Re: [math-fun] Help with some math
c2(x) := c(c(x)) - x² + 2 is fascinating. Continuous and real for x > -2, where c2(-2+) = - 4 sin(π √2)² ~ -3.7164323713376353833, followed immediately by a minimum at x = -√(2 - 2 sin(π √2)) ~ -1.981869083895239 where c2(x) = 4 sin(π √2) ~ -3.8556101313995. c2(x) first reaches its permanent value of 0 at x = 2 cos(π/√2) ~ -1.2113997341576, where the slope is discontinuous. All these magic numbers are technically conjectural, from a spectacular performance by Munafo's ries. As David suggests, Mathematica's simplifiers were embarrassed at several points but excelled at others. Likewise the numerics. There are some bugs to report! --rwg (Note: David is using b^3, e.g., to mean b(b(b())).) On 2017-03-23 15:40, David Wilson wrote:
In the fortuitous case of
b(x) = x^2 - 2
b is conjugate to
a(x) = x^2
by the simple mapping
f(x) = x + x^-1 f^-1(x) = (x + sqrt(x^2 - 4))/2
with
b^n = f o a^n o f^-1
Specifically, if
c(x) = b^(1/2)(x) = ((x + sqrt(x^2 - 4))/2)^sqrt(2) + ((x - sqrt(x^2 - 4))/2)^sqrt(2)
then
c(c(x)) = x^2 - 2.
It's a nice problem to give the c(x) formula and ask for c(c(x)). It might be fun to blindside a math forum with this problem. I can't get Wolfram Alpha to answer it.
At any rate, the form of this f(x) led me to postulate the power series form for the f mapping in the x -> x^2 + 1 case.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of rcs@xmission.com Sent: Thursday, March 23, 2017 12:42 AM To: math-fun@mailman.xmission.com Cc: rcs@xmission.com Subject: Re: [math-fun] Help with some math
Dan's worked with flows quite a bit.
Some of the rest of us have played around a little, looking for functional square-roots or real-indexed iterated functions. There are dragons lurking.
Long ago, I wrote several flow programs in Lisp, using the nice feature of exact rational arithmetic. I tried to find a power series for the half flow of x+x^2, expanded around 0. I used f(x) = x+x^2, g(x) = x + x^2/2 + tbd, and the equation f(g(x)) = g(f(x)), matching the power series on each side. This led to a relatively simple formula for computing terms of g(). Things looked good at the beginning, but I got greedy and went out to a hundred terms. The numbers looked nice for a while, but soon the numerators & denominators ballooned. To make sense of things, I floated the rational coefficients, and took the nth root of |g_n| to estimate the reciprocal of the radius of convergence. A nasty surprise: the numerical evidence was that the radius of convergence seemed to be heading for 0.
Some of our other experiments had odd results.
Good luck with your approach!
Rich
Thanks for those numbers Bill. I was trying to figure out the number where c(c(x)) - x² + 2x is first permanently zero, but without success. So your number x = 2 cos(π/√2) ~ -1.2113997341576 was just the ticket! On Fri, Mar 24, 2017 at 4:06 AM, Bill Gosper <billgosper@gmail.com> wrote:
c2(x) := c(c(x)) - x² + 2 is fascinating. Continuous and real for x > -2, where c2(-2+) = - 4 sin(π √2)² ~ -3.7164323713376353833, followed immediately by a minimum at x = -√(2 - 2 sin(π √2)) ~ -1.981869083895239 where c2(x) = 4 sin(π √2) ~ -3.8556101313995. c2(x) first reaches its permanent value of 0 at x = 2 cos(π/√2) ~ -1.2113997341576, where the slope is discontinuous. All these magic numbers are technically conjectural, from a spectacular performance by Munafo's ries. As David suggests, Mathematica's simplifiers were embarrassed at several points but excelled at others. Likewise the numerics. There are some bugs to report! --rwg
(Note: David is using b^3, e.g., to mean b(b(b())).) On 2017-03-23 15:40, David Wilson wrote:
In the fortuitous case of
b(x) = x^2 - 2
b is conjugate to
a(x) = x^2
by the simple mapping
f(x) = x + x^-1 f^-1(x) = (x + sqrt(x^2 - 4))/2
with
b^n = f o a^n o f^-1
Specifically, if
c(x) = b^(1/2)(x) = ((x + sqrt(x^2 - 4))/2)^sqrt(2) + ((x - sqrt(x^2 - 4))/2)^sqrt(2)
then
c(c(x)) = x^2 - 2.
It's a nice problem to give the c(x) formula and ask for c(c(x)). It might be fun to blindside a math forum with this problem. I can't get Wolfram Alpha to answer it.
At any rate, the form of this f(x) led me to postulate the power series form for the f mapping in the x -> x^2 + 1 case.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of rcs@xmission.com Sent: Thursday, March 23, 2017 12:42 AM To: math-fun@mailman.xmission.com Cc: rcs@xmission.com Subject: Re: [math-fun] Help with some math
Dan's worked with flows quite a bit.
Some of the rest of us have played around a little, looking for functional square-roots or real-indexed iterated functions. There are dragons lurking.
Long ago, I wrote several flow programs in Lisp, using the nice feature of exact rational arithmetic. I tried to find a power series for the half flow of x+x^2, expanded around 0. I used f(x) = x+x^2, g(x) = x + x^2/2 + tbd, and the equation f(g(x)) = g(f(x)), matching the power series on each side. This led to a relatively simple formula for computing terms of g(). Things looked good at the beginning, but I got greedy and went out to a hundred terms. The numbers looked nice for a while, but soon the numerators & denominators ballooned. To make sense of things, I floated the rational coefficients, and took the nth root of |g_n| to estimate the reciprocal of the radius of convergence. A nasty surprise: the numerical evidence was that the radius of convergence seemed to be heading for 0.
Some of our other experiments had odd results.
Good luck with your approach!
Rich
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I presume you are talking about evaluating c2(x) = c(c(x)) - x^2 + 2 with c(x) = ((x + √(x^2 - 4))/2)^√2 + ((x - √(x^2 - 4))/2)^√2 If you start with |x| >= 2, you end up with c(x) = y^√2 + z^√2 where y and z are real, so it is reasonable to take y^√2 and z^√2 real as well. If |x| < 2, however, we end up with c(x) = y^√2 + z^√2 Where y and z are complex numbers. It then seems that y^√2 and z^√2 take on an infinitude of non-real complex values, none better than another. So how do you compute y^√2 or z^√2 with y or z complex? Is there some principle value? At any rate, rwg and J. Buddenhagen seem to agree on an evaluation of c2(x) on |x| < 2. I would like to see a plot of c2(x) on the real line.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Friday, March 24, 2017 6:06 AM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Help with some math
c2(x) := c(c(x)) - x² + 2 is fascinating. Continuous and real for x > -2, where c2(-2+) = - 4 sin(π √2)² ~ -3.7164323713376353833, followed immediately by a minimum at x = -√(2 - 2 sin(π √2)) ~ -1.981869083895239 where c2(x) = 4 sin(π √2) ~ -3.8556101313995. c2(x) first reaches its permanent value of 0 at x = 2 cos(π/√2) ~ -1.2113997341576, where the slope is discontinuous. All these magic numbers are technically conjectural, from a spectacular performance by Munafo's ries. As David suggests, Mathematica's simplifiers were embarrassed at several points but excelled at others. Likewise the numerics. There are some bugs to report! --rwg
(Note: David is using b^3, e.g., to mean b(b(b())).) On 2017-03-23 15:40, David Wilson wrote:
In the fortuitous case of
b(x) = x^2 - 2
b is conjugate to
a(x) = x^2
by the simple mapping
f(x) = x + x^-1 f^-1(x) = (x + sqrt(x^2 - 4))/2
with
b^n = f o a^n o f^-1
Specifically, if
c(x) = b^(1/2)(x) = ((x + sqrt(x^2 - 4))/2)^sqrt(2) + ((x - sqrt(x^2 - 4))/2)^sqrt(2)
then
c(c(x)) = x^2 - 2.
It's a nice problem to give the c(x) formula and ask for c(c(x)). It might be fun to blindside a math forum with this problem. I can't get Wolfram Alpha to answer it.
At any rate, the form of this f(x) led me to postulate the power series form for the f mapping in the x -> x^2 + 1 case.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of rcs@xmission.com Sent: Thursday, March 23, 2017 12:42 AM To: math-fun@mailman.xmission.com Cc: rcs@xmission.com Subject: Re: [math-fun] Help with some math
Dan's worked with flows quite a bit.
Some of the rest of us have played around a little, looking for functional square-roots or real-indexed iterated functions. There are dragons lurking.
Long ago, I wrote several flow programs in Lisp, using the nice feature of exact rational arithmetic. I tried to find a power series for the half flow of x+x^2, expanded around 0. I used f(x) = x+x^2, g(x) = x + x^2/2 + tbd, and the equation f(g(x)) = g(f(x)), matching the power series on each side. This led to a relatively simple formula for computing terms of g(). Things looked good at the beginning, but I got greedy and went out to a hundred terms. The numbers looked nice for a while, but soon the numerators & denominators ballooned. To make sense of things, I floated the rational coefficients, and took the nth root of |g_n| to estimate the reciprocal of the radius of convergence. A nasty surprise: the numerical evidence was that the radius of convergence seemed to be heading for 0.
Some of our other experiments had odd results.
Good luck with your approach!
Rich
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I only looked at it with Maple, which finds c2(x) = c(c(x)) - x^2 + 2 real for x>=-2, and makes a nice plot, which I can put somewhere (e.g. dropbox) if anyone wants. On Fri, Mar 24, 2017 at 6:07 PM, David Wilson <davidwwilson@comcast.net> wrote:
I presume you are talking about evaluating
c2(x) = c(c(x)) - x^2 + 2
with
c(x) = ((x + √(x^2 - 4))/2)^√2 + ((x - √(x^2 - 4))/2)^√2
If you start with |x| >= 2, you end up with
c(x) = y^√2 + z^√2
where y and z are real, so it is reasonable to take y^√2 and z^√2 real as well.
If |x| < 2, however, we end up with
c(x) = y^√2 + z^√2
Where y and z are complex numbers. It then seems that y^√2 and z^√2 take on an infinitude of non-real complex values, none better than another. So how do you compute y^√2 or z^√2 with y or z complex? Is there some principle value?
At any rate, rwg and J. Buddenhagen seem to agree on an evaluation of c2(x) on |x| < 2. I would like to see a plot of c2(x) on the real line.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Friday, March 24, 2017 6:06 AM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Help with some math
c2(x) := c(c(x)) - x² + 2 is fascinating. Continuous and real for x > -2, where c2(-2+) = - 4 sin(π √2)² ~ -3.7164323713376353833, followed immediately by a minimum at x = -√(2 - 2 sin(π √2)) ~ -1.981869083895239 where c2(x) = 4 sin(π √2) ~ -3.8556101313995. c2(x) first reaches its permanent value of 0 at x = 2 cos(π/√2) ~ -1.2113997341576, where the slope is discontinuous. All these magic numbers are technically conjectural, from a spectacular performance by Munafo's ries. As David suggests, Mathematica's simplifiers were embarrassed at several points but excelled at others. Likewise the numerics. There are some bugs to report! --rwg
(Note: David is using b^3, e.g., to mean b(b(b())).) On 2017-03-23 15:40, David Wilson wrote:
In the fortuitous case of
b(x) = x^2 - 2
b is conjugate to
a(x) = x^2
by the simple mapping
f(x) = x + x^-1 f^-1(x) = (x + sqrt(x^2 - 4))/2
with
b^n = f o a^n o f^-1
Specifically, if
c(x) = b^(1/2)(x) = ((x + sqrt(x^2 - 4))/2)^sqrt(2) + ((x - sqrt(x^2 - 4))/2)^sqrt(2)
then
c(c(x)) = x^2 - 2.
It's a nice problem to give the c(x) formula and ask for c(c(x)). It might be fun to blindside a math forum with this problem. I can't get Wolfram Alpha to answer it.
At any rate, the form of this f(x) led me to postulate the power series form for the f mapping in the x -> x^2 + 1 case.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of rcs@xmission.com Sent: Thursday, March 23, 2017 12:42 AM To: math-fun@mailman.xmission.com Cc: rcs@xmission.com Subject: Re: [math-fun] Help with some math
Dan's worked with flows quite a bit.
Some of the rest of us have played around a little, looking for functional square-roots or real-indexed iterated functions. There are dragons lurking.
Long ago, I wrote several flow programs in Lisp, using the nice feature of exact rational arithmetic. I tried to find a power series for the half flow of x+x^2, expanded around 0. I used f(x) = x+x^2, g(x) = x + x^2/2 + tbd, and the equation f(g(x)) = g(f(x)), matching the power series on each side. This led to a relatively simple formula for computing terms of g(). Things looked good at the beginning, but I got greedy and went out to a hundred terms. The numbers looked nice for a while, but soon the numerators & denominators ballooned. To make sense of things, I floated the rational coefficients, and took the nth root of |g_n| to estimate the reciprocal of the radius of convergence. A nasty surprise: the numerical evidence was that the radius of convergence seemed to be heading for 0.
Some of our other experiments had odd results.
Good luck with your approach!
Rich
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
-
Bill Gosper -
David Wilson -
James Buddenhagen