## 7. The Seed & Root Equations

### Introduction

In this Notebook we delve into a content-based approach to the calculation of Data Stream Derivatives, discovering many amazing connections in the process. In the first section we look at some diagrams of the content-based approach that has dominated our discussion so far. In the second section, we establish that functions exist, which determine the Derivatives in terms of the Raw Data only. We immediately derive Function F, which determines the Initial Derivative, the Decaying Average, in terms of the Data alone. In the third section, we introduce some new operations to simplify the multiplying complexity, that occurs when looking at the higher derivatives in terms of the raw data alone. We use furling, raveling, and folding to derive a general expression for the higher Directionals in terms of Data alone. In this journey we come across a Root Equation and a Seed Equation. The Root Equation is at the basis of all the Directionals. The Seed Equation when unfolded, unraveled and unfurled reveals the entire equation needed to compute individual Directionals. This is a great success for the content-based approach. Simplification is achieved. In the next notebook, Time, a Fractal Response, we discuss the limitations and advantages of the content-based approach over the context-based approach.

1. History: A Context-based approach to Directionals

A, Directional Diagrams

2. Derivatives as a function of their content

A. Derivation: Each Derivative is a function of the Raw Data
B. Derivation of the simple  F function
C. An Interesting Sidelight: the total sum of XN's contribution to the Decaying Averages approaches XN as N becomes larger
D. An Unsuccessful Approach to the G Function of the First Directional

3. Furling, Unfurling & a Fractal Root

A. A Reductionist Approach: Furling & Unfurling
B. The Distributive Law for the Unfurling Function
C. Back to the First Directional: Introducing Questions

4. Double Unfurling & Unraveling

A. Defining a Double Unfurling
B. Defining an Unraveling
C. Deriving another General Equation for the Directionals

5. Folding, Raveling & the Seed Equation

A. Fractalizing functions, or Fractal Function Sets
B. Raveling our functions
C. The Folding Function and the H function
D. Testing the Seed Equation

Summary & Conclusions