In terms of transformations, I have always liked the idea behind the Laplace, X and Fourier transforms (also Mellin’s). I also loved the geometrical approach to tasks, hence last post about a tangible similarity between lattice coverage and primes (geometry).
Thus, I decided to work with the -function. As each number can be created as a sum of two smaller numbers (symmetrical due to the character of summation), I took dimensions of those numbers and mapped the output spectrum to the number. The word spectrum is somewhat new to numbers, but you can visualize it as a standard transformation of some sort of quantity representing the geometrical character of the numbers.
The origin of this research is as follows: when we talk about numbers, we usually refer to their quantitative aspect (or only). Still, there is merit to claiming that numbers themselves exist as objects defining the universe, ie. we cannot prove the existence of practical chaos, thus we might conjecture that such does not exist, which would imply determinism (see: Turing machine) that leads us to the place where we need to listen more carefully to signals. Not only the quantity- thus the reference to Langton’s ant or chaos theory. That simple.
An example: when speaking about statistics, we generalize the ouput and claim the statistical new value, so we basically claim the ability to foresee future without understanding of the mechanism. Works pretty good when it comes to invaluable things (finite). When it comes to more complex mechanism, every single one from the universe, we cannot generalize that simply, thus the approach from Tesla, mentioned a couple of times ago. And hence, this spectral analysis.
Given the fact that numbers are connected (their geometry) by eigenvalues represented by common divisors, I can now show you a bit of my approach to numbers. In case of physics, we – for instance – refer to mass as one way function that allows representation of goemetry of object (in given sense). In the meantime, we lose all the information about the object.
It might be intuitive that for primes the chart grows (probably) the same way (as every number is in some way “almost” a prime). Below a chart for primes. Number of frequencies goes the similar way.
Finally, a note, lets recall the known rule that summation of odd numbers gives squares. Geometrically, we could also say that infinite sumations would give us “areas” (in terms of dimensions). Thus, the example below showing the proposed approach.
The key to the analysis suggested first by Poincare (on movement from “standard” geometry to topology and on problems with understanding sets) and Tesla (on claiming curvature of time space without the (a priori) definition of mass and space well enough to even start talking rigorously. In order to solve problems, we need to focus on what we truly understand. What we currently have is a set of axioms, with potentially many leaks. Hence my blog.
We know that mathematics gives the best of human intelligence, which though went astray due to code coverage (space search) issue. There is often way more mathematics in painting than in mathematics itself, which is currently used by sofists (crypto, compression, optimization, economical modelling, use of probability (e.g. gaming), algorithms, physics, telecoms (RAKE, orthogonal and sparse matrices etc.), etc. . Mathematics itself is being able to listen well enough to different types of people who approach the problems differently enough to understand that mathematics must undergo a critical change.
A true mathematician cannot be selective when making decisions and his decisions will almost one hundred percent (in this world) will make him bump into a big wall, by Tesla’s claim (working for the future, not the present). In mathematics in-depth understanding of one thing is worth more than thousands of papers. Especially when we’re talking about understanding variance or time here.
Poetically, mathematics itself is like music of a hidden message. You can strive for learning it, but you need to be ready. When you decide that you want to be ready, doing it in many different ways (often, not connected with “counting” itself but rather deciding to either kill or defend human rights, whatever, confined to your current level of understanding of the world), you are paying the prize, ie. the convolution of goals of the humanity (and you there, in that group) is likely to be getting on the top of a single person, by Tesla’s claim (difference in goals, different reasonings).
Anyone here knows how Universe is shaped according to Dr. Perelman’s work on Ricci flow in terms of 3-manifolds?