I'm searching for the expansion, e.g. to second order in

>>8165451
what is p_i(t) ?? Or what can you tell us about it?

>>8165451

Is the repeated multiplication meant to be over the variable i or n? And what is pi(t)?

>>8165469
>>8165630

It looks like p is a function of i and t and the index should probably be over i instead of n...

... but we need to know what p is.

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>>8165469
>>8165630
The multiplication is supposed to be over the index i (not n).

For each i, we have that [math] p_i(t) [/math] is a function and I take it to be smooth.
Let's be more specific, although these are not fixed rules:
[math] p_i(t) [/math] should to be a function that is zero until a time [math] t=T_i^{introduction} [/math] and grows to some value in [math](0,1)[/math], e.g. 0.2

Background:
There are driver assistance systems [math]i[/math] (like Automatic parking for cars) that that ought to reduce the number of car accidents.
https://en.wikipedia.org/wiki/Advanced_driver_assistance_systems

People in this industry assign pretty random numbers, called potentials [math]p_i[/math], to those systems that ought to capture their effectiveness in doing so, and thus their value. Not all cars will have all systems implemented, and that's why one must make a choice/subset of the available ones.
For a fixed time span, say a month, the idea is that if there wasn't the system i, then there would be [math]n_z[/math] accidents, and with the system, [math]n_i[/math] (with [math]n_z < n_i[/math] because lives are saved etc.) of those would be prevented.
The basic deal is to define
[math] p_i = \dfrac{n_i}{n_z} [/math]
If e.g. [math] p_i = \dfrac{1}{10} [/math] or [math] n_i = n_z/10 [/math], then it means the system i prevents 1/10's of all the accidents.

What's empirically available is the number [math] n_r [/math] of real accidents that happened.
I have the police data for Austria, there are about 5 accidents per hour recorded or whatever.
We don't know [math] n_z [/math], the number of accidents in the world without i's.
For the prevented accidents, we should have
[math] n_i = n_z - n_r [/math].
So
[math] p_i = 1 - \dfrac{n_r}{n_z} [/math]
[math] 1 - p_i = \dfrac{n_r}{n_z} [/math]
[math] n_z = \dfrac{n_r}{1 - p_i} [/math]
for i.
cont.

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(cont.)

Now the preventable accidents [math] n_i [/math] are actually severely reduced by the fraction of cars that have it implemented. For some i, I have the introduction/invention date and the distribution for cars today.
Say i was introduced 1986 and [math] d_i = 0.004 [/math] in 2016, meaning 0.4% of cars have it implemented today.
Between that there will be a ramp, a growth function [math]g(t)[/math] from 0 to [math] d_i [/math] how many people use it. Somebody does literature research about this soon at the company, I think. Say it's linear or something else.

Then it seems natural to consider as potentials [math]p_i = g_i(t) p_i^{max} [/math] where [math] p_i^{max} [/math] is the potential if we forced all cars to implement it.

Yeah, so I'm tasked (kinda task myself - they pay me to publish any good mathematical modeling justifying their political lobbying for stricter regulations) to compute the "what if we did nothing" worse cases, given the real data and heuristics about the potentials, which are guessed by people, car producers, or rarely evaluated in experiment.

So finally, that product ought to give me the multiplication factor (assuming a collection of potentials with the same introduction times) for the best guess going from the real data to the "what if we hadn't introduced those systems curve".

(For a report by the European commission talking about the potentials, google
"benefit and feasibility of a range of new..."
Doesn't seem like people who know any math are involved in this. Which is arguably good, makes it more interesting and I can't fail too hard.)

I need to think about how to implement other time delays and some human factors (as people are known to be more reckless if they have more tech) - no idea yet how to do all this.

ALL IDEAS WELCOME

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examples for such potentials

http://udv.de/system/files_force/tx_udvpublications/RR_12__fas.pdf#57

>>8165681
Btw.
the formula

[math] n_z(t) = n_r(t) \, \prod_{i=1}^N \dfrac { 1 } { 1-p_i(t) } [/math]

or even

[math] n_z = \dfrac { n_r } { 1-p } [/math]

clearly isn't flawless, as for high potentials p, i.e. good system, and given some real [math] n_r [/math], it would imply the number of accidents [math] n_z [/math] diverges, while it's of course capped from above by a number [math] n_{max} [/math] of accidents that would have ever taken place.

Any ideas how to implement this into the formulas welcome too.

Can i ask here how to solve an integral?

A while ago someone asked how to generate positive integer solutions to 1/x+1/y = 1/z for a given x

I just figured it out. The solutions are y = x(p_n - 1) and z = x(1 - 1/p_n) where p_n is the nth prime factor of x, and p_0 = x

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>>8166380
I was actually the one with the series of answers in your thread.
PS looks fishy

>>8166292
Sure, I consider it a meaningful bump

>>8165737
> \dfrac
gross

>>8166814
It's pretty easy to prove. An obvious solution is y = x(x-1) and z = x-1. Then prove that every solution must be a multiple of this solution because if we take z = x-n, where n is a positive integer then x and z must be divisible by n for y to be a positive integer. So y = x(x-n)/n and z = x-n. Since x must be divisible by n, x/n must be a prime factor of x or 1. If x/n is a factor but not prime then we get duplicate solutions.

>>8166380
>>8166814
>>8166817

First, let's find rational solutions to
1/x = 1/z-1/y
where x is a given rational.
One point is: z= x-1, y= x*(x-1).
Find equation of line through point, slope m:
m = (z-x+1)/(y-x*(x-1)).
Find intersection of line and 1/x=1/z-1/y:
z=x-m*x^2
y=1/m-x
Now if y and z are to be positive integers,
need 1/m = n (integer) i.e. m=1/n, and n>x, so
y=n-x
z=x-x^2/n, where n divides x^2 and n>x

Example:
x=10, n=25, y=15, z=6.
1/10+1/15=1/6

This is more general than the case where
n=x*p and p divides x.

>>8166963
>>8166982
Looks like we did something very similar. I think mine is a little more general because my "thing" doesn't need to divide x, just needs to divide x^2.