The form factor is for the entire duration of the pulse
So whether the areas be equal or not the form factor of each pulse shown is 1
The amplitudes differ but form factor refers to EACH ONE taken in ISOLATION
Of course for a pulse preceded and followed by zero for alltime, the mean is zero!
0/0 is by mutual agreement, to keep us partially sane, considered to be meaningles.
I agree with Debjit
Normally used for alternating current
For your two pulses, their areas may be the same.
The rms current is the steady current that would deliver the same heat per second or power.
But your currents ARE steady - so the rms is the same as the peak current
And if the two areas (peak times duration) are equal, such a current in a resistor would deliver the same heat!
The form factor of each is 1
This reminds me of the prisoner who was told
"You will be executed one day this week but NOT if you tell us which day it will be"
So he says (like your k and k+1)
"Well I WIIL be executed one day this week
Today is Sunday, so that means BEFORE next Saturday
So if they leave it til Saturday, that is too late - for early Saturday morn I will KNOW it is "today" and they will not do it if I KNOW the day!
Similarly they cannot do it on Friday
So they CANNOT DO It
But Wednesday the jailer knocks on the door and says "It's today and you did not tell me!"
The thing about Nature is it is NOT linear.
The things going on are NOT the linear superposition of n happenings each working out as it would do ALONE (in the absence of the other things going on).
All that stuff we were taught at school in wrong!
Linears things are far less than 1% of things actually observed and going on.
If the Greeks, Indians and Etruscans had owned a computer they would NOT have stuck us in the blind alley of "linear systems"
To talk of "non linear systems" as though they were the exception is like talking of Biology as "Non-elephant biology"!
A good question!
Are you asking what they use - whether right or wrong?
Or are you asking what they ought to be using?
The thing about "standard distribution" is this:
It is a FORMULA that people would LIKE real measurements to APPROXIMATE
They adjust the numbers (mean and standard deviation) to "best fit" the available data.
The data fits well when near the mean. There are lots of results near there.
But it is a bodge
As far away as 7% the data are far fewer and the fit is bad - and nobody cares!
The farther we are from the mean the less we know and the fewer is the data!
A couple of standard deviations is probably as far as we usually should trust the numbers and the "normal distribution" idealisation.
What the hospital should do, politically, is ASSUME the misfits to the normal distribution are entirely "experimental errors"!!! That satisfies their bosses
What they ought to do is record ALL the birth weights and then SELECT the 7% lightest.
This will vary as the data varies from year to year and from state to state, of course.
Maybe last years results might HELP guess this year's results. Maybe not.
Last year's results in Illinois MIGHT be a better match to this year's in Michigan.
Depends on the weather and all else that maybe controls birth weights.
I think we now have ALL the solutions
For we have Z cubed=16
and every equation with ODD power (like we have power 3 here) has at least ONE real answer
The complex answers, like z=x+iy come ONLY IN PAIRS (because the square root of minus one is PLUS or MINUS i
Here is an interesting variant of your equation
Suppose iZ=(-4) power 2/3
Then squareroot(iZ) = minus cuberoot 16
Can you solve that one?
How many square roots does i have?
An interesting thing about this question:-
If you pull the ends of a string bent into a circle it will stretch and go straight.
But if you PUSH the ends of a piece of string you have NO IDEA what shape you will get.
It is amazing how the sort of maths we are taught at school says "everything is linear and reversible" whereas we all know that is nonsense (try mending a broken window by "reversing the forces".
Do you realise that if one of those daily numbers was a misprint then its error is exaggerated by taking its square.
Mean Absolute deviation is better
Indeed in experimental science (all data!) mean of the cube root of the deviation gives a more reliable answer as it gives more weight to the numbers whose deviation is smallest (more carefully measured?)
It might help you a lot to try to describe what your troubles are.
For example is it the symbols and notation you find difficult?
Is "complex analysis" either strange or pointless for you
I often find it helps to try to explain to others
(a) the bits you do understand
(b) what stops you on the rest of it
I don't believe there is. The satisfaction to be had when you work something out and the program it in and see it come to life is the sort of thing to make you run about the house shouting and cheering.
Dave, your icon (or whatever it is called) is great.
What equations give this endless never-repeating locus?
Are you a chaos enthusiast?
What you are twlling me is the brute-force approach, but I need a better approach than that to tackle larger numbers
The best method depends on the tools you have!
How big a number
How many milliseconds do you have?
How about all N up to 100,million?
I did not even optimise my search.
What is the point in spending one minute longer writing a better algorithm just to have the prog run in 0.0001 sec instead of 0.001
We can afford MANY seconds - our computers are ever willing ever stupid and ENJOY IT!
EXPERIMENTAL MATHS is FUN - the computer does all the boring bits!
No teeth being extracted at all!
Do I understand you want 4 squares that sum to 130
I must have got summat wrong (as usual!)
The "algorithm" is "Write a program that tries ALL the numbers up to 12 (13 is to big for on its own its square exceeds 130"
Writing the prog takes 60 seconds - running it takes milliseconds.
I assume you want all different integers
1 4 7 8
1 2 5 10
2 3 6 9
and so on
I must have completely misunderstood!
In qbasic you write your aon program and therefore UNDERSTAND the assumptions made!
Annealing is very simple
Use the RND list of pseudorandom numbers to adjust each parameter a very small amount.
Note in each case whether the solution you get is nearer to your desires or farther away.
Continue the way that promises most.
James, i'd be happy to try to help if you say what you are trying to anneal.
I have no experience of "Boltzman lattices" but lots of Southwell's Relaxation methods.
Southwell's Relaxation of Constraints is indeed a form of "annealing" and the wonderful thing about it is it FORGIVES errors (just takes longer if you make them!)
How can we put N points inside a sphere as evenly-spaced and far apart as possible?
Well with 2 points it is simple - poles apart.
With 3 it is easy too - an equilateral triangle
4 gets us thinking about the vertices of a regular tetrahedron
6 makes us think of the mid-points of its edges
But what about 5?
Well if we try one at each pole of a sphere and 4 on its equator, then the equator ones are nearer together than is any one of them to the poles.
So lets try a little trick:
REMOVE one of the points from the equispaced 6!
Ah, yes, so we have a "semiconductor crystal" with one vacancy - one "hole" for an electron to move into
There are 6 alternative positions for the vacancy - but NOT 6 alternative configurations.
Each of the 6 is the SAME, as can be shown by rotating it.
So did this trick solve the question "5 equispaced points"
Well, it depends what equispaced is.
They ARE equispaced in as much as EACH of the 5 has a nearest neighbour at distance d AND d is the same for every one of them!
OK so how about 7 points
Addding one to 6 is tough - where should we try to put it?
So remove one from 8!
8 points evenly spaced are the vertices of a cube (and 12 are the mid points of its edges)
So we remove one of the 8
Once again for each point remaining we can find a nearest neighbour the same distance away as for all other points.
Once again every one of the 8 "alternative" shapes is the SAME - a mere rotation of itself!
How much longer can we keep doing this "vacancy" trick
Can we remove 2, 3?
And HOW MANY alternative different "crystals" are there for each N?
How about N= 17?
If I were you I'd use the computer language QB64
Find out all about it on Google
It is a much empowered version of dartmout basic inventted by enthusiasts all those years ago who demanded it be SIMPLE TO USE.
An even simpler version is QB45 - but some Windows computers are prevented from running it.
I have found a way arounnd this, so get all my progs running on qb45 and THEN transfer to the faster, more powerful qb64 (requires no changes in my prog at all!)