Thoughts Guiding Systems Corp.,

Ottawa, Canada.

e-mail:karasik@sympatico.ca

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 *m _{1},..m_{n}*,
then it is represented as a sum of distributions:

n | |

∑ | d(m_{i}, 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.

n | ||

d(m, x, y, z) ⇒ | ∑ | d_{i}(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) ⇒ | ∑ | d_{i}(m_{i}, x, y, z) |

i=1 |

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

n | |||

∑ | d_{i}(m, x, y, z) |
⇒ | d(m, x, y, z) |

i=1 |

Replication of an object at points
*(x _{1}, y_{1}, z_{1}), ..., (x_{n}, y_{n}, z_{n})*
is given by the formula:

n | ||

d(m, x, y, z) ∗ | ∑ | δ(x-x_{i}, y-y_{i}, z-z_{i}) |

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
*(x _{1}, y_{1}, z_{1}), ..., (x_{n}, y_{n}, z_{n})* with a different material

n | ||

d(x, y, z) ∗ | ∑ | δ(m_{i}, x-x_{i}, y-y_{i}, z-z_{i}) |

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 *m _{i}* and located at points

n | n | n | ||||||

d(x, y, z) ∗ | ∑ | δ(m_{i}, x-x_{i}, y-y_{i}, z-z_{i}) |
⇒ | d( | ∑ | m_{i}, x,y,z)∗ |
∑ | δ(x-x_{i}, y-y_{i}, z-z_{i}) |

i=1 | i=1 | i=1 |

Please note that *m _{i}* migrated from δ-functions on the left to function

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

*
d _{1}(m_{1}, x-r, y-s, z-t) + d_{2}(m_{2}, 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:

*
d _{1}(m_{1}, x-r, y-s, z-t) + d_{2}(m_{2}, 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:

*
d _{1}(m_{1}, x-r, y-s, z-t) + d_{2}(m_{2}, x-u, y-v, z-w) ⇒
d_{1}(m_{1}, x-r, y-s, z-t) + d_{2}(m_{2}, 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:

*
d _{1}(m_{1}, x-r, y-s, z-t) + d_{2}(m_{2}, x-(r+ξ), y-(s+ζ), z-(t+θ)) ⇒
d_{1}(m_{1}, x-r, y-s, z-t) + d_{2}(m_{2}, x-(r+a), y-(s+b), z-(t+c))
*

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:

*
d _{1}(m_{1}, x-r, y-s, z-t) + d_{2}(m_{2}, x-(r+a), y-(s+b), z-(t+c)) ⇒
d_{1}(m_{1}, x-r, y-s, z-t) + d_{2}(m_{2}, x-(r+ξ), y-(s+ζ), z-(t+θ))
*

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.