do it, what it's for, and why it works.  I think I will mostly leave those alone.  How to do it - how to find $R(\omega)$ given $z(t)$ is found in any introductory treatment, and is fairly intuitive if you understand orthogonality.  Why it works is a rather deep question.  It's a consequence of the spectral theorem.
What it's for has a huge range.  It's useful in analyzing the response of linear physical systems to an external input, such as an electrical circuit responding to the signal it picks up with an antenna or a mass on a spring responding to being pushed.  It's useful in optics; the interference pattern from light scattering from a diffraction grating is the Fourier transform of the grating, and the image of a source at the focus of a lens is its Fourier transform.  It's useful in spectroscopy, and in the analysis of any sort of wave phenomena.  It converts between position and momentum representations of a wavefunction in quantum mechanics.  Check out this question on physics.stackexchange for more detailed examples.  Fourier techniques are useful in signal analysis, image processing, and other digital applications.  Finally, they are of course useful mathematically, as many other posts here describe.


Question:
I need to get started using Finite Element Methods. I am about to start reading Numerical solutions of partial differential equations by the finite element method by Claes Johnson, but it's dated 1987.
Two questions:
1) What newer good resources/textbooks/e-books/lecture notes on  this subject are out there?
2) How much am I missing by reading a 1987 book?
Thanks.

There are lots of modern finite element references, but I will just comment on a few books that I think are practical and relevant to applications, plus one containing more comprehensive analysis.

Wriggers Nonlinear Finite Element Methods (2008) is a good general reference, but will be most relevant to those concerned with applications in structural mechanics (including contact, shells, and plasticity).
Elman, Silvester, and Wathen Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics (2005) is less comprehensive on finite element discretization techniques, but has good content on incompressible flow and a certain class of iterative solvers. It also explains the IFISS package.
Donéa and Huerta Finite element methods for flow problems (2003) covers similar material, but includes ALE moving mesh methods and compressible gas dynamics.
Brenner and Scott The mathematical theory of finite element methods (2008 revision) contains a rigorous theoretical development of discretizations for linear elliptic problems, including associated multigrid and domain decomposition theory. It does not treat transport-dominated problems, "messy" nonlinearities like plasticity, or non-polynomial bases.

These resources fail to cover topics such as discontinuous Galerkin methods or $H(curl)$ problems (Maxwell). I think papers are currently a better resource than books for these topics, although Hesthaven and Warburton Nodal discontinuous Galerkin methods (2008) is certainly worthwhile.
I also recommend reading the examples from open source finite element software packages such as FEniCS, Libmesh, and Deal.II.


Question: This particular question has been of a great deal of interest to me, especially since it dives at the heart of abiogenesis.

In 2010, Dr. Craig Venter actually used a bacterial shell and wrote DNA for it.

Scientists have created the world's first synthetic life form in a landmark experiment that paves the way for designer organisms that are built rather than evolved.
(Snip)
The new organism is based on an existing bacterium that causes mastitis in goats, but at its core is an entirely synthetic genome that was constructed from chemicals in the laboratory.

Keep in mind, this is only a synthetic genome, not a truly unique organism created from scratch.  Although I am confident that the technology will become available in the future.  As has been pointed out, the entire genome wasn't built de novo, but rather most of it was copied from a baseline which was built up from the base chemicals with no biological processes, and then the watermarks were added (still damn impressive since they took inorganic matter and made a living cell function with it). But they are working at building a totally unique genome from scratch (PDF).
This is actually quite an emerging field, so much so that the MIT press has set up an entire series of journalsfor this.  As far as to the purpose of these artificial organisms, most research funded by companies are meant to be for specific purposes that biology hasn't solved yet (such as a bacteria that eats a toxic waste or something).  Although, a lot of people are concerned about scientists venturing into the domain of theology.
In terms of abiogenesis, there are many resources to learn more about this.  Here is a list of 88 papers that discuss the natural mechanisms of abiogenesis (this list is a little old, so I am sure that there are many, many more papers at this time).
I also found this list of links and resources for artificial life.  I cannot verify the usefulness of this since the field is a bit outside my area of expertise.  However, it does seem quite extensive.
EDIT TO ADD:  Now we have "XNA" (a totally synthetic genome) on the way.


Question:
The Mersenne Twister is widely regarded as good. Heck, the CPython source says that it "is one of the most extensively tested generators in existence." But what does this mean? When asked to list properties of this generator, most of what I can offer is bad:

It's massive and inflexible (eg. no seeking or multiple streams),
It fails standard statistical tests despite its massive state size,
It has serious problems around 0, suggesting that it randomizes itself pretty poorly,
It's hardly fast

and so on. Compared to simple RNGs like XorShift*, it's also hopelessly complicated.
So I looked for some information about why this was ever thought to be good. The original paper makes lots of comments on the "super astronomical" period and 623-dimensional equidistribution, saying

Among many known measures, the tests based on the higher dimensional
  uniformity, such as the spectral test (c.f., Knuth [1981]) and the k-distribution test, described below, are considered to be strongest.

But, for this property, the generator is beaten by a counter of sufficient length! This makes no commentary of local distributions, which is what you actually care about in a generator (although "local" can mean various things). And even CSPRNGs don't care for such large periods, since it's just not remotely important.
There's a lot of maths in the paper, but as far as I can tell little of this is actually about randomness quality. Pretty much every mention of that quickly jumps back to these original, largely useless claims.
It seems like people jumped onto this bandwagon at the expense of older, more reliable technologies. For example, if you just up the number of words in an LCG to 3 (much less than the "only 624" of a Mersenne Twister) and output the top word each pass, it passes BigCrush (the harder part of the TestU01 test suite), despite the Twister failing it (PCG paper, fig. 2). Given this, and the weak evidence I was able to find in support of the Mersenne Twister, what did cause attention to favour it over the other choices?
This isn't purely historical either. I've been told in passing that the Mersenne Twister is at least more proven in practice than, say, PCG random. But are use-cases so discerning that they can do better than our batteries of tests? Some Googling suggests they're probably not.
In short, I'm wondering how the Mersenne Twister got its widespread positive reputation, both in its historical context and otherwise. On one hand I'm obviously skeptical of its qualities, but on the other it's hard to imagine that it was an entirely randomly occurrence.

The initial Mersenne-Twister (MT) was regarded as good for some years, until it was found out to be pretty bad with the more advanced TestU01 BigCrush tests and better PRNGs.
This page lists the Mersenne-Twister features in detail:
Positive Qualities

Produces 32-bit or 64-bit numbers (thus usable as source of random bits)
Passes most statistical tests

Neutral Qualities

Inordinately huge period of $2^{19937} - 1$
623-dimensionally equidistributed
Period can be partitioned to emulate multiple streams

Negative Qualities

Fails some statistical tests, with as few as 45,000 numbers. Fails LinearComp Test of the TestU01 Crush and BigCrush batteries.
Predictable — after 624 outputs, we can completely predict its output.
Generator state occupies 2504 bytes of RAM — in contrast, an extremely usable generator with a huger-than-anyone-can-ever-use period can fit in 8 bytes of RAM.
Not particularly fast.
Not particularly space efficient. The generator uses 20000 bits to store its internal state (20032 bits on 64-bit machines), but has a period of only $2^{19937}$, a factor of $2^{63}$ (or $2^{95}$) fewer than an ideal generator of the same size.
Uneven in its output; the generator can get into “bad states” that are slow to recover from.
Seedings that only differ slightly take a long time to diverge from each other; seeding must be done carefully to avoid bad states.
While jump-ahead is possible, algorithms to do so are slow to compute (i.e., require several seconds) and rarely provided by implementations.

Summary: Mersenne Twister is not good enough anymore, but most applications and libraries are not there yet.


Question:
I'm learning DSP slowly and trying to wrap my head around some terminology:

Question 1: Suppose I have the following filter difference equation:
$$y[n] = 2 x[n] + 4 x[n-2] + 6 x[n-3] + 8 x[n-4]$$
There are 4 coefficients on the right-hand side. Are the "number of taps" also 4? Is the "filter order" also 4?
Question 2: I am trying to use the MATLAB fir1(n, Wn) function. If I wanted to create a 10-tap filter, would I set $n=10$?
Question 3: Suppose I have the following recursive (presumably IIR) filter difference equation:
$$y[n] + 2 y[n-1] = 2 x[n] + 4 x[n-2] + 6 x[n-3] + 8 x[n-4]$$
How would I determine the "number of taps" and the "filter order" since the number of coefficients differ on the left-hand and right-hand sides?
Question 4: Are the following logical if-and-only-if statements true?

The filter is recursive $\iff$ The filter is IIR.
The filter is nonrecursive $\iff$ The filter is FIR.

OK, I'll try to answer your questions:
Q1: the number of taps is not equal the to the filter order. In your example the filter length is 5, i.e. the filter extends over 5 input samples [$x(n), x(n-1), x(n-2), x(n-3), x(n-4)$]. The number of taps is the same as the filter length. In your case you have one tap equal to zero (the coefficient for $x(n-1)$), so you happen to have 4 non-zero taps. Still, the filter length is 5. The order of an FIR filter is filter length minus 1, i.e. the filter order in your example is 4.
Q2: the $n$ in the Matlab function fir1() is the filter order, i.e. you get a vector with $n+1$ elements as a result (so $n+1$ is your filter length = number of taps).
Q3: the filter order is again 4. You can see it from the maximum delay needed to implement your filter. It is indeed a recursive IIR filter. If by number of taps you mean the number of filter coefficients, then for an $n^{th}$ order IIR filter you generally have $2(n+1)$ coefficients, even though in your example several of them are zero.
Q4: this is a slightly tricky one. Let's start with the simple case: a non-recursive filter always has a finite impulse response, i.e. it is a FIR filter. Usually a recursive filter has an infinite impulse response, i.e. it is an IIR filter, but there are degenerate cases where a finite impulse response is implemented using a recursive structure. But the latter case is the exception.


Question:
As a novice electronics hobbyist, I have heard these terms and more being thrown around everywhere. At its root, I understand that they are all based on communication between devices, computers, peripherals, etc.
I have a basic understanding of how all of them work, but I get confused when I see so many of them and am having difficulty understanding how they relate to each other. For example, is UART a subset of USART? What is the difference between RS232 and Serial? What are the core differences between all of these communication methods: reliability, cost, application, speed, hardware requirements?
If you can imagine, I have all of these terms written on cards, scattered on the coffee table, and I need someone to just help me organize my understanding. Forgive me if this question is a little vague, but I really feel that is the nature of this question all together.

Serial is an umbrella word for all that is "Time Division Multiplexed", to use an expensive term. It means that the data is sent spread over time, most often one single bit after another. All the protocols you're naming are serial protocols.
UART, for Universal Asynchronous Receiver Transmitter, is one of the most used serial protocols. It's almost as old as I am, and very simple. Most controllers have a hardware UART on board. It uses a single data line for transmitting and one for receiving data. Most often 8-bit data is transferred, as follows: 1 start bit (low level), 8 data bits and 1 stop bit (high level). The low level start bit and high level stop bit mean that there's always a high to low transition to start the communication. That's what describes UART. No voltage level, so you can have it at 3.3 V or 5 V, whichever your microcontroller uses. Note that the microcontrollers which want to communicate via UART have to agree on the transmission speed, the bit-rate, as they only have the start bits falling edge to synchronize. That's called asynchronous communication.
For long distance communication (That doesn't have to be hundreds of meters) the 5 V UART is not very reliable, that's why it's converted to a higher voltage, typically +12 V for a "0" and -12 V for a "1". The data format remains the same. Then you have RS-232 (which you actually should call EIA-232, but nobody does.)
The timing dependency is one of the big drawbacks of UART, and the solution is USART, for Universal Synchronous/Asynchronous Receiver Transmitter. This can do UART, but also a synchronous protocol. In synchronous there's not only data, but also a clock transmitted. With each bit a clock pulse tells the receiver it should latch that bit. Synchronous protocols either need a higher bandwidth, like in the case of Manchester encoding, or an extra wire for the clock, like SPI and I2C.
SPI (Serial Peripheral Interface) is another very simple serial protocol. A master sends a clock signal, and upon each clock pulse it shifts one bit out to the slave, and one bit in, coming from the slave. Signal names are therefore SCK for clock, MOSI for Master Out Slave In, and MISO for Master In Slave Out. By using SS (Slave Select) signals the master can control more than one slave on the bus. There are two ways to connect multiple slave devices to one master, one is mentioned above i.e. using slave select, and other is daisy chaining, it uses fewer hardware pins (select lines), but software gets complicated.
I2C (Inter-Integrated Circuit, pronounced "I squared C") is also a synchronous protocol, and it's the first we see which has some "intelligence" in it; the other ones dumbly shifted bits in and out, and that was that. I2C uses only 2 wires, one for the clock (SCL) and one for the data (SDA). That means that master and slave send data over the same wire, again controlled by the master who creates the clock signal. I2C doesn't use separate Slave Selects to select a particular device, but has addressing. The first byte sent by the master holds a 7 bit address (so that you can use 127 devices on the bus) and a read/write bit, indicating whether the next byte(s) will also come from the master or should come from the slave. After each byte, the receiver must send a "0" to acknowledge the reception of the byte, which the master latches with a 9th clock pulse. If the master wants to write a byte, the same process repeats: the master puts bit after bit on the bus and each time gives a clock pulse to signal that the data is ready to be read. If the master wants to receive data it only generates the clock pulses. The slave has to take care that the next bit is ready when the clock pulse is given. This protocol is patented by NXP (formerly Phillips), to save licensing cost, Atmel using the word TWI (2-wire interface) which exactly same as I2C, so any AVR device will not have I2C but it will have TWI.
Two or more signals on the same wire may cause conflicts, and you would have a problem if one device sends a "1" while the other sends a "0". Therefore the bus is wired-OR'd: two resistors pull the bus to a high level, and the devices only send low levels. If they want to send a high level they simply release the bus.
TTL (Transistor Transistor Logic) is not a protocol. It's an older technology for digital logic, but the name is often used to refer to the 5 V supply voltage, often incorrectly referring to what should be called UART.

About each of these you can write a book, and it looks I'm well on my way. This is just a very brief overview, let us know if some things need clarification.


Question:
In our computer systems lecture we were introduced to the MIPS processor. It was (re)developed over the course of the term and has in fact been quite easy to understand. It uses a RISC design, that is its elementary commands are regularly encoded and there are only few of them in order to keep the wires simple.
It was mentioned that CISC follows a different philosophy. I looked briefly at the x86 instruction set and was shocked. I can not image how anyone would want to build a processor that uses so complex a command set!
So I figure there have to be good arguments why large portions of the processor market use CISC architectures. What are they?

There is a general historical trend.
In the olden days, memories were small, and so programs were perforce small. Also, compilers were not very smart, and many programs were written in assembler, so it was considered a good thing to be able to write a program using few instructions. Instruction pipelines were simple, and processors grabbed one instruction at a time to execute it. The machinery inside the processor was quite complex anyway; decoding instructions was not felt to be much of a burden.
In the 1970s, CPU and compiler designers realized that having such complex instructions was not so helpful after all. It was difficult to design processors in which those instructions were really efficient, and it was difficult to design compilers that really took advantage of these instructions. Chip area and compiler complexity was better spent on more generic pursuits such as more general-purpose registers. The Wikipedia article on RISC explains this in more detail.
MIPS is the ultimate RISC architecture, which is why it's taught so often.
The x86 family is a bit different. It was originally a CISC architecture meant for systems with very small memory (no room for large instructions), and has undergone many successive versions. Today's x86 instruction set is not only complicated because it's CISC, but because it's really a 8088 with a 80386 with a Pentium possibly with an x86_64 processor.
In today's world, RISC and CISC are no longer the black-and-white distinction they might have been once. Most CPU architectures have evolved to different shades of grey.
On the RISC side, some modern MIPS variants have added multiplication and division instructions, with a non-uniform encoding. ARM processors have become more complex: many of them have a 16-bit instruction set called Thumb in addition to the “original” 32-bit instructions, not to mention Jazelle to execute JVM instructions on the CPU. Modern ARM processors also have SIMD instructions for multimedia applications: some complex instructions do pay after all.
On the CISC side, all recent processors are to some extent RISC inside. They have microcode to define all these complex macro instructions. The sheer complexity of the processor makes the design of each model take several years, even with a RISC design, what with the large number of components, with pipelining and predictive execution and whatnot.
So why do the fastest processors remain CISC outside? Part of it, in the case of the x86 (32-bit and 64-bit) family, is historical compatibility. But that's not the whole of it. In the early 2000s, Intel tried pushing the Itanium architecture. Itanium is an extreme case of complex instructions (not really CISC, though: its design has been dubbed EPIC). It even does away with the old-fashioned idea of executing instructions in sequence: all instructions are executed in parallel until the next barrier. One of the reasons Itanium didn't take is that nobody, whether at Intel or elsewhere, could write a decent compiler for it. Now a good old mostly-sequential processor like x86_64, that's something we understand.


Question: Are there any theoretical machines which exceed Turing machines capability in at least some areas?

Yes, there are theoretical machines which exceed the Turing machines in computational power, such as Oracle machines and Infinite time Turing machines. The buzzword that you should feed to Google is hypercomputation.


Question:
Why the drain the source terminal of the MOSFET function differently while their physical structure is similar/symmetrical ?
This is a MOSFET:

You can see that the drain and source are similar.
So why do I need to connect one of them to VCC and the other to GND ?

Myth: manufactures conspire to put internal diodes in discrete components so only IC designers can do neat things with 4-terminal MOSFETs.
Truth: 4-terminal MOSFETs aren't very useful.
Any P-N junction is a diode (among other ways to make diodes). A MOSFET has two of them, right here:

That big chunk of P-doped silicon is the body or the substrate. Considering these diodes, one can see it's pretty important that the body is always at a lower voltage than the source or the drain. Otherwise, you forward-bias the diodes, and that's probably not what you wanted.
But wait, it gets worse! A BJT is a three layer sandwich of NPN materials, right? A MOSFET also contains a BJT:

If the drain current is high, then the voltage across the channel between the source and the drain can also be high, because \$R_{DS(on)}\$ is non-zero. If it's high enough to forward-bias the body-source diode, you don't have a MOSFET anymore: you have a BJT. That's also not what you wanted.
In CMOS devices, it gets even worse. In CMOS, you have PNPN structures, which make a parasitic thyristor. This is what causes latchup.
Solution: short the body to the source. This shorts the base-emitter of the parasitic BJT, holding it firmly off. Ideally you don't do this through external leads, because then the "short" would also have high parasitic inductance and resistance, making the "holding off" of the parasitic BJT not so strong. Instead, you short them right at the die.
This is why MOSFETs aren't symmetrical. It may be that some designs otherwise are symmetrical, but to make a MOSFET that behaves reliably like a MOSFET, you have to short one of those N regions to the body. To whichever one you do that, it's now the source, and the diode you didn't short out is the "body diode".
This isn't anything specific to discrete transistors, really. If you do have a 4-terminal MOSFET, then you need to make sure that the body is always at the lowest voltage (or highest, for P-channel devices). In ICs, the body is the substrate for the whole IC, and it's usually connected to ground. If the body is at a lower voltage than the source, then you must consider body effect. If you take a look at a CMOS circuit where there's a source not connected to ground (like the NAND gate below), it doesn't really matter, because if B is high, then the lower-most transistor is on, and the one above it actually does have its source connected to ground. Or, B is low, and the output is high, and there isn't any current in the lower two transistors.


Question:
Everyone discusses the Fourier transform when discussing signal processing. Why is it so important to signal processing and what does it tell us about the signal?
Does it only apply to digital signal processing or does it apply to analog signals as well?

This is quite a broad question and it indeed is quite hard to pinpoint why exactly Fourier transforms are important in signal processing. The simplest, hand waving answer one can provide is that it is an extremely powerful mathematical tool that allows you to view your signals in a different domain, inside which several difficult problems become very simple to analyze.
Its ubiquity in nearly every field of engineering and physical sciences, all for different reasons, makes it all the more harder to narrow down a reason. I hope that looking at some of its properties which led to its widespread adoption along with some practical examples and a dash of history might help one to understand its importance.
History:
To understand the importance of the Fourier transform, it is important to step back a little and appreciate the power of the Fourier series put forth by Joseph Fourier. In a nut-shell, any periodic function $g(x)$ integrable on the domain $\mathcal{D}=[-\pi,\pi]$ can be written as an infinite sum of sines and cosines as
$$g(x)=\sum_{k=-\infty}^{\infty}\tau_k e^{\jmath k x}$$
$$\tau_k=\frac{1}{2\pi}\int_{\mathcal{D}}g(x)e^{-\jmath k x}\ dx$$
where $e^{\imath\theta}=\cos(\theta)+\jmath\sin(\theta)$. This idea that a function could be broken down into its constituent frequencies (i.e., into sines and cosines of all frequencies) was a powerful one and forms the backbone of the Fourier transform.
The Fourier transform:
The Fourier transform can be viewed as an extension of the above Fourier series to non-periodic functions. For completeness and for clarity, I'll define the Fourier transform here. If $x(t)$ is a continuous, integrable signal, then its Fourier transform, $X(f)$ is given by
$$X(f)=\int_{\mathbb{R}}x(t)e^{-\jmath 2\pi f t}\ dt,\quad \forall f\in\mathbb{R}$$
and the inverse transform is given by
$$x(t)=\int_{\mathbb{R}}X(f)e^{\jmath 2\pi f t}\ df,\quad \forall t\in\mathbb{R}$$
Importance in signal processing:
First and foremost, a Fourier transform of a signal tells you what frequencies are present in your signal and in what proportions.

Example: Have you ever noticed that each of your phone's number buttons
  sounds different when you press during a call and that it sounds the same for every phone model? That's because they're each composed of two different sinusoids which can be used to uniquely identify the button. When you use your phone to punch in combinations to navigate a menu, the way that the other party knows what keys you pressed is by doing a Fourier transform of the input and looking at the frequencies present.

Apart from some very useful elementary properties which make the mathematics involved simple, some of the other reasons why it has such a widespread importance in signal processing are:

The magnitude square of the Fourier transform, $\vert X(f)\vert^2$ instantly tells us how much power the signal $x(t)$ has at a particular frequency $f$.
From Parseval's theorem (more generally Plancherel's theorem), we have
$$\int_\mathbb{R}\vert x(t)\vert^2\ dt = \int_\mathbb{R}\vert X(f)\vert^2\ df$$
which means that the total energy in a signal across all time is equal to the total energy in the transform across all frequencies. Thus, the transform is energy preserving.
Convolutions in the time domain are equivalent to multiplications in the frequency domain, i.e., given two signals $x(t)$ and $y(t)$, then if
$$z(t)=x(t)\star y(t)$$
where $\star$ denotes convolution, then the Fourier transform of $z(t)$ is merely
$$Z(f)=X(f)\cdot Y(f)$$
For discrete signals, with the development of efficient FFT algorithms, almost always, it is faster to implement a convolution operation in the frequency domain than in the time domain.
Similar to the convolution operation, cross-correlations are also easily implemented in the frequency domain as $Z(f)=X(f)^*Y(f)$, where $^*$ denotes complex conjugate.
By being able to split signals into their constituent frequencies, one can easily block out certain frequencies selectively by nullifying their contributions.

Example: If you're a football (soccer) fan, you might've been
  annoyed at the constant drone of the vuvuzelas that pretty much
  drowned all the commentary during the 2010 world cup in South Africa.
  However, the vuvuzela has a constant pitch of ~235Hz which made it
  easy for broadcasters to implement a notch filter to cut-off the
  offending noise.[1]

A shifted (delayed) signal in the time domain manifests as a phase change in the frequency domain. While this falls under the elementary property category, this is a widely used property in practice, especially in imaging and tomography applications,

Example: When a wave travels through a heterogenous medium, it
  slows down and speeds up according to changes in the speed of wave
  propagation in the medium. So by observing a change in phase from
  what's expected and what's measured, one can infer the excess time
  delay which in turn tells you how much the wave speed has changed in
  the medium. This is of course, a very simplified layman explanation, but
  forms the basis for tomography.

Derivatives of signals (nth derivatives too) can be easily calculated(see 106) using Fourier transforms.

Digital signal processing (DSP) vs. Analog signal processing (ASP)
The theory of Fourier transforms is applicable irrespective of whether the signal is continuous or discrete, as long as it is "nice" and absolutely integrable. So yes, ASP uses Fourier transforms as long as the signals satisfy this criterion. However, it is perhaps more common to talk about Laplace transforms, which is a generalized Fourier transform, in ASP. The Laplace transform is defined as
$$X(s)=\int_{0}^{\infty}x(t)e^{-st}\ dt,\quad \forall s\in\mathbb{C}$$
The advantage is that one is not necessarily confined to "nice signals" as in the Fourier transform, but the transform is valid only within a certain region of convergence. It is widely used in studying/analyzing/designing LC/RC/LCR circuits, which in turn are used in radios/electric guitars, wah-wah pedals, etc.

This is pretty much all I could think of right now, but do note that no amount of writing/explanation can fully capture the true importance of Fourier transforms in signal processing and in science/engineering


Question:
This cool design was "tattooed" on this leaf.
I found it on the windshield of my car.
What's up with it?

That is the work of a leaf miner. A leaf miner is the larval stage of an insect that feeds on the inside layer of leaves. Notice how the galleries (tunnels) start small and then get larger as the larva matures? Most leaf miners are moth larvae (Lepidoptera)


Question:
I am using Louvain clustering (1,2) to cluster cells in scRNAseq data, as implemented by scanpy.
One of the parameter required for this kind of clustering is the number of neighbors used to construct the neighborhood graph of cells (docs).
Larger values result in a more global view of the manifold, leading to lower number of clusters, while reducing the number of neighbors goes in the opposite direction. However, it is unclear how to choose this parameter.
The resolution parameter seems to work in the opposite way.
Do you know of any methodology and/or rule-of-thumb to define these parameters? E.g. depending on the size of the dataset?

Levine, Jacob H., et al. "Data-driven phenotypic dissection of AML reveals progenitor-like cells that correlate with prognosis." Cell 162.1 (2015): 184-197.
Blondel, Vincent D., et al. "Fast unfolding of communities in large networks." Journal of statistical mechanics: theory and experiment 2008.10 (2008): P10008.

A general rule of thumb is that in order to improve the variance $n$ times you need $n^2$ neighbours. This is only applicable if you consider the $n^2$ nearest neighbours of a cell to be biologically identical (i.e. "similar enough"); if your data includes 10 types of cells with 10 cells each, then using the 20 nearest neighbours for smoothing will obscure the data.
As far as I know, there is no single best answer to this question. I would suggest trying different numbers and sticking to what agrees more with the biology of the dataset.


Question:
For the following quantities respectively, could someone write down the common definitions, their meaning, the field of study in which one would typically find these under their actual name, and most foremost the associated abuse of language as well as difference and correlation (no pun intended):

Propagator
Two-point function
Wightman function
Green's function
Kernel
Linear Response function
Correlation function
Covariance function

Maybe including side notes regarding the distinction between Covariance, Covariance function and Cross-Covariance, the pair correlation function for different observables, relations to the autocorrelation function, the $n$-point function, the Schwinger function, the relation to transition amplitudes, retardation and related adjectives for Greens functions and/or propagators, the Heat-Kernel and its seemingly privileged position, the spectral density, spectra and the resolvent.

Edit: I'd still like to hear about the "Correlation fuction interpretation" of the quantum field theoretical framework. Can transition amplitudes be seen as a sort of auto-correlation? Like... such that the QFT dynamics at hand just determine the structure of the temporal and spatial overlaps?

The main distinction you want to make is between the Green function and the kernel.  (I prefer the terminology "Green function" without the 's.  Imagine a different name, say, Feynman.  People would definitely say the Feynman function, not the Feynman's function.  But I digress...)
Start with a differential operator, call it $L$.  E.g., in the case of Laplace's equation, then $L$ is the Laplacian $L = \nabla^2$.  Then, the Green function of $L$ is the solution of the inhomogenous differential equation
$$
L_x G(x, x^\prime) = \delta(x - x^\prime)\,.
$$
We'll talk about its boundary conditions later on.  The kernel is a solution of the homogeneous equation
$$
L_x K(x, x^\prime) = 0\,,
$$
subject to a Dirichlet boundary condition $\lim_{x \rightarrow x^\prime}K(x,x^\prime) = \delta (x-x^\prime)$, or Neumann boundary condition $\lim_{x \rightarrow x^\prime} \partial K(x,x^\prime) = \delta(x-x^\prime)$.
So, how do we use them?  The Green function solves linear differential equations with driving terms.  $L_x u(x) = \rho(x)$ is solved by
$$
u(x) = \int G(x,x^\prime)\rho(x^\prime)dx^\prime\,.
$$
Whichever boundary conditions we what to impose on the solution $u$ specify the boundary conditions we impose on $G$.  For example, a retarded Green function propagates influence strictly forward in time, so that $G(x,x^\prime) = 0$ whenever $x^0 < x^{\prime\,0}$.  (The 0 here denotes the time coordinate.)  One would use this if the boundary condition on $u$ was that $u(x) = 0$ far in the past, before the source term $\rho$ "turns on."
The kernel solves boundary value problems.  Say we're solving the equation $L_x u(x) = 0$ on a manifold $M$, and specify $u$ on the boundary $\partial M$ to be $v$.  Then,
$$
u(x) = \int_{\partial M} K(x,x^\prime)v(x^\prime)dx^\prime\,.
$$
In this case, we're using the kernel with Dirichlet boundary conditions.
For example, the heat kernel is the kernel of the heat equation, in which
$$
L = \frac{\partial}{\partial t} - \nabla_{R^d}^2\,.
$$
We can see that
$$
K(x,t; x^\prime, t^\prime) = \frac{1}{[4\pi (t-t^\prime)]^{d/2}}\,e^{-|x-x^\prime|^2/4(t-t^\prime)},
$$
solves $L_{x,t} K(x,t;x^\prime,t^\prime) = 0$ and moreover satisfies
$$
\lim_{t \rightarrow t^\prime} \, K(x,t;x^\prime,t^\prime) = \delta^{(d)}(x-x^\prime)\,.
$$
(We must be careful to consider only $t > t^\prime$ and hence also take a directional limit.)  Say you're given some shape $v(x)$ at time $0$ and want to "melt" is according to the heat equation.  Then later on, this shape has become
$$
u(x,t) = \int_{R^d} K(x,t;x^\prime,0)v(x^\prime)d^dx^\prime\,.
$$
So in this case, the boundary was the time-slice at $t^\prime = 0$.
Now for the rest of them.  Propagator is sometimes used to mean Green function, sometimes used to mean kernel.  The Klein-Gordon propagator is a Green function, because it satisfies $L_x D(x,x^\prime) = \delta(x-x^\prime)$ for $L_x = \partial_x^2 + m^2$.  The boundary conditions specify the difference between the retarded, advanced and Feynman propagators.  (See?  Not Feynman's propagator)  In the case of a Klein-Gordon field, the retarded propagator is defined as
$$
D_R(x,x^\prime) = \Theta(x^0 - x^{\prime\,0})\,\langle0| \varphi(x) \varphi(x^\prime) |0\rangle\,
$$
where $\Theta(x) = 1$ for $x > 0$ and $= 0$ otherwise.  The Wightman function is defined as
$$
W(x,x^\prime) = \langle0| \varphi(x) \varphi(x^\prime) |0\rangle\,,
$$
i.e. without the time ordering constraint.  But guess what?  It solves $L_x W(x,x^\prime) = 0$.  It's a kernel.  The difference is that $\Theta$ out front, which becomes a Dirac $\delta$ upon taking one time derivative.  If one uses the kernel with Neumann boundary conditions on a time-slice boundary, the relationship
$$
G_R(x,x^\prime) = \Theta(x^0 - x^{\prime\,0}) K(x,x^\prime)
$$
is general.
In quantum mechanics, the evolution operator
$$
U(x,t; x^\prime, t^\prime) = \langle x | e^{-i (t-t^\prime) \hat{H}} | x^\prime \rangle
$$
is a kernel.  It solves the Schroedinger equation and equals $\delta(x - x^\prime)$ for $t = t^\prime$.  People sometimes call it the propagator.  It can also be written in path integral form.
Linear response and impulse response functions are Green functions.
These are all two-point correlation functions.  "Two-point" because they're all functions of two points in space(time).  In quantum field theory, statistical field theory, etc. one can also consider correlation functions with more field insertions/random variables.  That's where the real work begins!


Question:
Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$.
It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ for some (real) inner product $\langle \cdot, \cdot \rangle$, then the parallelogram equality
$$ 2\lVert u\rVert^2 + 2\lVert v\rVert^2 = \lVert u + v\rVert^2 + \lVert u - v\rVert^2 $$
holds for all pairs $u, v \in V$.
I'm having difficulty with the converse.  Assuming the parallelogram identity, I'm able to convince myself that the inner product should be
$$ \langle u, v \rangle = \frac{\lVert u\rVert^2 + \lVert v\rVert^2 - \lVert u - v\rVert^2}{2} = \frac{\lVert u + v\rVert^2 - \lVert u\rVert^2 - \lVert v\rVert^2}{2} = \frac{\lVert u + v\rVert^2 - \lVert u - v\rVert^2}{4} $$
I cannot seem to get that $\langle \lambda u,v \rangle = \lambda \langle u,v \rangle$ for $\lambda \in \mathbb{R}$.  How would one go about proving this?

Since this question is asked often enough, let me add a detailed solution. I'm not quite following Arturo's outline, though. The main difference is that I'm not re-proving the Cauchy-Schwarz inequality (Step 4 in Arturo's outline) but rather use the fact that multiplication by scalars and addition of vectors as well as the norm are continuous, which is a bit easier to prove.
So, assume that the norm $\|\cdot\|$ satisfies the parallelogram law
$$2 \Vert x \Vert^2 + 2\Vert y \Vert^2 = \Vert x + y \Vert^2 + \Vert x - y \Vert^2$$
for all $x,y \in V$ and put
$$\langle x, y \rangle = \frac{1}{4} \left( \Vert x + y \Vert^2 - \Vert x - y \Vert^2\right).$$ We're dealing with real vector spaces and defer the treatment of the complex case to Step 4 below.
Step 0. $\langle x, y \rangle = \langle y, x\rangle$ and $\Vert x \Vert = \sqrt{\langle x, x\rangle}$.
Obvious.
Step 1. The function $(x,y) \mapsto \langle x,y \rangle$ is continuous with respect to $\Vert \cdot \Vert$.
Continuity with respect to the norm $\Vert \cdot\Vert$ follows from the fact that addition and negation are $\Vert \cdot \Vert$-continuous, that the norm itself is continuous and that sums and compositions of continuous functions are continuous.
Remark. This continuity property of the (putative) scalar product will only be used at the very end of step 3. Until then the solution consists of purely algebraic steps.
Step 2. We have $\langle x + y, z \rangle = \langle x, z \rangle + \langle y, z\rangle$.
By the parallelogram law we have
$$2\Vert x + z \Vert^2 + 2\Vert y \Vert^2  = \Vert x + y + z \Vert^2 + \Vert x - y + z\Vert^2 .$$
This gives
$$\begin{align*}
\Vert x + y + z \Vert^2 & = 2\Vert x + z \Vert^2 + 2\Vert y \Vert^2 - \Vert x - y + z \Vert^2 \\
& = 2\Vert y + z \Vert^2 + 2\Vert x \Vert^2 - \Vert y - x + z \Vert^2
\end{align*}$$
where the second formula follows from the first by exchanging $x$ and $y$. Since $A = B$ and $A = C$ imply $A = \frac{1}{2} (B + C)$ we get
$$\Vert x + y + z \Vert^2 = \Vert x \Vert^2 + \Vert y \Vert^2 + \Vert x + z \Vert^2 + \Vert y + z \Vert^2 - \frac{1}{2}\Vert x - y + z \Vert^2 - \frac{1}{2}\Vert y - x + z \Vert^2.$$
Replacing $z$ by $-z$ in the last equation gives
$$\Vert x + y - z \Vert^2 = \Vert x \Vert^2 + \Vert y \Vert^2 + \Vert x - z \Vert^2 + \Vert y - z \Vert^2 - \frac{1}{2}\Vert x - y - z \Vert^2 - \frac{1}{2}\Vert y - x - z \Vert^2.$$
Applying $\Vert w \Vert = \Vert - w\Vert$ to the two negative terms in the last  equation we get
$$\begin{align*}\langle x + y, z \rangle & = \frac{1}{4}\left(\Vert x + y + z \Vert^2 - \Vert x + y - z \Vert^2\right) \\
& = \frac{1}{4}\left(\Vert x + z \Vert^2 - \Vert x - z \Vert^2\right) +
\frac{1}{4}\left(\Vert y + z \Vert^2 - \Vert y - z \Vert^2\right) \\
& = \langle x, z \rangle + \langle y, z \rangle
\end{align*}$$
as desired.
Step 3. $\langle \lambda x, y \rangle = \lambda \langle x, y \rangle$ for all $\lambda \in \mathbb{R}$.
This clearly holds for $\lambda = -1$ and by step 2 and induction we have $\langle \lambda x, y \rangle = \lambda \langle x, y \rangle$ for all $\lambda \in \mathbb{N}$, thus for all $\lambda \in \mathbb{Z}$. If $\lambda = \frac{p}{q}$ with $p,q \in \mathbb{Z}, q \neq 0$ we get with $x' = \dfrac{x}{q}$ that
$$q \langle \lambda x, y \rangle = q\langle p x', y \rangle = p \langle q x', y \rangle = p\langle x,y \rangle,$$
so dividing this by $q$ gives
$$\langle \lambda x , y \rangle = \lambda \langle x, y \rangle \qquad\text{for all } \lambda \in \mathbb{Q}.$$
We have just seen that for fixed $x,y$ the continuous function $\displaystyle t \mapsto \frac{1}{t} \langle t x,y \rangle$ defined on $\mathbb{R} \smallsetminus \{0\}$ is equal to $\langle x,y \rangle$ for all $t \in \mathbb{Q} \smallsetminus \{0\}$, thus equality holds for all $t \in \mathbb{R} \smallsetminus \{0\}$. The case $\lambda = 0$ being trivial, we're done.
Step 4. The complex case.
Define $\displaystyle \langle x, y \rangle =\frac{1}{4} \sum_{k =0}^{3} i^{k} \Vert x +i^k y\Vert^2$, observe that $\langle ix,y \rangle = i \langle x, y \rangle$ and $\langle x, y \rangle = \overline{\langle y, x \rangle}$ and apply the case of real scalars twice (to the real and imaginary parts of $\langle \cdot, \cdot \rangle$).
Addendum. In fact we can weaken requirements of Jordan von Neumann theorem to
$$
2\Vert x\Vert^2+2\Vert y\Vert^2\leq\Vert x+y\Vert^2+\Vert x-y\Vert^2
$$
Indeed after substitution $x\to\frac{1}{2}(x+y)$, $y\to\frac{1}{2}(x-y)$ and simplifications we get
$$
\Vert x+y\Vert^2+\Vert x-y\Vert^2\leq 2\Vert x\Vert^2+2\Vert y\Vert^2
$$
which together with previous inequality gives the equality.


Question:
According to this article (and a lot more published today on the same topic), Kosovo electricity net production balance has decreased during the last weeks.
This has led to a small deviation of the European network’s frequency (from 50Hz to 49.996Hz).
In turn, this frequency deviation led to some electric clocks (like ones in ovens) being out of sync (up to 6 min since January).

How can a decrease in electricity production lead to a decrease of the frequency on the grid on the long term? Isn't the frequency a parameter controlled by the power plant at the end of the day?
If the loss of power from some countries causes a frequency deviation, shouldn't we also observe other impacts, like a drop of the output voltage? Does this mean that we've also been experiencing a drop of voltage for weeks here in Europe?
Why do some electric devices directly use the network frequency to sync their clocks, instead of a quartz crystal technology? This means the same oven needs two different firmwares for countries with different electric network frequencies, while, with a crystal (that should be needed anyway to run all the embedded circuits), the same device would run unmodified everywhere.

From the Reuter's article referenced:

SARAJEVO, March 7 (Reuters) - European power grid lobby ENTSO-E urged Serbia and Kosovo to urgently resolve a dispute over their power grid, which has affected the broader European network, causing some digital clocks on the continent to lose time.

Figure 1. The ENTSO-E System Operations Committee has 5 permanent regional groups based on the synchronous areas (Continental Europe, Nordic, Baltic, Great Britain, and Ireland-Northern Ireland), and 2 voluntary Regional Groups (Northern Europe and Isolated Systems). Source: ENTSO-E.
The European grid shares power across borders. AC grids have to be kept 100% in-sync if AC connections are used. Britain and Ireland, for example, are connected to the European grid by DC interconnectors so each nations grid can run asynchronously with the rest of Europe whilst sharing power.

The grid shared by Serbia and its former province Kosovo is connected to Europe’s synchronized high voltage power network.

As explained above.

ENTSO-E, which represents European electricity transmission operators, said the continental network had lost 113 gigawatt-hours (GWh) of energy since mid-January because Kosovo had been using more electricity than it generates. Serbia, which is responsible for balancing Kosovo’s grid, had failed to do so, ENTSO-E said.

The energy hasn't been lost. It was never produced.
According to NetzFrequenzMessung.de (you might want to translate) the 113 GWh shortfall averages out to about 80 MW continuous on a total of 60 GW capacity. That's a 0.13%. The scary thing is that we're actually maxed out and can't find an extra 0.13%!

The loss [sic] of energy had meant that electric clocks that are steered by the frequency of the power system, rather than by a quartz crystal, to lag nearly six minutes behind, ENTSO-E said.

"Steered" is probably a mistranslation. "Regulated" would be better.

Many digital clocks, such as those in alarm clocks and in ovens or microwaves, use the frequency of the power grid to keep time. The problem emerges when the frequency drops over a sustained period of time.

Figure 2. An electro-mechanical timeswitch of the style popular with the utility companies.
Analogue, motorised clocks do too. The day / night clock on my electricity meter is > 40 years old and it has a mains-powered clock with a self-rewinding clockwork UPS to keep it OK during power cuts!

ENTSO-E said the European network’s frequency had deviated from its standard of 50 Hertz (Hz) to 49.996 Hz since mid-January, resulting in 113 gigawatt-hours (GWh) [sic] of lost energy, although it had appeared to be returning to normal on Tuesday.

The frequency is not held constant to three decimal places for months on end. That might be an average figure. Here's the data for the last five minutes:

Figure 3. Note that frequency deviation will be much wider over a longer time period. Source: MainsFrequency.com.

Figure 4. Network time deviation has increased from -100 s to -350 s in three weeks. Source: MainsFrequency.com.

Figure 5. [WOW!] In our previous measurement operation (July 2011 to 2017), network time deviations of ± 160 seconds occurred (June 2013). But since January 3, 2018, the network time deviation is continuously decreasing. Changing the setpoint for the secondary control power on January 15 from 50,000 Hz to 50,010 Hz has not yet been able to reduce the mains time.  Source: MainsFrequency.com.
Secondary control power is activated when the system is affected for longer than 30 seconds or it is assumed that the system will be affected for a period longer than 30 seconds. Prior to this, deviations in the system are only covered through primary control. Source: APG.at.

“Deviation stopped yesterday after Kosovo took some steps but it will take some time to get the system back to normal,” ENTSO-E spokeswoman Susanne Nies told Reuters. She said the risk could remain if there is no political solution to the problem.

If they start generating and feeding into the grid it will speed up.

The political dispute centres mainly on regulatory issues and a row between Serbia and Kosovo over grid operation. It is further complicated by the fact that Belgrade still does not recognise Kosovo.
“We will try to fix the technicalities by the end of this week but the question of who will compensate for this loss has to be answered,” Nies said.

This doesn't make any sense to me. Energy flow is metered and billed accordingly. Each country pays for their imports.

ENTSO-E urged European governments and policymakers to take swift action and exert pressure on Kosovo and Serbia to resolve the issue, which is also hampering integration of the western Balkans energy market required by the European Union.
“These actions need to address the political side of this issue,” ENTSO-E said in a statement. The grid operators in Serbia and Kosovo were not immediately available to comment.
Kosovo seceded from Serbia in 2008. Both states want to join the European Union but Brussels says they must normalize relations to forge closer ties with the bloc.
Serbia and Kosovo signed an agreement on operating their power grid in 2015. However, it has not been implemented yet as they cannot agree on power distribution in Kosovo amid conflicting claims about ownership of the grid, built when they were both part of Yugoslavia. (Writing by Maja Zuvela; Editing by Susan Fenton)

I guess neither of the above are electrical engineers.

Answering the questions:

How a decrease in electricity production can lead to a decrease of the frequency on the grid on the long term? Isn't the frequency a parameter controlled by the power plant at the end of the day?

If demand is approaching peak capacity then we have to let either the voltage or the frequency droop if we wish to avoid disconnecting customers. Dropping the voltage will cause problems with certain loads and is to be avoided.
The Reuter's article fails to explain why the system average frequency has been low for so long. It can only be that it hasn't been able to run above 50 Hz for long enough to catch up. Off-peak seems the time to do this but there will be an upper limit on the frequency deviation - about 50.5 Hz (but I don't have a definite number).

If the loss of power from some countries causes a frequency deviation, shouldn't we also observe other impacts, like a drop of the output voltage? This means we've also been experiencing a drop of voltage for weeks here in Europe?

No, we reduce frequency to avoid the drop in voltage.

Why some electric devices directly use the network frequency to sync their clocks, instead of a quartz crystal technology?

They don't sync the clocks in the sense of adjusting or correcting the time. They maintain synchronisation by keeping the average frequency at exactly 50 Hz. One reason for this is the millions of electro-mechanical clocks in service. These are fantastically reliable, don't require batteries and do the job. Why replace them?

This means the same oven needs 2 different firmwares for countries with different electric network frequencies, while, with a crystal (that should be needed anyway to run all the embedded circuits), the same device would run unmodified everywhere.

Crystals will drift and the further complication of real-time clock with battery backup are required. Electrical utilities work on timescales of 20 to 50 years. How long do you think the electrolytic capacitors in your digital clock will last?

Links:

ENTSO-E transmission system map.
ENTSO-E FAQ on the matter.

Other interesting bits:

This grid time deviation is constantly balanced out. If the time deviation is more than twenty seconds the frequency is corrected in the grid. In order to balance out the time deviation again the otherwise customary frequency of 50 Hz (Europe) is changed as follows:
49.990 Hz, if the grid time is running ahead of UTC time
50.010 Hz, if the grid time is lagging behind UTC time
Source: SwissGrid.

Meanwhile on 2018-03-08:

NTSO-E has now confirmed with the Serbian and Kosovar TSOs, respectively EMS and KOSTT, that the deviations which affected the average frequency in the synchronous area of Continental Europe have ceased.
This is a first step in the resolution of the issue. The second step is now to develop a plan for returning the missing energy to the system and putting the situation back to normal.
Source: ENTSO-E.

Hmmm! They're referring to it as "missing energy".


Question:
Related:
Safe current/voltage limit for human contact?

From what I've heard:

110 V (or 220 V; household voltage pretty much) is dangerous (i.e. can kill you) I think there's consensus on this, no need to try :)
60 V (old telephone lines) is supposedly dangerous (never tried, only heard it once... probably won't try)

From what I know first-hand:

9 V is not dangerous (I've put a 9-V battery on my tongue, nbd... actually it kinda hurt!)
1.5 V can indeed be quite shocking with enough current (fell for one of those "Do you want some gum?" tricks back in high school...), but they sometimes do not use 1.5 V with the low amperage levels, some use a DC motor to vibrate and complete the trick.

So I guess there's two parameters here, voltage and current... but are there rough numbers on how much of each (or in combination, which I guess would be power) would be considered hazardous?
No old telephone lines have always been 48vDC well at least since from 1950s, if your skin is wet you can feel it slightly, like on your forearm.  Now the ring voltage is 90-110vAC with a 2 on 4 sec off cycle (USA).  It will ring your bell but good, should you be touching the wires when someone calls. The ring voltage rides on top of the 48vDC, so its present on the same two conductors that the voice voltage(DC) is on. Luckily it's 4 seconds off will give you a chance to get off the conductors with a scream (of pain).

How much voltage is dangerous is not really a static number as it depends on your body resistance, time of exposure and source "stiffness" (i.e. how much current it can supply). You get figures like 60V (or as low as 30V) which are an attempt at an average figure above which "caution should be taken".
However, depending on how "conductive" you are at any one time, sometimes e.g. 50V might be quite safe and other times it may kill you.
DC or AC (and what frequency) seem to make a difference too, female or male, etc - this table is very instructive:

Figures as low as 20mA across the heart are given as possibly capable of inducing fibrillation - here is another table from the same source that gives body resistance based on different situations:

You can see that as low as 20V may be dangerous given the right conditions.
Here is the reference the tables came from, I think it is quite accurate based on some experiments I have done myself measuring body resistances. The rest of the site seems to be generally very well informed and presented from the bits I have read, so I think this may be quite a trustworthy source.


Question:
I have heard several conspiracy theories regarding the origin of the new coronavirus, 2019-nCov. For example that the virus and/or SARS were produced in a laboratory or were some variant of Middle Eastern respiratory syndrome (MERS), shipped via laboratory workers.
I am well aware bioinformatics has debunked many conspiracy theories involving infectious diseases, an important one being polio vaccination programs in Africa were the origin of HIV, for example here. Likewise, numerous conspiracy theories on the man-made origins of HIV were similarly overturned using bioinformatic studies.
Do you have bioinformatics evidence that would debunk the current conspiracists about coronaviruses?

The scenarios are impossible and would be laughable if they were not so serious. The evidence is in the phylogenetic trees. Its a bit like a crime scene when the forensics team investigate. We've done enough crime-scenes often going to the site, collecting the pathogen, sequencing and then analysis - (usually neglected diseases) without any associated conspiracy theories.
The key technical issue is coronaviruses are zoonoses , pathogens spread to humans from animal reservoirs and phylogenetic tree really helps understand the  how the virus is transmitted.
Trees

The key thing about all the trees are bats. Bats lineages are present at every single point of the betacoronavirus phylogeny (tree), both as paraphyletic and monophyletic lineages, one example is this tree of betacoronaviruses here. Meaning the nodes connecting the branches of the tree to the "master-branch", represent common ancestors and these were almost certainly bat-borne cornaviruses. This is especially true for SARS and - here bat-viruses are EVERYWHERE.
The tree here also shows that SARS arose on independently on two occassions, again surrounded by bat lineages and 2019-nCov has emerged separately at least once, again associated with bats.
Finally the tree below is a figure from BioRxiv Zhou et al (2020) "Discovery of a novel coronavirus associated with the recent pneumonia outbreak in humans and its potential bat origin" shows the 2019-nCov lineage is a direct descendent of a very closely virus isolated from a bat (RaTG13*). This is a really conclusive finding BTW.

Note, I don't normally present inline images, but it is such a nice finding (hint to reviewers) and BioRxiv is open access.
Conspiracy theory 1: laboratory made virus
Literally it would require someone passaging a new virus, with unknown human pathogenicity, and independently introducing all the earlier passages enmass across bat populations of China. They would then hope each lineage becomes an indepedent virus population before, then introducing the virus to humans. Thus when field teams of scientists go around using mist nets to trap the bats, buy them from markets, isolate the virus and sequence it they would find a beautiful, array of natural variation in the bat populations leading up to the human epidemics, that perfectly matches vast numbers of other viral zoonoses. Moreover, this would have to have happen substancially prior to SARS and 2019-nCov, because the bat betacoronaviruses have been known about prior both epidemics, viz. its simply not feasible.
Biological explanation
General Bats are a reservoir host to vast numbers of pathogens, particularly viruses, including many alphaviruses, flaviviruses, rabies virus and beleived to be important in ebolavirus (I don't know about this) and even important to several eukaryotic parasites. It makes sense, they are mammals, so evolutionary much closer to us than birds for example, with large dispersal potential and roost in 'overcrowded' areas enable rapid transmission between bats.
Technical
The trees show bats are the common ancestor of betacoronaviruses in particular for the lineage leading into the emergence of 2019-nCov and SARS, this is seen in this tree, this one and the tree above. The obvious explanation is the virus circulates endemically in bats and has jumped into humans. For SARS the intermediate host, or possible "vector" was civet cats.
The theory and the observations fit into a seamless biological answer.
Conspiracy theory 2: Middle Eastern connection
I heard a very weird conspiracy theory attempting to connect MERS with 2019-nCov. The theory was elaborate and I don't think it is productive to describe here.
Biological explanation
All the trees of betacoronaviruses show MERS was one of the earliest viruses to diverge and is very distant from 2019-nCov, to the extent the theory is completely implausible. The homology between these viruses is 50%, so its either MERS or 2019-nCov. Its more extreme than mixing up yellow fever virus (mortality 40-80%) with West Nile virus (mortality <<0.1%), the two viruses are completely different at every level.
What about errors? Phylogeneticists can spot it a mile off. There are tell-tale phylogenetic signatures we pick up, but also we do this to assess 'rare' genetic phenomina. There is nothing 'rare' about the coronaviruses. The only anomaly is variation in the poly-A tail and that is the natural variation from in vitro time-series experiments. Basically we've looked at enough virses/parasites through trees, that have no conspiracy theories at all (often neglected diseases), and understand how natural variation operates - so a phylogenecist can shift the wheat from the chaff without really thinking about it.
Opinion
The conspiracy theories are deeply misplaced, and the only connection I can imagine is its China. However, the Chinese have loads of viruses, influenza in particular which causes major pandemics, but that is a consequence of their natural ecology (small-holder farming) allowing the virus to move between reservoir hosts. I've not visited small-hold farms in China, but I have in other parts of the world and when you see them, you get it. The pigs, chickens (ducks .. China), dogs, horses and humans all living within 10 meters of each other.
Conclusion
Shipping large numbers of bats to market, bat soup, raw meat from arboreal (tree-living) mammals such as civets that are sympatric to bats. Then consider the classical epidemiology in light of the phylogenetic data, which is very consistent, a single picture emerges that coronavirus is one of many zoonoses which has managed to transmit between patients.
Summary The fundamental point is the bioinformatics fit into the classical epidemiology of a zoonose.
*, Note The bat coronavirus RaTG13 predates the 2019-nCov outbreak by 7 years. It is not even clear whether the virus has been isolated, i.e. could just be a RNA sequence.
"They have found some 500 novel coronaviruses, about 50 of which fall relatively close to the SARS virus on the family tree, including RaTG13—it was fished out of a bat fecal sample they collected in 2013 from a cave in Moglang in Yunnan province."
Cohen, "Mining coronavirus genomes for clues to the outbreak’s origins" Feb, 2020 Science Magazine,


Question:
Say you cook up a model about a physical system. Such a model consists of, say, a system of differential equations. What criterion decides whether the model is classical or quantum-mechanical?
None of the following criteria are valid:

Partial differential equations: Both the Maxwell equations and the Schrödinger equation are PDE's, but the first model is clearly classical and the second one is not. Conversely, finite-dimensional quantum systems have as equations of motion ordinary differential equations, so the latter are not restricted to classical systems only.
Complex numbers: You can use those to analyse electric circuits, so that's not enough. Conversely, you don't need complex numbers to formulate standard QM (cf. this PSE post).
Operators and Hilbert spaces: You can formulate classical mechanics à la Koopman-von Neumann. In the same vein:
Dirac-von Neumann axioms: These are too restrictive (e.g., they do not accommodate topological quantum field theories). Also, a certain model may be formulated in such a way that it's very hard to tell whether it satisfies these axioms or not. For example, the Schrödinger equation corresponds to a model that does not explicitly satisfy these axioms; and only when formulated in abstract terms this becomes obvious. It's not clear whether the same thing could be done with e.g. the Maxwell equations. In fact, one can formulate these equations as a Dirac-like equation $(\Gamma^\mu\partial_\mu+\Gamma^0)\Psi=0$ (see e.g. 1804.00556), which can be recast in abstract terms as $i\dot\Psi=H\Psi$ for a certain $H$.
Probabilities: Classical statistical mechanics does also deal with probabilistic concepts. Also, one could argue that standard QM is not inherently probabilistic, but that probabilities are an emergent property due to the measurement process and our choice of observable degrees of freedom.
Planck's constant: It's just a matter of units. You can eliminate this constant by means of the redefinition $t\to \hbar t$. One could even argue that this would be a natural definition from an experimental point of view, if we agree to measure frequencies instead of energies. Conversely, you may introduce this constant in classical mechanics by a similar change of variables (say, $F=\hbar\tilde F$ in the Newton equation). Needless to say, such a change of variables would be unnatural, but naturalness is not a well-defined criterion for classical vs. quantum.
Realism/determinism: This seems to depend on interpretations. But whether a theory is classical or quantum mechanical should not depend on how we interpret the theory; it should be intrinsic to the formalism.

People are after a quantum theory of gravity. What prevents me from saying that General Relativity is already quantum mechanical? It seems intuitively obvious that it is a classical theory, but I'm not sure how to put that intuition into words. None of the criteria above is conclusive.

As far as I know, the commutator relations make a theory quantum. If all observables commute, the theory is classical. If some observables have non-zero commutators (no matter if they are proportional to $\hbar$ or not), the theory is quantum.
Intuitively, what makes a theory quantum is the fact that observations affect the state of the system. In some sense, this is encoded in the commutator relations: The order of the measurements affects their outcome, the first measurement affects the result of the second one.


Question: Most of the time RS-232 and UART come together in serial communication theories. Are they both the same? From my readings I concluded UART is a hardware form of the RS-232 protocol. Am I correct?

No, UART and RS-232 are not the same.
UART is responsible for sending and receiving a sequence of bits.  At the output of a UART these bits are usually represented by logic level voltages.  These bits can become RS-232, RS-422, RS-485, or perhaps some proprietary spec.
RS-232 specifies voltage levels.  Notice that some of these voltage levels are negative, and they can also reach ±15V.  Larger voltage swing makes RS-232 more resistant to interference (albeit only to some extent).
A microcontroller UART can not generate such voltages levels by itself.  This is done with help of an additional component: RS-232 line driver.  A classic example of an RS-232 line driver is MAX232.  If you go through the datasheet, you'll notice that this IC has a charge pump, which generates ±10V from +5V.

(source)


Question:
I just starting taking a course on Data Structures and Algorithms and my teaching assistant gave us the following pseudo-code for sorting an array of integers:
void F3() {
    for (int i = 1; i < n; i++) {
        if (A[i-1] > A[i]) {
            swap(i-1, i)
            i = 0
        }
    }
}

It may not be clear, but here $n$ is the size of the array A that we are trying to sort.
In any case, the teaching assistant explained to the class that this algorithm is in $\Theta(n^3)$ time (worst-case, I believe), but no matter how many times I go through it with a reversely-sorted array, it seems to me that it should be $\Theta(n^2)$ and not $\Theta(n^3)$.
Would someone be able to explain to me why this is $Θ(n^3)$ and not $Θ(n^2)$?

This algorithm can be re-written like this

Scan A until you find an inversion.
If you find one, swap and start over.
If there is none, terminate.

Now there can be at most $\binom{n}{2} \in \Theta(n^2)$ inversions and you need a linear-time scan to find each -- so the worst-case running time is $\Theta(n^3)$. A beautiful teaching example as it trips up the pattern-matching approach many succumb to!
Nota bene: One has to be a little careful: some inversions appear early, some late, so it is not per se trivial that the costs add up as claimed (for the lower bound). You also need to observe that swaps never introduce new inversions. A more detailed analysis of the case with the inversely sorted array will then yield something like the quadratic case of Gauss' formula.
As @gnasher729 aptly comments, it's easy to see the worst-case running time is $\Omega(n^3)$ by analyzing the running time when sorting the input $[1, 2, \dots, n, 2n, 2n-1, \dots, n+1]$ (though this input is probably not the worst case).
Be careful: don't assume that a reversely-sorted array will necessarily be the worst-case input for all sorting algorithms.  That depends on the algorithm.  There are some sorting algorithms where a reversely-sorted array isn't the worst case, and might even be close to the best case.


Question:
I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me.
The inverse limit. I tried to ask one of the guys in my office, and despite a very shady explanation he ended up muttering that "you usually take an already known construction."
The Wikipedia article presents two approaches, the algebraic and the categorical. While the categorical is extremely vague for me, the algebraic one is too general and the intuition remains hidden underneath the text in a place I cannot find it.
Since I am not too familiar with categories, the explanation most people would try to give me which is categorical in nature seems to confuse me - as I keep asking this question over and over every now and then.
Could anyone explain to me in non-categorical terms what is the idea behind an inverse limit? (I am roughly familiar with its friend "direct limit", if that helps)
(While editing, I can say that the answers given so far are very interesting, and I have read them thoroughly, although I need to give it quite some thinking before I can comment on all of them right now.)

I like George Bergman's explanation (beginning in section 7.4 of his Invitation to General Algebra and Universal Constructions).
We start with a motivating example.
Suppose you are interested in solving $x^2=-1$ in $\mathbb{Z}$. Of course, there are no solutions, but let's ignore that annoying reality for a moment.
We use the notation $\mathbb{Z}_n$ for $\mathbb Z / n \mathbb Z$.
The equation has a solution in the ring $\mathbb{Z}_5$ (in fact, two: both $2$ and $3$, which are the same up to sign). So we want to find a solution to $x^2=-1$ in $\mathbb{Z}$ which satisfies $x\equiv 2 \pmod{5}$.
An integer that is congruent to $2$ modulo $5$ is of the form $5y+2$, so we can rewrite our original equation as $(5y+2)^2 = -1$, and expand to get
$25y^2 + 20y = -5$.
That means $20y\equiv -5\pmod{25}$, or $4y\equiv -1\pmod{5}$, which has the unique solution $y\equiv 1\pmod{5}$. Substituting back we determine $x$ modulo $25$:
$$x = 5y+2 \equiv 5\cdot 1 + 2  = 7 \pmod{25}.$$
Continue this way: putting $x=25z+7$ into $x^2=-1$ we conclude $z\equiv 2 \pmod{5}$, so $x\equiv 57\pmod{125}$.
Using Hensel's Lemma, we can continue this indefinitely. What we deduce is that there is a sequence of residues,
$$x_1\in\mathbb{Z}_5,\quad x_2\in\mathbb{Z}_{25},\quad \ldots, x_{i}\in\mathbb{Z}_{5^i},\ldots$$
each of which satisfies $x^2=-1$ in the appropriate ring, and which are "consistent", in the sense that each $x_{i+1}$ is a lifting of $x_i$ under the natural homomorphisms
$$\cdots \stackrel{f_{i+1}}{\longrightarrow} \mathbb{Z}_{5^{i+1}} \stackrel{f_i}{\longrightarrow} \mathbb{Z}_{5^i} \stackrel{f_{i-1}}{\longrightarrow}\cdots\stackrel{f_2}{\longrightarrow} \mathbb{Z}_{5^2}\stackrel{f_1}{\longrightarrow} \mathbb{Z}_5.$$
Take the set of all strings $(\ldots,x_i,\ldots,x_2,x_1)$ such that $x_i\in\mathbb{Z}_{5^i}$ and $f_i(x_{i+1}) = x_i$, $i=1,2,\ldots$. This is a ring under componentwise operations. What we did above shows that in this ring, you do have a square root of $-1$.

Added. Bergman here inserts the quote, "If the fool will persist in his folly, he will become wise." We obtained the sequence by stubbornly looking for a solution to an equation that has no solution, by looking at putative approximations, first modulo 5, then modulo 25, then modulo 125, etc. We foolishly kept going even though there was no solution to be found. In the end, we get a "full description" of what that object must look like; since we don't have a ready-made object that satisfies this condition, then we simply take this "full description" and use that description as if it were an object itself. By insisting in our folly of looking for a solution, we have become wise by introducing an entirely new object that is a solution.
This is much along the lines of taking a Cauchy sequence of rationals, which "describes" a limit point, and using the entire Cauchy sequence to represent this limit point, even if that limit point does not exist in our original set.

This ring is the $5$-adic integers; since an integer is completely determined by its remainders modulo the powers of $5$, this ring contains an isomorphic copy of $\mathbb{Z}$.
Essentially, we are taking successive approximations to a putative answer to the original equation, by first solving it modulo $5$, then solving it modulo $25$ in a way that is consistent with our solution modulo $5$; then solving it modulo $125$ in a way that is consistent with out solution modulo $25$, etc.
The ring of $5$-adic integers projects onto each $\mathbb{Z}_{5^i}$ via the projections; because the elements of the $5$-adic integers are consistent sequences, these projections commute with our original maps $f_i$. So the projections are compatible with the $f_i$ in the sense that for all $i$, $f_i\circ\pi_{i+1} = \pi_{i}$, where $\pi_k$ is the projection onto the $k$th coordinate from the $5$-adics.
Moreover, the ring of $5$-adic integers is universal for this property: given any ring $R$ with homomorphisms $r_i\colon R\to\mathbb{Z}_{5^i}$ such that $f_i\circ r_{i+1} = r_i$, for any $a\in R$ the tuple of images $(\ldots, r_i(a),\ldots, r_2(a),r_1(a))$ defines an element in the $5$-adics. The $5$-adics are the inverse limit of the system of maps
$$\cdots\stackrel{f_{i+1}}{\longrightarrow}\mathbb{Z}_{5^{i+1}}\stackrel{f_i}{\longrightarrow}\mathbb{Z}_{5^i}\stackrel{f_{i-1}}{\longrightarrow}\cdots\stackrel{f_2}{\longrightarrow}\mathbb{Z}_{5^2}\stackrel{f_1}{\longrightarrow}\mathbb{Z}_5.$$
So the elements of the inverse limit are "consistent sequences" of partial approximations, and the inverse limit is a way of taking all these "partial approximations" and combine them into a "target object."
More generally, assume that you have a system of, say, rings, $\{R_i\}$, indexed by an directed set $(I,\leq)$ (so that for all $i,j\in I$ there exists $k\in I$ such that $i,j\leq k$), and a system of maps $f_{rs}\colon R_s\to R_r$ whenever $r\leq s$ which are "consistent" (if $r\leq s\leq t$, then $f_{rs}\circ f_{st} = f_{rt}$), and let's assume that the $f_{rs}$ are surjective, as they were in the example of the $5$-adics. Then you can think of the $R_i$ as being "successive approximations" (with a higher indexed $R_i$ as being a "finer" or "better" approximation than the lower indexed one). The directedness of the index set guarantees that given any two approximations, even if they are not directly comparable to one another, you can combine them into an approximation which is finer (better) than each of them (if $i,j$ are incomparable, then find a $k$ with $i,j\leq k$). The inverse limit is a way to combine all of these approximations into an object in a consistent manner.
If you imagine your maps as going right to left, you have a branching tree that is getting "thinner" as you move left, and the inverse limit is the combination of all branches occurring "at infinity".

Added. The example of the $p$-adic integers may be a bit misleading because our directed set is totally ordered and all maps are surjective. In the more general case, you can think of every chain in the directed set as a "line of approximation"; the directed property ensures that any finite number of "lines of approximation" will meet in "finite time", but you may need to go all the way to "infinity" to really put all the lines of approximation together. The inverse limit takes care of this.
If the directed set has no maximal elements, but the structure maps are not surjective, it turns out that no element that is not in the image will matter; essentially, that element never shows up in a net of "successive approximations", so it never forms part of a "consistent system of approximations" (which is what the elements of the inverse limit are).


Question:
I have been on a date recently, and everything went fine until the moment the girl has told me that the Earth is flat. After realizing she was not trolling me, and trying to provide her with a couple of suggestions why that may not be the case, I've faced arguments of the like "well, you have not been to the space yourself".
That made me think of the following: I myself am certain that the Earth is ball-shaped, and I trust the school physics, but being a kind of a scientist, I could not help but agree with her that some of the arguments that I had in mind were taken by me for granted. Hence, I have asked myself - how can I prove to myself that the earth is indeed ball-shaped, as opposed to being a flat circle (around which the moon and the sun rotate in a convenient for this girl manner).
Question: Ideally I want to have a proof that would not require travelling more than a couple of kilometers, but I am fine with using any convenient day (if e.g. we need to wait for some eclipse or a moon phase). For example, "jump an a plane and fly around the Earth" would not work for me, whereas "look at the moon what it is in phase X, and check the shape of the shade" would.
Trick is, I know that it is rather easy to verify local curvature of the Earth by moving away from a tall object in the field/sitting on the beach and watching some big ship going to the horizon. However, to me that does not prove immediately that globally the Earth has same/similar curvature. For example, maybe it's just a shape of a hemisphere. So, I want to prove to myself that the Earth is ball-shaped globally, and I don't want to move much to do this. Help me, or tell me that this is not possible and why, please. As an example, most of the answers in this popular thread only focus on showing the local curvature.
P.S. I think, asking how to use physics to derive global characteristics of an object from observing things only locally (with the help of the Sun and the Moon, of course) is a valid question, but if something can be improved in it, feel free to tell me. Thanks.
Update: I have not expected such a great and strong feedback when asking this question, even though it is indeed different from the linked ones. Them are still very similar, which was not grasped by all those who replied. I will thoroughly go over all the answers to make sure which one fits the best, but in the meantime if you would like to contribute, please let me clarify a couple of things regarding this question: they were in the OP, but perhaps can be made more obvious.

I do not have a goal of proving something to this date. I see that mentioning her might have been confusing. Yet, before this meeting I was certain about the shape of the earth - but her words (even though I think she's incorrect in her beliefs) made me realize that my certainty was based on the assumption I have not really questioned. So sitting on a beach with another friend of mine (both being ball-believers) we thought of a simple check to confirm our certainty, rather than to convince anyone else in us being right.
I am only looking for the check that would confirm the GLOBAL shape of the earth being ball-like. There were several brilliant answers to another question that worked as a local curvature proof, and I am not interested in them.
I am looking for the the answer that will show that the Earth is ball-shaped (or rather an ellipsoid), not that it is not flat. There are many other great shapes being neither ball/ellipsoid nor flat. I do still have an assumption that this shape is convex, otherwise things can go too wild and e.g. projections on the Moon would not help us.

I think point 1. shows why is that a valid physics/astronomy question, rather than playing a devil's advocate defending the flat Earth hypothesis, and I would also happily accept the answer like you cannot show this by not moving for 20k kilometers because A, B, C if there's indeed no simple proof. At the same time, points 2 and 3 should distinguish this question from the linked ones.

I love this question, because it's a very simple demonstration of how to do science. While it's true that in science one should never accept anything 'without evidence', it's also true that blind skepticism of everything and anything gets one nowhere - skepticism has to be combined with rational inquiry. Your date has gotten the 'skepticism' part of science, but she's failed to grasp the equally-crucial part where one looks at the evidence and thinks about what the evidence implies. You cannot just refuse to think or accept evidence. If your goal is to learn nothing, then nothing is what you'll learn.
There are many, many ways of verifying that the Earth is not flat, and most of them are easy to think about and verify. You certainly do not need to go to space to realize the Earth is round!
If the Earth is flat, why can't you see Mt. Kilimanjaro from your house?
Mt. Kilimanjaro is tall, probably taller than anything in your immediate neighborhood (unless you live in a very deep valley) and so the question is why wouldn't you be able to see it from anywhere on Earth? Or, for that matter, why can't you see it from even closer? You have to be really close, in planetary terms, to be able to see it. This wouldn't be true if the Earth were flat!
One might argue that this is just because of the scattering of the atmosphere. Distant objects appear paler, so probably after some distance you can't see anything at all.
So then let's think about things that are closer. Stand on the ground and the horizon appears only a few km away. Go to the top of a hill, or a large tower, and suddenly you can see things much farther away. Why is this the case if the Earth is flat? Why would your height above ground have anything to do with it? If I raise or lower my eyes with respect to a flat table, I can still see everything on that table. The 'horizon' of the table never appears closer.
If the Earth is flat, why do time zones exist?
Hopefully your date realizes that time zones exist. If not, it's pretty easy to verify by doing a video call with someone in a distant location. The reason for time zones, of course, is that the sun sets and rises at different times at different parts of the globe. Why would this be the case? On a flat Earth, the sun would rise and set at the same time everywhere.
If the Earth is flat, why is the Moon round?
The moon is round and not a flat disc, as you can see by the librations of the moon. What makes the Earth special, then?
Further, all the planets are round, although to verify this you need a good telescope. Again, what makes the Earth special?
If the Earth is flat, then what is on its 'underside'?
Hanging dirt and leaves? A large tree? Turtles? Those who reject the roundness of Earth either have no explanation or their explanation is based on much less solid grounding than the pro-round arguments (which, of course, is because the Earth is not flat).
If there is 'nothing' under the Earth, then lunar eclipses would make no sense as the Earth needs to be between the Moon and the Sun.
EDIT: As to the question of whether the Earth is round or some weird hemisphere/pear/donut shape, among other things those would all lead to a situation where gravity is wrong. For a hemisphere for example, gravity would not point down (towards the Earth) at any point on the Earth's surface unless if you were sitting right at the top of the hemisphere. Similar arguments can be made for the other shapes.
Sure, it's possible to make it 'work' by doing even stranger things like altering the distribution of mass and so on, but at that point you've gone very far into violating Occam's razor.


Question:
I am a big fan of the old-school games and I once noticed that there is a sort of parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece.  This parity is found with no other piece in the game.
Background: The Tetris playing field has width $10$.  Rotation is allowed, so there are then exactly $7$ unique pieces, each of which is composed of $4$ blocks.
For convenience, we can name each piece by a letter.  See this Wikipedia page for the Image ($\color{cyan}{\text{I}}$ is for the stick piece, $\color{goldenrod}{\text{O}}$ for the square, and $\color{green}{\text{S}},\color{purple}{\text{T}},\color{red}{\text{Z}},\color{orange}{\text{L}},\color{blue}{\text{J}}$ are the others)
There are $2$ sets of $2$ pieces which are mirrors of each other, namely $\color{orange}{\text{L}}, \color{blue}{\text{J}}$ and $\color{green}{\text{S}},\color{red}{\text{Z}}$ whereas the other three are symmetric $\color{cyan}{\text{I}},\color{goldenrod}{\text{O}}, \color{purple}{\text{T}}$
Language: If a row is completely full, that row disappears.  We call it a perfect clear if no blocks remain in the playing field.  Since the blocks are size 4, and the playing field has width $10$, the number of blocks for a perfect clear must always be a multiple of $5$.
My Question:  I noticed while playing that the $\color{purple}{\text{T}}$ piece is particularly special.  It seems that it has some sort of parity which no other piece has.  Specifically:

Conjecture:  If we have played some number of pieces, and we have a perfect clear, then the number of $\color{purple}{\text{T}}$ pieces used must be even.  Moreover, the $\color{purple}{\text{T}}$ piece is the only piece with this property.

I have verified the second part; all of the other pieces can give a perfect clear with either an odd or an even number used.  However, I am not sure how to prove the first part.  I think that assigning some kind of invariant to the pieces must be the right way to go, but I am not sure.
Thank you,

My colleague, Ido Segev, pointed out that there is a problem with most of the elegant proofs here - Tetris is not just a problem of tiling a rectangle.
Below is his proof that the conjecture is, in fact, false.