# the linear systems --- is the [[The Fourier Transform]] just one of the unique solutions of a differential equation that leverages the natural exponential I'd love to know more about the natural exponential, is there an actual function that estatmets it? Itself a fourier series? what makes a series a fourier series? ![[linear_systems_diff_eqns.png]] so here's how it starts ... I had a problem related to finding a stable solution on a landscape of continuous movement There is the ARR -- the annually recurring revenue -- you can think of it as a flowing water. Then there are things _about_ it like how fast it flows, what kind of water it is made of, and some interesting things like its color, or how it smells. In mathematical language, the flowing water and all the dimensions that describe it The question was, given this high dimensional flowing water, how can we combine it in a way with all other flows in the area such that it flows in this particular pattern we have in mind in mathematical visualisation ``` ∿ CONTINUOUS FLOW LANDSCAPE ∿ (high-dimensional water system) ↓ ~~~~~~ ~~~~~ ~~~~ ~~~~~~ ~~~~~~~~ ~~~~~~~ ~~~~~~~ ~~~~~~~~~ / / / / / ARR₁ → / ARR₂ → / ARR₃ → / ARR₄ → / / / / ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ↑ flow speed ↑ composition ↑ “color” ↑ “smell” (latent dims) ↓ combination operator Φ(flow₁, flow₂, …) ↓ TARGET PATTERN / FIELD ψ(x) (stable, directed flow pattern) ``` the part of mathematics that deals with these kinds of information systems is called _Differential Equations_. Equations that constitute _differentials_ as their core pillars. Nature is full of these systems: - The rate at which an apple falls towards the ground. - `m * d²x/dt² = -m*g - c*dx/dt` - The rate at which temperature flows from a cup of tea, not only to the surrounding environment, but also in time - `∂T/∂t = α*∇²T` - How electric and magnetic fields continiously generate each otehr - `∇²E = (1/c²) * ∂²E/∂t²` - How real fluids move in space - `ρ * (∂u/∂t + (u·∇)u) = -∇p + μ∇²u + f` These mathematical systems are ubiquitous in natural systems. Suggesting that they play an integral role in the [[The Cathedral of Mathematical Structure -- DRAFT]], and [[The Computational Universe|Computational Sciences]] so the problem that started all this thing is a form of a differential equation. Formulated that way, it asks: Given a dimension of Revenue and its sub-dimensions, help me identify the first and second derivative of the primary dimension; solve the solution bounded to the initial and boundary conditions I actually only formulated it that way, and solved it a different way. I simply mapped all dimensions on statistical space, and navigated them with statistical principles -- identifying the solution in a more efficient way, [[Logistic Regression]], [[Linear Algebra]], [[Linear Regression]], [[]] -- I was not subject to [[The Computational Universe|Computational Irreducibility]] in any case, I kept wondering about _Differential Equations_ looking more closely as to how they really work I then came across what the book decided to call _Linear Systems_ -- systems where the independent variables are only linearly combined (+ or -, no multiplications or divisions) -- kind of like [[The Cathedral of Mathematical Structure -- DRAFT|the Ring structure]]. Anyways there is this idea that you can sort of estimate any function you have, or any observation, using a linear combination of certain functions, this is called a Fourier Series. You can for example, decompose a certain music tone, as a combination of many other music tones now imagine if these other music tones, are themselves oscillations, themselves being the sum of other tones, the sum of other tones etc. ``` Any complex function f(x) can be approximated as: f(x) = a₀ + a₁sin(x) + a₂sin(2x) + a₃sin(3x) + ... + b₁cos(x) + b₂cos(2x) + b₃cos(3x) + ... Example: Music tone decomposition Complex Sound Wave: ∿∿∿∿∿∿∿∿∿∿∿∿∿ ↓ = ↓ ∿∿∿∿∿∿∿∿∿∿∿∿∿ (fundamental frequency) + ∿∿∿∿∿∿∿∿∿∿ (2x frequency, smaller amplitude) + ∿∿∿∿∿∿ (3x frequency, even smaller) + ∿∿∿∿ (4x frequency) + ... RECURSIVE DECOMPOSITION: Each tone is itself a sum of tones: Each of THOSE tones is a sum of tones: And so on... f(x) ├── tone₁ = Σ(sub-tones₁) │ ├── sub-tone₁₁ = Σ(sub-sub-tones₁₁) │ └── sub-tone₁₂ = Σ(sub-sub-tones₁₂) ├── tone₂ = Σ(sub-tones₂) │ ├── sub-tone₂₁ = Σ(sub-sub-tones₂₁) │ └── ... └── tone₃ = Σ(sub-tones₃) └── ... ∞ Infinite nested oscillations all the way down ∞ ``` _how cool_ I'm now wondering, about this thing called a [[The Riemann Hypothesis|Zeta Function]] which essentially describes how Prime Numbers appear on the number line in a way that is apparently oscillatory, and hence the wonder if this could be described by _a_ Fourier Series ``` ∿ THE MATHEMATICAL FOREST ∿ ↙ ↓ ↓ ↓ ↘ Fractals Primes Diff.Eq Groups Complex \ | | | / \ | | | / \_____|_______|_______|_____/ \ | / Harmonics | Fields \_____|_____/ ∿ Fourier Series f(x) = Σ aₙsin(nx) ∿ ζ(s) ∇²T ∿ same water different mind ∞ ``` _the continuity continues_ ... [[The Natural Exponential]], what a fascinating number ``` PATTERN HUNT ↓ d/dt(2^t) = 0.6931·2^t d/dt(3^t) = 1.0986·3^t d/dt(8^t) = 2.079·8^t ↓ 8 = 2³ → 2.079 = 3×0.6931 ↓ Pattern exists! ↓ KEY QUESTION ↓ d/dt(a^t) = ?·a^t where ? = 1 ↓ Answer: a = e ≈ 2.718 ↓ d/dt(e^t) = e^t ↓ REVELATION ↓ a^t = e^(ln(a)·t) ↓ d/dt(a^t) = ln(a)·a^t ↓ 0.6931 = ln(2) 2.079 = ln(8) = 3·ln(2) ✓ ↓ ∞ ``` ![[Invariants#**CATALOG OF MATHEMATICAL INVARIANTS**#The Natural Exponential `e`]] on the penetration of capitalism into the deeper market space ``` NETWORK TAKEOVER DYNAMICS: Step 1: IDEOLOGICAL VICTORY "Capitalism won" → Single legitimate model Alternative attractors delegitimized Step 2: STRUCTURAL ADJUSTMENT IMF/World Bank impose connectivity ├─ "Reforms" = opening closed nodes ├─ Debt = forced integration └─ "Efficiency" = capitalist logic required Step 3: ASSET PRIVATIZATION WAVE Public goods → Market commodities Power stations: ◎ (public) → ◊ (private equity) Water systems: ◎ → ◊ Telecoms: ◎ → ◊ Railways: ◎ → ◊ Healthcare: ◎ → ◊ Education: ◎ → ◊ The "◊ization" of everything Step 4: FINANCIALIZATION Not just ownership transfer Everything becomes INVESTABLE ASSET ├─ Public pensions → Stock market exposure ├─ Housing → Financial instruments ├─ Education → Debt instruments └─ Future earnings → Tradeable securities ```