A's Value: Influence upon Number String Graphs
The Graphs of the Number String Data Stream
When the Living Algorithm digests the Number String Data Stream, it generates the derivatives of the data stream. We can plot a graph of the Data over time – with each trial N. As the Number String Data Stream consists of a single value, A, the graph would appear as a horizontal line, starting from A on the y-axis. In similar fashion, we can plot the derivatives of the data stream over time. Remember each instant, each data point, is associated with a moment consisting of a cloud of derivatives. Each of the derivatives has its own unique graph. As a group, we will refer to them as the Number String Graphs.
1st & 2nd derivatives = Data * Constant; Assumption the higher derivatives follow suit
In the prior article, we derived general expressions for this data stream's 1st derivative, XN bar, and 2nd derivative, XN arrow. (Note K, the Scaling Factor, is a simple function of D.)
As exhibited, it turns out that they are both functions, albeit different, of 3 factors:
- 1) D, the Decay Factor
- 2) N, their position in data stream &
- 3) A, the value of the Number String.
Both D and N are constants in the sense that they are fixed at each point in the data stream. A can be anything as long as it is consistent for a long enough duration. Both proofs illustrated that both derivatives equal A multiplied by a constant – shown below. (The values of these constants are provided by equations H3 and H6, shown above.)
We suspect and the evidence suggests that the higher derivatives also equal the product of A and their own constant. Hence, although we have only proved the 1st and 2nd derivatives, we will assume that the following analysis applies to all the other derivatives as well.
A, as a positive number: Graphs' proportions shrink, magnified or same.
We've examined what happens to the Number String graphs if A is a positive number. No matter what value A is, the proportions of the graphs remain the same. When A is greater than 1, it magnifies the shape of the graphs. When A is between 1 and 0, it shrinks the shape of the graphs. When A equals 1, the graphs remains the same.
A = 0: Number strings graphs become horizontal straight lines at 0.
When A equals 0, (A=0) all the data stream derivatives also equal 0. All we have are horizontal lines – no pulses.
A is negative: Graphs are inverted; proportions shrink, magnified or same
When A is negative, the Number String graphs flip flop around the horizonta x-axis – becomes its inverse. For instance, instead of the positive Active Pulse, we get a negative Rest Pulse instead. If A equals '-1' (A = -1), the Rest Pulse is a negative mirror of the positive Active Pulse. If A is less than -1 (A< -1), then it magnifies the Rest Pulse. If A is between -1 and 0 (0>A> -1), then it shrinks the Rest Pulse. In either case, the proportions remain the same. Only the size differs. This analysis also applies to the Living Average. Negative numbers flip the positive proportions, but don't have any effect on the relative sizes.
Translations
The above analysis was based upon a system that has been zeroed out – a system that has been tared of extraneous noise. In other words, the following assumptions are at the foundation of the preceding proofs.
This assumption is adequate for a data stream consisting of a single Number String that generates a single Pulse. But it does not address the potentials of a data stream consisting of multiple Number Strings that generate multiple pulses. For instance, a single Number String addresses the potentials of the Pulse of Attention, but does not address the by now famous Triple Pulse. On a specific level, the equally famous Rest Pulse of 0s is left out of equation. As mentioned, when the Number String equals zero, flat lines are the result. To prevent confusion, negative numbers, not 0s, generated a Rest Pulse in the prior analysis. In other words, if the System starts at 0, then a Number String of 0s generates flat lines, not a Rest Pulse. However, Number Strings of negative numbers do generate the Rest Pulse.
What is the reason for this curiosity?
Simply speaking, a Number String of 0s only generates a Rest Pulse, if it follows a positive Number String. Let's see why.
At the end of a Number String (the end of the Pulse), the Living Average's constant, CN bar, approaches 1. Hence the Living Average, XN bar, approaches A, the Number String's value. Simultaneously, the Directional's constant, CN arrow, approaches 0. As such, the Directional, XN arrow, also approaches 0. (For validation see the mathematical proofs in the prior article.) This is true of all the higher derivatives. They all approach 0 at the end of the Pulse. To test what happens to a subsequent Number String after the end of a Pulse, let's make an alternate, more general, assumption. Let's assume that the initial Living Average equals A, instead of 0. We'll allow the other assumption to remain, i.e. that all the derivatives equal 0.