On Thursday, 28 April 2022 at 17:30:25 UTC+2, Archimedes Plutonium wrote:

> Calling on failure of Logic Dan Christensen with his failed machine of DC proof.

Please, Archimedes, do not feed the trolls.

Julio

On Wednesday, 27 April 2022 at 17:55:35 UTC+2, Julio Di Egidio wrote:
> On Wednesday, 27 April 2022 at 14:26:46 UTC+2, Julio Di Egidio wrote:
> > On Tuesday, 26 April 2022 at 20:27:44 UTC+2, FromTheRafters wrote:
> > > Julio Di Egidio presented the following explanation :
> > >
> > > > Eh, it's not even l'Hopital, those factors just simplify in the limit, but I
> > > > do not remember the name if any of that rule right now...
> > >
> > > Power rule?
> >
> > Nah, this is just simplification of factors by the "just how limits work" rule.
> >
> > I don't think I can explain it properly, anyway it's by definition of limit that
> > whatever is in the limit behaves as if the limit is not actually reached, hence
> > we can do e.g. lim_{x->oo} 2x/x = lim_{x->oo} 2 = 2. And that's where the
> > definition of limit is a higher order operation, i.e. to begin with more of an
> > algebraic operation than an arithmetic one... or something along that line.
> Let's try and broaden the spectrum here.
>
> Can we define the limit of a function as a specific operator on functions?
> (So not relying on, rather recovering, the standard, first-order, set-based
> formalization: indeed, that remains after the fact, we "guess" results to
> begin with, or something along that line...)
>
> Of course it's not just:
> Lim_{x->a from /dir/}(f( x, y, z...)) = f(Lim_{x->a from /dir/}(x), y, z...)
> but what is it if anything?

Beside the standard epsilon-delta definition, I only find limits in category
theory, but that looks quite beyond the present scope. Anyway, I can't find
any "algebraic" definition of limit, which I find very strange: maybe I am not
using the right keywords...

At the moment I am entertaining the idea that I could define it inductively
on the complexity of formulas: the non obvious part is the limit of irrational
functions, for which I am thinking I need a canonical form relative to which
every limit becomes the limit of (possibly the limit of a series of) rational
function(s)... which would not cover non-computable and all the more so
undefined and undefinable numbers, but I am not envisioning getting any
of those anyway... And, as a symbolic tool, probably something like:
<https://en.wikipedia.org/wiki/Big_O_notation>

Anyway, that's where we are: feedback of course remains welcome.

Julio

On 4/27/2022 4:47 AM, Julio Di Egidio wrote:
> On Wednesday, 27 April 2022 at 10:14:45 UTC+2, Chris M. Thomasson wrote:
>> On 4/26/2022 11:59 PM, Archimedes Plutonium wrote:
>>> Julio's thread caused me to consider parametric equations once again.
>>> I remember several years back, perhaps 5 perhaps 10 years ago that I raised
>>> the question of why have parametric equations at all. And some polite poster
>>> tried explaining the "why?". Although, he knew, and I knew I was not buying it.[...]
>>
>> I remember back when you inspired to me to create a parametric that
>> creates two semi-circles. Thanks again!:
> <snip>
>
>> void ct_alien_code_render(
>> int irecur,
>> int irecur_max,
>> float ox,
>> float oy,
>> int imax
>> ) {
>> if (irecur >= irecur_max) return;
>>
>> float angle_start = 1 / (imax - 1.0) * (PI*2);
>
> Whence (I'll have to surmise, in that sense your code is redundant/opaque)
> you just skip the case imax==1 (where you'd have to return an "undefined",
> just your function of corse returns void),

imax must be greater than one in this implementation. Well, I guess I
should of added a comment in my example code, sorry about that.

> which, if you notice, is the very
> motivating example for my question. Have you seen how I "fix" it? Would
> you be able extend your code to use that method? Would you like some
> help in that sense? I have nothing to do myself.

Actually, I forgot what the rendering looked like if I just used:

float angle_start = 1 / (imax) * (PI*2);

That handles imax = 1 case, also this is just a quick a dirty example.
Not meant to be robust, what about the imax = 0 case? Bad input. I just
failed to put in a comment about the legitimate values imax can be in
the ct_alien_code_render function.

On Thursday, April 28, 2022 at 10:42:44 AM UTC-5, ju...@diegidio.name wrote:
> On Thursday, 28 April 2022 at 17:30:25 UTC+2, Archimedes Plutonium wrote:
>
> > Calling on failure of Logic Dan Christensen with his failed machine of DC proof.
> Please, Archimedes, do not feed the trolls.
>
> Julio

Good advice, and advice taken.

In regards to trolls, do you know who James Kibo Parry is under the fake name of Michael Moroney, and is he paid to troll Usenet. And is he connected to Jan Burse and Dan Christensen, paid to stalk and troll Usenet? Are they government paid for trollers

On 4/28/2022 4:44 PM, Chris M. Thomasson wrote:
> On 4/27/2022 4:47 AM, Julio Di Egidio wrote:
>> On Wednesday, 27 April 2022 at 10:14:45 UTC+2, Chris M. Thomasson wrote:
>>> On 4/26/2022 11:59 PM, Archimedes Plutonium wrote:
>>>> Julio's thread caused me to consider parametric equations once again.
>>>> I remember several years back, perhaps 5 perhaps 10 years ago that I
>>>> raised
>>>> the question of why have parametric equations at all. And some
>>>> polite poster
>>>> tried explaining the "why?". Although, he knew, and I knew I was not
>>>> buying it.[...]
>>>
>>> I remember back when you inspired to me to create a parametric that
>>> creates two semi-circles. Thanks again!:
>> <snip>
>>
>>> void ct_alien_code_render(
>>> int irecur,
>>> int irecur_max,
>>> float ox,
>>> float oy,
>>> int imax
>>> ) {
>>> if (irecur >= irecur_max) return;
>>>
>>> float angle_start = 1 / (imax - 1.0) * (PI*2);
>>
>> Whence (I'll have to surmise, in that sense your code is
>> redundant/opaque)
>> you just skip the case imax==1 (where you'd have to return an
>> "undefined",
>> just your function of corse returns void),
>
> imax must be greater than one in this implementation. Well, I guess I
> should of added a comment in my example code, sorry about that.

Ideally imax should be unsigned int, but alas iirc Processing does not
support unsigned types. Iirc, it's Java based. Actually, if I were to
rewrite this, I might be tempted to pass in an angle. So, perhaps
something like:

Instead of:
___________________
void ct_alien_code_render(
int irecur,
int irecur_max,
float ox,
float oy,
int imax
) {
if (irecur >= irecur_max) return;

float angle_start = 1 / (imax - 1.0) * (PI*2);
[...]
___________________

why not just use:
___________________
void ct_alien_code_render(
int irecur,
int irecur_max,
float ox,
float oy,
float angle_start
) {
if (irecur >= irecur_max) return;
[...]
___________________

Humm...

>> which, if you notice, is the very
>> motivating example for my question.  Have you seen how I "fix" it?  Would
>> you be able extend your code to use that method?  Would you like some
>> help in that sense?  I have nothing to do myself.
>
> Actually, I forgot what the rendering looked like if I just used:
>
>  float angle_start = 1 / (imax) * (PI*2);
>
> That handles imax = 1 case, also this is just a quick a dirty example.
> Not meant to be robust, what about the imax = 0 case? Bad input. I just
> failed to put in a comment about the legitimate values imax can be in
> the ct_alien_code_render function.

On Tuesday, April 26, 2022 at 1:20:20 PM UTC-5, ju...@diegidio.name wrote:
> On Tuesday, 26 April 2022 at 00:14:17 UTC+2, Archimedes Plutonium wrote:
>
> > Math is easy, but given the centuries after Newton,
> > math was made more and more crazy with every math professor
> > wanting fame and leaving math a tangled mess of illogical methods.
> Newton is the one who started the mathematical treatment of physics
> in the modern sense, whence the need for, and his early contributions
> to calculus (together and in contrast with Leibniz, but that's another
> story), i.e. a solid foundation for taking what now is limits, derivatives
> and integrals.
> > with Planck's Quantum of Quantum Mechanics means math had to
> > be discrete, not continuous. Yet there are the bozo the clowns Cohen
> > pursuing ever more continuity when physics found discreteness.
> FWIW, here are my comments on this matter:
>
> Besides that physics has no import on mathematics, whatever either
> of the two is actually talking about, physics is and remains continuous
> all over the place, even quantum physics despite some popularizations
> seem to suggest otherwise. For example:
>
> - The levels of energy of a particle system, e.g. the hydrogen atom
> are indeed "quantized", but that specifically means that they come
> in integer multiples of some "ground energy", and that itself is not
> an integer.
>
> - The uncertainty principle (roughly speaking) says that we cannot
> measure complementary quantities with absolute (i.e. arbitrarily good)
> accuracy *at the same time*, but it puts no limits on the accuracy with
> which we can measure a single quantity at any time. In practice, we
> can indeed measure the position of a particle with as much accuracy
> as we like and can, just in the process we will make the momentum
> more and more uncertain. So, not even here there is any actual
> discreetness of quantities.
>
> - The Plank length/time/mass put limits on what we can observe/
> measure, not on the granularity of the physical reality itself. And,
> if I am not too mistaken, these limits have more to do with gravity,
> i.e. the fact that when we go to too minuscule a scale/high an
> energy of our probes, we rather blow the region into a black hole
> as big as the energy employed makes it.
>
> - The primary parameter of a dynamical system is time, which was,
> is, and forever will be a continuous quantity...!
>
> And there is also that thing called quantum information theory,
> whose "atoms" are the q-bits, but that's even more remote from
> reality, indeed patently something to do with (the limits of) what
> we can say about it: again, it's the granularity of our observations
> and descriptions, not that of reality per se. But, in fact, even
> there, a q-bit is quite more than just a 0/1 value...
>
> Have fun,
>
> Julio

In my attempts to get rid of parametric equations, I find an example in which I simply cannot, and thus have to reinstall them.

Maybe I can save some of "my face" by saying parametric equations was not "taught well". Seems as if they were pulled out of thin air.

Maybe a name change is in order. Why "parametric"? What if we call them "Double Functions". The function of time versus quantity, then a second function-- quantity yielding new quantity.

In seeking the cause of gigantic dragonflies in Devonian, I assembled a table of values. Convinced that the cause was the size of the planet Earth is a growing Earth with time.

The Faraday Law inside of each atom is a doubling over time of that same atom of hydrogen. So at t_0 we have one atom of hydrogen and at t_1 we have 2 atoms of hydrogen, and at the same interval of time t_2 we doubled the 2 to be 4 now. So a doubling in physics. So we write out a chart.

Number of Hydrogen atoms                      Doubling time interval             Math form
1                                                                              t_0                                         2^0
2                                                                             t_1                                          2^1
4                                                                             t_2                                         2^2
8                                                                             t_3                                         2^3
16                                                                           t_4                                         2^4
32                                                                          t_5                                          2^5
..                                                                                .                                              .
..                                                                                .                                              .
1,073,741,824                                                       t_30                                       2^30
2,147,483,648                                                      t_31                                       2^31
4,294,967,296                                                      t_32                                      2^32

Now I stop there because it is nearby to the total time covered of 4,500,000,000

And here is where I divide that time of Earth existence by the number 32 in order to get what the doubling time interval is all about.

4,500,000,000/ 32 = approx 140,000,000

So the function for Doubling is simply Y = 2x. But both x and y are quantity of matter. No time involved.

So here it appears that I cannot get rid of parametric equations because of time factor.

So a remedy appears to be Double functions in all of mathematics. Time t, f(t) -> x and then f(x) -> y.

Has anyone taught parametric equations as being a two function network?

So it looks as though my efforts and attempts to throw out parametric equations has failed and apparently what it truly is are Double Functions. However, I can still maintain the idea that the only valid functions are polynomials, even strengthen that idea for you cannot have time as anything other than a straightline polynomial Y= mx+b.

Maybe the whole entire quagmire of parametric equations was never a nice clear one example problem set that exposes the need of a time function onto a material quantity function as the above Dragonfly table exposes.

In all the calculus textbooks I have seen they interchange x axis and call it time t axis. But never do they say that every calculus problem is a Double Function problem. Not that time is a separate axis from a y and x axes. But that time is involved in the spacing of the x axis.

So this dragonfly example forces me to save parametric equations. Calling them Double Functions.

I simply cannot pull out the dragonfly table of the number 140,000,000 without a second function involved.

AP

On Thursday, 28 April 2022 at 17:44:27 UTC+2, Julio Di Egidio wrote:
> On Wednesday, 27 April 2022 at 17:55:35 UTC+2, Julio Di Egidio wrote:
<snip>
> > Of course it's not just:
> > Lim_{x->a from /dir/}(f( x, y, z...)) = f(Lim_{x->a from /dir/}(x), y, z...)
> > but what is it if anything?
>
> Beside the standard epsilon-delta definition, I only find limits in category
> theory, but that looks quite beyond the present scope. Anyway, I can't find
> any "algebraic" definition of limit, which I find very strange: maybe I am not
> using the right keywords...

For the record, I have found at least one "formal definition" of limit:
<https://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_definition>

Interestingly enough, it depends on the real numbers: so that already
stating "every sequence of real numbers that converges, converges to
a real number" is easily circular...

> At the moment I am entertaining the idea that I could define it inductively
> on the complexity of formulas: the non obvious part is the limit of irrational
> functions, for which I am thinking I need a canonical form relative to which
> every limit becomes the limit of (possibly the limit of a series of) rational
> function(s)... which would not cover non-computable and all the more so
> undefined and undefinable numbers, but I am not envisioning getting any
> of those anyway... And, as a symbolic tool, probably something like:
> <https://en.wikipedia.org/wiki/Big_O_notation>
>
> Anyway, that's where we are: feedback of course remains welcome.

Julio

On Tuesday, 26 April 2022 at 19:08:07 UTC+2, Julio Di Egidio wrote:
<snip>
> At this point, I can try and reformulate the question more precisely:
>
> Say I have a problem parametrized on b, a real number in [0, 1], and
> a derived quantity g_b = 1/sqrt(1-b^2), which blows up for b = 1.
> Assume g_b is the only quantity that directly blows up, while I can
> write all remaining quantities of my problem as (g_b^k)*Q, for k an
> integer (positive or negative) and Q an expression that stays finite
> for all values of b.
>
> Now, in order to make the whole thing finite, I just consider the highest
> power of g_b that appears in any quantity I have in the problem, call
> that K, then I scale down all quantities by g_b^K (sure, all quantities with
> k<K go to zero for b=1). In fact, I also now write g_b in [1, g_1]...
>
> But there's the problem, how to justify that division by g_b^K if g_b can
> blow up?! And now that I look at it again, I'm thinking I am just back to
> good old limits [...]

This sounds too good not to be it: :)
<https://en.wikipedia.org/wiki/Wonderful_compactification>

Anyway, module the details I still should check, whether one can
legitimately write an *interval* such as [1, g_1] to me remains an
open question.

Julio