Lets examine a very basic differential equation : y' - fn1(x) = 0 this is a test condition that we're interested in , we might even relax this to : | y' - fn1(x) | < epi , where epi is a small tolerance . We're quite familiar with this in fractint . Now suppose that we're able to generate y' and y , as iterated functions . Then when the aforementioned condition is satisfied we have a value for the integral of fn1(x) at x = Pixel ; this being y . As available , via an earlier post of mine , a general formula for finding the derivative of an iterated function . Initially this might just be examined as a fractal . At this stage I really don't know what this might produce ; maybe some interesting fractals .
Sounds intriguing! Could we have some examples, please? Tony Hanmer On 12 November 2017 at 07:29, Edward Montague <sciwiseg@gmail.com> wrote:
Lets examine a very basic differential equation :
y' - fn1(x) = 0
this is a test condition that we're interested in , we might even relax this to :
| y' - fn1(x) | < epi , where epi is a small tolerance .
We're quite familiar with this in fractint .
Now suppose that we're able to generate y' and y , as iterated functions . Then when the aforementioned condition is satisfied we have a value for the integral of fn1(x) at x = Pixel ; this being y .
As available , via an earlier post of mine , a general formula for finding the derivative of an iterated function .
Initially this might just be examined as a fractal .
At this stage I really don't know what this might produce ; maybe some interesting fractals . _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
Okay , this is a very basic example , doesn't look particularly interesting . It's as if integrals smooth away the fractal structure . The group might be able to do someting with this . f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.1 } On Sun, Nov 12, 2017 at 9:34 PM, Tony Hanmer <a.hanmer@gmail.com> wrote:
Sounds intriguing! Could we have some examples, please?
Tony Hanmer
On 12 November 2017 at 07:29, Edward Montague <sciwiseg@gmail.com> wrote:
Lets examine a very basic differential equation :
y' - fn1(x) = 0
this is a test condition that we're interested in , we might even relax this to :
| y' - fn1(x) | < epi , where epi is a small tolerance .
We're quite familiar with this in fractint .
Now suppose that we're able to generate y' and y , as iterated functions . Then when the aforementioned condition is satisfied we have a value for the integral of fn1(x) at x = Pixel ; this being y .
As available , via an earlier post of mine , a general formula for finding the derivative of an iterated function .
Initially this might just be examined as a fractal .
At this stage I really don't know what this might produce ; maybe some interesting fractals . _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
_______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
The group might be able to do something with this . I changed the bailout value to 4 and was able to see a few more iterations than when it was 0.1 . I'm in the process of trying different functions and color maps.
I set p1 and the function in the z screen. With p1=0 (making the bailout be 4), the function exp() get iterations up into the 30's. - Hal Lane ######################## # hallane@earthlink.net ######################## -------- Start of PAR file: ---------------- IntgrDerivOfMset { ; hhlane Colors from Jim Muth reset=2004 type=formula formulafile=basicer.frm formulaname=f1Mandelbrot passes=1 center-mag=-0.465582/0.0/1.0/1/ float=y maxiter=1000 inside=0 logmap= periodicity=6 colors=00054L65K76K87K98K99MAAOABQ9CS9EU9GW9IY8K_8\ La8Mc8NeAPcBQaDS`ETZGVYHWWIYVKZTL`SNaQOcPQdNRfMSgK\ UiJVjHXlGYmE_oD`pBaqA`oC`nD_mE_lFZkGZjIYiJYhKXgLXf\ MXeNWcPWbQVaRV`SU_TUZVTYWTXXSWYSVZSVZSV_SV_KWVCZQ4\ `L6aO7aQ9bTAbVBcXDc_EdaGddHefIehKfkMfmOfnQfnSfoU_o\ UZoVYoVXoWWpWVmXUjXTgYSdYQaZOZZPW_MT_JR`GQ`DM_BJYA\ F_6AY6BX6CW6CV7DT7ES8ER9FQAGPBGNBHMCILDIKEJJFKHFKG\ GLFHMEIMDJNBJOAKO9LP8MQ3PR5NQ7MQ9LQBJQDIQFHQGGQIEQ\ KDQMCQOAQQ9QS8QT7QU9PVAPWBPXCPYDPZFP_GP`HPaIPbJPcL\ PdMPeNPfOPgPPfSJgRMhQPiPSjOVkNYlM`mMcnLfoKipJlqIor\ HrsGusGwrIurJsrLqrMprOnrPlrQkrSiqTgqVeqWdqXbqZ`q__\ qaYqbWpdUpeTpfRphPpiOpkMplKpmJolInkHmjGmjGliFkhEjh\ EjgDifChfCgeBgdAfdAdc9ba8a_8_Y7YW6ZV3ZV5ZV6_W8_Y9`\ _B`aC`cE_eF_gHZhIZiKYjLYjMYjMWkLWkKWkKVkJVlJUlIUlI\ UlHUmHUmGUmGVmFWmEYnE_oDapDdrCgsCitBkuBmvAowAqx8ty\ 7wz6zz5zz3zz2zz1zz0zzTzzU } frm:f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel test = (4 + p1) u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < test } -------- End of PAR file ------------------ -----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of Edward Montague Sent: Monday, November 13, 2017 12:22 AM To: Fractint and General Fractals Discussion <fractint@mailman.xmission.com> Subject: Re: [Fractint] Integration under differentiation . Okay , this is a very basic example , doesn't look particularly interesting . It's as if integrals smooth away the fractal structure . The group might be able to do someting with this . f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.1 } On Sun, Nov 12, 2017 at 9:34 PM, Tony Hanmer <a.hanmer@gmail.com> wrote:
Sounds intriguing! Could we have some examples, please?
Tony Hanmer
On 12 November 2017 at 07:29, Edward Montague <sciwiseg@gmail.com> wrote:
Lets examine a very basic differential equation :
y' - fn1(x) = 0
this is a test condition that we're interested in , we might even relax this to :
| y' - fn1(x) | < epi , where epi is a small tolerance .
We're quite familiar with this in fractint .
Now suppose that we're able to generate y' and y , as iterated functions . Then when the aforementioned condition is satisfied we have a value for the integral of fn1(x) at x = Pixel ; this being y .
As available , via an earlier post of mine , a general formula for finding the derivative of an iterated function .
Initially this might just be examined as a fractal .
At this stage I really don't know what this might produce ; maybe some interesting fractals . _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
_______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
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As usual Hal.Lane has produced something remarkable . Thinking about what I'm attempting to do , align a particular iteration of the derivative of the mandelbrot set with a unique value of a function , using a constant for the function variable seems more appropriate. Eventually the constant might be replaced with a parameter. Also in the future , the comparison might involve a more complicated differential equation ; however first things first . At some stage I shall use maxima cas to examine the results . Here's the new formula : f1cMandel(XAXIS) {;sciwiseg , Edward Montague ; ; Integral via derivative of Mset . x = Pixel c=(0.5,0.5) u = fn1(c) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.01 } On Mon, Nov 13, 2017 at 10:11 PM, Harold Lane <hallane@earthlink.net> wrote:
The group might be able to do something with this . I changed the bailout value to 4 and was able to see a few more iterations than when it was 0.1 . I'm in the process of trying different functions and color maps.
I set p1 and the function in the z screen. With p1=0 (making the bailout be 4), the function exp() get iterations up into the 30's.
- Hal Lane
######################## # hallane@earthlink.net ########################
-------- Start of PAR file: ---------------- IntgrDerivOfMset { ; hhlane Colors from Jim Muth reset=2004 type=formula formulafile=basicer.frm formulaname=f1Mandelbrot passes=1 center-mag=-0.465582/0.0/1.0/1/ float=y maxiter=1000 inside=0 logmap= periodicity=6 colors=00054L65K76K87K98K99MAAOABQ9CS9EU9GW9IY8K_8\ La8Mc8NeAPcBQaDS`ETZGVYHWWIYVKZTL`SNaQOcPQdNRfMSgK\ UiJVjHXlGYmE_oD`pBaqA`oC`nD_mE_lFZkGZjIYiJYhKXgLXf\ MXeNWcPWbQVaRV`SU_TUZVTYWTXXSWYSVZSVZSV_SV_KWVCZQ4\ `L6aO7aQ9bTAbVBcXDc_EdaGddHefIehKfkMfmOfnQfnSfoU_o\ UZoVYoVXoWWpWVmXUjXTgYSdYQaZOZZPW_MT_JR`GQ`DM_BJYA\ F_6AY6BX6CW6CV7DT7ES8ER9FQAGPBGNBHMCILDIKEJJFKHFKG\ GLFHMEIMDJNBJOAKO9LP8MQ3PR5NQ7MQ9LQBJQDIQFHQGGQIEQ\ KDQMCQOAQQ9QS8QT7QU9PVAPWBPXCPYDPZFP_GP`HPaIPbJPcL\ PdMPeNPfOPgPPfSJgRMhQPiPSjOVkNYlM`mMcnLfoKipJlqIor\ HrsGusGwrIurJsrLqrMprOnrPlrQkrSiqTgqVeqWdqXbqZ`q__\ qaYqbWpdUpeTpfRphPpiOpkMplKpmJolInkHmjGmjGliFkhEjh\ EjgDifChfCgeBgdAfdAdc9ba8a_8_Y7YW6ZV3ZV5ZV6_W8_Y9`\ _B`aC`cE_eF_gHZhIZiKYjLYjMYjMWkLWkKWkKVkJVlJUlIUlI\ UlHUmHUmGUmGVmFWmEYnE_oDapDdrCgsCitBkuBmvAowAqx8ty\ 7wz6zz5zz3zz2zz1zz0zzTzzU }
frm:f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel test = (4 + p1) u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < test } -------- End of PAR file ------------------
-----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of Edward Montague Sent: Monday, November 13, 2017 12:22 AM To: Fractint and General Fractals Discussion < fractint@mailman.xmission.com> Subject: Re: [Fractint] Integration under differentiation .
Okay , this is a very basic example , doesn't look particularly interesting . It's as if integrals smooth away the fractal structure .
The group might be able to do someting with this .
f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.1 }
On Sun, Nov 12, 2017 at 9:34 PM, Tony Hanmer <a.hanmer@gmail.com> wrote:
Sounds intriguing! Could we have some examples, please?
Tony Hanmer
On 12 November 2017 at 07:29, Edward Montague <sciwiseg@gmail.com> wrote:
Lets examine a very basic differential equation :
y' - fn1(x) = 0
this is a test condition that we're interested in , we might even relax this to :
| y' - fn1(x) | < epi , where epi is a small tolerance .
We're quite familiar with this in fractint .
Now suppose that we're able to generate y' and y , as iterated functions . Then when the aforementioned condition is satisfied we have a value for the integral of fn1(x) at x = Pixel ; this being y .
As available , via an earlier post of mine , a general formula for finding the derivative of an iterated function .
Initially this might just be examined as a fractal .
At this stage I really don't know what this might produce ; maybe some interesting fractals . _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
_______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
_______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
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Then again , the previous formula is almost trivial as we're attempting to find y for y' = a constant . On Tue, Nov 14, 2017 at 3:50 PM, Edward Montague <sciwiseg@gmail.com> wrote:
As usual Hal.Lane has produced something remarkable .
Thinking about what I'm attempting to do , align a particular iteration of the derivative of the mandelbrot set with a unique value of a function , using a constant for the function variable seems more appropriate. Eventually the constant might be replaced with a parameter. Also in the future , the comparison might involve a more complicated differential equation ; however first things first .
At some stage I shall use maxima cas to examine the results .
Here's the new formula :
f1cMandel(XAXIS) {;sciwiseg , Edward Montague ; ; Integral via derivative of Mset . x = Pixel c=(0.5,0.5) u = fn1(c) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.01 }
On Mon, Nov 13, 2017 at 10:11 PM, Harold Lane <hallane@earthlink.net> wrote:
The group might be able to do something with this . I changed the bailout value to 4 and was able to see a few more iterations than when it was 0.1 . I'm in the process of trying different functions and color maps.
I set p1 and the function in the z screen. With p1=0 (making the bailout be 4), the function exp() get iterations up into the 30's.
- Hal Lane
######################## # hallane@earthlink.net ########################
-------- Start of PAR file: ---------------- IntgrDerivOfMset { ; hhlane Colors from Jim Muth reset=2004 type=formula formulafile=basicer.frm formulaname=f1Mandelbrot passes=1 center-mag=-0.465582/0.0/1.0/1/ float=y maxiter=1000 inside=0 logmap= periodicity=6 colors=00054L65K76K87K98K99MAAOABQ9CS9EU9GW9IY8K_8\ La8Mc8NeAPcBQaDS`ETZGVYHWWIYVKZTL`SNaQOcPQdNRfMSgK\ UiJVjHXlGYmE_oD`pBaqA`oC`nD_mE_lFZkGZjIYiJYhKXgLXf\ MXeNWcPWbQVaRV`SU_TUZVTYWTXXSWYSVZSVZSV_SV_KWVCZQ4\ `L6aO7aQ9bTAbVBcXDc_EdaGddHefIehKfkMfmOfnQfnSfoU_o\ UZoVYoVXoWWpWVmXUjXTgYSdYQaZOZZPW_MT_JR`GQ`DM_BJYA\ F_6AY6BX6CW6CV7DT7ES8ER9FQAGPBGNBHMCILDIKEJJFKHFKG\ GLFHMEIMDJNBJOAKO9LP8MQ3PR5NQ7MQ9LQBJQDIQFHQGGQIEQ\ KDQMCQOAQQ9QS8QT7QU9PVAPWBPXCPYDPZFP_GP`HPaIPbJPcL\ PdMPeNPfOPgPPfSJgRMhQPiPSjOVkNYlM`mMcnLfoKipJlqIor\ HrsGusGwrIurJsrLqrMprOnrPlrQkrSiqTgqVeqWdqXbqZ`q__\ qaYqbWpdUpeTpfRphPpiOpkMplKpmJolInkHmjGmjGliFkhEjh\ EjgDifChfCgeBgdAfdAdc9ba8a_8_Y7YW6ZV3ZV5ZV6_W8_Y9`\ _B`aC`cE_eF_gHZhIZiKYjLYjMYjMWkLWkKWkKVkJVlJUlIUlI\ UlHUmHUmGUmGVmFWmEYnE_oDapDdrCgsCitBkuBmvAowAqx8ty\ 7wz6zz5zz3zz2zz1zz0zzTzzU }
frm:f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel test = (4 + p1) u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < test } -------- End of PAR file ------------------
-----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of Edward Montague Sent: Monday, November 13, 2017 12:22 AM To: Fractint and General Fractals Discussion < fractint@mailman.xmission.com> Subject: Re: [Fractint] Integration under differentiation .
Okay , this is a very basic example , doesn't look particularly interesting . It's as if integrals smooth away the fractal structure .
The group might be able to do someting with this .
f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.1 }
On Sun, Nov 12, 2017 at 9:34 PM, Tony Hanmer <a.hanmer@gmail.com> wrote:
Sounds intriguing! Could we have some examples, please?
Tony Hanmer
On 12 November 2017 at 07:29, Edward Montague <sciwiseg@gmail.com> wrote:
Lets examine a very basic differential equation :
y' - fn1(x) = 0
this is a test condition that we're interested in , we might even relax this to :
| y' - fn1(x) | < epi , where epi is a small tolerance .
We're quite familiar with this in fractint .
Now suppose that we're able to generate y' and y , as iterated functions . Then when the aforementioned condition is satisfied we have a value for the integral of fn1(x) at x = Pixel ; this being y .
As available , via an earlier post of mine , a general formula for finding the derivative of an iterated function .
Initially this might just be examined as a fractal .
At this stage I really don't know what this might produce ; maybe some interesting fractals . _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
_______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
_______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
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As usual Hal.Lane has produced something remarkable . Thanks! I just diddle stuff over and over again until a Shakespeare sonnet appears...
Eventually the constant might be replaced with a parameter. I took the above as an immediate challenge!
I parameterized SciWise's formula's: c=(0.5,0.5) to: c=real(p1) + imag(p1) so I could use Fractint's Evolver to vary the real and imaginary parts of c separately. I set the evolver to create a grid of images with their real values of c varying in X, and their imaginary values of c varying in Y. I again set the bailout to 4, and the function to: exp() and used the "default color map," since it has high contrast colors in low iteration locations. Here's the Evolver's output: (The black bars at the edges are expected.) http://tinyurl.com/SintMnd3 or: http://www.emarketingiseasy.com/Sciwise/2017-11-13/SintMnd3.gif The ranges of X and Y values are not round numbers like I started with, since I used the Evolver's Zoom and Pan functions. These are the values the Evolver wound up with for c=(X-parameter, Y-parameter): http://tinyurl.com/Int-Diff-Mand-Evolve-P1-P2 or: http://www.emarketingiseasy.com/Sciwise/2017-11-13/Int-Diff-Mand-Evolve-P1-P 2.jpg For documentation, here is my Z-screen -- showing the values of c in real(p1) and imag(p1) (not the ones I started with), the different bailout value I chose: 4, and the function: exp(): http://tinyurl.com/Int-Diff-Mand-Z-screen or: http://www.emarketingiseasy.com/Sciwise/2017-11-13/Int-Diff-Mand-Z-screen.jp g SciWise created another winner of a formula! Further investigation is warranted. - Hal Lane ######################## # hallane@earthlink.net ######################## -----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of Edward Montague Sent: Monday, November 13, 2017 9:50 PM To: Harold Lane <hallane@earthlink.net> Cc: Fractint and General Fractals Discussion <fractint@mailman.xmission.com> Subject: Re: [Fractint] Integration under differentiation . As usual Hal.Lane has produced something remarkable . Thinking about what I'm attempting to do , align a particular iteration of the derivative of the mandelbrot set with a unique value of a function , using a constant for the function variable seems more appropriate. Eventually the constant might be replaced with a parameter. Also in the future , the comparison might involve a more complicated differential equation ; however first things first . At some stage I shall use maxima cas to examine the results . Here's the new formula : f1cMandel(XAXIS) {;sciwiseg , Edward Montague ; ; Integral via derivative of Mset . x = Pixel c=(0.5,0.5) u = fn1(c) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.01 } On Mon, Nov 13, 2017 at 10:11 PM, Harold Lane <hallane@earthlink.net> wrote:
The group might be able to do something with this . I changed the bailout value to 4 and was able to see a few more iterations than when it was 0.1 . I'm in the process of trying different functions and color maps.
I set p1 and the function in the z screen. With p1=0 (making the bailout be 4), the function exp() get iterations up into the 30's.
- Hal Lane
######################## # hallane@earthlink.net ########################
-------- Start of PAR file: ---------------- IntgrDerivOfMset { ; hhlane Colors from Jim Muth reset=2004 type=formula formulafile=basicer.frm formulaname=f1Mandelbrot passes=1 center-mag=-0.465582/0.0/1.0/1/ float=y maxiter=1000 inside=0 logmap= periodicity=6 colors=00054L65K76K87K98K99MAAOABQ9CS9EU9GW9IY8K_8\ La8Mc8NeAPcBQaDS`ETZGVYHWWIYVKZTL`SNaQOcPQdNRfMSgK\ UiJVjHXlGYmE_oD`pBaqA`oC`nD_mE_lFZkGZjIYiJYhKXgLXf\ MXeNWcPWbQVaRV`SU_TUZVTYWTXXSWYSVZSVZSV_SV_KWVCZQ4\ `L6aO7aQ9bTAbVBcXDc_EdaGddHefIehKfkMfmOfnQfnSfoU_o\ UZoVYoVXoWWpWVmXUjXTgYSdYQaZOZZPW_MT_JR`GQ`DM_BJYA\ F_6AY6BX6CW6CV7DT7ES8ER9FQAGPBGNBHMCILDIKEJJFKHFKG\ GLFHMEIMDJNBJOAKO9LP8MQ3PR5NQ7MQ9LQBJQDIQFHQGGQIEQ\ KDQMCQOAQQ9QS8QT7QU9PVAPWBPXCPYDPZFP_GP`HPaIPbJPcL\ PdMPeNPfOPgPPfSJgRMhQPiPSjOVkNYlM`mMcnLfoKipJlqIor\ HrsGusGwrIurJsrLqrMprOnrPlrQkrSiqTgqVeqWdqXbqZ`q__\ qaYqbWpdUpeTpfRphPpiOpkMplKpmJolInkHmjGmjGliFkhEjh\ EjgDifChfCgeBgdAfdAdc9ba8a_8_Y7YW6ZV3ZV5ZV6_W8_Y9`\ _B`aC`cE_eF_gHZhIZiKYjLYjMYjMWkLWkKWkKVkJVlJUlIUlI\ UlHUmHUmGUmGVmFWmEYnE_oDapDdrCgsCitBkuBmvAowAqx8ty\ 7wz6zz5zz3zz2zz1zz0zzTzzU }
frm:f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel test = (4 + p1) u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < test } -------- End of PAR file ------------------
-----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of Edward Montague Sent: Monday, November 13, 2017 12:22 AM To: Fractint and General Fractals Discussion < fractint@mailman.xmission.com> Subject: Re: [Fractint] Integration under differentiation .
Okay , this is a very basic example , doesn't look particularly interesting . It's as if integrals smooth away the fractal structure .
The group might be able to do someting with this .
f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.1 }
On Sun, Nov 12, 2017 at 9:34 PM, Tony Hanmer <a.hanmer@gmail.com> wrote:
Sounds intriguing! Could we have some examples, please?
Tony Hanmer
On 12 November 2017 at 07:29, Edward Montague <sciwiseg@gmail.com> wrote:
Lets examine a very basic differential equation :
y' - fn1(x) = 0
this is a test condition that we're interested in , we might even relax this to :
| y' - fn1(x) | < epi , where epi is a small tolerance .
We're quite familiar with this in fractint .
Now suppose that we're able to generate y' and y , as iterated functions . Then when the aforementioned condition is satisfied we have a value for the integral of fn1(x) at x = Pixel ; this being y .
As available , via an earlier post of mine , a general formula for finding the derivative of an iterated function .
Initially this might just be examined as a fractal .
At this stage I really don't know what this might produce ; maybe some interesting fractals . _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
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Sorry. I omitted my PAR file in my previous post where I tried out SciWise's second version of his formula for integrating the differential of the Mandelbrot equation: I discovered that Richard's Fractint for Windows beta 5 (not for general distribution) that I use changes upper case letters in function names to lower case when writing parameter values out, (with the <b> key) -- so I changed the name of SciWise's function (f1cmandel) to all lower case letters to avoid this problem. --------- Start of PAR: ---------- IntgrDerivOfMset2a { ; Hal Lane; SciWise; Default color map ; SciWise's 2nd ver of integrating ; the deriv of the Mand equation. ; Bailout = 4; fn1 = exp; c=(0.5,0.5) ; Fractint Version 2099 Patchlevel 8 reset=2099 type=formula formulafile=171113.par formulaname=f1cmandel function=exp passes=1 center-mag=0.866196/2.44249e-015/0.9461387 params=0.5/0.5/4/0 float=y maxiter=1000 inside=0 periodicity=6 colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\ zLzzz000555<3>HHHKKKOOO<3>ccchhhmmmssszzz00z<3>z0z\ <3>z00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zVV<3>zz\ V<3>VzV<3>Vzz<2>Vbzhhz<3>zhz<3>zhh<3>zzh<3>hzh<3>h\ zz<2>hlz00S<3>S0S<3>S00<3>SS0<3>0S0<3>0SS<2>07SEES\ <3>SES<3>SEE<3>SSE<3>ESE<3>ESS<2>EHSKKS<2>QKSSKSSK\ QSKOSKMSKK<2>SQKSSKQSKOSKMSKKSK<2>KSQKSSKQSKOSKMS0\ 0G<3>G0G<3>G00<3>GG0<3>0G0<3>0GG<2>04G88G<2>E8GG8G\ G8EG8CG8AG88<2>GE8GG8EG8CG8AG88G8<2>8GE8GG8EG8CG8A\ GBBG<2>FBGGBGGBFGBDGBCGBB<2>GFBGGBFGBDGBCGBBGB<2>B\ GFBGGBFGBDGBCG000<6>000 } frm:f1cmandel(XAXIS) {; sciwiseg, Edward Montague ; ; Integral via derivative of Mset. x = Pixel c=real(p1) + imag(p1) bailoutTest = p2 u = fn1(c) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < bailoutTest } --------- End of PAR file ------------- - Hal Lane ######################## # hallane@earthlink.net ######################## -----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of Edward Montague Sent: Monday, November 13, 2017 9:50 PM To: Harold Lane <hallane@earthlink.net> Cc: Fractint and General Fractals Discussion <fractint@mailman.xmission.com> Subject: Re: [Fractint] Integration under differentiation . As usual Hal.Lane has produced something remarkable . Thinking about what I'm attempting to do , align a particular iteration of the derivative of the mandelbrot set with a unique value of a function , using a constant for the function variable seems more appropriate. Eventually the constant might be replaced with a parameter. Also in the future , the comparison might involve a more complicated differential equation ; however first things first . At some stage I shall use maxima cas to examine the results . Here's the new formula : f1cMandel(XAXIS) {;sciwiseg , Edward Montague ; ; Integral via derivative of Mset . x = Pixel c=(0.5,0.5) u = fn1(c) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.01 } On Mon, Nov 13, 2017 at 10:11 PM, Harold Lane <hallane@earthlink.net> wrote:
The group might be able to do something with this . I changed the bailout value to 4 and was able to see a few more iterations than when it was 0.1 . I'm in the process of trying different functions and color maps.
I set p1 and the function in the z screen. With p1=0 (making the bailout be 4), the function exp() get iterations up into the 30's.
- Hal Lane
######################## # hallane@earthlink.net ########################
-------- Start of PAR file: ---------------- IntgrDerivOfMset { ; hhlane Colors from Jim Muth reset=2004 type=formula formulafile=basicer.frm formulaname=f1Mandelbrot passes=1 center-mag=-0.465582/0.0/1.0/1/ float=y maxiter=1000 inside=0 logmap= periodicity=6 colors=00054L65K76K87K98K99MAAOABQ9CS9EU9GW9IY8K_8\ La8Mc8NeAPcBQaDS`ETZGVYHWWIYVKZTL`SNaQOcPQdNRfMSgK\ UiJVjHXlGYmE_oD`pBaqA`oC`nD_mE_lFZkGZjIYiJYhKXgLXf\ MXeNWcPWbQVaRV`SU_TUZVTYWTXXSWYSVZSVZSV_SV_KWVCZQ4\ `L6aO7aQ9bTAbVBcXDc_EdaGddHefIehKfkMfmOfnQfnSfoU_o\ UZoVYoVXoWWpWVmXUjXTgYSdYQaZOZZPW_MT_JR`GQ`DM_BJYA\ F_6AY6BX6CW6CV7DT7ES8ER9FQAGPBGNBHMCILDIKEJJFKHFKG\ GLFHMEIMDJNBJOAKO9LP8MQ3PR5NQ7MQ9LQBJQDIQFHQGGQIEQ\ KDQMCQOAQQ9QS8QT7QU9PVAPWBPXCPYDPZFP_GP`HPaIPbJPcL\ PdMPeNPfOPgPPfSJgRMhQPiPSjOVkNYlM`mMcnLfoKipJlqIor\ HrsGusGwrIurJsrLqrMprOnrPlrQkrSiqTgqVeqWdqXbqZ`q__\ qaYqbWpdUpeTpfRphPpiOpkMplKpmJolInkHmjGmjGliFkhEjh\ EjgDifChfCgeBgdAfdAdc9ba8a_8_Y7YW6ZV3ZV5ZV6_W8_Y9`\ _B`aC`cE_eF_gHZhIZiKYjLYjMYjMWkLWkKWkKVkJVlJUlIUlI\ UlHUmHUmGUmGVmFWmEYnE_oDapDdrCgsCitBkuBmvAowAqx8ty\ 7wz6zz5zz3zz2zz1zz0zzTzzU }
frm:f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel test = (4 + p1) u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < test } -------- End of PAR file ------------------
-----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of Edward Montague Sent: Monday, November 13, 2017 12:22 AM To: Fractint and General Fractals Discussion < fractint@mailman.xmission.com> Subject: Re: [Fractint] Integration under differentiation .
Okay , this is a very basic example , doesn't look particularly interesting . It's as if integrals smooth away the fractal structure .
The group might be able to do someting with this .
f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.1 }
On Sun, Nov 12, 2017 at 9:34 PM, Tony Hanmer <a.hanmer@gmail.com> wrote:
Sounds intriguing! Could we have some examples, please?
Tony Hanmer
On 12 November 2017 at 07:29, Edward Montague <sciwiseg@gmail.com> wrote:
Lets examine a very basic differential equation :
y' - fn1(x) = 0
this is a test condition that we're interested in , we might even relax this to :
| y' - fn1(x) | < epi , where epi is a small tolerance .
We're quite familiar with this in fractint .
Now suppose that we're able to generate y' and y , as iterated functions . Then when the aforementioned condition is satisfied we have a value for the integral of fn1(x) at x = Pixel ; this being y .
As available , via an earlier post of mine , a general formula for finding the derivative of an iterated function .
Initially this might just be examined as a fractal .
At this stage I really don't know what this might produce ; maybe some interesting fractals . _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
_______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
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I'm going between maxima cas and fractint in an attempt to improve my understanding of differential equation , with a little more success . Feels like there's a dynamic tension between fractint , which is mostly numerical and maxima cas which also has symbolic capabilities . If you look at my previous post on differentiating iterated functions , you'll see that there are considerably more candidates available for solving a differential equation . Or , at least , examining possible series alignments . Here's another example with a more complicated differential equation . f1fMandel(XAXIS) {;sciwiseg , Edward Montague ; ; Integral via derivative of Mset . x = Pixel y = 1 : u = fn1(x) z = u - y*y + exp(y) y= 2*x*y+1 x = x*x + Pixel |z| < 1 && |x| < 4 } . On Wed, Nov 15, 2017 at 1:35 AM, Harold Lane <hallane@earthlink.net> wrote:
Sorry. I omitted my PAR file in my previous post where I tried out SciWise's second version of his formula for integrating the differential of the Mandelbrot equation:
I discovered that Richard's Fractint for Windows beta 5 (not for general distribution) that I use changes upper case letters in function names to lower case when writing parameter values out, (with the <b> key) -- so I changed the name of SciWise's function (f1cmandel) to all lower case letters to avoid this problem.
--------- Start of PAR: ---------- IntgrDerivOfMset2a { ; Hal Lane; SciWise; Default color map ; SciWise's 2nd ver of integrating ; the deriv of the Mand equation. ; Bailout = 4; fn1 = exp; c=(0.5,0.5) ; Fractint Version 2099 Patchlevel 8 reset=2099 type=formula formulafile=171113.par formulaname=f1cmandel function=exp passes=1 center-mag=0.866196/2.44249e-015/0.9461387 params=0.5/0.5/4/0 float=y maxiter=1000 inside=0 periodicity=6 colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\ zLzzz000555<3>HHHKKKOOO<3>ccchhhmmmssszzz00z<3>z0z\ <3>z00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zVz<3>zVV<3>zz\ V<3>VzV<3>Vzz<2>Vbzhhz<3>zhz<3>zhh<3>zzh<3>hzh<3>h\ zz<2>hlz00S<3>S0S<3>S00<3>SS0<3>0S0<3>0SS<2>07SEES\ <3>SES<3>SEE<3>SSE<3>ESE<3>ESS<2>EHSKKS<2>QKSSKSSK\ QSKOSKMSKK<2>SQKSSKQSKOSKMSKKSK<2>KSQKSSKQSKOSKMS0\ 0G<3>G0G<3>G00<3>GG0<3>0G0<3>0GG<2>04G88G<2>E8GG8G\ G8EG8CG8AG88<2>GE8GG8EG8CG8AG88G8<2>8GE8GG8EG8CG8A\ GBBG<2>FBGGBGGBFGBDGBCGBB<2>GFBGGBFGBDGBCGBBGB<2>B\ GFBGGBFGBDGBCG000<6>000 } frm:f1cmandel(XAXIS) {; sciwiseg, Edward Montague ; ; Integral via derivative of Mset. x = Pixel c=real(p1) + imag(p1) bailoutTest = p2 u = fn1(c) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < bailoutTest } --------- End of PAR file -------------
- Hal Lane
######################## # hallane@earthlink.net ########################
-----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of Edward Montague Sent: Monday, November 13, 2017 9:50 PM To: Harold Lane <hallane@earthlink.net> Cc: Fractint and General Fractals Discussion < fractint@mailman.xmission.com> Subject: Re: [Fractint] Integration under differentiation .
As usual Hal.Lane has produced something remarkable .
Thinking about what I'm attempting to do , align a particular iteration of the derivative of the mandelbrot set with a unique value of a function , using a constant for the function variable seems more appropriate. Eventually the constant might be replaced with a parameter. Also in the future , the comparison might involve a more complicated differential equation ; however first things first .
At some stage I shall use maxima cas to examine the results .
Here's the new formula :
f1cMandel(XAXIS) {;sciwiseg , Edward Montague ; ; Integral via derivative of Mset . x = Pixel c=(0.5,0.5) u = fn1(c) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.01 }
On Mon, Nov 13, 2017 at 10:11 PM, Harold Lane <hallane@earthlink.net> wrote:
The group might be able to do something with this . I changed the bailout value to 4 and was able to see a few more iterations than when it was 0.1 . I'm in the process of trying different functions and color maps.
I set p1 and the function in the z screen. With p1=0 (making the bailout be 4), the function exp() get iterations up into the 30's.
- Hal Lane
######################## # hallane@earthlink.net ########################
-------- Start of PAR file: ---------------- IntgrDerivOfMset { ; hhlane Colors from Jim Muth reset=2004 type=formula formulafile=basicer.frm formulaname=f1Mandelbrot passes=1 center-mag=-0.465582/0.0/1.0/1/ float=y maxiter=1000 inside=0 logmap= periodicity=6 colors=00054L65K76K87K98K99MAAOABQ9CS9EU9GW9IY8K_8\ La8Mc8NeAPcBQaDS`ETZGVYHWWIYVKZTL`SNaQOcPQdNRfMSgK\ UiJVjHXlGYmE_oD`pBaqA`oC`nD_mE_lFZkGZjIYiJYhKXgLXf\ MXeNWcPWbQVaRV`SU_TUZVTYWTXXSWYSVZSVZSV_SV_KWVCZQ4\ `L6aO7aQ9bTAbVBcXDc_EdaGddHefIehKfkMfmOfnQfnSfoU_o\ UZoVYoVXoWWpWVmXUjXTgYSdYQaZOZZPW_MT_JR`GQ`DM_BJYA\ F_6AY6BX6CW6CV7DT7ES8ER9FQAGPBGNBHMCILDIKEJJFKHFKG\ GLFHMEIMDJNBJOAKO9LP8MQ3PR5NQ7MQ9LQBJQDIQFHQGGQIEQ\ KDQMCQOAQQ9QS8QT7QU9PVAPWBPXCPYDPZFP_GP`HPaIPbJPcL\ PdMPeNPfOPgPPfSJgRMhQPiPSjOVkNYlM`mMcnLfoKipJlqIor\ HrsGusGwrIurJsrLqrMprOnrPlrQkrSiqTgqVeqWdqXbqZ`q__\ qaYqbWpdUpeTpfRphPpiOpkMplKpmJolInkHmjGmjGliFkhEjh\ EjgDifChfCgeBgdAfdAdc9ba8a_8_Y7YW6ZV3ZV5ZV6_W8_Y9`\ _B`aC`cE_eF_gHZhIZiKYjLYjMYjMWkLWkKWkKVkJVlJUlIUlI\ UlHUmHUmGUmGVmFWmEYnE_oDapDdrCgsCitBkuBmvAowAqx8ty\ 7wz6zz5zz3zz2zz1zz0zzTzzU }
frm:f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel test = (4 + p1) u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < test } -------- End of PAR file ------------------
-----Original Message----- From: Fractint [mailto:fractint-bounces@mailman.xmission.com] On Behalf Of Edward Montague Sent: Monday, November 13, 2017 12:22 AM To: Fractint and General Fractals Discussion < fractint@mailman.xmission.com> Subject: Re: [Fractint] Integration under differentiation .
Okay , this is a very basic example , doesn't look particularly interesting . It's as if integrals smooth away the fractal structure .
The group might be able to do someting with this .
f1Mandelbrot(XAXIS) {;sciwiseg , Edward Montague ; ; | x' - fn1(Pixel) | < 0.1 ; ; Integral via derivative of Mset . x = Pixel u = fn1(x) y = 1 : y= 2*x*y+1 z = u - y x = x*x + Pixel |z| < 0.1 }
On Sun, Nov 12, 2017 at 9:34 PM, Tony Hanmer <a.hanmer@gmail.com> wrote:
Sounds intriguing! Could we have some examples, please?
Tony Hanmer
On 12 November 2017 at 07:29, Edward Montague <sciwiseg@gmail.com> wrote:
Lets examine a very basic differential equation :
y' - fn1(x) = 0
this is a test condition that we're interested in , we might even relax this to :
| y' - fn1(x) | < epi , where epi is a small tolerance .
We're quite familiar with this in fractint .
Now suppose that we're able to generate y' and y , as iterated functions . Then when the aforementioned condition is satisfied we have a value for the integral of fn1(x) at x = Pixel ; this being y .
As available , via an earlier post of mine , a general formula for finding the derivative of an iterated function .
Initially this might just be examined as a fractal .
At this stage I really don't know what this might produce ; maybe some interesting fractals . _______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
_______________________________________________ Fractint mailing list Fractint@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/fractint
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participants (3)
-
Edward Montague -
Harold Lane -
Tony Hanmer