[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 539
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And then iterate (draw a graph of the functions y=x^(n+1)-1 and y=A(x-1) to see):
x(0) = 1 - 1 / A
x(k+1) = ( x(k)^(n+1) - 1 + A ) / A
Check at n = 10, A = 5:
x(0)=0.8, and the 12th iteration results in a number of 0.823679 to the nearest 0.000001.
Checking shows that the left side of the equality differs from the right side by 0.000005.
Thank you.
I have a problem too))
solve a system of 4 equations to the 4th degree with 4 variables))
Ai - parameters, values from the corresponding admissible area.
Something nothing comes to mind, may be someone will see a simple solution... And by the way I do not remember how such systems are solved numerically?
Forgot
ai >0
-1 <= bi <= 1
... and most likely even 0 < bi <= 1
In general, the first two should suffice.
If anything, the fourth greatly simplifies the third.
There are two material points, one is moving relative to the other, how do you figure out which one is moving? )
(no way))) motion is relative.
There are two material points, one is moving relative to the other, how do you figure out which one is moving? )
It is possible to determine that stars in space are moving and even their speed relative to each other, how do astronomers do it?
The relative speed of the two objects can be calculated. There is no problem here. For example, with a speedometer or radar, the speed of a car relative to a traffic cop.
As for stars, they are not moving, they were moving once. Because the speed of light is finite and starlight reached us after a very long period of time and it is possible that it was also distorted during this very time. Many astronomical objects that we or astronomers observe no longer exist at the present time, or are located in completely different places than we observe. When astronomers report a supernova nucleation, what they are really referring to is supernova information.
The speed of stars towards or away from us is calculated from the radiation spectrum, i.e. the Doppler effect.
... The speed of stars towards or away from us is calculated from the radiation spectrum, i.e. the Doppler effect.
... thus relying on the constancy of the speed of light and the linearity of measuring distances(velocities) with it.
However, as you, Yuri, rightly point out it is possible that light was distorted for the time (or distance) of its course to us. To date it is known for certain that light is distorted by gravity, but only distorted light or distorted space through which it travels, have not been determined.
We are used to measuring everything and everything with reference "inviolable" units - meters, grams, seconds, ... And all because, within laboratory conditions, their impermanence can be neglected within the measurement error. Transferring this habit to cosmic distances is inexcusable folly. At such distances, trying to simply measure the velocity of an object by Doppler shift radiation turns into an equation with at least two unknowns.
It's a shame humanity will never get off jet propulsion in our lifetime.