Dr David McGrath

Spine Physician

MB BS (Hons) FAFOM, RACP, FAFMM
Master of Pain Medicine

GRAPHICAL CONCEPTUALISATION

1.Disturbances from the vertex set V(X,Y,Z) can spread to any other vertex

2.Linear disturbance trails, developed by the operation of concatenation, are considered.

3.Trails can be open/closed and originate from any vertex, in any order.

4.The usual terminology is V1(e1)V2(e2)V3(e3)V4(e5)V5 eg X(e1)Y(e2)Z(e3)X(e4)Y(e5)Z

5.There is no loss of meaning by using the terminology (XYZXYZ)

6.The upper case is replaced by lower case (xyzxyz) to create an n-tuple disturbance trail.

7.Any trail (>2) can be represented by a series of local 3-tuple paths, where the later two elements are the first two elements of the subsequent 3-tuple.

eg xyzxyz=xyz+yzx+zxy+xyz The 3-tuples are defined as neighbourhoods.

8.A binary operation on any two trails is defined as follows: One trail (head) loses a later part of its trail and concatenates the later part of another trail (tail). This operation is allowed by an auto-convergent bonding at a vertex. The operation itself is called auto-convergence and indicated by the symbol@ followed by the converging vertex.

eg (x1y1z1x1 @Y x2z2y2x2) =x1y1y2x2. ( xyzx=head, xzyx=tail )

9.Auto-convergence can occur at any vertex capable of bonding a precedent disturbance (y1) with an adjacent later (y2). The upstream trail to (y1) is concatenated to the downstream trail of (y2) in future occurrences of the head trail. The head and tail trails must be sufficiently separated in time and location to not destructively interfere prior to the vertex bonding.

10. Auto-convergence can be considered a local operation between neighbourhoods at a vertex. In the example above, we could construct the bonding as

(x1y1z1 @Y z2y2x2) =x1y1y2x2. Without the ordinal labels, write (xyz @Y zyx) =xyyx

11. Autodisturbance segments of a trail such as (y1y2...yn) are only considered as multiple Y disturbances if they arise from autoconvergence. If consecutive y1 and y2 arise from the a single origin, they are coallesced into a single y denotation.

12.Multiple autodisturbances (y1y2...yn) can be written to express the convergent origin. All yn after the first are replaced by the convergent origin symbol and placed in parenthesis. eg y*(xzxxzz)

ALTERNATE CONCEPTUALISATION

Consider a three element set M(X,Y,Z) Each element X,Y,Z contains a family of disjoint disturbances, where X=(x1,x2....) Y=(y1,y2...) Z=(z1,z2....)

This is derived as follows:

1. Consider a disturbance medium M(m0,m1,m2...)

2. Consider a subset Y(y0,y1...) of M with the following properties.

a. There is a surjective mapping from the set to itself. Only (y0) representing the undisturbed state maps to itself.

b. This can be seen as a binary operation on the set defined as (yiyj)=yj Simple disturbances are defined as (yiyo) Consecutive operations (from L to R) such as (yiyj)(yjyk)(..)(..)(.y0) are Compound Y disturbances rewritten as (yiyj......yo) This operation is the closed homeostasis operator. YxY=Y

3. Consider a subset V(v1,v2...) of (M-Y), defined as all elements of (M-Y) which map to Y (surjective and complete)

4. Consider a subset W(w1,w2...) of (M-Y) defined as all elements of (M-Y) which are mapped from Y. (surjective and partial)

We now have the possibility of trails on Y,V,W length>2. (viyi)(yiwi) abbreviated as (viyiwi).

5. If some elements of W map to V closure back to V can occur. (viyiwjvj). Define (V+W) defined as a total environment.

6. Consider subsets Zi(z1,z2,..) of V and Zo(z1,z2...) of W, allowing ordered triples (vzy) and (yzw). This is equivalent to filtering the input V and output W through a special class of disturbance. (social). Note some elements of Zo may map to Zi. (as some W maps to V)

Define Zi+Zo=Z and (V-Z1)+(W-Zo)=X

7.We can now recognise four Y neighbourhood disturbance configurations relating Y,M-Y elements.

1.xyx 2. xyzx 3. xzyx 4. xzyzx and //////1. zyz 2. zyxz 3. zxyz 4. zxyxz 5.zyxyz and 1. yxy 2. yxzy 3. yzxy 4. yzxzy 5. yxzxy

This is equivalent to bidirectional interactions on a triangle with vertices X,Y,Z. The vertices X,Z represents the non-social and social environment respectively. A quadrilateral form maintains Z1,Z2 as separate vertices. (not integrated through a single person)

We now develop local Y neighbourhood structure.

1. Define the ordered triple set Ty(t) containing elements t=(ayb) where (a,b) come from (X,Z) and (y) belongs to (Y)

2.Define a relation on elements t=(ayb) as follows:

(a1y1b1) relates to (a2y2b2) if and only if (a1,a2) belong to the same set (X or Z) and (b1,b2) belong to same set (X or Z). This is an equivalence relation.

The set Ty(t) is partitioned into four subsets. T(xyx,xyz,zyx,zyz)

3. Elements of (Ty) may irreversibly create new elements by a binary operation defined as disjoint autoconvergence. (@)

The transformation can be expressed as:

(a1.y1.b1) @ (a2.y2.b2)= (a1.y1.y2.b2) + (a2.y2.b2)

eg xyz @ zyx = xyyx + zyx (replacing a,b and deleting the labels)

4.The set Y(y1,y2....) now contain compound disturbances (y1y2)

5. This transformation can be continued recursively creating Y disturbances of increasing length.

eg xyyx @ zyyz = xyyyyz + zyyz

Disjoint Autoconvergence Diagram with Front Recursion
PRE	a1	y1	b1		a1	y1	y2	b2		a1	y1	y2	y3	b3		a1	y1	y2	y3	y4	b4
PRE		a2	y2	b2			a3	y3	b3				a4	y4	b4					a5	y5	b5

POST	a1	y1	y2	b2	a1	y1	y2	y3	b3	a1	y1	y2	y3	y4	b4	a1	y1	y2	y3	y4	y5	b5
POST		a2	y2	b2			a3	y3	b3				a4	y4	b4					a5	y5	b5

Disjoint Autoconvergence Diagram with Back Recursion
PRE	a2	y2	b2		a3	y3	b3			a4	y4	b4				a5	y5	b5
PRE		a1	y1	b1		a2	y2	y1	b1		a3	y3	y2	y1	b1		a4	y4	y3	y2	y1	b1

POST	a2	y2	y1	b1	a3	y3	y2	y1	b1	a4	y4	y3	y2	y1	b1	a5	y5	y4	y3	y2	y1	b1
POST		a1	y1	b1		a2	y2	y1	b1		a3	y3	y2	y1	b1		a4	y4	y3	y2	y1	b1

6. A variation on disjoint autoconvergence is (y) overlap autoconvergence.

This variation allows the possibility of (Y) compound trail length reduction.

(y)Overlap Autoconvergence
PRE	a5	y1	y2	y3	y4	y5	b5
PRE		a6	y6	b6

POST	a5	y1	y6	b6
POST		a6	y6	b6

7. Auto convergence is only possible in the absence of destructive interference prior to autodisturbance bonding.

This imposes a limit on overlap possibilities.

Auto-Convergence Limit
PRE		a5	y1	y2	y3	y4	y5
PRE	a5	y6	y7	y8	b9

POST		a5	y7	y8	b9
POST	a5	y6	y7	y8	b9

8. Disjoint autoconvergence is a closed associative operation. The operation is irreversible and no idenity exists. This defines a semigroup structure on T.

Associative Property with Disjoint Autoconvergence (also true for overlap)
1	a1	y1	b1				a1	y1	b1
2		a2	y2	b2				a2	y2	b2
3			a3	y3	b3				a3	y3	b3

1@2	a1	y1	y2	b2		1	a1	y1	b1
@3			a3	y3	b3	2@3		a2	y2	y3	b3

1@2@3	a1	y1	y2	y3	b3	1@2@3	a1	y1	y2	y3	b3

(This is somewhat academic, as the operations are deemed irreverisible)

9. There are 16 possible transformations on the Transformation Set.

Transformation Caley Table
t1 @ t2	xyx=1	xyz=2	zyx=3	zyz=4
xyx=1	1	2	1	2
xyz=2	1	2	1	2
zyx=3	3	4	3	4
zyz=4	3	4	3	4

(Continued on Further Pages)