Wednesday, May 7, 2025

The Euclidean Algorithm and the Path to Continued Fractions

The Euclidean Algorithm is actually the means to obtain continued fractions.  What is the Euclidean algorithm? A general theorem can be stated thus:

If a is any integer, and b is any integer greater than zero, then we can always find an integer q such that: 

a = b*q + r

where r is an integer satisfying the inequality:

0 < r < b


A general task is to find the “greatest common divisor” of a and b, called d(a,b). If d = (a,b)then positive or negative integers k and m can always be found such that:

d = ka + mb

and this proceeds by successive divisions with remainders.

Consider using the Euclidean algorithm for finding:

 (a,b) = (61, 24). 

Here gcd or the greatest common divisor is 1 and we find, repeating the same process over until an equality emerges:

61 = 2*24 + 13

24 = 1*13 + 11

13 = 1*11 + 2

11 = 5*2 + 1

2 = 2*1 + 0

From the 1st equation we have: 13 = 61 - 2*24

From the second:

11 = 24 – 13 = 24 - 61 - 2*24 = -61 + 3*24

From the third:

2 = 13 – 11 = (61 – 2*24) – (-61 + 3*24) = 2*61 – 5*24

From the fourth:

1 = 11 – 5*2 = (-61 + 3*24) – 5(2*61 – 5*24) = -11
*61 + 28*24


Example Exercise: Carry out the Euclidean algorithm for finding the greatest common divisor of (187, 77)

Solution:

To use the Euclidean algorithm on (187,77)

Take (a,b) = (187, 77)  = 2 R33 [e.g. 187 – 154]

= 2*77 + 33   or  just (77, 33)

So take:

(77)/ 33 = 2 R11 [e.g. 77 – 66 = 11]

= 2*33 + 11 = (33, 11) = 3*11 + 0

Hence: (187, 77) = (33,11)


Another example problem:

Reduce (245, 193) via the Euclidean Algorithm:

(a,b) = (245, 193)

=> (245) / 193 = 1 R52 [e.g. 245 – 193 = 52]

= 1*193 + 52 = (193, 52)

ð (193)/ 52 = 3 R 37 [e.g. 193 – 156 = 33]

= 3*52 + 37 = (52, 37)

But: (52, 37) = (52)/ 37 = 1 R 15 [e.g. 52 – 37 = 15]

= 1*37 + 15 = (37, 15)

But (37, 15) => (37)/ 15 = 2 R 1 = 2*7 + 1

= (7, 1)

7/1 = 7 Hence

(245, 193) = (193, 52) = (52, 37) = (37, 15) = (15, 7) = (7, 1)


Now, let’s look more closely at continued fractions:

                            Fig. 1 - Continued fractions

We see in the graphic above how 2/5 is converted into a continued fraction. Next consider 169/70 and seek the continued fraction development:

We have:

169/70 = 2 + 29/70 = 2 + 1/70/29

70/29 = 2 + 12/29 = 2 + 1/ (29/12)

29/12 = 2 + 5/18 = 2 + 1/ (12/5)

12/5 = 2 + 2/5 = 2 + 1/(5/2)

And: 5/2 = 2 ½ = 2 + ½

Thus 169/70 is finally developed as shown below with result = 2.414


The key flag that the process is at an end is the final fraction is proper and also simple. No further processing is needed and the continued fraction form is now complete. 


Suggested Problem: 

1) Using the Euclidean Algorithm show the continued fraction for:

(237/ 139)

2) Show how the continued fraction for 840/ 611 results in C1 = 1.375  (Refer to top of Fig. 1)

Monday, May 5, 2025

Francis' Successor Needs No "Doctrinal Clarity" - It's Irrelevant Except To Radical Conservo Absolutists

 

                                        Pope Francis - soon after ascent to papacy


Predictably Pope Francis' death April 21, has resulted in numerous media appeals for his successor - whomever that might be - to return to "doctrine'. We can start with the Friday May 1 WSJ column ('What the Church Needs in the Next Pope', p. A13) by Archbishop Emeritus Charles J. Chaput, who insists: 

 "Francis papacy...was more process than substance and reopened debates from the Second Vatican Council (1962-65) that had been addressed and settled by previous pontificates. The unintended but corrosive result is the spirit of fracture that now infects millions of otherwise faithful Catholics. Internal conflict is the menu of the day.  This is the legacy that the cardinal -electors of the next pope urgently need to address.

They must attempt to restore clear teaching - to build on the good Francis accomplished- while purging the confusing, incoherent and bad."

And what, pray tell, constitutes the "confusing, incoherent and bad".  It's not hard to find so we can look at a couple examples. None of which, I will argue, need to be "purged."

As  a point of entry we can reference an earlier WSJ column (Pope Francis' Legacy of Question Marks', April 22, p. A15) by Francis X. Maier who wrote:

"Confusion on matters of doctrine is lethal for a creedal community like the Church."

So let's begin by noting Maier's' unmentioned violation of  his 'creeds' which have caused so much consternation among the Church's conservatives.  I referenced one in my earlier post after Francis death.  Therein I noted a homily Francis delivered in December, 2013 at the Chapel of the Domus Santa Marta speaking thusly: 


"The Lord has redeemed all of us, all of us, with the Blood of Christ: all of us, not just Catholics. Everyone! 'Father, the atheists?' Even the atheists. Everyone! And this Blood makes us children of God of the first class. We are created children in the likeness of God and the Blood of Christ has redeemed us all. And we all have a duty to do good. And this commandment for everyone to do good, I think, is a beautiful path towards peace. 

If we are each doing our own part, if we do good to others, if we meet there, doing good, and we go slowly, gently, little by little, we will make that culture of encounter: We need that so much. We must meet one another doing good. 'But I don't believe, Father, I am an atheist!' But do good: We will meet one another there."

Headlines across the world immediately proclaimed, "Even Atheists Can Go To Heaven.". 

And the conservo cardinals, especially the recently retired Ratzinger, went batshit nuts.  No Hell for atheists?  What is wrong with this guy? Of course atheists must go to Hell! Comedian Bill Maher on his 'Real Time' show opined the new Pope "is really an atheist".   But he missed the point.  

That is, the Pope was really referencing an ancient teaching from the time of Origen of Adamantius, i.e. of  being a  Universalist or embracing the potential of 'Universal Salvation' .  I had my first encounter with this teaching during a debate with an Anglican priest in 1982.  But in truth one needn't go to historical teachings at all to dismiss the fantasy notion of a Hell. I have already done this in previous blog posts, e.g.

Indeed, on being invited to attend one Catholic students seminar at the University of Alaska-Fairbanks in February, 1986 - while doing a research assistantship in Space Physics-  I interjected similar questions.  When discussions turned to the afterlife and punishments, I asked the gathering of mainly freshmen and sophomore UAF students:

"Okay if God is infinite, how can there be a Hell? Wouldn’t Hell have to be a part of God?  

If not, then how can God be infinite if there is a place-abode (“Hell”), even supernatural,  He isn’t?"

To which their heads nearly exploded, some crying and pleading to the priest in charge to "correct this heretic".  But he merely calmed them down and answered:

“I believe your concept of God is the same as mine, the All. The One.”

 This is basically what the fuss is about. The chance that Francis' homily remark might neutralize or deflate the Church's final card to keep its minions in check: Hell.  But if there is only one reality, ultimately, one universal mind -if you will - then clearly whatever exists now on the material plane must be subsumed into it.

More to the point, some years after Francis' homily, RC fossils really went ballistic after he himself rejected the very notion of "Hell", e.g.

Brane Space: Pope Denies Existence Of "Hell" - Why The Big Fuss?


Now, in the interest of honest and full disclosure,  let me say again: I do not believe there is any "heaven" or "hell". We each create our own heavens and hells on this Earth, in the here and now. Basically "salvation" is accomplished on one's own terms to redeem his or her life here on Earth to become the best human he can be...with the resources allotted him over a finite time.  In other words, there is no hereafter of the kind the orthodox religious types - including Catholics - believe.

Once the Hell bogey is punctured, none of the other RC doctrinal obsessions amount to a hill of beans. That leads us to the next bugbear, first mentioned by Francis X. Rocca in a WSJ column 9 years ago:

"What has changed since 1968? Pope Francis has laid increased emphasis on the role of an individual’s conscience in discerning factors that mitigate culpability in particular circumstances."

This specifically referred to Francis' relaxing standards for artificial birth control during the Zika virus plague in Brazil and Africa. To fix ideas, Pope Francis suggested that artificial contraception "may be morally acceptable to avoid spreading the Zika virus."  He opened the doctrinal 'door' just a tad, and in the process killed the absolutist basis for the idiotic proscription against using any artificial birth control.  Since a 'mortal sin' and "Hell" eternal punishment was visited upon violators, he ditched them as well. Again!

We needn't get into all the particulars of why the birth control doctrine was bogus, but suffice it to say, it relied on the antiquated bunkum known as "natural law'. 

As a matter of theological record the anti-artificial birth control dogma was based on "natural law". The problem is that there is no genuine theological or other (scientific) basis for this antiquated belief, given it rests entirely on reducing human sexuality to the state of lower animals- with their "natural" reproductive cycles. 

But it ignores that, unlike the lower animals, humans have the intellectual capacity and sense of novelty to introduce a vast variety of pleasure-play into their sex relations - none of which require conception. Thus humans aren't yoked to  primitive instincts to simply mount and hump at specific times. As observed by Biologist Elizabeth A. Daugherty,  ( in Contraception and Holiness, pp. 96-97):

"Humans are free from physiologically determined sexual desires so we possess a more or less permanent sexuality from adolescence to old age."

So this lays waste to the whole natural law moral underpinning, which is more subjective moralism.   This is tied to an emotional reaction to an act, or behavior, rather than any objective measure or index. As Ethics professor Cheryl Mendelson has pointed out  ('The Good Life', p. 157), this mindset:

"confuses the disgusting with the wrong and retains an infantile fear of things sexual. Its rationality is overcome by emotion, fantasy, wish and projection,"

And as I wrote in an earlier post:

"Recall the Church once believed it "natural" that some men be enslaved because they were “unable to manage their own affairs”  (ibid.). So why be surprised when the same Church seeks to ordain all her members abide by a sexuality more fitting of lower primates? "

So in the end, natural law is another 'macguffin' like Hell. Indeed, all the various doctrines fall like a house of cards, because that is what they are. If one more or less serious doctrinal canard is punctured, they all are.  And lest we forget, the RC Church lost whatever moral credibility it had once the sexual abuse scandal erupted and the two popes preceding Francis (John Paul II and Benedict) sought to conceal the extent.

What Catholics need then, is for the successor to Francis to continue along the path he's set, diminishing the import of rules and doctrines, in favor of reflecting on and embracing the actual values Yeshua demonstrated. 

See Also:


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And:

by Alex Henderson | May 4, 2025 - 5:36am | permalink

— from Alternet

Following the death of 88-year-old Pope Francis, the Vatican is considering its possibilities for a successor. President Donald Trump, on Tuesday, April 29, joked that his "number one choice" for the Catholic Church's next pope is himself.

Later that day on X, formerly Twitter, Sen. Lindsey Graham (R-South Carolina) humorously posted, "I was excited to hear that President Trump is open to the idea of being the next Pope. This would truly be a dark horse candidate, but I would ask the papal conclave and Catholic faithful to keep an open mind about this possibility!"

Although Trump and Graham were joking, some MAGA Republicans are taking the idea seriously. Trump, however, isn't Catholic; he is a Mainline Protestant who was raised Presbyterian in Queens.

MSNBC's Ali Velshi examined the "Trump for Pope" campaign in a Saturday, May 3 commentary, laying out some reasons why he finds the idea ridiculous.

» article continues...

Friday, May 2, 2025

Mensa Ellipse Geometry Solution

 


The ellipse is first re-drawn as shown with the major axis = 2a.  Then draw the minor axis and let its length be 2b.  The area of the ellipse A (e)= p ab.  Note that the green region comprises two unabridged quadrants and two right triangles. 

The area of the two quadrants is:  ½ (p ab).   The area of the two right triangles = 2 (½ ab)  = ab.  

Then the rea A of the green region is: ½ (p ab) + ab.   Given there is only one Pythagorean triple (9, 12, 15) in which the largest number is 15 we must have: a =12, b = 9. 

 Then: A = ½ (p · 12 · 9) + 12 · 9 =   54 p   + 108

Or:   54 (p   + 2)




Obtaining Soliton Solutions For The Korteweg deVries Equation

 The  Korteweg de Vries (KdV) equation:

(- v o  + c s   + v)   v / x  -  m  2 v / x2  +  a  3 v / x3 =  0

is a well-known example of a soliton equation admitting nonlinear superposition, see e.g. a graphical representation of a soliton solution here:


In the KdV equation c s is the ion sound speed, and v o the  electron thermal speed.  In the form shown, note the appearance of the dissipative term:

 (m  2 v / x2)  

For a soliton to evolve into a shock a dissipative mechanism is needed.  In the more common situation steepening of the wave balances dispersion and we obtain a wave form such as shown in the graphic

  On integrating once, and excluding the shock evolution term (in m),  we obtain:

a  d2 V/ dx2   -    (v o   -   c s ) v  -   v 2/ 2   =  0 

Which has the same mathematical form as Newton's 2nd law of motion, e.g. 

 m x" =  F(x)  =   -   x V(x)  

Where  x"  =  d2 x/dt 2

Where V(x) is the potential energy. With some further manipulation we find:

a d2 V/ dx2   =   -   v  V' (x)  =

-  ¶ v [(c s   -  v o) v 2/ 2   +     v3/ 6 ]

For a particle 0f mass a   moving under a potential field given by the quantity in brackets.  We call this quantity the pseudo potential and designate it:

 F (v) =  [(c s   -  v o) v 2/ 2   +     v3/ 6 ]

Which is also known as the Sagdeev potential.  It is left to the industrious reader to do a simple plot of  F (v) vs.  v,   with   v max   shown on v -axis.

For the criteria on  F  to obtain a soliton solution we have:

i)    F / ¶ V ] v= 0  =  0;   2 F/ ¶ v2   <   0

ii)  F   <   0,   For   0   <   v   <   v max

iii)  d F / d V ] v= v max  >  0

We  note here that two graphs of  F   vs.  v are possible,  one for  c s   -  v o   >   0,  the other for  c s   -  v o   <   0.  Since we demand only a localized wave form then it will always be the latter form used, i.e. in further analyses.  One such is to obtain a soliton solution for the KdV equation:

a  d2 V/ dx2   -    (v o   -   c s ) v  -   v 2/ 2   =  0 

This may be solved exactly i.e. with c s   -  v o   <   0.   The procedure is then to multiply the KdV by v'  and then integrate to obtain:

a/ 2  (v' 2)  =  (v o   -   c s ) v 2/ 2  -    v3/ 6

And we choose the constant of integration to be zero because we want v' = 0  when  v = 0.   Working through the process the final solution is found to be:

v   =   3 (v o   -   c s ) sech 2  [(v o   -   c s  / 4 a) ½     x' ]

Suggested Problem:

Integrate  the KdV equation:

a  d2 V/ dx2   -    (v o   -   c s ) v  -   v 2/ 2   =  0 

And show how the soliton solution:

v   =   3 (v o   -   c s ) sech 2  [(v o   -   c s  / 4 a) ½     x' ]

Is obtained.