[Fis] Paradigms and Machines

[Karl Javorszky]

Cycles 101

2023 12 06



Dear Krassimir,



As an introduction, let me repeat my honest gratitude towards your
enlightened attitude what regards my work. Since some twenty years ago, the
Journals you are the Chief Editor of have published some five or six
articles of mine, the last one, a good summary of the matter, this past
Summer. In view of this trust, let me comment on two of your points, where
we appear not to sing from the same sheet.



In your mail of today, you wrote:



*… the question remains open as to how the new paradigm will be implemented
on the computers we have**.*



This is what friends of soccer call in rhetoric a “match ball”. May I
answer as clearly and pointedly as your question was phrased:

*… the question is no more open, because the algorithms have been found
that allow implementing the new paradigm on the computers we have. *



To explicate, let me bring forward a message you have sent a few weeks ago:

*Thousands of math and computer science students study combinatorics, which
means that permutations and cycles (loops) are part of first-year studies
and exercises.*

Would you do me a favor and arrange for the next semester’s exercises in
combinatorics to include the following simple introductory task:





*Combinatorics 101 – Cycles*

*Task 1: Establish definition of cycles by deictic method*

   1. Take 12 pieces of paper. Write the letters ‘A, B, … , K, L’ on them.
   Shuffle the papers well.
   2. Write the numbers ‘1, 2, …, 11, 12’ on them. The result is a random
   assignment of a letter to a number. The papers are presently ordered
   according to the numbers. Line up the papers on your desk in the present
   sequence.
   3. Now you reorder the papers into the alphabetical order. Keep a
   protocol about the push-away incidents that occur as you replace a paper
   with a different one. If a paper that you move to its target place finds
   that target place empty, then this cycle has become closed. Keep track of
   which paper has moved where till its cycle got closed-
   4. Tabulate the cycles. The number of cycles in a reorder is called the
   fragmentation of the reorder. The number of members in a cycle is named the
   length of the cycle. The sum of the distances that a paper has been moved
   is called the run of the cycle. The term carry is self-evident but will not
   be applied here.
   5. Compare the results of the students. Do these random results appear
   to cluster? (In this case, one could speak about spontaneous self-creation
   of structures.) (Aside: this is the stage we are at presently in Fis.
   The learned friends have not yet given their results in.)

*Task 2: Formalize and discuss convergence*

   1. Regenerate the exercise on your computer, calling the alphabet values
   from the experience *variable a, *the numeric symbols *variable b*. Use *a,b
   ≤ d; a ≤ b.*
   2. Generate the values *Rd *(for *fRagmentation of the cohort with
   d(d+1)/2 members), *for the number of cycles in cohorts *(a,b) *during
   reorders [ab ↔ ba] in the set of pairs *(i,j), 1 ≤ i ≤ j ≤ n* for the
   first few 100 *d. *Check against *oeis.org/A235647
   <https://urldefense.com/v3/__http://oeis.org/A235647__;!!D9dNQwwGXtA!QtkY1D5Nz_veuqBM3Cwwqb5GyRwQBKZa8yEcs4sghfv-UDyABC9KgsAlawAuBexlknWoaF71ptCMQ1yeDNdugL75wbQ$ >. *
   3. Generate aspects of *(a,b) : c = a+b; k = b – 2a; u = b- a; t = 2b –
   3a; q = a – 2b; s = (d + 1) – (a + b); w = **2a – 3b, *where *d *is the
   number of different *a,b *in that cohort. Generate 72 sorting orders,
   based on one of the aspects *{a,b,c,k,u,t,q,s,w} *being the first,
   outer, senior sorting criterium, and a different one of the same collection
   of aspects being the second, inner, junior sorting criterium. Resort from
   72 ‘predecessor’ into 71 ‘successor’ sorting orders. Redo the extension
   exercise for the first few 100 arguments of *d *with these *72*71 *aspects,
   which we call the *catalogued reorders,* noting the fragmentation,
   length, run, carry attributes. Do you find convergence?

*Task 3: Self-creation of structures*

   1. Set* d = 16, → n = 136. *
   2. Repeatedly generate random permutations of the *n* pairs of*
(a,b). *Discuss,
   whether any two random permutations *RP1, RP2 *are closer to each other
   than any *RPi* is close to any of the catalogued reorders’s statements
   of facts.

*Task 4: Create spaces*

   1. Find such among the catalogued reorders which share axes that
   generate a plane in which there are *45* cycles of length *3* and *1* of
   length *1*.
   2. Assemble two Descartes-type spaces of these planes.
   3. Note the two planes that transcend the two Descartes-type spaces.
   4. Find the coordinates of the Central Elements and discuss whether
   there are two, three or four Central Elements.



This Introduction to Order Combinatorics course should not take more than
three – five units, days, weeks or months. You do have the executive
authority to ask a junior lecturer to include this exercise within his
didactic material in probability, number theory, combinatorics or whatever
name the general course has.

By the method of charging a subordinate you save yourself the inner fight
whether you will or you will not part with your long-standing belief that
if you count something that is made of natural numbers, there can basically
never be a surprise, because all neighborhoods are of unitary nature.
Whichever way you count a matrix, there will always remain that many rows
and columns as you started with. Believe it or not, this is an outdated
idea. Turns out, if you follow the melee exactly that comes from ordering
and reordering, you will find values for a membrane that separates values
from surely existing to maybe existing, which also means that the number of
rows and columns becomes a value dependent on the succinctness of how the
fact is expressed in the matrix. The very idea of redundance emerges at
first, but where there is redundance, there is information.

Let your assistant work with the students, the task is kosher, and you and
your team will serve in the sense of the taxpayer if you cause them to
solve abstract problems of where is what, when and how much of it.

Then, after your students and assistants will have done the exercises and
credibly report having found nothing remarkable, then you repeat your
assertion that one cannot teach an old mathematician any new tricks of data
processing, specifically not in the field of combinatorics of cycles. You
may, however, incline to consider that the economics of and within the
combinatorics of cycles could contain some juicy news. (Aside: Francesco,
you hear?) The bet is still on, this is solid stuff and good business. One
wonders why no business-oriented people make any remarks in this chatroom.
Mendel tries to explain to you that there are clear logical-numerical rules
to future events, Leo Szilard explains to you one possible method of how
the future is formed (in the present case, it is not chain reaction but
loop reaction), and it is okay that fellow learned friends are by no means
eager to understand someone else’s good ideas, but practically minded
people would recognize the business of genetic and the scope of the
Manhattan project, presently called AI project. The key question was in
1985: how does the DNA translate from linear into multidimensional and
back? Here is the answer in Combinatorics 101 – Cycles (the spaces turn in
three phases). This should be worth something in commercial circles, isn’t
it. As a bonus there are the corollaries for the natural sciences, being
necessary half-steps in the argumentation, how a linearly placed symbol
determines the properties of a specific subspace in a complex melee of
planes and spaces.

To summarize:

The bet is still on. The solution you say is impossible to implement on
today’s computers, but which you also say is everyday knowledge in
introductions to combinatorics, is indeed here and is indeed a new
methodology and is furthermore easily implementable on today’s computers.
In every other point, I share your views (and your good-natured humor).



Respectfully, with friendly greetings:

Karl
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