# Encapsulation theory: the minimised, uniformly hidden radial branch.

# Edmund Kirwan^{*}

# www.EdmundKirwan.com

# Abstract

This paper investigates how the violational elements of a single, radial branch uniformly distributed in hidden elements should be distributed so as to minimise the branch's potential coupling.

# Keywords

Encapsulation theory, radial encapsulation, branch, potential coupling.

# 1. Introduction

As was shown in [3], the potential coupling of a radial branch is not minimised when all its hidden and violational elements are uniformly distributed over all its disjoint primary sets and instead if the violational elements are uniformly distributed then there exists a function which dictates how to non-uniformly distribute the hidden elements such that the potential coupling falls below that of a uniformly-distributed branch.

Similarly, there exists a function which dictates how to distribute the violation elements of a branch so as to minimise its potential coupling given a uniform distribution of hidden elements.

This paper investigates this second, violational distribution function.

This paper considers sets of radial information-hiding only.

# 2. Selected branch potential couplings

Consider the branch shown in figure 1, which was also investigated in [3], showing a branch of three disjoint primary sets, where the information-hiding and the information-hiding violation of the sets are shown within their representative symbols.

*Figure
1: A branch of three disjoint primary sets.*

Thus,
taking the root set, *K*_{1}, for example, we see
that *K*_{1} has 10 violational elements,*=10*,
and 5 hidden elements,*=5.*

As
was shown in [3], the branch's potential coupling is:
5*10*+*360+210=1080*.

We
know that, in the absolute encapsulation context, potential coupling
is minimised (ignoring A.M.C.s) when both the violational and hidden
element are distributed uniformly over all sets. Let us test to see
whether this also holds in the radial encapsulation context: let us
move a violational element from *K*_{1} to *K*_{3},
see figure 2.

*Figure
2: A branch of three disjoint primary sets, one element migrated.*

Let us calculate the potential coupling of the branch in figure 2.

The
potential coupling of *K*_{3} is the sum of its
internal and external potential couplings. The internal potential
coupling of *K*_{3} is the total number of
contained elements multiplied by this number minus one, i.e.,
*16x15=240*. The external potential coupling of *K*_{3}
is the total number of contained elements multiplied by the total
number of violational elements it can see "below" it, i.e.,
*16x(10+9)=304*. So the potential coupling of *K*_{3}
is *240+304=544*.

The
internal potential coupling of *K*_{2} is
*15x14=210*. *K*_{2} can only see *9*
violational elements "below" it, so its external potential
coupling is *15x9=135*. So the potential coupling of *K*_{2}
is *210+135=345*.

The
internal potential coupling of *K*_{1} is
*14x13=182 *and
*K*_{1} has not external potential coupling so
its potential coupling is just *182*.

The
branch's total potential coupling has now fallen from *1080* to
*544+345+182=1071*.

Thus by moving a violational element, thereby rendering the branch non-uniformly distributed in violational elements, we have reduced the branch's potential coupling. We note that the branch is still uniformly distributed in hidden elements. This then raises the obvious question: given a branch that is uniformly distributed in hidden elements, what distribution of violational elements will minimise its potential coupling?

It
can be shown that, to attain the minimum potential coupling in this
case, the number of violational elements in the *i*^{th}
disjoint primary set is given by the following equation (see
proposition 10.18):

With this equation we can calculate the number of violational elements in each of the three sets examined above to minimise the potential coupling.

*K*_{3}
should contain the following number of violational elements:

*K*_{2}
should contain the following number of violational elements:

*K*_{1}
should contain the following number of violational elements:

The branch thus configured is shown in figure 3.

*Figure
3: A branch non-uniformly distributed in violational elements with
minimised potential coupling.*

Calculating the potential coupling of this minimised branch we see the following.

The
potential coupling of *K*_{3} is *20x19+20x15=680*.

The
potential coupling of *K*_{2} is *15x14+15x5=285*.

The
potential coupling of *K*_{1} is* 10x9=90.*

The
branch's total potential coupling is *680+285+90=1055*,
down from the original, uniformly distributed figure of *1080*.

# 4. Conclusion

When a branch is uniformly distributed in hidden elements, its minimum potential coupling is not achieved by also uniformly distributing its violational elements. Instead, an equation describes the violational element distribution to achieve this minimum potential coupling and this equation suggests placing more violational elements towards the higher part of the branch than the lower.

# 5. Appendix A

## 5.2 Definitions

[D10.1]
Given a branch *B* ofdisjoint
primary sets, the number of elements in *B*, denoted,
is the sum of the number of hidden and violational elements in each
disjoint primary set constituting the branch, or:

[D10.2] Given a branch *B*
ofdisjoint
primary sets, the number of violational elements in *B*,
denoted,
is the sum of the number of violational elements in each disjoint
primary set constituting the branch, or:

## 5.2 Propositions

### Proposition 10.1

Given
a branch *B* ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the number of elements in the *i*^{th} primary
set,,
is given by:

*Proof:*

By definitions [D1.2], [D1.6]
and [D1.7] of [1], the elements in *K*_{i} are
those in the intersection of *K*_{i} with *H*
and *V*, or:

(i)

By definition [D1.1] in [1], *H*
and *V* are disjoint so taking the cardinality of (i) gives:

(ii)

We presume that each disjoint primary set has an information-hiding of, that is:

(iii)

Substituting (iii) into (ii) gives:

*QED*

### Proposition 10.2

Given
a branch *B* ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the internal potential coupling of the *i*^{th}
primary set,,
is given by:

*Proof:*

By definition [D1.2] in [1],
given a primary set *Q*_{i} the internal
potential couplingis
given by:

(i)

By proposition 10.1, the number
of elements in the *i*^{th} primary set,,
is given by:

(ii)

Substituting (ii) into (i) gives:

=

*QED*

### Proposition 10.3

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the following holds:

*Proof:*

From
definition [D10.1], the number of elements in *B*,
denoted,
is the sum of hidden and violational elements in each disjoint
primary set constituting the branch, or:

(i)

By proposition 10.1, the number
of elements in the *j*^{th} primary set,,
is given by:

(ii)

Substituting (ii) into (i) gives:

=

= (iii)

Taking an arbitrary *i*
such that,
then it follows that:

(iv)

Substituting (iv) into (iii) gives:

*QED*

### Proposition 10.4

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the external potential coupling of the *i*^{th}
primary set,,
is given by:

*Proof:*

Consider the root primary set
*Q*_{1}.

As *Q*_{1}
is the root primary set, however, all other disjoint primary sets in
this branch are subsets of *Q*_{1}, thus the
external potential coupling of the root disjoint primary set is 0.

The external potential coupling
of the *Q*_{2} is then the number of ordered
pairs that may be formed towards *K*_{1}.

The external potential coupling
of the *Q*_{3} is the number of ordered pairs
that may be formed towards *K*_{1} and *K*_{2},
etc.

Thus:

= (i)

By proposition 10.3, the following holds:

(ii)

Substituting (ii) into (i) gives:

(iii)

By proposition 10.1, the number
of elements in the *i*^{th} primary set,,
is given by:

(iv)

Substituting (iv) into (iii) gives;

=

*QED*

### Proposition 10.5

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the external potential coupling of the *i*^{th}
primary set,,
is given by:

*Proof:*

By proposition 10.4, the
external potential coupling of the *i*^{th}
primary set,,
is given by:

(i)

By definition:

(ii)

Substituting (ii) into (i) gives:

*QED*

### Proposition 10.6

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the potential coupling of the *i*^{th} primary
set,,
is given by:

*Proof:*

By proposition 10.2, the
internal potential coupling of the *i*^{th}
primary set,,
is given by:

(i)

By proposition 10.5, the
external potential coupling of the *i*^{th}
primary set,,
is given by:

(ii)

By proposition 8.7 in [3], the
potential coupling of *Q*_{i} is the sum of the
external potential coupling sets of *Q*_{i} and
the internal potential coupling of *Q*_{i}, or:

(iii)

Substituting (i) and (ii) into (iii) gives:

=

*QED*

### Proposition 10.7

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the number of violational elements in the first disjoint primary
set,,
is given by:

*Proof:*

From
definition [D10.1], the number of elements in *B*,
denoted,
is the sum of hidden and violational elements in each disjoint
primary set constituting the branch, or:

(i)

By proposition 10.1, the number
of elements in the *j*^{th} primary set,,
is given by:

(ii)

Substituting (ii) into (i) gives:

=

= (iii)

By definition:

(iv)

Substituting (iv) into (iii) gives:

*QED*

### Proposition 10.8

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the square of the number of violational elements in the first
disjoint primary set,,
is given by:

*Proof:*

By proposition 10.7, the number of violational elements in the first disjoint primary set,, is given by:

(i)

Squaring (i) gives:

=

*QED*

### Proposition 10.9

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the potential coupling of the first primary set,,
is given by:

*Proof:*

By definition [D1.2] in [1],
given primary set *Q*_{1} the internal potential
couplingis
given by:

(i)

By proposition 10.1, the number of elements in the first primary set,, is given by:

(ii)

Substituting (ii) into (i) gives:

= (iii)

By proposition 10.7, the number of violational elements in the first disjoint primary set,, is given by:

(iv)

By proposition 10.8, the square of the number of violational elements in the first disjoint primary set,, is given by:

(v)

Substituting (iv) and (v) into (iii) gives:

(vi)

By definition, the first disjoint primary set has no external potential coupling, thereforeand therefore (vi) is also the equation for the potential coupling of the first disjoint primary set.

*QED*

### Proposition 10.10

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the potential coupling of *B *is minimised when:

*Proof:*

To find the number of
violational elements that minimises the potential coupling of branch
*B* we must differentiate the potential coupling of *B*
with respect to the total number of violational elements and set
equal to zero, that is:

(i)

By definition [D10.2], the
number of violational elements in *B*, denoted,
is the sum of the number of violational elements in each disjoint
primary set constituting the branch, or:

(ii)

Substituting (ii) into (i) gives:

(iii)

By definition I SHOULD PROVE THIS ????????????????????????xxx, the potential coupling of branch B is equal to the sum of the potential coupling of all its primary sets, or:

(iv)

Substituting (iv) into (iii) gives:

(v)

=

*QED*

### Proposition 10.11

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the sum of the potential coupling of the first primary set *Q*_{1}
with respect to the number of violational elements in all disjoint
primary sets of *B* is given by:

*Proof:*

By proposition 10.9, the potential coupling of the first primary set,, is given by:

(i)

As there are no *v(K*_{1}*)*
terms (i) even after expansion then differentiating this with respect
togives:

(ii)

For *i > 1*,
differentiating this with respect togives:

(iii)

By proposition 10.7, the number of violational elements in the first disjoint primary set,, is given by:

(iv)

Substituting (iv) into (iii) gives:

(v)

Taking the sum of the potential
coupling of *Q*_{1} with respect to the number of
violational elements in all disjoint primary sets of *B *gives:

(vi)

Substituting (ii) and (v) into (vi) gives:

=

=

=

*QED*

### Proposition 10.12

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
differentiating the potential coupling of the *m*^{th}
primary set,,
where *m > 1*, with respect the number of violational
elements in any randomly chosen *i*^{th} disjoint
primary gives:

*Proof:*

By proposition 10.6, the
potential coupling of the *m*^{th} primary set,,
is given by:

(i)

We now wish to differentiate (i)
with *m > 1* with respect to the number of violational
elements in any randomly chosen *i*^{th} disjoint
primary set, but there is a problem: the solution is not unique, and
instead depends on how *i* relates to *m*. Thus we must
consider three separate solutions.

Firstly, differentiating (i)
with respect to the number of violational elements in the *i*^{th}
disjoint primary set such that *i < m* gives:

(i)

Differentiating (i) with respect
to the number of violational elements in the *i*^{th}
disjoint primary set such that *i = m* gives:

(ii)

By proposition 10.3, the following holds:

(iii)

And from (iii) it follows that:

(iv)

Substituting (iv) into (ii) gives:

(v)

Finally, differentiating (i)
with respect to the number of violational elements in the *i*^{th}
disjoint primary set such that *i > m* gives:

(vi)

*QED*

### Proposition 10.13

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the sum of the potential coupling of all primary sets
where *m > 1*, with respect to the number of violational
elements in the *i*^{th} disjoint primary set
such that *i = m* gives:

*Proof:*

By equation (v) of proposition
10.12, differentiating the potential coupling of the *m*^{th}
primary set,,
where *m > 1*, with respect to the number of violational
elements in the *i*^{th} disjoint primary set
such that *i = m* gives:

(i)

As we are only interested in the
case of , then taking the sum over all *K*_{i}_{
}will just result in those terms where *i = m*; in other
words, there is only one *i*^{th} term per *m*
in which we are interested, or:

(ii)

Taking the sum fromof (ii), however, gives:

=

=

=

= (iii)

But:

And:

(iv)

Substituting (iv) into (iii) gives:

(v)

By equation (iii) of proposition 10.7:

(vi)

Substituting (vi) into (v) gives:

=

*QED*

### Proposition 10.14

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the sum of the potential coupling of all primary sets
where *m > 1*, with respect to the number of violational
elements in the *i*^{th} disjoint primary set
such that *i > m* gives:

*Proof:*

By equation (vi) of proposition
10.12, differentiating the potential coupling of the *m*^{th}
primary set,,
where *m > 1*, with respect to the number of violational
elements in the *i*^{th} disjoint primary set
such that *i > m* gives:

(i)

We are interested in evaluating
(i) over all *i, m > 1*. We note, however, that *i > m*
only for values of *m* that are less than *r*_{B},
so the upper value of *m* is *r*_{B}* - 1*.
That is, we need to calculate:

Let us first take the sum of (i)
over all *i > m,* giving:

(ii)

As the neither of the terms on
the right-hand side of (i) is a function of *i* and there are
*(r*_{B}* - m)* terms in which *i > m*,
then:

= (iii)

Taking the sum fromof (iii) gives:

=

= (iv)

By equation (iii) of proposition 10.7:

(v)

Substituting (v) into (iv) gives:

=

=

*QED*

### Proposition 10.15

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the sum of the potential coupling of all primary sets
where *m > 1*, with respect to the number of violational
elements in the *i*^{th} disjoint primary set
where *i > 1* gives:

*Proof:*

By definition:

(i)

By proposition 10.12:

(ii)

By proposition 10.13, the sum of
the potential coupling of all primary sets
where *m > 1*, with respect to the number of violational
elements in the *i*^{th} disjoint primary set
such that *i = m* gives:

(iii)

By proposition 10.14, the sum of
the potential coupling of all primary sets
where *m > 1*, with respect to the number of violational
elements in the *i*^{th} disjoint primary set
such that *i > m* gives:

(iv)

Substituting (ii), (iii) and (iv) into (i) gives:

= (v)

By definition:

=

(vi)

Substituting (vi) into (v) gives:

= (vii)

By equation (iii) of proposition 10.7:

(viii)

Substituting (viii) into (vii) gives:

=

*QED*

### Proposition 10.16

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the number of violational elements in the first disjoint primary set
that minimises the branch's potential coupling is given by:

*Proof:*

By proposition 10.10, the
potential coupling of *B *is minimised when:

(i)

By proposition 10.15, the sum of
the potential coupling of all primary sets
where *m > 1*, with respect to the number of violational
elements in the *i*^{th} disjoint primary set
where *i > 1* gives:

(ii)

By proposition 10.11, the sum of
the potential coupling of the first primary set *Q*_{1}
with respect to the number of violational elements in all disjoint
primary sets of *B* is given by:

(iii)

Substituting (ii) and (iii) into (i) gives:

= (iv)

By definition:

= (v)

Substituting (v) into (iv) gives:

=

=

=

=

=

*QED*

### Proposition 10.17

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the number of violational elements in the *i*^{th}
disjoint primary set that minimises the branch's potential coupling
is given by:

*Proof:*

By definition I SHOULD PROVE THIS ????????????????????????xxx, the potential coupling of branch B is equal to the sum of the potential coupling of all its primary sets, or:

(i)

Choosing an arbitrary *i*
such thatwe
can re-write (i) as:

(ii)

Taking the derivative of both sides of (ii) with respect togives:

(iii)

Setting (iii) equal to *0*
will yield the minimum potential coupling of branch B with respect to
the number of violational elements in the *K*_{i}.

By proposition 10.12,
differentiating the potential coupling of the *m*^{th}
primary set,,
where *m > 1*, with respect the number of violational
elements in any randomly chosen *i*^{th} disjoint
primary gives:

(iv)

By equation (v) in proposition 10.11,

(v)

Substituting (iv) and (v) into (iii) gives:

=

=

=

=

*QED*

### Proposition 10.18

Given
a branch *B *ofelements
and ofdisjoint
primary sets, with each disjoint primary set having an
information-hiding of,
the number of violational elements in the *i*^{th}
disjoint primary set that minimises the branch's potential coupling
is given by:

*Proof:*

By proposition 10.16, the number of violational elements in the first disjoint primary set that minimises the branch's potential coupling is given by:

(i)

By proposition 10.17, the number
of violational elements in the *i*^{th} disjoint
primary set that minimises the branch's potential coupling is given
by:

(ii)

Substituting (i) into into (ii) gives:

=

=

=

*QED*

# 6. References

[1] - "Encapsulation theory fundamentals," Ed Kirwan, www.EdmundKirwan.com/pub/paper1.pdf

[2] - "Encapsulation theory: the minimised, uniformly hidden radial branch," Ed Kirwan, www.EdmundKirwan.com/pub/paper10.pdf

[3] “Encapsulation theory: the minimised, uniformly violational radial branch," Ed Kirwan, www.EdmundKirwan.com/pub/paper9.pdf

*©
Edmund Kirwan 2010. Revision
1.0, March 28^{th}
2010. (Revision 1.0: March 28^{th}
2010.) All entities may republish, but not for profit, all or
part of this material provided reference is made to the author and
title of this paper. The latest revision of this paper is available
at [2].