Last time we sent out a warning email along the lines of:
We never ask for your username and password. If you get an email that looks like: “There is an issue with your account. Please reply with your username and password and we will rectify it” You should never reply to these messages with your details.
50 people replied with their usernames and passwords.
"Virtually all mammals experience REM sleep. However, REM sleep is a risky time, because it renders animals paralyzed and helpless. To make matters worse, during REM sleep the body consumes nearly as much energy as when it’s awake, which is a marked contrast to the other, thriftier stages of sleep. This suggests that REM has some fundamental importance, or a superior race of non-dreamers would have evolved millions of years ago"
Consider a large balloon of 100 metres in diameter. Imagine this large balloon in a football stadium. The balloon is so large that it lies on top of many members of the crowd. Because they are excited, these fans hit the balloon at different times and in different directions with the motions being completely random. In the end, the balloon is pushed in random directions, so it should not move on average. Consider now the force exerted at a certain time. We might have 20 supporters pushing right, and 21 other supporters pushing left, where each supporter is exerting equivalent amounts of force. In this case, the forces exerted towards the left and the right are imbalanced in favor of the left; the balloon will move slightly to the left. This type of imbalance exists at all times, and it causes random motion of the balloon. If we look at this situation from far above, so that we cannot see the supporters, we see the large balloon as a small object animated by erratic movement. Consider the particles emitted by Brown’s pollen grain moving randomly in water: we know that a water molecule is about 0.1 by 0.2 nm in size, whereas the particles which Brown observed were of the order of a few micrometres in size (these are not to be confused with the actual pollen particle which is about 100 micrometres). So a particle from the pollen may be likened to the balloon, and the water molecules to the fans, except that in this case the balloon is surrounded by fans. The Brownian motion of a particle in a liquid is thus due to the instantaneous imbalance in the combined forces exerted by collisions of the particle with the much smaller liquid molecules (which are in random thermal motion) surrounding it.
Basically, Brownian motion of a body is what happens when a very large number of similar forces act on the same body from all possible directions. If the forces were uniform and applied at the same time to the body then there wouldn’t be any movement but since there are slight differences in the direction, strength and time of application of the forces, the body begins to move like in the simulation below of a large dust particle (yellow) experiencing brownian motion due to collision with the smaller gas molecules around it.
The clever thing is that the same can be said of stock price fluctuations where millions of tiny trading exchanges between people (small brown gas particles) push the stock price (large yellow dust particle) into a form of Brownian motion called Geometric Brownian motion. Hundreds of billions of dollars are moving similar to the lowly dust in the air around you right now. Self-similarity, once again.
The smooth motion of rotating circles can be used to build up any repeating curve even one as angular as a digital square wave. Each circle spins at a multiple of a fundamental frequency, and a method called Fourier analysis shows how to pick the radiuses of the circles to make the picture work. Decomposing signals like this lies at the heart of a lot of signal processing. [more] [code]