Partitions: Exact Approximations

… there is something strange going on with Primes
Paul Erdös


Never mind the mock theta, Ramanujan’s gap, Namagiri dreams.


When Srinivasa Ramanujan wrote to G. H. Hardy in the 16th of January 1913, he had some remarkable formulas in that letter.  So remarkable are some of his formulas that mathematicians have been studying Ramanujan’s notebooks of formulas for new mathematical insights to this day, more than a hundred years later.
I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras… I have no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as “startling”. 
Hardy invited him to England because some of the formulas “had to be true, because no one could have the imagination to make them up”.   But there were no proofs.  However, when this poor vegetarian Indian Hindu came to England, eventually Hardy showed Ramanujan (thru Littlewood) that his formula on Primes was not EXACTLY correct. So Ramanujan had to bend to Hardy and work on his proofs of some of his formulas, so when they tackled the function of Partitions P(n), Ramanujan with the help of Hardy got to point where they “cracked” Partitions (and could prove it). They developed a direct formula that computed the number of partitions pretty accurately, and at the limit (infinity) it was “perfect” — and, could by truncating the number for high partition number to an integer could guarantee to be exact: since the number of partitions of integers is an whole number (i.e., the real number series “formula” converges with an deceasing error rate). Together they “cracked” the problem where neither man could do it alone. Ramanujan supplied the “intuition” (the finding of the hidden pattern) and Hardy provided the rigor to explain why the pattern is true.  The method they created, in this instance, was called the “circle method” — and it has been used ever since by numerous mathematicians for various other results.

Is it unitary?
Paul Dirac

No Tegmark or Linde, but Verlinde in name. It’s all but Feynman’s streams,
and weigh.


At Pocono, however, Feynman did not have a “complete formal derivation” for his equation.  He recalled that Dirac interrupted him and asked him, “Is it unitary?” As he was explaining “how [he] was going to work out positrons and so on,” Feynman remembered Schwinger’s trick and said, “Perhaps it will become clear as we proceed,” “But Dirac was not put off, and like the Raven kept saying ‘Is it unitary?’ ”  Not being quite sure what Dirac meant, Feynman asked, “Is what unitary?”, To which Dirac replied, “Is the matrix that carries you from past to future unitary?” Feynman wasn’t quite sure what Dirac meant by “the matrix” nor exactly what Dirac understood by unitary in connection with the equation. — QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga, by Silvan S. Schweber.

Sometimes it is easier to understand what you are not, than it is to understand what you are.

Hurwitz Gauge


The Love of Form

Such a Prime rank, any such Milnor’s exotic sank.
No mess, no Stress, but Strain. Tensors Bohm and bain.
It’s Held together.


to be continued…

3 thoughts on “Prime

  1. Yours Truly November 16, 2017 / 9:05 pm

    I have been perusing your other website–– and have been enjoying the information, which strikes me as fresh in its exploration of complexity in general and dissipative structures specifically. I am here because I wanted to ask you a question but could not find an email address with which to contact you directly. In your post, ‘The Relations between Replication and Dissipation…’ I came across the following statement: ‘…replication is dissipation of order, and dissipation is replication of disorder…’ and it gave me pause. Are you sure this statement should not read: ‘replication is dissipation of DISorder, and dissipation is replication of disorder’? This seems to make more sense to me. Of course, you have given more thought to these ideas than I. Just looking to gain some clarity. Thank you in advance and thank you for the cogent material.


    • Keirsey November 16, 2017 / 10:00 pm

      Thanks for your interest. You were perusing edgeoforder.ORG. So directly answer you question: I meant the statement to be as stated. Replication and Dissipation (by my definition) are “exact” polar opposites.

      Everything in the universe is “entangled” — every “thing” or “entity” or even “stuff” (what ever delineation you care to make) is dissipative AND replicative. Replication (copying the same information) “makes” multiple “independent” copies — and these various copies must be “distributed(dissipated) by embedding” into the object or onto the context.

      One could use your definition, it is perfectly reasonable — however I am interested from an information theoretic perspective.


      • Yours Truly November 16, 2017 / 10:34 pm

        Thank you very much for your quick and detailed response. One word brought instant clarity to your meaning: Distributed [as a synonym of dissipated]. I get it now.

        If I understand correctly, one of the central [philosophical] themes in your work seems to be situated around the idea of what I conceptualize as ‘Ideology Of Opposites vs Ideology Of Spectrums’. That is, in what seems to be an inherent human need to make distinctions between ‘different things’, we refer to one thing– eg. chaos– as the ‘opposite’ of another– eg. order.m– when, in fact, there are no true ‘opposites.’ All ‘things’ exist on a Spectrum, in so that no one ‘thing’ is entirely absent of any other ‘thing’. If this principle is accepted, we immediately find ourselves in the world of percentages [eg. statistical mechanics] and likelihoods [probability theory]. And, so, it seems that at least one of your aims is to try and reconcile this universal yet admittedly qualitative principle of ‘Spectrums’ with the more quantative methodologies employed within the sciences. No easy task, especially when one cannot seem to rely solely on reductionism [which seems indelibly linked to distinction/differentiation] to arrive at conclusions. I commend your efforts. Please continue the pursuit. I am doing the very same.

        Thank you again for the info and clarity.


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