Mathematical modeling of system transformations with the help of spatial distributions of material

Y. B. Karasik,
Thoughts Guiding Systems Corp.,
Ottawa, Canada.
e-mail:karasik@sympatico.ca

Introduction

Physical objects can be modelled by spatial distributions of materials they are made of. If it is made of one material m then spatial distribution d(m,x,y,z) of this material completely describes the object. Recall, that d(m,x,y,z) is defined as the density of material m at point (x,y,z). Obviously d(m,x,y,z) = 0 if point (x,y,z) does not belong to the object. Distributions that are the same for any material will be denoted as d(x,y,z) with m omitted, as they do not depend on it.

If an object is made of several materials m1,..mn, then it is represented as a sum of distributions:

n
d(mi, x, y, z)
i=1

The purpose of this article is to show that various basic system transformations (fragmentation/defragmentation, joining/disjoining, replication, etc.) can be modelled with the help of distributions.

Partitioning (or Fragmentation, or Segmentation)

Dividing an object into parts in the terms of distributions looks as follows:

n
d(m, x, y, z) ⇒ di(m, x, y, z)
i=1

provided all parts are made of the same material. If the parts are made of different materials then the formula looks as follows:

n
d(m, x, y, z) ⇒ di(mi, x, y, z)
i=1

Merging (or Defragmentation, or Desegmentation)

This operation is opposite to the above one. Hence it is described by the formula

n
di(m, x, y, z) d(m, x, y, z)
i=1

Replication

Replication of an object at points (x1, y1, z1), ..., (xn, yn, zn) is given by the formula:

n
d(m, x, y, z) ∗ δ(x-xi, y-yi, z-zi)
i=1

where d(m, x,y,z) is the material density distribution of the original object and ∗ is a mathematical operation called convolution, and δ(x,y,z) is the δ-function.

Replication of an object at points (x1, y1, z1), ..., (xn, yn, zn) with a different material mi at every point is given by the formula:

n
d(x, y, z) ∗ δ(mi, x-xi, y-yi, z-zi)
i=1

Consider now how the transformation dealt with in the editorial is modelled by distributions. Specifically, I am talking about replacement of a set of geometrically identical objects made of different materials mi and located at points {(xi, yi, zi)} by a set of identical objects each of which is made of the mixture the above materials, and located at the same points. This transformation is modelled by the following formula:

n n n
d(x, y, z) ∗ δ(mi, x-xi, y-yi, z-zi)  ⇒  d( mi, x,y,z)∗ δ(x-xi, y-yi, z-zi)
i=1 i=1 i=1

Please note that mi migrated from δ-functions on the left to function d(...) on the right. Hence, use of distributions also points to some transposition here, although this pointer is not as clear as in the notation used in the editorial.

Combining

Each object has degrees of freedom. When objects are not combined into a system they on the whole have more degrees of freedom than when they are combined. Actually combining is synonymous to reducing the number of the total degrees of freedom of objects.

The total distribution of two independent objects is

d1(m1, x-r, y-s, z-t) + d2(m2, x-u, y-v, z-w)

It has 6 translational degrees of freedom (r,s,t,u,v,w), each of which are independent variables. The combined objects have only 3 translational degrees of freedom:

d1(m1, x-r, y-s, z-t) + d2(m2, x-(r+a), y-(s+b), z-(t+c)),

where a,b, and c are constants rather than variables. Hence combining can be described by the following formula:

d1(m1, x-r, y-s, z-t) + d2(m2, x-u, y-v, z-w) ⇒ d1(m1, x-r, y-s, z-t) + d2(m2, x-(r+a), y-(s+b), z-(t+c))

If we introduce variables ξ = r + u, ζ = s + v, and θ = t + w, then the above formula can be rewritten as follows:

d1(m1, x-r, y-s, z-t) + d2(m2, x-(r+ξ), y-(s+ζ), z-(t+θ)) ⇒ d1(m1, x-r, y-s, z-t) + d2(m2, x-(r+a), y-(s+b), z-(t+c))

which immediately reveals that combining is a result of inversion where variables (ξ, ζ, θ) are replaced by constants (a, b, c).

Taking out

This operation is opposite to combining. Taking out increases the number of degrees of freedom. Hence it can be modelled by the inversion of the above formula:

d1(m1, x-r, y-s, z-t) + d2(m2, x-(r+a), y-(s+b), z-(t+c)) ⇒ d1(m1, x-r, y-s, z-t) + d2(m2, x-(r+ξ), y-(s+ζ), z-(t+θ))

Conclusion

Modelling system transformations with the help of distributions is feasible for many more transformations than modelling with the help of other notations considered in this issue. However, it is least suitable to highlight inversions and transpositions behind these transformations.