modular arithmetic (sometimes called clock arithmetic) is a system of arithmetic for integers, where numbers ”wrap around” upon reaching a certain value—the modulus.
An example of this is the clock which starts repeating itself when it has reached 12. Therefore it is said that the clock follows anarithmetic modulo 12.
Ex: If you add 7 hours to 8 o’clock the result is not 15 since when thee clock reaches zero it starts over from zero once again so the result is 7+8=3 in this arithmetic.
For a positive integer n, two integers a and b are said to be congruentmodulon,
a=b mod(n)
if the the remainder is the same when a and b are divided by n.
if their difference a − b is an integer multiple of n. The number n is called the modulus of the congruence.
In the evening today February the 15th. an asteroid flies by close to the Earth only about 27 600 km away. It measures about 40 m in length and is similar to the one responsible for the Tunguska (Siberia) impact in 1908. It deforestated everything within 30 km.
The force of gravity determines the trajectories of the celestial bodies. Mathematical analysis reveals that there are three types of trajectories possible for a body moving in a gravitational field determined by Newton’s law of gravity.
If one of the bodies has very high speed relative to the other the moving body traces out a hyperbola. The equation for this is x 2/a2 – y 2/b2 =1
If the speeds of the bodies are beelow a certain threshold value they move in an elliptical curve. x 2/a2 + y 2/b2=1
The limiting case between the hyperbola and the ellipse is the parabola. y=ax2
Those geometrical objects can be illustrated by slicing the cone according to the figure above.
A collection of rembrandts self-portraits serve as an illustration of the passage of time
When left to itself snow spontaneously would never build a snowman.
It will only form different kinds of heaps . This can be undestood as the consequence of statistics: There are more possible arrangements of the atoms constituting the snow leading to unordered heaps than arrangements resulting in the much more ordered state of a snowman,.
This cicumstance can be described with the concept ofstatistical weight.
The statistical weight (is the number of microscopic states(ways of arranging the atoms) %omega;corresponding to a certain macroscopic state(the large scale object).
The entropy, S, is defined as
S = k ln Ω
It is a measure of the disorder of the system. In all spontaneous processes the entropy increases because statistically the state with the greatest statistical Weight is more probable.
Such a process is irreversible.
In an adiabatic process no Heath is exchanged with The surroundings.
Världens största ånglok, Big Boy, är exempel på en värmemaskin vars effektivitet beror på temperaturskillnaden mellan vattenångan och den omgivande luften.
A spacecraft enters the atmosphere with a velocity of appr. 40 000 km
The temperature is a measure of the heat content of a body.
Heat or thermal energy is a form of energy that moves from hotter to cooler bodies.
The average kinetic energy of a molecule with an absolute temperature T is given by
E = 3kT/2.
Since temperature is a measure of heat heat is a measure of the kinetic energy of the constituents of the medium of a certain temperature.
If the temperature increases the movements of the individual atoms also gets more violent until the chemical bondings are broken up and the atoms become free.
This is how an element changes from solid to liquid to gaseous form.
Those are the three states of aggregation of matter. In stars a fourth state is possible: the plasma state.
Atoms and molecules never rest (this is the third law of thermodynamics) and when they bounce together electrons in the atoms are excited and within nanoseconds they deexcite by emission of low-energetic electromagnetic radiation so-called Infra-red radiation or heat-radiation.
According to Heisenberg’s uncertainty-principle it’s impossible for an atom to be at rest because this would imply that we have exact knowledge about the speed of the atom. The Heisenberg uncertainty principle δp˙δx>h
then postulates that the uncertainty in the position of the atom must be infinite.
This radiation from e.g. the Sun is perceived as heat and it drives the atoms of the skin into oscillations whereby the temperature in the skin is increased.
All materials having a temperature in excess of absolute zero emits heatradiation. This process occurs until the temperature of the body and its surroundings are equal.
Then the body emits and absorbs the same amount of heat energy and an equilibrium has been established.
The first postulate of thermodynamics states that the energy of an isolated system is constant. Energy can not be created or destroyed only transformed between different forms.
kinetic energy
potential energy
nuclear energy
radiation energy
Värmekapaciteten. Kan uttryckas med hjälp av frihetsgraderna med hjälp av ekvipartitionsprincipen. Den anger hur mycket energi som krävs för att öka temperaturen 1C. För ett kg av ämnet.
C=0,5(ntr + nrot + nvib2)R.
Van der Waals gaslag
(P+a/v)(V-b)=kT.
a/V anger växelverkan mellan gasmolekylerna medan b är ett mått på den volym gasmolekylerna upptar.
For your service and convenience; I have added an RSS-feed from Daily Telegraph as a widget in the meny list to the right. Just click on the link and the latest Science and Tech-News will flow into your computer.
Today I read that the ten most important British inventions are:
→Taylor-expansion is a method of approximating a function f(x) around a point a with a polynomial of the argument x in the vicinity of a. The polynomial itself consists of the derivatives of the function of various orders.
Tn(x) = f(a) + f ‘ (a) (x-a) + f ”(a) (x-a)2/2! + f(3)(x) (x-a)3/3! + …. + R(x). The Taylor-expansion of the exponential function is:
To prove the validity of this statement consider the special case of MacLaurin-polynomials were a function is expanded around x=0. Observe the polynomials
p(x) = a +bx + cx2 + dx3
p'(x) = b + 2cx + 3dx2
p”(x) = 2c + 2 *3dx.
x= 0 in each of those equalities and get expressions for the coefficients a,b,c and d. p(0) = a → a=p(0)
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