TRANSFORMATIONS WHICH RENDER ... - Wiley Online Library

R

RB-68-l5

(

S TRANSFO~iATIONS

t L ~ E T

WHICH RENDER CURVES PARALLEL

Michael V. Levine University

o~

Pennsylvania

I

N

This Bulletin is a

dra~t ~or

intero~fice

circulation.

Corrections and suggestions for revision are solicited. TIle Bulletin should not be cited as a the

speci~ic

re~erence

permission of the author.

cally superseded upon

~ormal

publication

It is automatio~

Educational Testing Service Princeton, New Jersey March 1968

without

the material.

TRANSFORMATIONS WHICH RENDER CURVES PARALLELl

Michael V. Levine

2 University of PennSYIVania

Michael V. Levine Research Building Room 139 Educational Testing Service Princeton, New Jersey 08540

1 This research was undertaken as a consequence of some conversations with R. Duncan Luce in 1963 at the University of Pennsylvania. It is a pleasure to acknoi'lledge his advice and encou.ragement. I a.m also indebted to David Krantz} M. Frank Norman and John Pfanza,gl for many pleasant and profitable conversations about this work. Dr. Norman made detailed comments on an earlier version of this paper which were extremely helpful. This research was supported by grant NSF GB 1462 from the National Science Foundation and grant ~~ 1222 from the National Institutes of Health to the University of Pennsylvania. The final manuscript was prepared at the Educational Testing Service.

2 Now on leave to Educational Testing Service.

ABSTRACT

Eight examples are given to indicate the desirability of having a general theory treating sets of curves which can be transformed into parallel curves.

With this theory it is shown that a set of curves can

be rendered parallel if and only if every pair of functions in the completion of a particular group associated 'Nith the curves is uncrossed. Under general conditions any two transformations rendering a set of curves parallel are related by a linear transformation. transformations are proposed.

Methods for calculating

Several structural properties of sets of

curves which can be rendered parallel are proven.

-3-

TRANSFORr-1ATIONS vlHICH RENDER CURVES PARALLEL This paper contains a discussion of the transformations which render a set of curves parallel.

It begins ,nth a formal statement of what it

means to render a set of curves parallel.

Then some examples and problems

in psychology to which the theory may be applied are listed to motivate the theory.

An outline or survey of the theory is given.

Finally the

theory itself is presented. I

1.1

INTRODUCTION

Formal Statement Two curves or subsets of the cartesian plane are defined as parallel

when a translation along the x-axis maps one onto the other. words, two curves are parallel if for some

k

the mapping

In other

< x,y > ~

< x + k,Y > carries one curve onto the other. A set of curves is called parallel if each pair of curves in the set is parallel. A real valued function

u

of a real variable renders a set of two

or more curves parallel if the mapping

< x,y >

~

< u(x),y > carries

the set of curves into a set of parallel curves. Real valued functions of a real variable are parallel if their graphs are parallel as curves.

Parallel sets of functions and transformations

rendering sets of functions parallel are defined in the obvious way. Details are lITitten out in section 11.1. 1.2

Motivation for the Theory In this section several examples are given to demonstrate that it is

common (and possibly natural) for psychologists to assume that sets of functions which are crucial in their theorizing can be rendered parallel. This paper gives necessary and sufficient conditions for the validity of such assumptions.

In addition, there is a general discussion of sets of

functions which can be rendered parallel. stantive

Nontrivial results of a sub-

nature follow from this discussion.

For example, it can be

shown that under very general conditions with three psychometric functions and the value of a fourth psychometric function at a particular stimulus as date,the value of the fourth psychometric function at every stimulus can be computed.

(See examples 3 and 6 below for a discussion of psy-

chometric fWlctions and section

v.4

for the facts from which this asser-

tion follows.) There is an enormous mathematical literature dealing with structures like those considered in this paper.

(For a bibliography and a discussion

of related results, see Acz~l, 1966, chapter 6.)

However, the major

results and the organization of this theory appear to be original.

In

addition, thi,s theory has been developed with applications to psychology in mind and written for mathematical psychologists with little specialized mathematical training.

-5-

The alternative of developing and presenting a particular psychological theory such as a theory of visual brightness or a mental test theory rather than a general mathematical theory was rejected for two reasons. Visionists have special concepts, experimental problems and ways of talking about data which are foreign to mental test theorists.

A well

written theory for one group would be unnecessarily difficult and filled with irrelevancies for the other.

In the second place, the deepest and

most useful results will be available at the proper level of generality. The author feels that this is the level he has chosen. The examples given in this section are intended only to motivate the

th~ory

by illustrating the ubiquity and diversity in psychological

theory of transformations which render curves parallel. The exposition of these examples is not self-contained. theory are

Example 1:

nO'Jl

being developed and

Shepard

·w~ll

Detailed applications of the be reported in later papers.

(1965) observes that the shape of a generalization

gradient is dependent upon the scale used for the independent variable. He observes that a :nonlinear transformation of the independent variable can change a convex gradient to a conCave or linear gradient.

However,

if several overlapping gradients are studied then there is generally essentially no more than one transformation of the independent variable such that the plotted generalization gradients all have the same shape. Two generalization gradients have the same shape when they are parallel.

Shepard has already given us effective techniques for finding

-6-

the transformations which render generalization gradients parallel.

It

is hoped that this work will complement his by giving necessary and sufficient conditions for the existence of such transformations, by discussing the uniqueness of the transformations and by giving additional techniques for finding the transformations which render gradients parallel.

Example 2:

Suppes l

(1959) stochastic learning models for probability

learning with a continuum of stimuli and responses suggest

a novel

behavioristic way to study generalization and measure sensation.

The

subject in Suppes' theory is characterized by his rate of learning, his initial distribution of response probabilities and a function Suppes calls a smearing distribution. as a generalization function since which reinforcement at in the future at

y

k

which

This function may be thought of

k(x,y)

varies with the degree to

increases the subject's tendency to respond

x

Over moderately large ranges, perceived length seems to be approximately linear with physical length. length that

k

Suppes assumes for continua

~~ll be translation invariant; that is, that

the generalization from

y

to

x

will be a function of

y

k(x,y) minus

Translation invariance also means that the cumulated generalization functions

K y

given by

Ky(x)

=

~.X -00

like

k(t,y) dt will be parallel.

or x

-7-

To carry Suppes 1 reasoning one step further, for an experiment with a continuum like intensity of light such that sensation is a nonlinear function

u

of the physical measure of the continuum, the translation

invariance assumption may lead to an accurate description of the experiment if the experiment is reported in sensation units

This would mean that the function

in physical units

x

render the curves

K

functions

Ky

u(x)

parallel.

y

rather than u

'ITould

Levine (1965) has shown that the

can be calculated from the data of a single sUbject in

a single noncontingent

probability learning experiment.

in principle, analyze learning data to find

u

Thus one can,

and measure sensation.

This method of studying generalization and sensation has some advantages:

extremely accurate estimates of

many values of

x

and

y

K (x) y

can be obtained for

and inter-trial dependencies are explicitly

dealt with to get information about the subject. Example 3:

Thurstone (19 27a, 19270) considers experiments in which pairs

of stimuli are compared to obtain psychometric functions

Fy(X)

probability (as approximated by relative frequency) that stimulus judged greater than stimulus seems to be this.

y.

Each stimulus

greater than

S y

x

x

is

corresponds, not to a particular S

x

When two st imuli

is judged greater than

(or greater than

x

The essence of Thurstone's psychophysics

sensation, but to a random variable are being compared,

giving the

S

y

y

whenever

x S

x

and

is

plus some threshold constant).

y

More precisely, the probability that

x

is judged greater than y ,

F (x) , is the probability that S - S is positive (or greater than y . x y some threshold constant). For Thurstone, all the random variables uted.

Thus for a fixed set of stimuli

functions

Fy(X)

and

were normally distrib-

X a complete account of the

is obtained when three other functions are specified.

These functions are viation of

Sx

S , and x

m(x) - the mean of

S ,

sex) - the standard de-

x

r(x,y) - the correlation coefficient between

S x

S". J Since the theory in this form has more unknown parameters (m, s , r)

(F)

than data points tions.

when

X

is finite, Thurstone made simplifying assump-

For example,he assumed that

standard deviations, sex) and

Sx

and

Sy

are independent and that the

s (y) , are equal for all x

to apply the results of this paper is to asswne

Sx

and

and S y

y •

One way

are independent,

that all the standard deviations are equal, but to relax Thurstone's normality assumption.

If

S x

is normal, then

S x

is equal to

m(x) + sN

the normal random variable with mean zero and variance one. zation, we just assume that

(or

Fy (x) sD

plus

is the probability that SOWE

In this generali-

m(x) - m(y)

threshold constant) where

Then it follows

is greater than

sD

D is the random variable

obtained as the difference between two independent random variables which both have the

sa~e

N is

N is simply some unknown random variable which,

like the normal random variable, has a positive density. that

where

distribution as the unknown random variable

N

If

-9-

the set of stimuli

is already physically measured so that i t can be

considered to be the set of real numbers and if the function strictly increasing fronl

m is continuous,

X onto the reals, then the unknown function

m

mapping stimuli to the modal or mean sensation renders the set of psychometric functions

{Fy(,)Jy is in X}

parallel.

Under these conditions, it is

straightfon.,rard to apply the theory given in this paper to obtain necessary and sufficient conditions for the existence of a fu.nction unobserved ra..'1dom variable

N satisfying the model.

calculated from the functions

(:F'~/. )}.

The function

m can be

In fact) it is possible to show' by

applying this theory that lillder very general conditions the function be calculated from ,just three of the psychometric functions Exanwle !+:

and an

Tn

m

C~)

F (.) •

y

In signal detection theory's account of the two-alternative

forced choice experiment) it is sometimes assumed that each alternative is associated 'ITi th a random variable which is normally distributed along the decision axis.

If these random variables are independent and have equal

variances then it folloYTs that the

ROC

,.;hen plotted on probability paper.

Ttlith the theory presented in this

paper one can ans',rer the question:

does there exist some function

such that the mapping

HOC

(x,y) ---~~(u(x),u(;y»

curves into parallel straight lines?

Hhich is

nO\'I

curves are parallel straight lines

u

carries a given set of

A generalization of this theory

being prepared for publication ,.,rill permit one to drop the

restriction that the rescaled

ROC

curves be para-Ile1 straight lines.

-10-

5:

Example jnd

A Weber function is a function which gives the size of the

as a function of the comparison stimulus.

function

u

such that

the number of between

y

solution

jnd' s

and u

u(x) - u(x') between

y' •

and

Xl

u(y) - U(yl)

if and only if

is equal to the number of

jnd' s

Alternatively, a Fechner function may be defined as a

to the functional equation

u(f(x»

where

x

equals

A Fechner function is a

f(x)

is

x

- u(x)

= constant

plus the Weber function evaluated at

x.

Luce and Edwards (1958) show that it is illogical to integrate a Weber function to obtain a Fechner function. equation defining f

Fechn~r

They obtain a solution to the functional

functions by studying the derivatives of the function

near zero. Since a Fechner function can be shown to be simply a function which

renders

f

and the identity function parallel, the theory presented in

this paper can be used to stUdy all the strictly increasing, continuous Fechner functions.

A graphical method for "cumulating jnd'sll (that

is, finding Fechner functions) is sketched in section 111.2.

Finally,

with the results presented in this paper it is easy to obtain necessary and sufficient conditions for the existence of continuous, strictly increasing solutions for the equation defining Fechner functions.

-11-

Example 6:

In extending Fechner's theoretical work in scaling, Luce

and Edivards (J.958)

(see also Luce, 1963, page 206 ff) express as a

functional equation the psychological rule of thumb:

equally often

noticed differences are equal unless always or never noticed. p(x,y)

denotes the probability that a stimulus

than a stimulus

x

If

is judged greater

y, then an increasing one-to-one function mapping a set

of stimuli onto the real numbers is a sensation scale if for all stimuli x , y , x' , y'

o
,»

and real valued function

For details see Luce and Suppes (1965, page 334).

-12-

Example 7:

In mental test theory it is sometimes assumed that a single

unidimensional ability underlies test performance in the sense that there is a real valued function

f

from the set of individuals being tested

such that the probability that individual can be >orritten

P.(f(s» ~

s

knows the a.11.S>Oler to item

for some cumulative distribution

i

In

developing models with such assumptions, the distributions

have in

the past been taken to be normal ogives (Lord and Novick) 1968, particularly chapter 16) or logistic distributions (Birnbaum, 1968)~ These distributions have the advantage of having two parameters so that one can classify items according to their difficulty or according to their discriminating power (i.e.) the rate at which populations are separated or the steepness of the curve

1

is some distribution a

In both cases there

P. ). This is achieved in the follow"ing ,'lay:

and

b

P

such that for each item

such that for all real

x

P.(x)=P(ax-b) l

the normal ogive in the normal case ~~d logistic Case.

b

l

is

Pis

(1 + e-x)-l a

in the

and the

in both cases.

If all the items have the P.

p(x)

The discriminating power varies with

difficulty with

functions

there are constants

i

sa~e

will be parallel.

discriminating power, then the

A generalization now being prepared

for publication will permit consideration of items 'Nith different discriminating powers.

It is hoped that the theory presented here will

facili tate theoretical 'tlork by permitting researchers to analyze two parameter models ,dthout making normality or logistic assumptions.

-13-

Example 8:

Jameson and Hurvich (1964) offer a theoretical account of

brightness contrast which predicts the qualitative feature of our enormous body of exceedingly complicated data.

However, as J-ameson and Hurvich

note, sufficient quantitative discrepancy between the theory and data remains to permit the possibility of errors in quantification.

The

theory presented in this paper permits a reformulation of the Jameson and Hurvich induced brightness theory which may bring it into closer agreement with data by freeing it from an assumption which appears (to this author) superfluous. The essence of induced brightness theory seems to be this.

There

is an initial invariant response of the visual system to a small patch of light.

This initial response is not sensation.

Sensation depends upon

preceding and concurrent activity in the visual system so that under fixed conditions of stimulation, sensation is related to luminance by the equation

Sensation

where

x

response,

is luminance, a

u(.)

= au(x)

- b

is the function giving the invariant local

is a multiplicative sensitivity constant and

b

is the

inhibition of the local response due to sensation elsewhere. Where there is only one source of inhibition, the amount of inhibition is assumed to be proportional to the amount of sensation elsewhere in the visual system.

Thus if two small lights having luminances

presented to an observer, the resulting sensations satisfy

x

and

yare

-14au(x) - kS (1)

Y

a'u(y) - k'Sy

where

S

x and Sy

are the brightness sensationsl a

sensitivity constants,

u

and

is the invariant function and

aI k

are and

k'

are

inhibitory constants. The invariant function

u

is taken by Jameson and Hurvich to be the

cube root function since S. S. Stevens has shown that the geometric mean of magnitude estimates of patches of light

for groups of subjects is

approximately proportional to the cube root of luminance.

This speciali-

zation does not appear (to this author) to be an essential part of the induced brightness theory. theory to calculate

u

In fact, it is possible to use induced brightness

from brightness matching data for individual subjects.

If another pair of lights having luminances

Xl

and

Yl

is compared

with the first pair considered, they have sensations

alu(xl ) - klS

Yl

a l' u(Yl) -. k'S

Yl

(2) 1

where the subscripted variables have the same interpretation as the analogous unsubscripted variables in (1). if

x

If

Y and

Yl

are held constant,

is systematically varied by the experimenter and if the observer adjusts

the remaining luminance

Xl

so as to obtain a brightness match between

-15..

and

S x

then a mapping (called a brightness matching ~unction) between

luminances

x

(1)

and eliminating terms in

and

(2)

and

Setting

is obtained.

S

x

and

S y

equal to

in xl it follows that the S

conditions for a match are

I~

the conditions for observing the two pairs of light are

similar, then we may assume can be

~Titten

where

K is independent

inverse

u

-1

k

1

=k

and

suf~iciently

k' = k' •

1

Then (3)

in the form

o~

xl

and

x.

Finally if

u

has a left

(as the cube root ~unction does) then (3) can be written in

the form

~or

and

the brightness matching

~unctions.

As the contrasting luminances

yare varied one obtains a fanuly of functional equations for

Yl u.

These equations are very similar to the equations studied in this paper.

.

By a generalization of the theory given in this paper

which is now being

prepared for publication one can solve these equations for the u.

invarim~

If it can be assumed that the invariant is continuous and strictly

-16-

increasing then the solution

~dll

be unique up to a linear

and the invariant can be calculated

~rom

trans~ormation

the empirical brightness matching

~unctions.

In this manner brightness induction theory can be pendently

1.3

o~

inde-

the controversial magnitude estimation results.

Survey and Outline

o~

the Theory

The theory focuses on the problem rendering a given set ~ctions

quanti~ied

o~ ~ctions

o~ ~inding

parallel.

all the

~unctions

Only strictly increasing

mapping the reals onto the reals are considered.

These functions

will henceforth be called scales as in measurement scale or psychological scale.

The theory can be easily generalized to deal with strictly

increasing mappings

~rom

one open interval onto another open interval.

A scale which renders a set of scales parallel is called a solution ~or

the set.

A set

~mich

has a solution is called a parallel system.

The theory is organized around three problems: 1.

Find necessary and sufficient conditions for a set of scales to be a parallel system

2.

Discover the relationship between two solutions

3.

Find methods

~or

calculating solutions.

Simultaneously an attempt is made to classify parallel systems, to provide techniques for their study and simply to give the psychological theorist a better

~eeling

for what they are like.

-17The theory begins with a detailed study of two scales. and sufficient condition that a set

{F, G)

A necessary

of two scales be a parallel

system is that the two scales be uncrossed; that is, for all G(x) , for all

x,

F(x)

G(x)

=

or for all

x,

F(x)
G(x)

A set of scales is called uncrossed if every pair of scales in the set is uncrossed.

Every parallel system is uncrossed, but an uncrossed

set of more than two scales may fail to be a parallel system.

The

condition that a set of scales be uncrossed is progressively strengthened throughout the paper until a necessary and sufficient condition for an arbitrarily large set of scales is obtained. The set of all solutions for two scales us surveyed. set is enormous and unvneldy.

In one sense this

This is so because every continuous, strictly

increasing, real valued function on a certain closed interval has a ill1ique extension to a solution. simple.

For if

u

In another sense, however, the set is small and

and

such that the function

v

are solutions, then there are constants

av(.) + b - u(.)

zero on an unbounded set.

a

and

is a bounded function and is

Thus, any two solutions can be linearly

transformed so as to "alrnost 11 coincide.

A graphical method for calculating

solutions is proposed. Each set of scales is associated with a unique group of scales in section IV. The first use of associated groups is to find the exact condHions under i'ihich the results on hlO scales can be applied.

If a

set of scales has solutions, then a pair of scales which has exactly the

b

-18-

same solutions can be found if and only if the associated group is monogenic (i.e.) cyclic).

Necessary and sufficient conditions for the associated

group to be monogenic and methods for finding equivalent sets of two scales for sets of scales with monogenic associated groups are proposed. The first

strengt~lening

of "uncrossed" is introduced in section IV.

A set of scales is strongly uncrossed if every pair of scales in its associated group is uncrossed.

This is a necessary condition on a set

of scales for the existence of a solution. relation for uncrossed groups of scales: all

x

There is a natural order F > G if

F(x) ~ G(x)

for

It is shoTtm that each u-l1crossed group of scales is isomorphic

as an ordered group to a subgroup of the additive reals vdth the usual ordering.

This isomophism is used to present some elementary facts about

uncrossed groups which were needed for later proofs or seemed important for applications.

Particular attention is given to the problem of finding

a pair of scales having the same solutions as a given set of scales. In section V a final strengthening of "uncrossed" is made to obtain a necessary and sufficient condition for parallel systems.

The completion

of a set of scales is the collection of all scales which are (pointvnse) limits of sequences of scales in the set.

A set of scales is completely

uncrossed if the completion of its associated group is

uY~rossed.

It

is proven that a set of scales is a parallel system if and only if it is completely

~crossed.

-19With the results established before section V, it is clear that a set vnth a monogenic associated group is a parallel system if and only if it is strongly uncrossed.

Furthermore, it is easy to show that for

strongly uncrossed monogenic groups, the completion of the group is simply the group itself. considered

Thus, completions and completely uncrossed are

only for nonmonogenic groups.

To illuminate the meaning of completely uncrossed, and to give new necessary and sufficient conditions, equivalent conditions are offered. It is proven that if nonmonogenic

F

is a strongly uncrossed set of scales with

associated group

G

then

F

is completely uncrossed

if and only if (1) every function in the completion of tinuous, (2)

G

scale,

= x, (3)

is in

f(x)

(;t)

~ is con-

has a sequence of scales decreasing to the identity the orbit for some

x

(ioe., the set

is dense in the reals, (4) the orbit of every

and, of course, (5) ~ has a solution.

{f(x)!f x

is dense,

This is the main mathematical

result of the paper. Constructions and results relevant to calculating solutions are collected and informally

discus~ed

in section V.4.

The paper concludes with a uniqueness theorem for sets with nonmonogenic groups. group and if

u

If a set of scales has a nonmonogenj.c is a solution, then

if for some positive

a

and real

b,

v

associated

is a solution too if and only u(.)

= av(.)

+ b

Using this

and some earlier results, an algorithm can be given for finding a parallel system with no more than three scales which has the same solution as a given parallel system with finitely many scales.

-20II

PRELIMINARIES

This section contains some

abbreviations and some

de~initions,

elementary facts about the objects defined which are needed throughout the theory. 11.1

A scale is

de~ined

to be a strictly increasing

~unction

of the

reals onto the reals. Clearly scales are continuous and one-to-one. all scales forms a group with composition operation.

The identity

o~

o~ ~unctions

this group is the identity

reals and will henceforth be denoted bye.

F

will be denoted by

F

or

-1

•

F

-n

F

F

n

o~

as the group ~unction

on the

The inverse of a scale

F

Otherwise, the usual multiplicative

notation for groups will be used so that followed by the mapping

The collection

FG

will be the mapping

Idll be the

n

th

iterate of

G

F

= F-n , and F0 = e

11.2

If

F

is a scale, then a function

transformation of

F

for all real

G(x)

x

if for some positive

of a scale is a scale. it will be said that For each positive

= aF(x) If

F

a

and real

b

such that

Clearly a linear transformation

+ b

G is a linear transformation of

and a

G is said to be a linear

F

J

then

G are related by a linear transformation.

and real

b

,there is a mapping of the set

of all scales onto itself; namely, the mapping which sends each scale into the scale

aF( .) + b

The collection of all such mappings is

F

-21.-

easily seen to be a group when the product of mappings is defined by the composition of mappings. the pairs

al,b

and

l

Thus the product of mappings corresponding to

a 2,b 2

is the mapping sending

F

a a F(.) +

into

l 2 (a b 2 + b ) , the identity is the identity mapping on the set of all l l

scales and the inverse of the mapping corresponding to the pair the mapping corresponding to the pair

l/a,-b/a

these mappings form a group, it folloTtTS that _ of

II.3

a,b

is

From the fact that is a linear transformation

is an equivalence relation on the set of all scales. ~~o

subsets of the plane are called parallel when for some

the mapping < x,y

>~


The graph of a fQ~ction

F

k

carries one onto the other.

is the set

(< x,F(x) > I x is real }

Two functions are parallel when their graphs are parallel. A function

is said to render

u

a

set of functions

F

parallel

when the mapping < x, y >-+< u(x), y > maps the graphs of any hro functions of

~ into parallel sets.

As a consequence of these definitions, it follows that a scale renders a set of functions G in

F,

Fu-

l

( < u(x) + k,F(x) >

and

I

F Gu- l

x iR real

parallel if and only if for each are parallel.

F

For

equals ( < u(x),G(x) >

I x

is real )

if and only if

( < x + k,Fu-lex) >

I

u

1 x is real) equals ( < x,Gu- (x»

I

x is

re8~}

and

-22-

rr.4

F

~

If

is a set of scales and

u

is a scale, then - u renders

parallel - is equivalent to each of the folloWing conditions: (1) For each

F and

G in

F

there is some for all real

such that

x

•

(2) For each' F and

G in

F

there is some

k

such that

(3) For each

G in

F

there is some

k

such that

F

u(x) For

k

and

= uF-lG(X)

(< x + k,FU-l(x) >

+ k , for all real

x

I

x is real} equal& (

R

= the

abb~eviations

are used throughout

the set of all integers set of all real numbers

(a, b)

=

{

x in R

a 0

,

-36"r(x) = Iu(x) - vex) I =

maximum

(u(x), vex») - min (u(x) , vex)}

< max (u(FG(O» =

Thus, for all u

and

v

x,

, v(FG(O»} - min (u(O) , v(O)}

PG(O) w(x) < PG(O)

is bounded by

and the absolute value of the difference This completes the proof.

PG(O)

This result suggests that it is possible to take a sequence of scales

{F} n

such that

{G,F ,F2 ,F , ... } will l 3

F

b~

n

jjG and FnG(O) decreases to zero such that

a parallel system with essentially one solu-

In Levine (1966) this is shown to be true.

tion.

Now we conclude with

a result which only slightly extends the preceding results, but brings the relationship between solutions for two scales into sharper focus. (Theorem) Let

III·5·B

Let

F

G be two different, uncrossed scales.

and

k = maximum (FG(O) , GF(O)}.

u(O) = 0

and u(k) = k

related to

u

Then there is a solution

If v

v,

This solution

such that

is any other solution, then v

is

in the following way:

There is a unique solution w such that of

u

w is a linear transformation

w(O) = 0 and w(k) = k • w is close to 1. 2.

w(x) = u(x)

u

in the sense that for infinitely many values of x

w(x) - u(x)1 < k

for all

x.

-37-

f

In other words, if F solution

u

such that a linear transformation of any other solution is

uniformly close to Proof: k

Let

= maximum

G form a parallel system, then there is a

F

u

and

G be two different uncrossed scales and

(FG(O) , GF(O)}

Clearly FG(O)

f

0

because

F

f

G &ld

positive because

FG(O) < 0

if and only if

niteness, assume

FG(O) > 0

and

Since

we have Let w(k)

=

v

= au'(')

a

u'

k

is

For defi-

Thus, putting

k

(k) - u' (0) >

b = -

au ' (0)

0

is a solution with

be any solution.

k , then

0 < GF(O)

a = u'

+ b

Thus,

k = FG(O)

F//G, there is a solution) say

u(·)

F//G

v(·)

If

=

u(O) = 0

aw(·) + b , w(O)

v(k) - v(O) > 0

can only be

and

b

u(k) = k

and

=0

and

can only be

k

v(O)

Thus) if w is a solution, it is the only solution having

these properties.

Since

wee)

1

= a v(.) -

b

a

and

1

a

= v(k) ~

v(O) > 0

w is a solution.

Since

u(O) = w(O) = 0

lu(FG)n(O) - w(FG)n(o)1

and

= lu(o)

uFG(O) = wFG(O) = FG(O) = k > 0 , - w(O)j

hypotheses in the preceding theorem.

=0

Thus,

) as was shovm under these u(x)

= w(x)

for each

x

..38of form

Under these same hypotheses it was sho\~

(FG)n(O)

w-(x)j < FG(O)

=

k

•

This completes the proof.

lu(x)-

-39PARALLEL SYSTEMS WITH MONOGENIC ASSOCIATED GroUPS

IV

IV.I

Associated Groups If

F ,G

u in

]Er ,

is a solution for

]Er

then by 11.4

each pair of scales

satisfies an equation of form

This fact naturally leads to useful necessary conditions and to the stUdy of certain groups by an argument which will now be given. For fixed ·u, the set of scales of the form subgroup of the group of all scales. g(o)

= u-l(u(.)

u-l(u(o) + k

l

+~) , then f-l(.)

+ k ) 2

For if

= u-l(u(o)

= u-l(u(')

- k ) l

and

is a

+ k ) l

and

=

fg(.)

The group so obtained is isomorphic to the group

of additive real numbers.

For the mapping

ditive, it is obviously onto, and since k

f(·)

u-l(u(.) + k)

u-l(uC·) + k) ~ k

u-l(u(o) + k) = e(o)

is adimplies

is zero, it is one-to-one. This does not yet yield a useful necessary condition because the

group is defined with respect to an unknown solution respect to the known scales of

~

u

rather than with

However, if we take the subgroup

of the group of scales generated by the scales group defined in terms of the functions of

F-IG, we do obtain a

~

As a necessary con-

dition, we have that this group is isomorphic to a subgroup of the additive reals.

Thus, for example, we have the condition that

(F-IG)(H-IJ) = (H-IJ)(F-IG)

for any F, G , H , J

in

F

if

F

- 40is a parallel system since the subgroups of the additive reals are abelian. This motivates the definition and study of associated groups. The associated group ot a set of scales. ~ by the scales

F-IG

for

F, G in

~

In the remainder of this section IV.l proven about associated groups.

is the group generated

some elementary facts are

This discussion will be used to motivate

a generalization of "uncrossed" which will give a new necessary condition.

~

The associated group of a set of scales first choosing a scale

F

in

F

can be obtained by

and then considering the set of all

scales of the form

... where

r :: I , and, for

is in

~ .r

i

=::

The relation

(F-l--)-I(F-IG) rI

so obtained is independent of the choice of

G.].

is an integer and

1 , 2 ,. ••• r ,

shows that the set F

each of the generators of the associated group.

and that this set contains Since this set is

obviously a group of scales, it is the associated group. From this characterization of associated groups it follows at once that a group of scales is its group of scales, then

e

F

is in

is generated by the elements

associated group.

OvID

e

-1

F

=::

For if ~

and the associated group of F

of

is a

F

F

One of the most useful facts about associated groups is the following:

-41IV .leA

If two sets of scales have the same associated group, 'then they

have the same solutions. Proof:

Let

F;

If

and

If

u be a solution for is in

G

Since

G

be sets of scales with associated group

G

there are scales

If

F

and

is also the

•.• G

G are in

2

as~ociated group for ~

all in

r

~} then

'F;

,

and integers m

-IG ) r ••. (F1 r

such that u for

is a solution for i

= 1,2, •.. ,r

.

p~ for some constants k , 2' , i Since the set of scales

u -l(u(') + k) a solution for a solution for

Thus,

F2 F;

u

renders

F, G in

Similarly, if

u

F~lGi(');

{u-l(u(.) + k)lk

a subgroup of the set of all scales, there is some

F2

Since

k

such that parallel and

is a solution for

If

is real} F-IG(.) u

is

=

is

it is

This completes the proof.

As a consequence of this fact, a set of scales and its associated group have the same solutions.

For a set of scales and its associated

group have the same associated group. It follows at once that the associated group of a parallel system is a parallel system.

Hence, the associated group of a parallel system

is uncrossed, for any pair of scales in a parallel system constitute a parallel system and are thus uncrossed. We now have two apparently distinct necessary conditions for parallel systems . 1.

ttle associated group is uncrossed

2.

the associated group is isomorphic to a subgroup of the additive reals.

The second condition does not linear functions mapping

(ae(.)la

i~ply

the first for the group of all increasing

is positive}

is isomorphic to the reals under the

ae ( • )----.~.log a , but every pair of scales in it cro ss.

The first

condition will be shown to imply the second. In the remainder of section IV sets of scales with uncrossed associated groups are studied.

The conditions under which the results of the

previous section can be used to generate all the solutions for a set of scales will be given and some general results needed for later sections ·...,ill be proven. IV.2

Strongly Uncrossed Scales and Uncrossed Groups A set of scales is strongly uncrossed if its associated group is

uncrossed. Parallel systems are strongl:y uncrossed) as was just shown.

The

converse is "almost" true; strongly uncrossed sets of scales admit a representation which is a multidimensional generalization of the representation characterizing parallel systems.

Details will be given in a later paper.

Strongly uncrossed is a strictly stronger condition than uncrossed since if F

and

G are crossed scales in

crossed scales in the associated group of

x + 2 and if G(x) ::: x + 3

if x

F F

is negative and

then

e ,

However, if

F-IG

are

F(x) =

2x + 3 otherwise,

then

{e, F ,G}

is uncrossed.

e-lG(x) :;: G(x) iff x :;: 1,

But since

{e, F ,G}

(e-~)2(x):;: F2 (x) :;: x

l~ =

+

is not strongly crossed.

The associated group of every strongly uncrossed set of scales is isomorphic to a subgroup of the additive reals.

To prove this we define

an order relation on uncrossed groups and then use Holder's qUalitative characterization of the subgroups of the additive ordered group of real numbers. IV.2.A


-45From Holder's theorem and the preceding result we have, IV.2.B

(Theorem)

The associated group of a strongly uncrossed set of

scales is isomorphic as an ordered grOUP to a subgroup of the additive reals.

< to denote anyone of three

Henceforth, we will use the symbol partial orderings:

the usual ordering of the reals, the archimedean or-

dering on a group of uncrossed scales and the ordering on functions of the reals into the reals given by

The symbols

2:, > ,
e

find

If

h

u

G

G

and

G

doesn't have a least

such that

for

F,

k ,

u(·)=uh(·) + k

2

{e}

doesn't have a least positive element, we can

is a solution for

uh (0) + 2k

t

Without loss of generality, then, we may assume

Since in

G with exactly the same solutions

G i s monogeni c.

Suppose on the contrary G positive scale.

and

{F, G} , then by hypothesis,

its associated group

and

G

and

{e, h}

This implies that

extended to solutions for uniqueness theorem III.4.A

{F, G} for

is a solution

Thus, for some

u(O) = uh(O) + k

2 u(O) - uh(O) ::: uh (0) - uh(O).

tinuous strictly increasing functions on

•

u

:::

Thus, there are con-

[0 , F-1G(0)]

"Thich cannot be

Since this contradicts the

{F, G} , the associated group of

~

must be monogenic.

rv.4

Finite Sets of Strongly Uncrossed Scales The problem of exploiting the results of section III by finding a

pair of scales with the same solutions as a given set of scales is reconsidered.

The criterion given in the preceding section ignored the

algebraic structure of the associated group and considered only the ordering on the group.

Now the algebraic structure is studied to find

generators and criteria for monogenicity.

Particular attention will be

paid to parallel systems of three scales because of the special role they play in this theory. Let group

G

{e}

be an uncrossed group of scales other than the trivial

~ to compute one of

• We use only the order relation on

the order-preserving isomorphism that exist by IV.2.B. a fixed positive scale in

~

archimedean group for each

f

p

is exactly one integer

1

be

G and for each positive integer there

is obtained by using only ordinal information on ~

g > e

is a linearly-ordered

there corresponds a unique sequence of integers

Now let

Let

g

is positive,

~(g) is also. For each

~ the following statements are equivalent

in

p~(g) ~

n$(f) < (1

+

p) $(g)

p ~ n e

m

rational.

f

nand Since

0

m such that

rm = hnm = gn

f ;;:: h

n ,

g ;;:: h

m

we apply 4., to show

Since

t

is

This completes the proof.

Since the associated group of three strongly uncrossed scales (Fl ' F2 ' F ) 3

is generated by the scales in case

F~~2' F~lF3 (and generated fails to be positive) we can

now take any strongly uncrossed set of three scales, and calculate an equivalent pair of two scales, if there is one, by finding a generator for the associated group.

These results are now extended to any finite

set of scales. If the associated group of a finite set of scales is nlonogenic, then clearly genic.

the associated group of any subset of three scales will be monoConversely, if every subset of three scales of strongly uncrossed

finite set of scales has a monogenic associated group, then the associated group of the finite set is monogenic and a generator can be obtained by considering no

n~re

than three scales at once.

This is now stated more

exactly and proven.

Iv.4.B

For

r 2 4 ,let

uncrossed set of for

r

1 < i < j ~ r,

then Proof:

(;t

F;;::

{F1 ' F2 ' ••• Fr}

be a strongly

different scales with associated group ~

iFl, Fi ' F j

)

If

has a monogenic associated group,

is monogenic.

Denote the funct ion

F1Fi

by

fi

and the subgroups of

G

generated by

(f'., f' .)

that G

G :;:

f', ••• h

].

in

G

by

G(f'1,f'2' ••• f'r)

Under the hypothesis

1 < i < j So r

is monogenic, f'or

J

G(f', ••• h).

, we prove that

is monogenic by constructing a chain of'monogenic

subgroups

...

Gc ~c 4

G i +l

will be generated by

Clearly, system of' some

G

r

h,

and

G

J

=G

(f'2,f )

, Gh )

3

(f2 , f'4)

Since

Since both

f2 =

h~

h

3

and and

3

= h

-f

4

Since the iterates of' G(h ) 34 If'

h

h

34

G i

~

because

is a

is monogenic, f'or

Similarly for some

f

and

4 h

G

G(h)

are in

4

(h ) 4

=

G

G(ii)

we may

,they are uncrossed.

there are integers

m -f 0

Clearly,

we may use the preceding result to conclude and to find a generator

=j

i

G(f'2'f',)

3

f'2

(h )

is in both G

m such that

unless

f'.

Since

take Since

-f

].

dif'f'erent scales.

in G

(h 4 ) =

and the elements of'

f'i+l f' .

cG:;::G r

G

-f e

nand

because

f

(h y h )

is monogenic

4

2

•

Thus

such that

include

34

= G(fl , f 2 , f 3, f 4 ) r

= 4, G =

that G

(h ) 5

G(h ) 5

we have

is monogenic.

G(f'2 ' f ) 5

G

(h

34

' h ) 5

Since

Otherwise we take f

2

is in both

h

5

G

> e (h

34

such )

and

is monogenic f'or the same reason as

can be chosen such that

before and

We continue in this manner until we use up the generators.

G

Thus,

is monogenic. IV.5

Idempotents An idempotent fUnction is a function

Constant functions and

e

h

are idempotent.

such that

hh = h

A fact that will prove

important in the sequel about idempotent function and uncrossed groups is now given. Every non-monogenic uncrossed group of scales has a sequence of positive scales decreasing to an idempotent function. Proof:

If a sequence of scales 2 e < 1'1 < 1'1 n+l n

in an uncrossed group has the property

{h ) n

for each

then it converges to an idempotent.

n First this is proven and then it is

proven that every uncrossed non-monogenic group contains such a sequence. If the sequence {h) in an uncrossed group has the property, then n 2 e(x) < h n +l(x) < hn+ lex) -< 1'1n (x) for each x. Thus, the sequence of numbers {hn(x») is monotonic and bounded and therefore convergent. Thus, for all i t is

converges pointwise to a function

[1'1 }

n

n

,

x ::: hex)

non-decreasing and

Since hex)

1'1 ::s

h

Since

x < 1'1 (x) n

is the limit of increasing functions, hh(x)

for all

x

-55To conclude h

h

hh(x) S. h(x)

is idempotent we show

Since the

are continuous

n

I

hh(x) :;;; inf {h nhm(x)

n,m > l} -

hex) If

n :;;;

m:;;;l+l

h

(h ) n

1 , then by the hypothesis that

2

h nhm(x) :;;; hi+(x) < ht(X) • 1-

has the property, and

l~

for any

Hence

hh(x) ~ hex)

is idempotent.

The theorem is proven when it is shown that each non-monogenic uncrossed group has a sequence

such that

{h ) n

~

clearly sufficient to show that if group and

G

h

is positive in

such that

g

2

< -

h

e < h

2

It is

n+ 1 -< h n is a non-monogenic uncrossed

~ then there is some positive

(Recall

that

{e}

is defined to be monogenic

so non-monogenic, uncrossed groups have positive elements.) any positive element of non-monogenic uncrossed has no least positive element there is some 2

h

< g2

Since uncrossed groups are abelian,

h g

e < g < h

Either

the positive scales than

h

g

~ h

hg -1

or

or

g

2 -2

in

g

g If

G in

Let

~

be

G

Since such that

2 h < g ,then

:;;; (hg- l )2

h

2 -2 h g < h

and at least one of

is such that its square is no greater

This concludes the proof.

Since a monogenic uncrossed group has no nontrivial convergent sequences, this result gives a third characterization of the monogenic groups.

-56V

V.l

pARALLEL SYSTEMS WITH NON -MONOGENIC ASSOCIATED GROUPS

Completions and Completely Uncrossed Sets of Scales The following proposition is offered to introduce the final

strengthening of "uncrossed ll • If

f

is the limit of a sequence of scales in the associated

group of a parallel system with solution

Proof:

For each sequence

{f}

real constants and

u

{k} n

f (.) = u-l(u(.) + k )

are continuous and since

its limit must be

f(.)

(k} n

= u-l(u(.)

Since for each scale

k,

u, there is a corresponding sequence of

such that

(pointwise) if and only if

then for some

of scales in the associated group of a

n

parallel system with solution

u

n

k

Since

n

n

equals

does.

uf

n

u-l(O)

Furthermore, if

{f } n {f } n

u

-1

converges converges

+ lim k ) n

u, the set of functions of form

u-l(u(.) + k)

is a parallel system we have at once a new necessary condition on parallel systems. V.l.B

Limits of sequences of scales in the associated group of a

parallel system are uncrossed scales. This motivates some definitions. The completion of a set of scales is the set of all functions which are (pointvnse) limits of sequences of scales in the set.

A set of scales

is completely uncrossed if the completion of its associated group is uncrossed.

The relation between strongly uncrossed and completely uncrossed is somewhat delicate.

Completely uncrossed implies strongly uncrossed

because the completion of a group clearly contains the group.

Thus a set

of scales with a non-monogenic group is completely uncrossed if and only if it is strongly uncrossed.

If a set of scales is uncountable, then

it is completely uncrossed if and only if it is strongly uncrossed.

How-

ever, there are finite and countable sets of scales which are strongly uncrossed but are not completely uncrossed.

Examples will be given on

request or in a later paper. The meaning of completely uncrossed should be much clearer after the following section V.2.

Here we show that a set of scales is completely

uncrossed if and only if it is a parallel system.

During the proof a

number of other necessary and sufficient conditions are obtained.

The

remainder of section V contains uniqueness results for non-monogenic parallel systems and informal remarks on the calculation of solutions. V.2

Necessary and Sufficient Conditions for the Existence of Solutions Some

interrelated necessary conditions are derived and then these

conditions are shown to be sufficient. Since the completion of the associated group of a parallel system is a set of uncrossed scales, it follows that each function in the completion is continuous. V.2.A

If

Using this fact we obtain a new necessary condition.

~ is a stronglY uncrossed set of scales such that each

function in the completion of the associated group is continuous, then

~ is monogenic or it has a sequence

either the associated group of of positive scales decreasing to

e.

Since parallel systems satisfy the hypothesis we have another necessary condition. As ShOIVll in section IV.5

Proof:

either the associated group is mono-

genic or there is a sequence of positive scales idempotent function to establish

h

h(XO)

= Xo

The set

We use the hypothesis that

h

is strictly increasing,

n

x

Let

x ::: hex)

= hh(XO) =

h(xl )

is continuous

is non-decreasing and

be freely chosen.

o

Put

h(xO)

h

Xl

= Xl

= h(xO)

We show

Since

h

is

.

h~l(Xl) since h~l(Xl) > x implies

Xl > hI (x) > h 2 (X) > hex)

:::

Xl

decreasing,

Thus

sup {hex)

h(h(x»

X < x2

Thus

I

{hex)

I

= hex)

implies

x < X2 ) ~ x 2 Since

Let

x2

h(x2 )::: Xl

= inf

h(x2 )

hex) > x 2 ' then A

h

A

is bounded below by

x 2 ::: Xo

X

to complete the proof.

idempotent,

decreasing to an

n

=e

Since each for all real

h.

{h)

be the infimum of Since

x€A) ~ Xl ::: h(x 2 )

h(~)

= Xl

which implies

This gives

Xl

h

=e

If x

is in

= h(x2 ) =

x 2 ::: Xo ~ Xl ~ x 2 ; i.e.,

is arbitrary,

Clearly

is continuous and non-

and

= Xl

hex) ~ x 2 Thus

h

A

h(XO)

=

, and the proof is complete.

-59We extend this chain of necessary conditions by one final link in the next theorem. V.2.B

~ be a group of uncrossed scales containing a sequence

Let

of scales decreasing to

e

Then for each x, the set A(x)

is dense in

R

Proof:

x

Let

there is some Let

and z

y be real.

in

A(x)

{hex)

=

Let

such that

n

be positive.

Iy - z

B

=

Thus for some

n, if

The subscript

n

I

We show that

~ E

~ to

e

Since

(h} n

e, the sequence converges uniformly

B,

on the closed interval

n

E

heG}

be a sequence decreasing in

{h}

converges monotonically to continuous

h

I

A(x)

(zl z

Ix is in

zl

< Ix -

B then

yl

+ E)

h (z) < e(z) + n

=Z

E

remains fixed throughout this proof so

h

+

E

instead of

will be written. Since

e< 11 {h,e}

h

is in a sequence of scales decreasing to

Thus for all

z

in

is a parallel system.

h(-) = u-1(u(.) + k)

k

B,

z < h ( z) < z +

is positive since

u-l(u(.) + rk) , the numbers

hr(x)

to plus or minus infinity as

r

e > h

we have

Since

E

Thus there isa scale

e

u

h > e ,

such that Since

hr(x) =

are strictly increasing and diverge

tends to plus or minus infinity

-w Thus f' or exactly one i n t eger

· 1y. respec t 1ve Since

h

lhr(x) If

r

is in