The Navier-Stokes equations describe the motion of fluid substances and are fundamental in fluid mechanics. They are derived from the principles of conservation of mass, momentum, and energy. Here, I'll summarize how these equations are derived step-by-step. ### 1. Conservation of Mass (Continuity Equation) The principle of mass conservation states that the mass of a fluid in a control volume must remain constant over time if no mass enters or leaves the volume.

FreeS/WAN is an open-source software implementation of the IPsec (Internet Protocol Security) protocol suite, which is used to secure Internet Protocol (IP) communications through encryption and authentication. The name "FreeS/WAN" stands for "Free Secure Wide Area Network." Developed in the late 1990s, FreeS/WAN was one of the first IPsec implementations available for Linux, allowing users to create secure virtual private networks (VPNs).

Hmmm, he does not know how to spell guerilla? sic? www.quora.com/What-is-the-correct-spelling-guerilla-or-guerrilla

Note to self: if you are going to commit a crime, don't publish your plans online.

Ross Ulbricht's diaries come to mind.

That's how Russian shadow library maintainers do it, they know how to crime good old Russians. Maybe there is a good thing about having dictatorships in the world that give zero fucks about American copyright laws. There will always be some random Russian academic who will implement this and not go to jail. Maybe it's even state sponsored.

This is a quick tutorial on how a quantum computer programmer thinks about how a quantum computer works. If you know:a concrete and precise hello world operation can be understood in 30 minutes.

Although there are several types of quantum computer under development, there exists a single high level model that represents what most of those computers can do, and we are going to explain that model here. This model is the is the digital quantum computer model, which uses a quantum circuit, that is made up of many quantum gates.

Beyond that basic model, programmers only may have to consider the imperfections of their hardware, but the starting point will almost always be this basic model, and tooling that automates mapping the high level model to real hardware considering those imperfections (i.e. quantum compilers) is already getting better and better.

The way quantum programmers think about a quantum computer in order to program can be described as follows:

To operate a quantum computer, you follow the step of operation of a quantum computer:

Each time you do this, you are literally conducting a physical experiment of the specific physical implementation of the computer:and each run as the above can is simply called "an experiment" or "a measurement".

The output comes out "instantly" in the sense that it is physically impossible to observe any intermediate state of the system, i.e. there are no clocks like in classical computers, further discussion at: quantum circuits vs classical circuits. Setting up, running the experiment and taking the does take some time however, and this is important because you have to run the same experiment multiple times because results are probabilistic as mentioned below.

Unlike in a classical computer, the output of a quantum computer is not deterministic however.

But the each output is not equally likely either, otherwise the computer would be useless except as random number generator!

This is because the probabilities of each output for a given input depends on the program (unitary matrix) it went through.

Therefore, what we have to do is to design the quantum circuit in a way that the right or better answers will come out more likely than the bad answers.

We then calculate the error bound for our circuit based on its design, and then determine how many times we have to run the experiment to reach the desired accuracy.

The probability of each output of a quantum computer is derived from the input and the circuit as follows.

First we take the classic input vector of dimension N of 0's and 1's and convert it to a "quantum state vector" $\overrightarrow{q_{in}}$ of dimension $2^n$:

We are after all going to multiply it by the program matrix, as you would expect, and that has dimension $2^n\times 2^n$!

Note that this initial transformation also transforms the discrete zeroes and ones into complex numbers.

For example, in a 3 qubit computer, the quantum state vector has dimension $2^3=8$ and the following shows all 8 possible conversions from the classic input to the quantum state vector:

This can be intuitively interpreted as:

Now that we finally have our quantum state vector, we just multiply it by the unitary matrix $Q$ of the quantum circuit, and obtain the $2^n$ dimensional output quantum state vector $\overrightarrow{q_{out}}$:

And at long last, the probability of each classical outcome of the measurement is proportional to the square of the length of each entry in the quantum vector, analogously to what is done in the Schrödinger equation.

For example, suppose that the 3 qubit output were:

Then, the probability of each possible outcomes would be the length of each component squared:i.e. 75% for the first, and 25% for the third outcomes, where just like for the input:

All other outcomes have probability 0 and cannot occur, e.g.: 001 is impossible.

Keep in mind that the quantum state vector can also contain complex numbers because we are doing quantum mechanics, but we just take their magnitude in that case, e.g. the following quantum state would lead to the same probabilities as the previous one:

This interpretation of the quantum state vector clarifies a few things:

As we could see, this model is was simple to understand, being only marginally more complex than that of a classical computer, see also: quantumcomputing.stackexchange.com/questions/6639/is-my-background-sufficient-to-start-quantum-computing/14317#14317 The situation of quantum computers today in the 2020's is somewhat analogous to that of the early days of classical circuits and computers in the 1950's and 1960's, before CPU came along and software ate the world. Even though the exact physics of a classical computer might be hard to understand and vary across different types of integrated circuits, those early hardware pioneers (and to this day modern CPU designers), can usefully view circuits from a higher level point of view, thinking only about concepts such as:as modelled at the register transfer level, and only in a separate compilation step translated into actual chips. This high level understanding of how a classical computer works is what we can call "the programmer's model of a classical computer". So we are now going to describe the quantum analogue of it.

Bibliography:

The drag equation is a mathematical formula used to calculate the drag force experienced by an object moving through a fluid, such as air or water. The drag force (F_d) is the resistance experienced by the object due to the fluid surrounding it.

One of the dudes from the AtomSea & EMBII Bitcoin-based file upload system.

The Euler–Tricomi equation is a second-order partial differential equation (PDE) that arises in various fields, including fluid dynamics and mathematical physics. It is named after the mathematicians Leonhard Euler and Francesco Tricomi.

Faxén's law describes the force experienced by a spherical particle suspended in a fluid when it is subjected to an external oscillating field, such as a pressure gradient or a fluid flow. It is particularly relevant in the study of colloidal suspensions and the behavior of particles in non-Newtonian fluids.

The Hadamard–Rybczynski equations describe the motion of a fluid in a gravitational field, particularly in the context of fluid dynamics. These equations are important in studying the behavior of inviscid and incompressible fluids, especially when analyzing potential flow around bodies. The Hadamard–Rybczynski equations relate the velocity potential or stream function to the shape of the body and the flow conditions around it.

Rayleigh's equation in fluid dynamics refers to a fundamental principle that describes the stability of a fluid flow. It is often associated with the stability analysis of boundary layers and the onset of turbulence and instabilities in various fluid flow situations. One common context in which Rayleigh's equation is discussed is in the study of stability of various flow regimes, particularly in relation to the growth of instabilities in a shear flow. The equation is typically derived from the Navier-Stokes equations under specific assumptions and conditions.

You modify the DNA of a cell and stick a fluorescent protein right before or after another protein. Then when it gets translated, the GFP is stuck to the protein of interest, which hopefully hasn't lost its function as a result, then you can just see the protein of interest.

The Buckley–Leverett equation is a fundamental equation in petroleum engineering and reservoir engineering that describes the movement of two-phase fluids (typically oil and water) in porous media. It models the flow behavior of immiscible fluids in a reservoir when one fluid displaces another, commonly used to analyze waterflooding operations during oil recovery. The equation is derived from the conservation of mass principle and reflects the dynamics of the interfaces between the two fluids.

The Boussinesq approximation is a mathematical simplification used in fluid dynamics, particularly in the study of weakly non-linear and dispersive wave phenomena, such as water waves. Named after the French physicist Joseph Boussinesq, this approximation is particularly useful for analyzing the behavior of surface waves in fluids where the amplitude of the waves is small compared to the wavelength.

This section discusses the pre-photon understanding of the polarization of light. For the photon one see: photon polarization.

polarization.com/history/history.html is a good page.

People were a bit confused when experiments started to show that light might be polarized. How could a wave that propages through a 3D homgenous material like luminiferous aether have polarization?? Light would presumably be understood to be analogous to a sound wave in 3D medium, which cannot have polarization. This was before Maxwell's equations, in the early 19th century, so there was no way to know.