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The consistently intriguing posts of The Everything Seminar has landed it on the sidebar. The latest post about summing divergent series includes the generating function
$T(z) = 1 + z + 2 z^2 + 5 z^3 + 14 z^4 + 42 z^5 + \cdots$
for the Catalan numbers. M theorists will recognise the coefficients as the number of vertices on an associahedron: two vertices for a line segment, five for a pentagon, and so on. And lest there be any doubt he is thinking about trees, he refers to this paper, which maps 7-tuples of (planar rooted binary) trees to trees. A commenter (with a blog called God Plays Dice) pointed out that a different set of trees gave another isomorphism between 5-tuples and singlets.
Apparently the reason the number 7 works for general trees is because 7 = 6 mod 1 and by dividing trees into left and right halves we get an equation $T = 1 + T^2$, where the 1 stands for the root. So planar rooted trees are associated to a sixth root of unity, and the fourth root case is about trees with either 0, 1 or 2 children at each vertex. Its generating function $M$ yields the series
$M(1) = 1 + 2 + 4 + 9 + 21 + 51 + · · · = −i$
It seems there are lots of cute ways of writing down complex numbers as infinite sums, so long as one uses series derived from trees! Here's a cool TWF on this stuff, with a link to this helpful, and seminal, paper by Fiore and Leinster. Oh, I can't wait to go and play some more...
$T(z) = 1 + z + 2 z^2 + 5 z^3 + 14 z^4 + 42 z^5 + \cdots$
for the Catalan numbers. M theorists will recognise the coefficients as the number of vertices on an associahedron: two vertices for a line segment, five for a pentagon, and so on. And lest there be any doubt he is thinking about trees, he refers to this paper, which maps 7-tuples of (planar rooted binary) trees to trees. A commenter (with a blog called God Plays Dice) pointed out that a different set of trees gave another isomorphism between 5-tuples and singlets.
Apparently the reason the number 7 works for general trees is because 7 = 6 mod 1 and by dividing trees into left and right halves we get an equation $T = 1 + T^2$, where the 1 stands for the root. So planar rooted trees are associated to a sixth root of unity, and the fourth root case is about trees with either 0, 1 or 2 children at each vertex. Its generating function $M$ yields the series
$M(1) = 1 + 2 + 4 + 9 + 21 + 51 + · · · = −i$
It seems there are lots of cute ways of writing down complex numbers as infinite sums, so long as one uses series derived from trees! Here's a cool TWF on this stuff, with a link to this helpful, and seminal, paper by Fiore and Leinster. Oh, I can't wait to go and play some more...
posted by Kea | 11:37 AM
6 Comments:
Dear Kea,
is it possible to explain for a layman why division and subtraction are so difficult to categorify?
I looked at the article and found that the trick was to transform subtraction and division to sum and product. This inspires the question whether one could start categorifying of division and subtraction from p-adics with unit norm.
a) In this kind of situation -1 equals to (p-1)(1+p+p^2+...). If positive integers and their sum have a category theoretic representation and if the infinite sum makes sense then one could categorify p-adic subtraction.
b) The inverse of a p-adic number of unit norm is also constructible as a similar infinite sum so that also p-adic division of units might be representable category theoretically. Any rational with unit p-adic norm would be categorifiable.
c) One should also define negative powers of p category-theoretically and here I do not have any proposal.
d) One also consider categorifying subset of algebraic extensions of p-adic numbers by categorifying the quantum phase exp(i2pi/p) associated with primes: their products give general quantum phases. Perhaps using a trick analogous to that discussed in the article. If the roots of any algebraic equation could be represented category-theoretically, one would have category theoretic representation of p-adics and their algebraic extensions.
Where this argument fails: in the introduction of infinite sum or?
why division and subtraction are so difficult to categorify?
Hi, Matti. Yes, p-adics are interesting. When one finds an Euler characteristic the Euler way, there is an alternating sum of counting numbers, so minus signs and hence subtraction. And in Leinster's Euler characteristic one can get minus signs, too. The problem is: what sort of an operation is subtraction? It is really an addition after all, only for negative quantities, and this is where the problem lies. What does the number -3 count? It doesn't count the number of elements in any ordinary set. It doesn't measure the dimension of any ordinary space. And after playing around with categories long enough, it is quite upsetting to realise that one is working with objects that don't have a nice categorical context, because everything should only make sense in context.
Now alternating sums work wonderfully well, and lots of invariants satisfy useful axioms, suitably functorial etc. ... so it comes down to the question of whether we should be extreme or not about categorifying everything. And I'm guessing you would agree that the physics is demanding precisely that when it comes to numbers.
Thanks a lot for the explanation.
I begin to dimly understand: it seems that you must have an object which represents the number in some manner.
You cannot define -1 as any geometric or set theoretic notion, it is essentially algebraic notion. I however wonder why simple abelian group theory does not help in case of -1: inverse of an operation is something very geometric? Why cannot operations (arrows) serve as objects?
What I had in mind that -1 in p-adic sense is expressible as an infinite real number number (p-1)(1+p+p^2+...), which is finite p-adically (of unit norm) and manifestly positive.
The question is whether one can have a geometric structure for which number of points is in real sense infinite but in p-adic sense finite, say -1=(1-p)(1+p+p^2+...).
p-Adic fractals suggest themselves as a natural geometrization of p-adic numbers with norm not larger than 1. For instance, to represent 1/1-p-1 =p+p^2+... appearing in -1 as key piece, one can start from vertices of p-polygon, put around each vertex scaled down p-polygon, put .... . This structure indeed has 1/1-p-1 = p/(1-p)(<0 in real sense) but infinite points in p-adic sense.
p-Adic cutoff as an approximation is also very natural. Generalization is obvious.
Also quite many algebraic numbers in real sense exist as p-adic numbers: for instance, half of integers 0 \le n \le p have p-adic square roots representable as p-adic fractals. Of course, this might be a trickery.
If the often stated claim that reals are in some sense obtained at limit when p goes to infinity make sense, one might dream of category theoretic realization of reals by realizations of all p-adic number fields.
Still a comment about representation of p-adics as a fractals. The earlier representation was still somewhat clumsy.
A canonical representation of p-adic numbers as fractals is to assign to n:th power of p a polygon of size scale p^-n: if coefficient of p^n is k, you paint k vertices of polygon black (the representation is not unique).
This allows to represent all p-adics, in particular all rationals, as fractal sets, and you can represent p-adic arithmetical operations geometrically as "set theoretic" operations for these fractals.
Also many p-adic variants of algebraic numbers say half of square roots have a geometric representation as a p-adic fractal. One might ask whether any algebraic number might have a representation as a p-adic fractal for some values of p.
Also transcendentals like e^(px) =1+xp+x^2p^/2!+... and log(1+px)=1+xp+... would have a representation as p-adic fractals.
Consciousness theorist might also ask whether p-adic fractals might be the manner how p-adic cognitive representations of numbers are realized at the fundamental level. These fractals could be realized as subset sets of rational intersections of p-adic and real space-time sheets obeying same algebraic equations. p-Adic cutoff as an approximation would emerge also naturally. At the limit when cognizer has access to p-adic space-time sheets with all values of p, she would become an infinitely intelligent number theorist!
Hi there,
It's Jason D. Padgett again. Well one more semester is finished and the first part of my theory has been proved! It's very exciting. I recently found a function for Pi, based on my fractal drawings. The first way I wrote it was based on a unit circle and the equation is...f(x)= x times the square root of 2-2cos(360/x) all over two. (Sorry I had to write it out but my computer doesn't have math symbols. In this equation,x stands for the number of sides to a circle, and as x approaches infinity, f(x) approaches Pi. Unfortunately our calculators call the cosine of a very small number zero so it eventually will give an output of zero. This is because of the calculator, not from the equation. A more stable version of the equation is f(x)= xsin(180/x) and as x approaches infinity, f(x) aproaches pi. I thought I was the first to discover this but it turns out that archimedes and Newton both also discovered equations that are equivalent of this. BUT, they didn't know about the Planck constant (or more importantly, when defining Pi, the Planck length). As you inscribe polygons with more and more sides within a circle (inner bounds of Pi) you find that mathematically you can keep doing this forever. So the value of Pi would go to an infinite number of decimals. But since we know that the Planck length (difined as the upper and lower bounds of uncertainty)is the smallest length that exists (relative to our universe), and we know that Planck lengths only exist as whole multiples (there are no fractions of a Planck anywhere in our universe). Then you eventually get to a point where all the 'sides' to a circle are one Planck length, and THAT is the point where Pi ends (physically). So basically, Pi goes to infinity mathematically but in the real world of physics Pi ends when each side of a circle is one Planck length.
It is important to note that there could be measurements that are fractions of a Planck length but the are not 'relative' to our universe or us because they cannot be observed in ANY way.
Well have a great day and I will let you know how my theory works out. Now that I finally can prove the part of my theory that has to do with Pi it will be much easier to prove the rest, but I think it will still take couple more years to finish it all. But as it stands now...grab you graphing calculators and see the function of Pi (also analyze its first and second derivative if your in the mood). One last comment, when you graph f(x)=xsin(180/x) and its reflection f(x)= - xsin(180/x) you will see what looks amazingly similar to little Mandlebrot's as x approches 0+ and 0-. Have fun with it and let me know what you think. Have a great holiday season.
Jason D. Padgett
email- PLANETFUTON1996@YAHOO.COM
Jason Padgett here again to clarify one last thing. There are some that say that the Planck length may not be the smallest length. That is not what is most important. As long as there is ANY miminum measurement that exists in th universe then Pi will end (relative to our universe) when each 'side' of a circle is equal to whatever that minimum distance turns out to be.
Jason D. Padgett
email- PLANETFUTON1996@YAHOO.COM
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