Mathematics

@robjohn yeah and so, how can I get this $\forall \quad x\quad \in \quad ]-1;1[\quad (f^{ -1 })^{ ' }(x)\quad =\quad \frac { 1 }{ \sqrt { 1-x^{ 2 } } }$ from the relation you had mentionned

Alexander

To check wheter a point is asymptotic stable or not for a differential system, do I have to solve the differential system?

KaliMa

how can i use the pythagorean generator for 5w^2 to output gcd(1) terms only?

Walkerneo

Anyone want a trip back in time with some of my calc homework? math.stackexchange.com/questions/250194/…

Old John

@pourjour Yep - that's me

@pourjour that photo was taken as I was looking at a camel I was about to try to ride :)

KaliMa

@OldJohn I think I got Will Jagy's relation working

Old John

7:54 PM

@KaliMa Good - you might also look at a solution I did a while ago which is relevant to exactly the same equation

KaliMa

ah

however, it seems like this parameterization generates nonprimitives too, correct?

user19161

@OldJohn Still getting used to your lovely new photo!

KaliMa

what i'd like to do is generate all unique primitive solutions to a^2+b^2=5c^2 where a<b<c

how can I accomplish this with Jagy's parameters?

i.e. how to properly scan through q and r?

Old John

@WillHunting not sure I would ever describe it as lovely ... but it is OK

user19161

@OldJohn Maybe it looked better than the previous one!

Old John

8:01 PM

@KaliMa Will's process certainly generates precisely all the primitive solutions (probably have to iterate over $q,r$ which are coprime) - you can then sort them into order if you wish by bubblesort or similar

KaliMa

it generates more than primitives though

since many of the results have gcd>1

Old John

@KaliMa probably get rid of those by taking $q,r$ coprime

KaliMa

i feel like there has to be an easier way

the original pyth. triplet generator doesn't need to do that

Charlie

@WillHunting Hello dear friend!

Old John

@KaliMa I very much doubt it - I would recommend studying Will's method carefully (actually identical to the method I used in my solution I posted a few minutes ago - I had forgotten it was exactly the same conic!). Will Jagy is very competent indeed at quadratic forms - you could learn a lot ...

KaliMa

8:06 PM

is there a systematic way to generate all r coprime to q?

Old John

@KaliMa the original Pthyag parametrisation certainly DOES have to do that - to get primitive solutions you have to ensure that $m,n$ are coprime AND of opposite parity - please check

KaliMa

or does that take more effort than just going through all q,r?

Charlie

@pourjour hello!

KaliMa

ahhh you are right

def triples(r):
  for n in xrange(1, r):
    if (n*n + (n+1)*(n+1) > r): break
    for m in xrange(n+1, r, 2):
      if (n*n + m*m > r): break
      if (gcd(n, m) > 1): continue
      yield (m*m - n*n, 2*m*n, m*m + n*n)

hmmm with jagy's generator checking for gcd(q,r)=1 is not enough

Old John

@KaliMa maybe opposite parity as well?

KaliMa

8:13 PM

hm nope

did this check

if gcd(q,r)>1: continue
if q%2==r%2: continue

Old John

@KaliMa there is almost certainly some simple modulus condition on $q,r$ which will guarantee solutions are primitive - try tabulating and looking for patterns

KaliMa

seems like they can be odd/even, even/odd, or odd/odd

but not even/even

that isnt enough though to ensure primality

Old John

@KaliMa try looking at other moduli other than 2 - maybe 3 or 5 or ...

KaliMa

tried that, nothing odd

Old John

try tabulating - and shade the squares which do not give primitive solutions and spot a pattern

KaliMa

8:21 PM

i dont know what that means

Old John

draw a square grid with $q$ and $r$ taking values 1, 2, 3, ...

Alexander

Can I write as the solution of $x(t)=A(t)x(t)$ the equation $x(t)=X(t,t_0)x_0+\int_{t_0}^{t}X(t,s)A(s)ds$ where "X(t,t_0)" is the principal matrix solution ?

KaliMa

[2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]

[1, 0, 1, 0, 1, 0, 1, 0, 5, 0, 1, 0, 1]

[10, 1, 0, 1, 2, 0, 2, 1, 0, 1, 10, 0, 2]

[1, 0, 5, 0, 1, 0, 1, 0, 1, 0, 1, 0, 5]

[2, 1, 2, 1, 0, 1, 2, 1, 2, 0, 2, 1, 2]

[1, 0, 0, 0, 1, 0, 5, 0, 0, 0, 1, 0, 1]

[2, 1, 2, 5, 2, 1, 0, 1, 10, 1, 2, 1, 2]

[5, 0, 1, 0, 1, 0, 1, 0, 1, 0, 5, 0, 1]

[2, 1, 0, 1, 2, 0, 2, 5, 0, 1, 2, 0, 10]

[1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1]

[2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 0, 5, 2]

[1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1]

[10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 0]

0's = the gcd was >1 so i just put 0, the others show the raw gcd

lvella

I have an angle as a real number; knowing it is greater than Pi, how to get this same angle between Pi and -Pi ?

KaliMa

er no that's wrong

oh worry, right, i meant the 0's meant the GCD of u,v was too high so we can ignore it

the values otherwise show gcd of the triplet

Charlie

8:31 PM

@lvella Não entendi...

Old John

@lvella that doesn't make any sense - can you explain

lvella

ok, not an angle

I have a rotation, in radians

supose 66.345 rad

Charlie

@lvella okay

KaliMa

filled in the matrix with 1's when the triplet was primitive, 0 otherwise; no pattern

[0, 0, 0, 1, 0, 1, 0, 1, 0]
[1, 0, 1, 0, 1, 0, 1, 0, 0]
[0, 1, 0, 1, 0, 0, 0, 1, 0]
[1, 0, 0, 0, 1, 0, 1, 0, 1]
[0, 1, 0, 1, 0, 1, 0, 1, 0]
[1, 0, 0, 0, 1, 0, 0, 0, 0]
[0, 1, 0, 0, 0, 1, 0, 1, 0]
[0, 0, 1, 0, 1, 0, 1, 0, 1]
[0, 1, 0, 1, 0, 0, 0, 0, 0]

lvella

and suppose a clock pointer has rotated that amount of radians

Charlie

8:36 PM

@lvella yes...

lvella

what is the angle between its final position and its initial position?

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$Mathematics$ Mathematics