QuantumBullet

It may be easier to work with if you first rearrange the fraction: (1-e^x / 1+e^x) = 1 - (2e^x / 1+e^x). Since the integral of this latter fraction is more easily calculable, and is equal to x - 2ln(1+e^x)

In that example, {a,b,c} is both a distinct element, and a subset, of the Power Set.

The Planck length is a necessary component of String Theory. In fact the Mathematics of String Theory ensures that this is the smallest possible length, since any 'measurement' taken that is within an area with a radius smaller than the Planck length, is mathematically indistinguishable from a measurement taken within an area with a radius equal to the reciprocal of the smaller length. Essentially, if you can imagine (which of course, you can't) a universe which has a radius equal to 1/10 the Planck length (10^-62 m), this would be mathematically indistinguishable from a Universe with a radius equal to the reciprocal of this - approximately 10^62 m, which is much much larger than the size of the current observable universe. it is this reciprocality which prevents any distance smaller than the Planck length from occuring; since the dual to this measurement will be larger than the Planck length.

You can think of a particle as having a Wave function, which extends to every spatial point in the Universe. Particular points on this wave correlate to the probability of finding the particle at a particular point in space. Once you make an observation on the particle, you are effectively interfering with this wave function, and cause it to decohere, and hence it appears to be in a single point in space. It can be said that, prior to this decoherence (and hence observation of the particle), the undisturbed wave function of the particle gives way to a small (but finite) probability that the particle can be found at any single arbitrary point in space.

Quantum teleportation doesn't re-arrange atoms; it involves scanning an atom or a particle, and then relaying the same quantum information to another atom or particle at a more distant location. So it is really creating a replica of an atom, not teleporting it. And considering at this stage the best that has been managed is to quantum-teleport a photon over a distance of around 50km, I'd say it would be quite some time before we have the ability (if ever) to perform the same feat on something as large as a spoon.

Due to the HUP, you cannot have an area of 0 energy. On a macro-scale, it may seem that the vacuum contains 0 energy. However, due to HUP, the smaller and smaller the scale on which you examine the vacuum, the quantum fluctuations and uncertainty increases. This is why, as it is commonly expressed, on the smallest of scales (i.e. the Plank length), the space-time fabric is extremely distorted, due to virtual particles etc. As you zoom out to a macro-scales, the very same region of space-time appears to be smooth, and hence can handle 0 energy. As a side, but very crucial, point, the Laws of Thermodynamics strictly disallow any system to attain 0 K. This is because temperature arises from kinetic energy, so a particle with a temperature of 0K would have no motion. I.e. we would know with 100% precision the location and energy of said particle. This is in compelte contradiction with the HUP. So there are 2 inherent Laws that prevent a 0 Energy system./

What got me hooked was learning about Newtonian mechanics, and being able to calculate the exact trajectory and final positon of an object in motion. It is pretty standard stuff, but i found it fascinating that you could work out exactly where something would land, if you shot it at a certain speed and angle.

A photon is the carrier particle for the Electromagnetic force. Quite the same as Gluons are the carrier particles for the Strong Nuclear force. In the extremely early universe, all 4 universal forces (Gravity, Strong/Weak Nuclear, and Electromagnetic) were unified into 1 signle force. As symmetry breaking occured (within the first fraction of a fraction of a fraction of a fraction of a second after the Big Bang), one by one the forces split off from each other, resulting in 4 distinct forces to govern the physical universe. It is widely accepted that Gravity was the first the separate, followed by the Strong Nuclear, then the Weak Nuclear forces. It would have been the latter symmetry breaking that created the EMF, and hence the propogation of photons. Note that this was prior to the existence of any baryonic matter (i.e protons, neutrons..). As for the the first 'atoms', it wasnt until around 380,000 years AFTER the Big Bang, when the temperature of the Universe has cooled to around 4,000K that it was cool enough for atoms to be able to condense. These newly forms atoms were then able to snare the rogue electrons flying around the cosmos, and create the first Hydrogen atoms. With the fog of electrons now thinner, photons were able to travel alot further before being absorbed, and hence the Universe ceased to be opaque for the first time since the beginning. And as for the first generation of stars would have been a considerable amount of time even after this; the newly formed atoms would need to condense into large enough clumps to initiate the gradual building of material necessary to make a star. It would need to have a large enough size/mass the ignite, and commence fusion....and considering the temperature of the Universe was around 4,000K when the Universe was 380,000 years old, which is comparable to the surface temperature of a star, again it would have been a considerable amount of time after this before the first stars ignited. So in short, no, the first photons did not appear along with the first generation of stars.,

Richard Feynman showed that there can be exceptions to the Law of Conservation of Matter/Energy. For example, in a vacuum electron/positron pairs can spontaneously appear and annihilate each other (due to quantum fluctuations in the vacuum), producing a pair of photons with a total energy equal to the energy of the original lepton pair. This would seemingly indicate that matter can appear from nothing; however, the above can only hold given that the photons then dissipate in a very short time frame. In fact, the higher the energy of the photons, the shorter the time period they can exist for (i.e they are inversely proportional)

It then includes the set of all elements. Since all the elements are the set, isn't this saying that set A is a subset of itself? The set A is a subset of the Power Set A. It is also an element of the Power Set A.

For the function f(x), finding d/dx f(x) = f'(x) will yield the gradient of the curve at the point x. Simply take the negative reciprocal to this value to find the gradient of the normal to the curve at this point. (i.e. m = - 1/f'(x) )