Note:All three of the above mentioned functions are multiplicative,which means that:
Euler's formula
Note: is not always the smallest number for which the above identity can hold.
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[/align]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 .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
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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