This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.


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1 Topology

Boundary operator: boundary_operator_binary.xlsx


Link to the file: boundary_dim_2.xlsx

2 Vector fields


For motion simulation, do one step at a time:

Excel options for motion.png

Flow simulation

Dimension 1

The settings for formulas are:

  • Manual,
  • 1 Iteration.
  • Push: Calculate now
Vector field dim 1.png

Link to the file: vector field_dim_1.xlsx

Dimension 2

Same settings.

Vector field dim 2.png

Link to the file: vector field_dim_2.xlsx Link to the file: vector field_dim_2.xlsx

3 Advection

4 Differential forms

4.1 Settings

For some simulations iterations are required:

Excel options.png

4.2 Dimension 1

Boundary operator:

Boundary dim 1.png

Link to the file: boundary_dim_1.xlsx

Exterior derivative:

Exterior derivative in dimension 1 Excel.png

Link to the file: Exterior_derivative_dim_1.xlsx

Hodge duality:

Hodge duality dim 1.png

Link to the file: Hodge_duality_dim_1.xlsx

Hodge duality with geometry:

Hodge duality dim 1 w geometry.png

Link to the file: Hodge_duality_dim_1_w_geometry.xlsx


Laplacian dim 1.png

Link to the file: Laplacian_dim_1.xlsx

Laplacian with geometry:

Laplacian dim 1 w geometry.png

Link to the file: Laplacian_dim_1_w_geometry.xlsx

4.3 Dimension 2

Warning: you will need to use a buffer sheet because Excel's row-by-row manner of evaluation will cause a skewed pattern:


5 Diffusion

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:

Diffusion dim 1.png

The result after $1,500$ iterations is shown next:

Diffusion dim 1 iterated.png


Dimension $2$.

A simulation of heat transfer from single point is shown below:

Heat transfer from single point.png


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: $$RC= RC - k*\bigg( .5*\left(RC-RC[-1]\right) + .5*\left(RC-RC[1]\right) + 2*\left(RC-R[-1]C\right) + 2*\left(RC-R[1]C\right) \bigg).$$ 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):

Diffusion w rectangles.png

6 Wave propagation

Dimension $1$:

The Excel formula is:

  • =R1C5*R[-1]C[-1]+2*(1-R1C5)*R[-1]C+R1C5*R[-1]C[1]-R[-2]C

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:

Wave equation Excel simplest.png

Dimension $2$:

Wave from single point.png

7 Social choice

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.