Boundary operator: boundary_operator_binary.xlsx
Link to the file: boundary_dim_2.xlsx
For motion simulation, do one step at a time:
The settings for formulas are:
- 1 Iteration.
- Push: Calculate now
Link to the file: vector field_dim_1.xlsx
For some simulations iterations are required:
Link to the file: boundary_dim_1.xlsx
Link to the file: Exterior_derivative_dim_1.xlsx
Link to the file: Hodge_duality_dim_1.xlsx
Hodge duality with geometry:
Link to the file: Hodge_duality_dim_1_w_geometry.xlsx
Link to the file: Laplacian_dim_1.xlsx
Laplacian with geometry:
Link to the file: Laplacian_dim_1_w_geometry.xlsx
Warning: you will need to use a buffer sheet because Excel's row-by-row manner of evaluation will cause a skewed pattern:
The recursive formula for the simulation: $$U(a, t+1):= U(a,t) + \Big[- k(a) \left( U(a) - U(a-1)\right) + k(a+1) \left( U(a+1) - U(a)\right)\Big].$$ The Excel setup is shown below:
The result after $1,500$ iterations is shown next:
A simulation of heat transfer from single point is shown below:
What if the horizontal walls are longer than the vertical ones? Then more heat should be exchanged across the vertical walls than the horizontal ones: $$RC= RC - k*\bigg( .5*\left(RC-RC[-1]\right) + .5*\left(RC-RC\right) + 2*\left(RC-R[-1]C\right) + 2*\left(RC-RC\right) \bigg).$$ As a result, the material does travel farther -- as measured by the number of cells -- in the vertical direction (second image) than normal (first image):
The Excel formula is:
Here cell R1C5 contains the value of $\alpha$.
The simplest propagation pattern is given by $\alpha=1$. Below we show the propagation of a single bump, a two-cell bump, and two bumps:
To experiment with PageRank, use Link to the file: Spreadsheets.this spreadsheet: pagerank.xlsx.
To experiment with decycling comparison elections, use this spreadsheet: decycled_elections.xlsx.