Yi Lin. General Systems Theory: A Mathematical Approach

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304

Chapter 10

there is an embedding mapping due to Theorem 10.5.11(ii). Hence,

condition (ii) also holds. We will check condition (iii). In the present case,

(10.185)

Choose an equivalence class [

]

from each

for i = 1,2, . . . , n. The definition

implies

for some

(10.186)

Suppose Since

for each i,

(10.187)

Hence, Consequently,

(10.188)

Since, in general,

implies

(10.189)

Since e

is injective, is empty or singleton. A similar argument works

for the collection

Hence, we have showed condition (iii).

Sufficiency. Suppose

satisfies conditions (i)–(iii). Let

defined by

(10.190)

where we identify singleton sets with the elements. Let X' = D

(

) and Y' = R

(

)

and mappings

and

be defined by

(10.191)

for each

and

Claim.

The mapping

= (

, k

) is an isomorphism from

S'.

Systems of Single Relations 305

In fact, let

be arbitrary. Theorem 10.5.9 implies that

Since condition (iii) implies that

(10.192)

we have That is,

is a modeling relation from

to S'. To show

k is one-to-one, we let It follows from the definition of k

that

condition (iii) implies that

(10.193)

Hence, k

is injective.

The same argument can be used to show that k

is also

injective.

To show that

is surjective from

onto S', let

be arbitrary. Then the definition of S' implies that there exists such that

(10.194)

Consequently, for any

and

i.e.,

and

Therefore,

(10.195)

This completes the proof that k is an isomorphism from to S'.

We now let S = S

× S

× . . . × S

. Since there is an embedding

from

, an injective modeling relation from

to S is

(10.196)

This ends the proof of Theorem 10.5.12.



Theorem 10.5.13. If a system

has a surjective complex system rep-

resentation S with order n over a class then the

following conditions hold:

(i)

has n input–output coverings

(10.197)

306

Chapter 10

(ii) There is a surjective mapping from S

to an associated input–output system

of C

Proof:

Let

h = (

) :

→

be a surjective modeling relation, and

the projection. For each

let

(10.198)

Then

is a covering of

In fact, let

be arbitrary. Since

is surjective,

for some

(10.199)

That is,

(10.200)

Next let C

be the subset of C' defined by deleting empty sets and duplicated sets

and write

(10.201)

Then

is an input–output system covering. In fact, let

and

(10.202)

and

(10.203)

Then X

and Y

are coverings of the sets

and

respectively. Next let the desired

relation between

and

b e

, and let

be the identity mapping on

. Then (ii)

holds.



307

Systems of Single Relations

Theorem 10.5.14. Suppose that is a system and {S

⊂ X

1, 2, . . . , n} is a family of systems. If the following conditions are true,

has a

surjective complex system representation with order n, S

⊂ X × Y, where

and

(10.204)

and

(10.205)

(i)

has n

input–output coverings

(10.206)

(ii)

There is a surjective modeling relation from S

to an

associated input–output system

of C

(iii) Let

and

be the input and

output spaces of respectively. Then for any

and

(10.207)

(10.208)

Proof: Suppose (i)–(iii) hold and I

is an associated input–output system of

Define a system as follows:

(10.209)

where each singleton set is identified with its element. Let X' = D (

) and

Y'=R

(

S')and

and

be the mappings defined by

(10.210)

308

Chapter 10

and

Condition (iii) and the definition of S' ensure that

and

are well defined.

We now show that h =

(

) is a surjective modeling relation from S to S'. If

(x,y)

∈ S'

then the definitions of

and

imply that (

(

))

∈

and hence

h is a modeling relation. Suppose is arbitrary. For each i,

(10.211)

The definition of an input–output system covering implies that for some

(10.212)

Consequently,

and

that is,

(10.213)

The modeling relation h is surjective. We now define and

: S → S' by

and

(10.214)

Then

σ is a surjective modeling relation from S to



Theorem 10.5.15. A system has an isomorphic complex system rep-

resentation S of order n over a family of systems

if and only if there are partitions P

i = 1

,. . . , n, of such that the following

conditions hold:

(i) P

is an input–output system partition.

(ii)

There is an isomorphism

from an associated input–output system of

to S'

, where

⊂

and

⊂

...

(iii) Condition (iii) in Theorem 10. 5.14 is true.

The proof is similar to those of Theorems 10.5.12 and 10.5.14 and is left to the

reader.

Systems of Single Relations

309

10.6. Some References for Further Study

The concepts of chaos and attractors of single-relation systems were introduced

by Zhu and Wu (1987). They named the concepts panchaos and panattractor,

and systematically studied them under the name pansystems theory. The theory

appeared in the late 1970s. For more information consult Wu’s work (1981b;

1982a; 1982b; 1982c; 1983a; 1983b; 1984b; 1984c; 1984d; 1985).

The concept of feedback systems has been a major research topic in systems

theory and has been studied in many different ways by, for example, Mesarovic

and Takahara (1975), Bunge (1979), Rugh (1981), Deng (1985), Klir (1985), and

Saito and Mesarovic (1985). The definition of feedback systems, given in this

chapter was introduced by Saito and Mesarovic (1985), and some results are from

Lin and Ma (1990).

General systems of a single relation originated with Mesarovic, who began

to study this concept in 1964 (Mesarovic, 1964). Its final form was contained

in the book by Mesarovic and Takahara (1975), in which the concept of general

linear systems was studied in detail. Theorem 10.2.1 is a combination of a result in

(Lin, 1989b) and a fundamental theorem given by Mesarovic and Takahara (1975).

Feedback transformation has been a really hot topic in recent years. Because all

living and many nonliving systems are feedback systems, the method of studying

these systems is important. The concepts of feedback transformation and feedback

decoupling shed some light on the matter, because the concepts tell when the system

under consideration can be divided into smaller systems. For recent research, see

Takahara and Asahi (1985) and Saito (1986; 1987).

The study of decomposability conditions enlightens the research of relations

between “parts” and “whole.” Section 10.5 is mainly from Takahara’s excellent

article (1982). The concept of decomposition of systems shows its power in

Section 10.4.

CHAPTER 11

Calculus of Generalized Numbers

As before, boldface letters are used for GNS-related concepts and lightface

letters for real-valued concepts. For example, ƒ(

) represents a real-valued func-

tion, and f

(

) a function from a subset of

GNS

to a subset of

GNS

. Without loss of

generality, we will always assume that each real-valued function is from

into

and each generalized number-valued function is from GNS into GNS

, where

the set of all real numbers and GNS stands for the set of all generalized numbers.

A comprehensive understanding of mathematical analysis is needed to understand

this chapter.

As shown in Chapter 6, it is necessary to develop a more mature theory of math-

ematics of levels. In this chapter the system of generalized numbers is treated

rigorously, such that the concepts of limit, continuity, differential calculus, in-

tegral calculus, and their applications in the well-known Schwartz distribution

theory are introduced. Further research topics are also listed.

11.1.

Introduction

Let be the set of all integers, the set of all real numbers, and the set

of all natural numbers. A generalized number x is a function from

into

such

that there exists a number k = k

(

)

∈

called the level index of x, satisfying the

condition

(11.1)

For convenience, the zero function, denoted by 0, from

into

will also be

considered a generalized number, and each such x is expressed as

(11.2)

311

312

Chapter 11

where 1

(

)

stands for the function from

into such that

if s ≠ t

(11.3)

and x

(

) = x

. The set of all generalized numbers will be written as GNS. The

operations of addition, subtraction, scalar multiplication, multiplication, division,

and ordering relation of the generalized numbers are defined as follows:

(1) Addition and subtraction. Let x and y be arbitrary from GNS

. Define

(11.4)

(2)

Ordering. The ordering relation between x, y

∈ GNS is the lexicographical

ordering; i.e., x < y iff there is an integer k

such that

and

(3)

Scalar multiplication. Let c ∈

and

∈

GNS

, and define

(11.5)

(4)

Multiplication. Let

, y ∈ GNS. The product of x and y, denoted x × y or

xy, is defined as

(11.6)

(5)

Division. For x, y

∈ GNS, if there exists a z ∈ GNS such that

× z = x

(11.7)

then the generalized number z is called the quotient of x divided by y,

denoted by x

÷ y or x

y. As shown in Chapter 6, if y ≠ 0 ∈ GNS, then

÷ y always uniquely exists.

Theorem 11.1.1.

(

GNS

, +, × , <) forms a non-Archimedean field.

There exists 0 ∈ GNS such that x + 0 = x, for any x ∈ GNS.

Proof:

It suffices to show that (

GNS

, +, × , <) satisfies the following axioms:

A1 x

for any

x, y

∈

GNS.

A2 x + (

) = (

) +

for any

x, y, z

∈

GNS.

Calculus of Generalized Numbers

313

For any

∈

GNS

, there exists y ∈ GNS such that x + y = 0.

M1 x

× y = y × x, for any x

, y

∈

GNS

(

y × z) = (x × y

)

for any

, y

, z

∈

GNS

M3 There exists 1 ∈ GNS such that x × 1 = x

, for any

∈

GNS

For any

≠

∈

GNS

, there exists y ∈ GNS such that x × y = 1.

0 ≠ l.

x × (

+ z) = (x + y) + (x × z

), for any

x , y,z

∈

GNS

x < x can never be true, for any x ∈ GNS.

For any

, y

, z

∈

GNS

, x < y and y < z imply x < z .

For any x

, y

∈

GNS

, one of the following must be true:

x < y, x = y, x > y

For any

, y

, z

∈

GNS

, x < y implies x + z < y + z

For any

, y

, z

∈

GNS

, x < y and z > 0 imply x × z < y × z

For each natural number

∈

there exists x ∈ GNS such that n < x.

It is straightforward to check all these axioms except, M4 and R5, if one

observes the following facts: In GNS 0 is the zero function from

defined

by 0

(

) = 0 for all

∈

GNS 1

is the unit element

( 0 )

of the zeroth level, and for

each natural number

∈

treat

as the generalized number

= (. . . , 0,

, 0, . . . )

such that k(n) = 0.

Proof of Axiom M4: Assume that the fixed and unknown generalized numbers

and

have expressions

Then the equation x × y = 1 to be satisfied by x and y is equivalent to

(11.8)

where

(0)

. That is, 1

(

)

= 0, if u ≠ 0, and l(0) = 1. If u = 0 in Eq. (11.8),

we see that k

(

) can be defined as –k

(

), and

314

Chapter 11

If u = 1, Eq. (11.8) becomes

Hence,

can be defined by

By mathematical induction, the

generalized number y is defined.

It can also be shown that y is unique. To this end, it suffices to show that

there is no other way to define the level index

(y). By contradiction, assume that

k (

) = –k

(

x) –d, for some d ∈

Replacing u by –

d, in Eq. (11.8), we get

This contradicts the fact that

This ends the proof of Axiom M4.

Proof of Axiom R5: The assumption that x < y and z > 0 implies that there

is t

∈

such that x

(

) = y

(

t ) for all t < t

) < y(

) and z(k(z)) > 0. To

compare x

× z and y × z, let k

+ k(z). It then suffices to show that for all

s < k

(

) (s) = (

) (

) and

(

x × z) (

) < (y × z) (k

). To this end, for

s < k

, one has

since v ≥ k(z) implies u < t

and

(

u) = y

(

). If s = k

= t

+ k

(

), x

(

) < y

(

)

implies

since if v =

k(z), u has to equal t

, so

while other terms in the summation

are the same as the corre-

sponding terms in This ends the proof of axiom R5.

Theorem 11.1.2.

Let

If h,k ∈ satisfy k < h,

then the generalized numbers in I

(k)

are infinities compared with those in I

(h)

, and

the generalized numbers in I

( h)

are infinitesimals compared with those in I

( k)