**Note**:*All three of the above mentioned functions are multiplicative,which means that:*

**Euler's formula**

**Note**:

The intensity per unit wavelength of the radiation of a black body of absolute temperature is given by

[align=center]

[/align]where is Plancks constant, is the speed of light in a vacuum, and is Boltzmanns constant.

From this, both **Stefans law** (which states that the intensity of the radiation of a black body is proportional to the 4[sup]th[/sup] power of its absolute temperature) and **Wiens displacement law** (which states that the maximum wavelength of the radiation of a black body is inversely proportional to its absolute temperature) can be derived.

To derive Stefans law, hold

constant and evaluate . For Wiens displacement law, set .]]>

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1. If *G* is a group of order *p[sup]n[/sup]k* (where *p* is a prime, *n* and *k* are positive integers and *p* is coprime with *k*) then *G* has a subgroup of order *p[sup]n[/sup]*, called a Sylow *p*-subgroup.

2. If *P* and *Q* are Sylow *p*-subgroups of *G*, there exists *g* ∊ *G* such that *gPg*[sup]−1[/sup] = *Q*.

3. The number of Sylow *p*-subgroups must divide the order of *G* and be congruent to 1 modulo *p*.

Newton's laws of motion.First law:- Every body continues in its state of rest or uniform motion unless compleeled by an external force to change its position.

Interestingly, this is *not* Newton's first law of motion. It is actually a restatement of his second law, *F*=*ma*.

Strangely enough, Newton made the oft repeated error when he stated the first law. However, in the pages of *Principia* immediately preceding his statement of the first law, he described the law as something substantially different.

The first law should be (and was *described* by Newton as):

*There exists a frame of reference in which an object continues in its state of rest or uniform motion unless a net external force is applied to that object.*

In fact, without this first law, the second law need not hold.

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Where: m = mass of a particle

k = Boltzmann constant (1.38 x 10[sup] -23 [/sup]JK-¹)

T = temperature

c = velocity of particle, defined as

When plotted on excel this produces a nice probability distribution

It's nice to see a model that uses pi and e

]]>The order of any subgroup *H* of a finite group *G* divides the order of *G*. The integer |*G*|⁄|*H*| is called the index of the subgroup *H* in the group *G* and is usually denoted |*G*:*H*|. Indeed, the index of *H* in *G* is precisely the number of left or right cosets of *H* in *G*.

For a star, there is a connection between the maximum wavelength of the radiation it emits, and its temperature.

Where lambda represents the wavelength, C a constant with the value 2.898 x 10[sup]-3[/sup] mK, and T is the temperature in Kelvin.

**Stefan Boltzmann's Law**

This law decribes how the luminosity of a star depends on its temperature and surface area. (assuming the star is roughly spherical)

Where L is the luminosty in Watts, r is the radius of the star in metres, T is the temperature in Kelvin, and sigma is Stefan's constant - with the value of 5.67 x 10[sup]-8[/sup] W m[sup]-2[/sup]K[sup]-4[/sup]

**Intensity**

The intensity (power per square metre) emmited by a star can be calculated using the following formula:

Where I is the intensity measured in Watts per metre squared, L is the luminosity, and D is the distance between the star and the Earth (or the radius of the sphere of radiation emitted)

]]>An expression of the conservation of energy in the steady flow of an incompressible, inviscid fluid; it states that the quantity (p/ρ) + gz + (v²/2) is constant along any streamline, where p is the fluid pressure, v is the fluid velocity, ρ is the mass density of the fluid, g is the acceleration due to gravity, and z is the vertical height. Also known as Bernoulli equation; Bernoulli law.

]]>The theorem that the equation

has no solutions in positive integers a, b, c if n is an integer greater than 2. It was stated as a marginal note by Pierre de Fermat around 1630 and not proved until 1994 by the British mathematician Andrew Wiles (born 1953).]]>

The unestablished conjecture that every even number except the number 2 is the sum of two primes.

]]>The theorem that the sum of the squares of the lengths of the sides of a right triangle is equal to the square of the length of the hypotenuse.

]]>Fermat's little theorem states that if p is a prime number, then for any integer a, (a^p − a) will be evenly divisible by p. This can be expressed in the notation of modular arithmetic as follows:

A variant of this theorem is stated in the following form: if p is a prime and a is an integer coprime to p, then (a^(p − 1) − 1) will be evenly divisible by p. In the notation of modular arithmetic:

Another way to state this is that if p is a prime number and a is any integer that does not have p as a factor, then a raised to the p-1 power will leave a remainder of 1 when divided by p.

Fermat's little theorem is the basis for the Fermat primality test.

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