Brian,
> I was disappointed that I could not play more with Andrews functions
> because I get the following error, as well.
>
> viewrgb 8 ccEnhPh sq 128
> |length error: ic
> | MAT=:(h,w)$_3 ic,|.@(4&{.)("1)_8]\3 ic,x
> JVERSION
> Engine: j806/j64/darwin
> Release: commercial/2017-11-06T10:20:33
> Library: 8.06.09
> Platform: Darwin 64
> Installer: J806 install
> InstallPath: /users/brian/j64-806
> Contact: www.jsoftware.com
I am surprised that Andrew's verbs did not work for you. The line,
viewrgb 8 ccEnhPh sq 128
reproduces Andrew's first graph "z, identity" running on the same latest
stable official J but running on Windows/JQt; that is,
JVERSION
Engine: j806/j64nonavx/windows
Release: commercial/2017-11-06T10:01:33
Library: 8.06.09
Qt IDE: 1.6.2/5.6.3
Platform: Win 64
Installer: J806 install
InstallPath: j:/program files/j
Contact: www.jsoftware.com
using my truncated copy of the script.
I do not recall modifying Andrew's script in any way (apart from using just
a portion). Perhaps, other members can verify whether it works, or does
not, on Linux and Windows.
Incidentally, the first graph "z, identity" (]) illustrates how the domain
coloring technique displays the complex plane (the straight lines are the
isophase lines (separating eight shades of the colors of the spectrum, from
red to violet, in the usual anticlockwise phase/angle order); the
circumferences are the isomagnitude lines. Together, the isomagnitude and
isophase lines, form a mesh similar to a spider web.
This graph can be used as a key when trying to interpret the domain
coloring of other complex functions; for example, Andrew's second graph
"Exponent" (^) can be thought of as transforming the spider web coloring of
the complex plane into a quadrangular mesh: the horizontal lines must be
the isophase lines (because they separate (eight) shades of the spectrum);
consequently the vertical lines must be the iso magnitude lines (and they
are evenly spaced because,
NB. Magnitude blocks are logarithmic and have light gradient toward
NB. increase.
). (It is interesting to note that the transformed isomagnitude and
isophase lines are still perpendicular at their intersections.)
The ccEnhPhGray graphs also use shading intensity to convey information
about the image of a complex function; for example, the neibourhood around
the pole (in the center) of Andrew's second ccEnhPhGray graph looks
different from the zeros.
I hope that you can be able to run eventually Andrew's script without any
errors and these hints can help you when your spouse is not around.
On Thu, May 10, 2018 at 1:29 PM, Brian Schott <[email protected]>
wrote:
> Pepe,
>
> Thank you very much for your links. I am color defective (red-green) so I
> have never attended to colors much, though I find them quite fascinating.
> To give you an idea, I have no idea what the colors magenta and cyan are,
> but my spouse is helping me with it all. Anyhow the color wheel and its
> sequence are foreign to me so I have to really study the graphics.
>
> I gather that the imaginary component of a complex number is associated
> with angles (measure in radians perhaps), which is very hard to understand.
> So are these graphs like graphs of polar coordinates (which I never could
> internalize, either)? What sense does that make?
>
> I have been playing with Andrew's jwiki link and now understand more about
> the graphs.
>
> I have still not understood how (_2 3 _4^-)s etc produce straight lines of
> finite length. What values of s are used and is the result the argument for
> either zeta or eta functions so that it's the zeta or eta function that
> produces the straight line, maybe?
>
> I was disappointed that I could not play more with Andrews functions
> because I get the following error, as well.
>
> viewrgb 8 ccEnhPh sq 128
> |length error: ic
> | MAT=:(h,w)$_3 ic,|.@(4&{.)("1)_8]\3 ic,x
> JVERSION
> Engine: j806/j64/darwin
> Release: commercial/2017-11-06T10:20:33
> Library: 8.06.09
> Platform: Darwin 64
> Installer: J806 install
> InstallPath: /users/brian/j64-806
> Contact: www.jsoftware.com
>
> Oh, and while I am thinking of it, can you or anyone clarify the following
> sentence from wikipedia. I think there are missing words or something in
> the sentence.
>
> https://en.wikipedia.org/wiki/Domain_coloring#A_structured_color_function
>
> "Therefore, in a strictly monotonic continuous function that stretches the
> whole range compromises the resolution of smaller changes in magnitude."
>
>
> Don't feel a need to answer my questions. I am just out of my depth again.
>
> On Tue, May 8, 2018 at 6:31 PM, Jose Mario Quintana <
> [email protected]> wrote:
>
> > > I've been completely silent on the J forums for the past few years
> >
> > You were silent for too long Ewart ;) Welcome back!
> >
> >
> > > I am partially confused by Ewart's definition of the "critical line"
> >
> > Brian, the "critical line" is used in the context of the Riemann
> > hypothesis; see [0].
> >
> > One way to get some insight into the behavior of complex functions is via
> > domain coloring; see [1] and the first couple of graphs on the page [2].
> > One can produce this kind of graphs using J thanks to Andrew Nikitin; see
> > [3]; for example, try,
> >
> > viewrgb 8 ccEnhPh (zetahat"0) 20 * sq 256
> >
> > (where zetahat is from Ewart's script and the rest from Andrew's script)
> > and compare to the two graphs mentioned above; see [2].
> >
> > References
> >
> > [0] Zeros, the critical line, and the Riemann hypothesis
> >
> > https://en.wikipedia.org/wiki/Riemann_zeta_function#Zeros,_
> > the_critical_line,_and_the_Riemann_hypothesis
> >
> > [1] Domain Coloring
> > https://en.wikipedia.org/wiki/Domain_coloring
> >
> > [2] Riemann zeta function
> > https://en.wikipedia.org/wiki/Riemann_zeta_function
> >
> > [3] Andrew_Nikitin/Phase_portraits
> > http://code.jsoftware.com/wiki/User:Andrew_Nikitin/Phase_portraits
> >
> >
> >
> ----------------------------------------------------------------------
> For information about J forums see http://www.jsoftware.com/forums.htm
>
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