Alternating / symmetric polynumbers: a missing chapter of Algebra | Math Foundations 234 | N J W
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- čas přidán 27. 07. 2024
- We introduce subtraction into the world of arithmetic with integral polynumbers. This presupposes prior familiarity with negative numbers, which in earlier videos we introduced via the basic duality between msets and anti msets, or just amsets. The fundamental reflection symmetry denoted by sigma between natural numbers and their negatives allows us to define symmetric and anti symmetric or alternating polynumbers. This gives us an arena for very fundamental yet elementary investigations into Algebra which have largely been missed by our educational system focused, perhaps too much, on functions due to their role in analysis.
We introduce the basis B_n of symmetric polynumbers, as well as closely related basis A_n of alternating polynumbers. The algebraic relations between these become important when we investigate a third class C_n of central polynumbers which play a big role in many areas of mathematics, including q-series, representations of SU(2), hypergroups, quantum groups, and in physics also.
Video Contents:
00:00 The subtraction operation
04:46 Fundamental reflection symmetry
06:32 Symmetric Binomials B_n
09:17 Product formula for symmetric binomials
11:24 Alternating binomials A_n
14:14 Products of Alternating & Symmetric
19:40 Central polynumbers C_n
22:33 The Central Importance of central polynumbers
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Thankyou. The electro weak connected to random walks is just intuitive. Thankyou
Thank you for sharing your research. 😀
Loving this, Euler's Doubly Infinite Identity has been a deep and very productive inspiration! :)
With central polynumbers we can construct a genuine "quantum metric", a huge simplification of the current mathematical structure of coordinate system dependent Hilbert spaces and real complex numbers, which are corrections of corrections of certain heuristic choises taken en route, starting from trying to solve incompleteness of Greek pure continuous geometry with neusis method expanded to methodology of Cartesian coordinate system.
Construction of quantum metric requires also "7th degree of freedom", the outwards-inwards transformation. Linearly, this is very easy to do. Let's write mark < for A and > for B. By concatenating their mediant we get < > for A C B. Inwards-outwards transform can this way be expressed as simple Boolean NOT from outwards < > to inwards >
This perspective is great! There’s a lot we can do with these alternating and symmetric polynumbers. It appears that B(k)^n is like a random walk with step size k and n number of steps. Here’s an interesting array I got by taking B(1)^n and then changing bases to C. To change bases from B to C for the binomial coefficients of B(1)^n take differences rather than sums of the coefficients.
1c(1)
1c(2) 1c(0)
1c(3) 2c(1)
1c(4) 3c(2) 2c(0)
1c(5) 4c(3) 5c(1)
1c(6) 5c(4) 9c(2) 5c(0)
1c(7) 6c(5) 14c(3) 14c(1)
1c(8) 7c(6) 20c(4) 28c(2) 14c(0)
1c(9) 8c(7) 27c(5) 48c(3) 42c(1)
1c(10) 9c(8) 35c(6) 75c(4) 90c(2) 42c(0)
1c(11) 10c(9) 44c(7) 110c(5) 165c(3) 132c(1)
1c(12) 11c(10) 54c(8) 154c(6) 275c(4) 297c(2) 132c(0)
The next row is easy to find. The sums of the coefficients for each row seem to correspond to the center and off-center column of Pascal’s symmetric triangle.
P.S. Sorry for the formatting, CZcams didn't allow subscripts or superscripts.
I'm currently learning probability, but it appears on one hand that the linear combinations of B's represents the probability of positions for a one dimensional random walk with n steps. On the other hand I'm not sure what the linear combination of C's mean. It's very interesting that both represent a distribution when the weights are added together. I posted this in my excitement. This perspective is so profound! Thank you Norman Wildberger for all of your work.
Recall that if S(n,x) is the n-th spread polynomial in x, then both
{S(m,x) S(n,x) S(m+n,x)}
and
{S(m,x) S(n,x) S(m-n,x)}
are spread triples. This generalizes to spread n-tuples. This was first noticed, as far as I know, in a comment to one of Wildberger's older videos.
It seems this phenomenon is somehow related to reflections and symmetry, considering the geometric construction of the spread polynomials as the spread obtained from a reflection sequences of lines (see chapter 8 of Divine Proportions).
I agree that it appears related to reflections and symmetry. After watching Norman's Random Walk video in the Diffusion Algebra series, I think it can be related to reflections and translations too. Very interesting.
The Fibonacci sequence has similar properties. Per example, the sum of the first Fibonacci numbers of same parity is looking similar to your lemma at 17:00.
Hi Catalin, Can you elaborate on this interesting comment?
@@njwildberger Combining Fibonacci numbers with Lucas numbers (and inverting) is an interesting point of view on your work.
This part of symmetric poly numbers reminds me of Singal decomposition into even and odd signals in Singal and systems. Something I studied in electrical engineering that deals with properties of signals (graph of functions) and how use such properties to study the systems.
it would be interesting to see how an "eval" or "substitude" function that takes a polynumber and returns a new polynumber would look like. Because that is what people often associate with polynomials (applying a polynomial at x=6 for example).
Mind-boggling 😮🤔🎶
Consider for notation, vis a vis the central polynumbers: 1 = B_{0^+}
Was confusing for a moment the way these two arrived in my email but this is a fun new set of lectures!
YES! YES! YES! The next step for me is to integrate with The Finite Simple Groups (e.g. Octonions, Normed Division Algebras, Albert Algebras, Parker Surfaces) understanding with the Standard Model embedded in GR. A Conway Monster Sporadic Mesh!
Sir, I really love the way you explain mathematics. Sir could you please make lectures on tensors.
👀 Polynumbers, Kinda reminds me of euclids planar numbers when substituting in functions instead of numbers functionality-wise. A kinda planar function of sorts, with how it acts. Yet... Even then, It still manages to avoid abstraction via avoiding variables and only considering concrete values (Keeping in mind of the more Metrological aspects of the ACTUAL foundations of math).
It's like mathematical cheat codes at this point. I love it. XD
Who knows...
Perhaps there is ties between the 2?
Between Euclids Planar Numbers + applied statistical methods, and polynumbers (statistical methods to translate between the 2 ofc)? 🤔
Hmmm... Interesting thought i just had there.
Anyways.... Imma stop pondering now in this comment.
Thanks for the dope polynumber stuff as usual. 🖤
ahh man, you're gonna love it.
How to get centered mereological polyrationals from Euler's Doubly Infinite Identity?
Relational operators < for 'increases' and > for 'decreases', both bounded by the Halting problem.
Starting from
0 = ...x^3 + x^2 + x^1 + 1 + 1/x^1 + 1/x^2 +1/x^3... = 0
let's do this weird turn:
1/0 + x^-1 + 1/0
and simplify to by interpreting x' indices similarly to their exponents
1/0 + ^> + 1/0
Conjecture:
< > = 1/0 + 0/1 + 1/0 by concatenating mediants / Farey addition, in terms of numerical comparison.
Starting from < > as the top of the exponent tower, we can add new rows as the logarithms of the exponent tower growing from the top to bottom.
< >
< > : Let's interprete < and > as numerator elements with value 1/0 and as the denominator element with value 0/1.
< >
< >
0/1 2/1 1/1 1/2 0/1 1/2 1/1 2/1 0/1
Checks. As suggested by semantics of > as 'decreases', we can interprete the left side as the positive side and right side as the negative side, but we can rotate each way we wish. The structure of chiral symbols is palindromic also geometrically, and makes the binary tree of white space blanks separating the strings into words nicely visible. Nice place to construct "irrationals" as path information of L and R turns. We can write L also as < and R as >, which is esthetically pleasing and intuitive.
As e.g. on the 4th row of generation the exponent size is already
Sir ! I am from Nepal . Please upload videos of Parametric and Non Parametric tests conceptual videos . Full concept of hypothesis testing
Those lemmas are so pleasant
Dear Professor Wildberger,
Sharing many of the views you've expressed in your lectures, but not realizing that you retired from UNSW two years ago, I sent a letter on the subject to your university email address about a week ago. The purpose of the letter was both to introduce myself, and to solicit your review of a fast approximate factorization algorithm I have developed. Attached to the letter is both the abstract of a dissertation I am writing and an example of the algorithm at work. If the idea of using an analytic technique to accomplish an algebraic task intrigues you, then perhaps we could collaborate on a multivariate generalization of the algorithm that I am developing.
Sincerely,
Russell L. F. Paul
Hi Russell, I did get your email but am so busy with many things. If you don’t hear from me in a few weeks, please send a reminder
@@njwildberger Thank you, sir, I will.