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DISCORD IN POSSIBILITY THEORYJAMES F. GEER a & GEORGE J. KLIR aa Department of Systems Science , Thomas J. Watson School, State University of New York ,Binghamton, New York, 13902-6000, U.S.APublished online: 06 Apr 2007.

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Printed in the United Kingdom

DISCORD IN POSSIBILITY THEORY

JAMES F. GEER AND GEORGE J. KLIR

Department of Systems Science, Thomas J. Watson School, State University ofNew York, Binghamton, New York 13902-6000, U.S.A.

(Received January 28, /99/; inflna/form March 5, /99/)

A function referred to as discord was recently introduced as a justifiable measure of one type of un­certainty in the Dempster-Shafer theory. uncertainty (hat results from conflicts among evidential claims.In this paper, we investigate mathematical properties of this function within the domain of possibilitytheory. We show, in particular, thaI the largest value of discord in possibility theory is about 0.892.This result implies that discord is negligible (in comparison with nonspecificity) in many practical ap­plications of possibility theory, especially when large bodies of evidence are involved.

INDEX TERMS: Possibility theory, uncertainty, discord, nonspecificity.

I. INTRODUCTION

In this paper, we assume the knowledge of basic notions of the Dempster-Shafertheory and possibility theory. We employ the notation introduced in a previous, closelyrelated paper. 8

It is well known that two types of uncertainty are involved in the Dempster-Shafertheory. One is connected with nonspecificity of evidential claims, the other withconflict among evidential claims. A well justified measure of nonspecificity wasproposed in the early 1980's' and its uniqueness was proven under appropriate ax­ioms in 19877

.9 Some measures of conflict were also proposed in the early 1980's,'·12

but each seemed to be deficient in some way. After examining these deficiencies,Klir and Ramer developed a new measure of conflict that is intuitively fully satis­factory." They also showed that the measure possesses all properties one would ex­pect of it. 10 The measure was given the name discord; it is expressed as a functionD given by the formula

[IA n BI]

D(m) = - L m(A)log2 L m(B) I I 'AEF BEF B

(I)

where m denotes the basic assignment defined on the power set of a finite universalset, F denotes the set of all focal elements, and IA n BI, IBI denote the cardinalitiesof the respective sets.":"

The purpose of this paper is to investigate mathematical properties of the functionD within the domain of possibility theory, i.e. for nested bodies of evidence.r" Toconvert equation (I) into a convenient possibilistic form, let us introduce appropriatenotation ..

Assume that X = {x" X2, ... , xn} and let AI C A2 C ... C An, where Ai = {XI'X2, ... , X;}, i = I, 2, ... , n, be a complete sequence of nested subsets that contain

119

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120 JAMES F. GEER AND GEORGE J. KLIR

all focal elements of a possibility measure. That is, meA) = 0 for each A rt {A"A2, ... , An}. Let m, = meA,) and r, = r(xi) for all i = I, 2, ... n. Then the n-tuples

m = (m" m-, ... , mn), r = (r l , r20 ... rn) (2)

fully characterize the basic assignment and possibility distribution, respectively, bywhich the possibility measure is defined. As a result of the nested structure of thefocal elements, the possibility distribution r is ordered in the sense that r, ;;; r i + 1 forall i = I, 2, ... , n - I. It is well known that

ri =I m, and m, =r, - ri + l , (3)k=i

for all i = I, 2, ... , n, where rn+, = 0 by convention.i"Using this notation, it is easy to derive the following possibilistic form of the

measure of discord:

Furthermore, using the replacements defined by equation (3), we obtain

n-I [n ]D(r) = - ~ (r, - ri+I)log2 I - i J~' rJ(j(j - I» . (5)

In our analysis of possibilistic discord, we employ this form.Let us remark that possibility theory can also be formulated in terms of normalized

fuzzy sets. '3 In this interpretation of possibility theory, focal elements associated witha possibility distribution correspond to distinct a-cuts of the fuzzy set from which

. the distribution is induced." The discord may then be expressed by the form

r' [ r' lAo n Ba·1 ]D(r) = Jo log, Jo IBa.,1 do' da, (6)

where Aa and B; denote a-cuts of the fuzzy set (assumed to be defined on a finiteuniversal set X):'

Aa = {x E Xlr(x) ;;; a}, Ba · = {x E Xlr(x) ;;; a'}.

2. DEFINITIONS

For any positive integer n ;;; '2 and for real numbers {r,} satisfying I = r, ;;; r2 ;;;... ;;; r; ;;; 0, we define

n-I

D; = Dn(r" r2, ... , rn) := -I: (r, - ri+I)log2(1 - 'Yn.,.) > 0,'j=1

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where

DISCORD IN POSSIBILITY THEORY

•'Yn.1 := i 2: rJ(j(j - I)).

j=i+1

121

We then define the vector r :'= (r,; r2, ... , rn) and define the subset '2Iln of R" by

<ziJn := [r , with I = r l ~ r2 ~ ... ~ r, ~ O}

Finally, we define the constants D: by

D: := Max. D.(r).rE~",

(Note: Since Dn is a continuous function of each component rj for r E '2Iln and since'2Iln is a compact subset of R", the function D; actually attains its maximum valuesomewhere either in the interior of <ziJ n or on the boundary of <ziJn • Thus the constantsD: are well defined.)

3. SOME GENERAL PROPERTIES OF THE FUNCTIONS o,

We now prove some general properties of D; and D:. To do this, we first prove thefollowing lemma.

LEMMA I The quantities 'Yn.1 which appear in the definition of D'; satisfy the fol­lowing properties:

a. 0;;; rn(l - i/n) ;;; 'Yn.';;; r'+I,(l - i/n) ;;; I - ifn, for all i = I, 2, ... , n - I;

b. 0;;; -log2(l - rn(l - i/n)) ;;; -log2(l - 'Yn.'> ;;; -log2(l - r,+I(l - i/n)) ;;;-[og2(i/n), for all i = 1,2, ... , n - I;

c. 'Yn+1.i = 'Yn.' + irn+,/(n(n + I», for all i = 1,2, ... , n;

Proof (a) From the definition of Y«, we have, since rj ~ rj+1 for each j,

•'Yn.' =;2: rJ«j(j - I));;; ir,+, 2: I/(j(j - I»

j=i+1 j=i+J

= r,+,(1 - i/n) ;;; I - ifn,

and

•'Yn.' =i 2: rJ«j(j - I)) ~ ir; 2: I/(j(j - I» =rn(l - i/n) ~ 0,

j=i+ 1 j=i+ I,

for i = I, 2, ... , n - I.

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122 JAMES F. GEER AND GEORGE J. KLIR

(b) From part (a) we can write

i/n = I - (I - i/n) ;§; I - r,+l(l - i/n) ;§; I - 'Y.,.' ;§; I - r.,(1 - i/n) ;§; I

and hence

°= -log2( I) ;§; -log2(1 - r.,(1 - i/n)) ;§; -log2(1 - 'Y.,);§; -log2(1 - ri+l(I - i/n)) ;§; -log2U/ n)..

(c) Using the definition of 'Yn.' with n replaced by n + I we find

n+l

'Y.,+I.' =;2: rJ(j(j - I))j=i+ I

= i 2: rJ(j(j - I)) + ir.,+,/(n(n + I)))=;+1

= 'Y.,.' + ir.,+,/(n(n + I)).Q.E.D.

We now use this lemma to prove the following properties of D; and D':.

PROPERTY I D':+, > D':, for n = 2, 3, ....

Proof We first show that D':+I ~ D::', for n = 2,3, .... Using the definition ofD; we can write, for any r E 9)"+1'

.,-1

Dn+ 1 - D; = - 2: tr, - r,+I)]og2(1 - 'Y.,+I) + '2: (r, - r'+I)]ogz{1 - 'Yn,;)i=1 ;=1

.,-1

-2: tr, - r'+I)log2[(1 - 'Y.,+I)/(I - 'Y.,.,)];=1

.,-1

-2: tr, - r,+,)!og2[1 - irn+1/(n(n + 1)(1 - 'Y.,))];=1

- (r" - rn+I)log2( 1 - r"+I/n).

In obtaining this last equality we have used part (c) of Lemma I. Also, using part(a) of Lemma I, we see that °;§; irn+I/(n(n + 1)(1 - 'Yn)) ;§; ir.,+,/(n(n + I)U/n))= r.,+.!(n + I) < I. Thus, each term in the above expression is non-negative andhence

Dn+ 1 - D., ~ 0, or D.,+, ~ D';..

Now let D': = Dn(rt, rt, ... , r':). Then for r = (rt, rt, ... , r':, r.,+ I) (which is in

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the set S!lI.+ I) we have

and hence

DISCORD IN POSSIBILITY THEORY 123

D:+, := Max. D.+,(r) ~ D:.rES\,+1

Next, we show that D.+,(r" r" ... , r., 0) = D.(r" r" ... , r.). To see this, fromthe definition of D.+, we can write

.-,D.+ 1 -'2: (r, - ri+,)log,(l - 'Y.+1.,) + ir, - r.+I)log,(l - 'Y.+,.,.).

'i=1

But from Lemma I, part c), we see that

while from the definition of 'Y.+ I.. we see that

'Y.+1.. = r.+,/(n + I) = 0, when r.+, = O.

Thus the last term in our expression for D.+ 1 vanishes and we find

.-,D.+,I".,_. = -'2: (r, - ri+,)log,(l - 'Y•.,) = D ...

;=1,

In particular, D.+,(rf, rt . ... , r:, 0) = D.(rf, r1', ... , rn = D:.Finally, we note that

ilD.+I/ilr.+I!. _. = 2: [(r, - r,+,)/log.(2)](l - 'Y.+,,;)-'i/(n(n + I)) > O.,+,i=1

Using this result, it follows easily by induction on the index n that r: > O. ThusD.+ ,(rf, r1:, ... , r:, r.+ I) > D.+ ,(rf, r1, ... , r:, 0) = D: for all sufficiently smallvalues of r.tl satisfying 0 < r.+, ~ r:. Hence, D:+ , > D:.

Q.E.D.

PROPERTY 2 D: ~ log,(e) = 1.443, for all n ~ 2.

GEN, SYS,-B

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124 JAMES F. GEER AND GEORGE 1. KLIR

Proof Using the definition of Dn as well as part b) of Lemma 1, we can write

n-I

D; = -'2: ir. - ri+I)logz(1 - 'Yn,i)i=1

n-I

~ -'2: tr. - ri+I)logz(1 - ri+l + iri+l/n)i=1

n-I

~ -'2: (r, - ri-l-1)logz(1 - ri+I)'i=l,

We now observe that -logz(1 - ri+l) ~ -logz(1 - x) for all r i+1 ~ x ~ r., i = 1,2, ... , n - 1. Thus we can write

- fn 10gz(1 - x)dx ~ - fn 10gz(1 - ri+l)dx = -(ri - ri+I)logz(1 - ri+I)"n+1 "+1

Using this relation and the inequality above involving Ir., we have

D. ~ -~ f. )Og2(l - x)dx

= - fl 10gz(1 - x)dxr«

= [(1 - r n)(1 - logil - rn»]/logi2)

~ 1/logi2) = logz(e).

Since D'; ~ logz(e) for any choice of the parameters {r} satisfying 1 = rl ~ rz ~'

. " ~ r, ~ 0, we also must have that D; ~ log-te).Q.E.D.

Thus Property 2 establishes that the maximum discord D; associated with a possi­bility distribution with n components is bounded from above for all values of n.Also, we note that, since by Property 1 above the constants {D;} form a monoton­ically increasing sequence of real numbers and by Property 2 they are uniformlybounded from above, the sequence {D;} approaches a unique limit, say D'j;, as n ~00. Thus, we can write

D'j; := lim D; ~ logz(e) = 1.443.

4. NUMERICAL RESULTS

The problem of determining the constants D; can be solved in a purely numerical

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Table I

DISCORD IN POSSIBILITY THEORY 125

D*n

rrrtrtrtrtrt

n=20.20885

0.540

30.30651

0.6100.457

40.36715

0.6510.5130.416

50.40998

0.6780.5510.4620.390

60.44255

0.6980.5790.4950.4280.372

70.49000

0.7140.6010.5210.4580.4050.359

manner by using any of a number of standard optimization techniques. In Table Ibelow we have recorded the values of D;, as well as the values of the parameters{rf} where these maximum values are attained, for a few small values of n. Theseresults were obtained using an iterative gradient search technique known as the Fletcher­Reeves method." The numerical results displayed in Table 1 are believed to be ac­curate to the number of decimal places shown.

In Figure 1 we have plotted D; as a function of n for 2 ~ n ~ 21, while in Figure2 we have plotted the values of rf for 2 ~ j ~ n, as n varies over this same range.It is interesting to observe how the values of D; appear to be "leveling off" as nincreases, while the values {rf} appear to be settling into a definite distribution as nincreases. In particular, the {r/'} appear to be approaching a distribution which is

D: 1.0 ~------.-------..........------...--------....,...----.

0.8

0.6

0.4

0.2 o

o

oo

5

oo

o 0o 0

o

10

o 0 0o

15 20 n

Figure 1 The maximum discord D: plotted as a function n for 2 ~ n ~ 21. The values of D: wereobtained by numerically optimizing D; subject to the constraints that 0 ~ r; ~ rn - 1 ~ '" ~ r 1 = 1.

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126

n

JAMES F. GEER AND GEORGE]; KLIR

CXXlOOOOOOOO 00 0 0 0 0 0 0 0

20 00000000000 0 0 0 0 0 0 0 o

00000000000 0 0 0 0 0 0 0

000000000 0 0 0 0 0 0 0 0

000000000 0 0 0 0 0 0 0

00000000 0 0 0 0 0 0 0

15 0000000 0 0 0 0 0 o o

10

5

o

00000000000 0 0

00000000000 0

000000000 0 0

000 0 0 0 0 0 0 0

o 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0

0 0

0

0.5 1.0

rJ

Figure 2 The optimal values rt of r j (2 ~ j ~ n) which maximize Ir., plotted as a function n for 2~ n ~ 21. These values were obtained by numerically optimizing D; subject to the constraints that 0~ rn ~ rn-l ~ ••• rl = 1..

asymptotically dense in some interval [a, b], where 0 ~ a < b ~ I, as n approachesinfinity.

Using the computed values of DC; we estimate that

D"!, = lim D; = 0.892.

This result was obtained by using the least squares method to fit a curve of the formD: = a + bn- 1

! 2 + cn- 1 to the last ten data points and then extrapolating the resultto n = 00, i.e. to lin = o.

5. SPECIAL POSSIBILITY DISTRIBUTIONS

We now examine two special possibility distributions for which we can make somedefinite analytical statements regarding D; and D:.

Example 1: r1 = I and r, = E, for 2 ~ i ~ n, where 0 ~ E ~ 1.In this case we find

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DISCORD IN POSSIBILITY THEORY

From this expression it follows that

Dn(O) = 0 = D~(1),

(dldE)DnCE) = (1/loge(2))[(a - x)/(1 - x) + log.t l - x)],

and

127

where x = E(1 - lin) and a = 1 - lin. We note that 0 ~ x ~ a and 1/2 ~ a <1. Thus, for this special case, we see that (dldE)2Dn < 0 and hence we find that

Dn := max Dn(E) = -(1 - E*)log2(l - E*(1 - lin)),O~€~I,

where E* = x* 1(1 - 1In) and x* is the unique solution to the equation

(a - x*)/(1 - x*) + 10ge(1 - x*) = o.

We note that, as n ~ 00, a ~ 1 and hence 10ge(1 - x*) ~ -1. Thus x* ~ 11[e, which implies that E* -~ 1 - 1[e. Thus,

In particular, since D; :;::::; o, for each value of n, we observe that

D'j;, = lim D; ~ lim Dn = [elogeC2)r 1, = 0.532,

n~OO n----;)oo

which provides us with a lower bound for D'j;,.In Figure 3 we have plotted Dn(E) for 0 ~ E ~ 1 for various values of n. The

figure clearly illustrates that the {Dn} are uniformly bounded for all values of n.

Example 2: rl = 1 and the remaining {r} (for 2 ~ j ~ n, with n > 2) are uniformlydistributed between two real numbers a and b satisfying 0 ~ a ~ b ~ 1.

In this case we can write

rj = b - (j - 2)d, for j = 2, 3, ... , n, where d = (b - a)/(n - 2).

Then, for this case,

n-l

Dn = -(1 - b)logll - 1'1) - 'L tr. - ri+ 1)log2(l - I'i)i=2

n-l

= -(1 - b)logll - 1'1) - ((b - a)/(n - 2)) 'L log2(l - I'i),i=2

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128 JAMES F. GEER AND GEORGE J. KLIR

o 1-1Ie

€

Figure 3 A plot of D.(E) in Example I for n = 2, 5, 10, 25, 50 and 100 as E varies between 0 andI. The horizontal dashed line represents the value (elog,(2))-', which the maximum values of D.(E)approach asymptotically as n approaches infinity.

where

n n

y; = iL rj(j(j - I)) = i L (b - (j - 2)d)1(j(j - I)),

or

]=;+1 j=;+1

n

Y, = a{3; + b(1 - (3; - if n), with {3; = (i/(n - 2))· L (j - 2)/(j(j - I)).j=i+ I

Thus, we can write D; as

Dn = (b - 1)log2[1 - a{3, - b(l - (3, - lin)]

n-'+ «a - b)/(n - 2))L log2[1 - a{3; - b(l - (3i - iln)].

;=2

We now wish to use this expression to obtain an improved lower bound for D:ias n ~ 00, i.e. we wish to choose a and b so that D'; is a maximum. To this end,

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DISCORD IN POSSIBILITY THEORY

we first make use of the fact that

!~ [~ I/j - IOg,(n)] = 'Y = 0.5772156649 ... ,

where 'Y is Euler's constant,' to write, for a fixed value of i,

n

0< {3i = (i/(n - 2» L [I/j - I/j(j - \)] < (i/(n - 2» L I/j)=i+1 j=l

< (i/(n - 2))[log,(n) + 'Y + I],

129

for all sufficiently large n. Thus, for any fixed value of i, (3i ---+ 0 as n ---+ 00. Also,we can write

n

2: l/j = L I/j - L I/j = -Iog..(i/n) + 0(1), as i, n -.,> 00,

)=;+1 )=1 )=1

where 0(1) represents a term which approaches zero as n ---+ 00. Thus, we can write

n

{3i = (i/n)(n/(n - 2» L [(I/}) - 1/(jU - 1»1J=i+l

= -(i/n)log..(i/n) + 0(1), as i, n ---+ 00.

Using this estimate for {3i we can write, as n ---+ 00,

D n = (b - I )log2(1 - b)n-I

+«a - b)/(n - 2» L log2[1 - b + (i/n)(b + (a - b)log..(i/n))] + 0(1).;=2

Thus,

D~ = (b - 1)log,(1 - b) +

(a - b) fa' log-]] - b + x(b + (a - b)log,(x»]iU := Dro(a, b).

As an application of this case, we can set b = I-a. Then our expression foro: becomes

D'; = -alog2(a) + (2a - I) fa' log2[a + x(l - a + (2a - I)log..(x»]dx.

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130 JAMES F. GEER AND GEORGE J. KLIR

Letting a -+ 0+, we find

o: = - Llog,(x(l - log,(x)))dx = [I - Llog,(l - log,(x»dx]/10&(2)

But

L10g,(1 log,(x»dx = fOO e'-Ulog,(u)du

= fOO ei-u(l/u)du = eE,(l) = e·(0.219383934 ... ).r.

where £,(1) denotes the exponential integral' £iz) , with n = I and z = I. Thus,

D'; = (I - e£,(1»/log,(2) = 0.582346.

and hence

D~ ii;; Ir; = 0.582346 ....

This provides us with a (somewhat) improved lower bound for D~ than we foundfrom our first example..

In general, the problem of determining a and b, with 0 ~ a ~ b ~ I, so thatDn(a, b) is a maximum can be solved numerically in a straight forward manner. Inparticular, again using the Fletcher-Reeves method, we estimate that the optimalvalues of a and b which maximize Doo(a, b) are

a* = 0.000 and b* = 0.817, with Doo(a*, b*) = 0.766.

This result, when used with Property 2 above, provides us with an interval estimatefor D~, namely

0.776 s D~ ~ 1.443.

We note that we have not established that either of the limits in this expression is"sharp." That is, by the way in which it was obtained, the lower limit of 0.776 isa lower bound for D~, but we have not shown that it is the greatest lower boundfor D~. An anologous statement applies to the upper limit of 1.443. We also notethat the estimate of D~ '" 0.892, obtained by extrapolating the numerical resultspresented in Section 4, is consistent with this interval estimate.

6. CONCLUSIONS

Possibilistic bodies of evidence are often called consonant. This name is a result ofa common supposition that evidential claims focusing on nested subsets are free of

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DISCORD IN POSSIBILITY THEORY 131

conflict. When conflict is measured by discord, a function that is well justified onboth intuitive and mathematical grounds, we see that this supposition is not correct.Possibilistic bodies of evidence involve conflict as well. Although an evidential claimfocusing on a set does not conflict with other claims focusing on its various supersets,it conflicts with those focusing on any of its proper subsets. Hence, there is noconflict if and only if the whole evidence focuses on exactly one set.

Although we realize now that possibility theory is not fully conflict free, resultsof this paper indicate that the conflict is rather modest. Our major results are thatthe maximum value of discord increases with the number of nonzero components inthe possibility distribution, n, and converges to a constant, estimated as 0.892 as n--+ 00. This estimate, which is based on an extrapolating numerical results calculatedfor n ~ 21, is safeguarded by an analytic result (Property 2), according to whichthe maximum discord cannot exceed the value 1/log,(2) = 1.443 for any n.

We may conclude from these results that the discord, whose range in the Demps­ter-Shafer theory is [0, !Og2n], is severely constrained within the domain of possi­bility theory. We may say that possibility theory is almost conflict-free. For largepossibilistic bodies of evidence, at least, discord is likely to be negligible when com­pared with the other type of uncertainty-nonspecificity. Neglecting discord, whenjustifiable, may substantially reduce computational complexity in dealing with largepossibilistic bodies of evidence.

REFERENCES

I. Abramowitz, M., and Stegun, I., Handbook of Mathematical Functions. Dover, New York. 1970.2. Dubois. D.. and H. Prade, Possibility Theory. Plenum Press, New York, 1988.3. Higashi, M.• and G. J. Klir, "Measures of uncertainty and information based on possibility distri­

butions." Intern. J. oj General Systems. 9. No. I. 1983. pp. 43-58.4. Hohle, U.. "Entropy with respect to plausibility measures." Proc. 12th IEEE Intern. Symp . on

Multiple-Valued Logic. Paris, 1982, pp. 167-169.5. Jeter, M. W .• Mathematical Programming. Marcel Dekker. New York. 1986.6. Klir, G. J .. and T. A. Folger. Fuzzy Sets. Uncertainty. and Information, Prentice Hall, Englewood

Cliffs. N.J. 1988.7. Klir, G. J. and M. Mariano, "On the uniqueness of possibilistic measures of uncertainty and in­

formation." Fuzry Sets and Systems. 24, No.2. 1987. pp. 197-219.8. Klir, G. J. and A. Ramer, "Uncertainty in the Dempster-Shafer theory: a critical re-examination."

Intern. J. of General Systems. 18, No.2, 1990, pp. 155-166.9. Ramer. A. "Uniqueness of information measure in the theory of evidence." Fuzzy Sets and Systems,

24. No.2. 1987. pp. 183-196.10. Ramer. A.. and G. J. Klir. "Measures of conflict and discord." Information Sciences (to appear).II. Shafer. G .. A Mathematical Theory oj Evidence. Princeton Univ. Press, Princeton, N.J .. 1976.12. Yager. R., "Entropy and specificity in a mathematical theory of evidence." Intern J. of General

Systems. 9. No.4. 1983, pp. 249-260.13. Zadeh. L. A.• "Fuzzy sets as a basis for a theory of possibility." Fuzzy Sets and Systems. 1. No.

I. 1978, pp. 3-28.

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132 JAMES F. GEER AND GEORGE J. KLIR

Dr. James Geer is a full professor in the Systems Science Department of theThomas J. Watson School of Engineering, Applied Science, and Technologyat the State University of New York at Binghamton, where he also holds thetitle of professor in the Department of Mathematical Sciences. He has servedas a visiting scholar at Stanford University, M.1.T., University College London,and the Courant Institute of Mathematical Sciences at New York University.Prior to joining SUNY-Binghamton, Dr. Geer worked as a staff mathematicianfor IBM Endicott (Glendale Laboratories). He is a mathematical consultant toseveral industries and also to the National Aeronautics and Space Administra­tion at the Langley Research Center. He has numerous publications in the areaof the analysis of physical systems, including mathematical physics, engineer­

ing mathematics, perturbation methods, numerical analysis, and differential equations. Dr. Geer regu­larly teaches graduate level applied mathematics courses in these areas, as well as in the areas of prob­ability and statistics, operations research, mathematical modeling and simulation, and was recently awardedthe first University Award for Excellence in Graduate Teaching. He has also been very active in inte­grating the use of personal computers into classroom instruction. Dr. Geer has presented special lecturesand seminars at oyer forty universities in this country and in Europe. He is a member of the Societyfor Industrial and Applied Mathematics and was recently elected to the American Academy of Mechanicsfor his work in fluid mechanics. Dr. Geer received his Ph.D. in applied mathematics from New YorkUniversity.

For biography and photograph of George J. Klir, please see Vol. 17, Nos. 2-3. 1990, p. 275, of rhisjournal.

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