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Mathematical models

See modelling.

[toc]

Spectra & Dimensions

Values can be:

  • Singular. E.g. nothing.

  • Discrete: Binary or categorical. E.g. left or right.

  • Continuous. E.g. real numbers.

models-categories

Type Question Action
Unity What happens in a different perspective? Change the perspective.
Binary What if neither option is applicable? Refine boundary conditions.
Categorical What sits between categories? Find intermediate steps.
Continuous What are the limits? What affects the function? Reflect.

Questionioning dimensions

  • E.g. focus on X or de-focus.
  • E.g. in organizations: focussing on specialization (efficiency) vs focussing on features (quality) vs focussing on simplicity.

From binary to categorical

binary-category

Independent or dependent dimensions

Tradeoffs of independent dimensions

dimensions

Correlation

Objects can be grouped into clusters, which behave as objects. See statistics.

particles-experssion-assemblage

Junctions

Options

  1. Continue as planned. Optimize.
  2. Deviate or expand. Diversify.
  3. Do something radically different

junction-change-deviate-rest

Signals

plot-waves-disturbance-chaotic

Plots in two dimensions

This generalizes to multiple dimensions.

plot-waves-disturbance-chaotic

Peaks may form clusters, which in turn may form form greater peaks.

plot-waves-disturbance-clusters-2d

Core-Context

The core is most important quality of a subject. Second, there are complementing qualities that make it more diverse / flexible. Third, the context maps the subject to other subjects.

core-complement-context

Pyramid

Each layer is contingent upon the previous layers.

communication-pyramid

pyramid-capability-production-product

Growth

Growth of populations. See statistics.

  • Linear: constant increase in size.
  • Exponential: relative increase in size.
  • Hyperbolic: nonlinear increase in size.
  • Logistic: diminishing returns.

plot-ODE-growth

Compounding

Exponential growth can result in strong compounding.

  • This shows how powerful continuous improvement can be.

plot-compounding

Entropy

plot-entropy-complexity

Series

Statistics A random variable (r.v.) X can be approximated in several levels of detail, which are called moments.

  1. Mean or expected value. E[X]
  2. Variance. Var[X]. See also covariance.
  3. Skewness or asymmetry.
  4. Kurtosis or tailed-ness.

Taylor Series Taylor series. Knowing all higher order derivatives at a certain point f(x) allows you to infer the whole function f(x+a).

In physics, the following terms are used:

  1. Position. The current state of the system
  2. Velocity. The change of the system over time (or space).
  3. Acceleration. How fast the system is changing.

Fourier Series Fourier series.