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Page 1: PImiMU EPSILON

P I m i M U EPSILON

I - I 'I

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THE PME MOTTO: MEANING A N D ETYMOLOGY

Panos D. Bardis

P ro fes so r o f Sociology and Ed i to r of S o c i a l Science

The Un ive r s i ty o f Toledo, Toledo, Ohio, USA

"Sos i c ra t e s s a y s t h a t P i t t a c u s , a f t e r c u t t i n p o f f a smal l fragment, dec l a red t h a t t h e h a l f was niore t han t h e whole ."i:

Dioeenes L a e r t i u s , P i t t a c u s , 75.

Ten paedeusin cae t a nathemat ica epispeudein: t o i romote scho la r- s h i p and mathematics.

1. Ten: feminine tender, s i n g u l a r number, a c c u s a t i v e ( o b j e c t i v e ) c a s e o f d e f i n i t e , o r p r e p o s i t i v e , a r t i c l e &, he, to: t h e .

In h i s D e f i n i t i o n s , Y I V , 1-6, Hero of Alexandria w r i t e s : " In pre- s e n t i n g t o you a s b r i e f l y a s p o s s i b l e , 0 most i l l u s t r i o u s Dionysius, an o u t l i n e of b a s i c t e c h n i c a l terms i n geometry, I w i l l t a k e a s t h e s t a r t i n g p o i n t , and w i l l base , the e n t i r e o r g a n i z a t i o n on, the t e a c h i n g o f Euc l id , t h e au tho r o f ~ l e m e n t r d e a l i n ~ wi th t h e o r e t i c a l geometry" ( s e e Panos 3. Bardis , "~e-~a Vinci of Ancient Alexandr ia : His Aeolosphaera and Other Invent ions ," School Science and Mathematics, June 1965, pp. 535-5142).

2. Paedeusin: feminine gender , s i n g u l a r number, a c c u s a t i v e case of noun paedeusis : c u l t u r e , educa t ion , educa t iona l system, i n s t r u c t i o n , l e a r n i n g , s c h o l a r s h i p , t r a i n i n g .

P l a t o s t a t e s t h a t "an e x c e l l e n t breedinc and educat ion, i f emphasized c o n s t a n t l y , gene ra t e s good na tu re s i n t h e p o l i t y" (Republ ic , b2lla).

From paes: c h i l d .

King Nostor, t h e m i l i t a r y f e n i u s and ga r ru lous Methuselah of Pylos , s a i d t o King Diomedes of Argos, i h e second g r e a t e s t Greek hero- - - af ter Achilles---durinp "ie Trojan War: "Besides, you a r e s o younp, you could even be my e, my youngest born" (Homer, I l i a d , I X , 57-58].

Engl ish words: pedagog, pedagogy ( c h i l d l ead in? ) , p e d i a t r i c i a n , ~ e d i a t r i c s ( c h i l d medicine) . pedobaptism, p e d o c e p h a l i c pedogenesis ( in t roduced by Von Baer i n 1828). pedology, pedo"orphism, and many o t h e r s .

3. z: conjunct ion: and. Also t r a n s l i t e r a t e d 5, I n E u c l i d ' s Elements. XIII, Scholium 1. we r ead t h e fol lowing regard-

ing Theaete tus (415-369 0 . C. ), t h e Athenian as t ronomer , mathematician, phi losopher , and one of t h e p u p i l s of Soc ra t e s : " In t h i s book, namely, t h e 1 3 t h , a r e desc r ibed t h e so- ca l l ed f i v e P l a t o n i c s o l i d s , which,

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ON NOT SEEING THROUGH THE FOREST FOR THE TREES

P r o f . D a v i d C. K a y , T h e U n i v e r s i t y of O k l a h o m a ( T a l k p r e s e n t e d t o t h e O k l a h o m a A l p h a c h a p t e r )

however, a r e not h i s , bu t t h r e e o f t h e aforementioned f i v e s o l i d s a r e of t h e Pythagoreans, t h a t is, t h e cube & t h e pyramid & t h e dodecahedron, t h e octahedron & t h e icosahedron being of Theaete tus ."

Engl ish word: t r i aka idekaphob ia ( f e a r o f t h e number 13 ) .

I n a moment I am going t o acqua in t you wi th a problem which w i l l l e a d u s q u i t e n a t u r a l l y i n t o two d i f f e r e n t f i e l d s o f mathematics---- Minkowski geometry, and number theo ry . It is a n extremely e n t e r t a i n i n g problem, and it may a c t u a l l y have an a p p l i c a t i o n o u t s i d e mathematics which r e n d e r s it p a r t i c u l a r l y heart-warming. The problem is i n t h e form o f two ques t ions , t h e second of which makes sense on ly i f t h e f i r s t has a "No" answer.

4. Ta: n e u t e r gender , p l u r a l number, a c c u s a t i v e c a s e o f d e f i n i t e a r t i c z ho, he, to: t h e .

The famous C a t t l e Problem o f Archimedes ( ? I . which t h e noted inven to r and mathematicran of Syracuse solved i n epigrams and s e n t t o t h e mathema- t i c i a n s of Alexandr ia i n a l e t t e r t o E ra tos thenes , c l o s e s wi th t h e s e f o u r l i n e s : "0 s t r a n g e r , i f you f i n d o u t t h e s e t h i n g s and add them up i n your mind, g i v i n g a l l t h e r e l a t i o n s among t h e s e q u a n t i t i e s , you w i l l d e p a r t g l o r i o u s l y and v i c t o r i o u s l y , knowing t h a t you have been adjudged g r e a t i n t h i s kind o f wisdom."

Consider t h e xy-plane and t h e s e t o f a l l l a t t i c e p o i n t s , p o i n t s o f t h e form (m,n) where m and n a r e i n t e g e r s . Now imagine an i n f i n i t e f o r e s t whose t r e e s a l l have t h e same r a d i u s , s a y r , and cen te red a t t h e l a t t i c e p o i n t s (m,n). Now I f e l l t h e t r e e a t (0,O) and s t and on t h e stump. The ques t ion is, can I s e e through t h e f o r e s t ? Geometr ical ly o f cou r se , I am ask ing whether I can pas s a l i n e through (0,1) which does no t t ouch any c i r c l e cen te red a t a l a t t i c e p o i n t and having r a d i u s r

5. Mathematica: n e u t e r gender , p l u r a l number, a c c u s a t i v e case of a d j e c t i v e mathematicus: fond of l e a r n i n g , mathemat ical .

Mathematice epis teme: mathemat ical s c i ence .

I have prepared a l i t t l e ske t ch o f t h e s i t u a t i o n f o r r = 1/10 which I ' l l l e t you see . Now i f we examine t h e problem c a r e f u l l y f o r t h i s c a s e we observe s e v e r a l s imp l i fy ing p r i n c i p l e s . One is t h a t a s I r o t a t e about t h e o r i g i n seeking a l i n e of v i s i o n o u t o f t h e f o r e s t , it i s not r e a l l y neces sa ry t o cons ide r a l l d i r e c t i o n s . I f I merely answer t h e ques t ion f o r t h e r a y s whose a n g l e s w i th t h e x- axis r ange from 0 t o n/4 , I w i l l be a b l e t o app ly t h a t answer t o a l l t h e o t h e r remaining s e c t o r s , s i n c e t h e s e c t o r o f r a y s from n/4 t o n/2 c o n t a i n s t h e r e f l e c t e d image i n t h e l i n e y = x o f t h a t conta ined by t h e s e c t o r from 0 t o n/4. It is t h e n obvious t h a t , having answered t h e ques t ion f o r t h e f i r s t

I n t h e Nicomachean E t h i c s , A r i s t o t l e a s s e r t s t h a t , " As f a r a s con- duc t is concerned, t h e b a s i c p r i n c i p l e co inc ides with t h e pursued g o a l , which is analogous t o t h e hypotheses (here t h e phi losopher means prop- s i t i o n 9 o f mathematics" (VII , v i i i , 17-18).

From manthanein: t o comprehend, t o know, t o l e a r n , t o pe rce ive .

Mathema: knowledge, something l ea rned , l e s son , s c i ence . Also, mathemat ical s c i ence , e s p e c i a l l y , a r i t h m e t i c , astronomy, geometry ( s e e Panos D. Ba rd i s , "Symmentrical Consonance o f Play, Rhythm, and Harmony: An Essay on P l a t o ' s Mathematics,"School Science and Mathematics, January 1963, pp. 52-67).

Engl ish words: mathemat ical , mathematician, mathematics, polymath, and s o f o r t h .

6. Epispeudein: i n f i n i t i v e o f v e r b epispeudo: I has t en onward, I promote, I urge on.

From + a n d y e u d e i n .

.. a . E&: p r e p o s i t i o n : by, on, ove r , upon, and t h e l i k e .

I n d e a l i n g wi th t h e Pythagorean theorem, Eucl id r e f e r s t o two of t h e l i n e s o f t h e c e l e b r a t e d "windmill" f i g u r e wi th t h e s e words: "two s t r a i g h t l i n e s , namely, AG, AE, no t l y i n g on t h e same s i d e , make t h e ad jacen t ang le s equa l t o two r i g h t onesf9 E l e m e n t s , I , 47).

Eng l i sh words: e p i b l a s t , ep i ca lyx , ep i can thus , epicardium, ep icene , e p i c e n t e r , e p i c r i s i s , epidemic, epidermis epididymis , and myriad o t h e r s . In mathematics: e p i c y c l o i d , e p i t r o c h o i d , e p i t r o c h o i d a l , and many more.

b. Speudein: i n f i n i t i v e o f v e r b speudo: I has t en , I p r e s s on, I promote, I quicken. quadrant , I have answered t h e ques t ion f o r a l l quadrants . So it s u f f i c e s

t o cons ide r on ly t h e s e c t o r from 0 t o 17/4 a s i n d i c a t e d by t h e d o t t e d l i n e s i n t h e diagram I have g iven you. Herodotus informs u s t h a t Queen Tomyris o f t h e Massagetae s e n t t h e

fol lowing message t o King Cyrus when he was marching a g a i n s t h e r king- dom: "0 king o f t h e Medes, cease from promoting what you a r e promoting" f o r vou cannot know i f t h e s e t a s k s w i l l be f o r your advantage when

Be making a few shrewd c a l c u l a t i o n s , we can a c t u a l l y observe an upper bound i n t h e c a s e r = 1/10, and it is approximately 9 ~ 8 . A few o f t h e longe r l i n e s o f v i s i o n a r e shown, l a b e l l e d wi th t h e i r approximate l eng ths . We may c o n j e c t u r e , t h e r e f o r e , t h a t t h e answer must be "no."! I cannot s e e through t h e f o r e s t f o r t h e t r e e s . But might we not be j u m x t o conc lus ions based on i n s u f f i c i e n t evidence? What i f r is exceedingly sma l l and t h e t r e e s i n t h e f o r e s t a r e merely ve ry skinny too thp icks? Is it t h e n conceivable I might s e e through t h e f o r e s t i n some c a r e f u l l y chosen d i r e c t i o n ? I t is obvious t h a t I s h a l l be a b l e t o s e e a l o t f a r t h e r t han I could when r = 1/10, bu t how much f a r t h e r ?

f in i shed" ( H i s t o r i e s , I , 206).

NOTE

* A 1 1 quoted passages have been t r a n s l a t e d by t h e p re sen t au tho r .

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274 The second ques t ion is , assuming t h a t my v i s i o n is blocked by some

t r e e i n every d i r e c t i o n , what is t h e f o r t h e s t I can see? I d e a l l y , t h e s o l u t i o n t o both p r o ~ l e m s would be found by ob ta in ing an expres s ion f o r t h e f u n c t i o n f ( o , r ) which measures t h e d i s t a n c e from t h e o r i g i n t o t h e f i r s t t r e e which t h e l i n e y = ox touches , i f any. I f none touch we could d e f i n e f ( o , r ) = - f o r t h a t o and t h a t r. But when we t r y t o w r i t e down what we t'r.ow al ' ru t f ( o , r ) it becomes c l e a r t h a t t h e r e must be an e a s i e r way.

But t h e , ano the r thought pops r e a d i l y i n t o mind. Suppose we r ephrase t h e ques t ion s l i g h t l y . Given a l i n e y = ox, O<ocl, does a l a t t i c e p o i n t (m,n) come a r b i t r a r i l y c l o s e t o t h i s l i n e , o r c l o s e r t han r , by making some cho ice o f t h e p o s i t i v e i n t e g e r s m and n? The answer is obv ious ly "yes" i f o i s r a t i o n a l . But suppose o is i r r a t i o n a l . Let u s r,et an idea how c l o s e t h e l a t t i c e p o i n t (m,n) is from t h e l i n e y = ox. We can e a s i l y s e e t h a t t h e d i f f e r e n c e between t h e o r d i n a t e s om and n, o r

i s a measure o f how c l o s e ( m . n ) i s t o y = o x . I f t h i s is l e s s t han r then t h e l i n e y = ox w i l l d e f i n i t e l y be c u t o f f bv t h e t r e e cen te red a t (m,n). So o u r problem becomes t h a t o f choosing p o s i t i v e i n t e g e r s m and n s o a s t o minimize

om - n l

where o is a n arbitrarily g iven i r r a t i o n a l , O<o<l.

A s an example o f t h e t r u e n a t u r e o f t h e problem, I have made a few c a l c u l a t i o n s f o r o = 6 NOW t h e r a t i o n a l

approximates</5"(which is 1.41421435.. .) t o w i th in .0001435,.. So

t h e q u a n t i t y we a r e i n t e r e s t e d i n , e q u a l s .1435.... which is not ve ry sma l l . Thus, c l o s e r and c l o s e r approximat ions t o o by r a t i o n a l numbers

2- does no t n e c e s s a r i l y make lmo - nl sma l l . The choice n = 99, m = 70

happens t o work b e t t e r f o r t h i s example, f o r t h e n ,

which g i v e s u s a d i f f e r e n c e of .005 approximately , a s compared wi th

.143. But how can we o b t a i n va lues f o r n and m which make t h i s sma l l e r

t han , s ay 1 0 - 7

Curiously enough t h i s problem appears t o be solved i n Niven's book "An In t roduc t ion t o t h e Theory o f Numbers" fol lowing a d i s c u s s i o n o f Farey f r a c t i o n s . Perhaps you have heard o f t h e s e f r a c t i o n s . For each p o s i t i v e i n t e g e r n one forms a l l p o s s i b l e f r a c t i o n s p/q between ze ro and u n i t y , i n c l u s i v e , such t h a t q i n , and reduce t o lowest terms, p l ac ing them i n numerical o rde r . For example, f o r n = 5 we have

iind p l ac ing thorn i n numerical o r d e r , we have

Th i s would be c a l l e d t h e 5 t h sequence o f Farey f r a c t i o n s . A l l p receding sequences may be obta ined by d e l e t i n g c e r t a i n f r a c t i o n s . For example, t h e 4 t h sequence would be

The 3 rd , and 2nd a r e :

Given a p o s i t i v e i n t e g e r n , however, I can form t h e n t h sequence independ- e n t o f t h e o t h e r s .

Farey f r a c t i o n s have a number o f i n t e r e s t i n g p r o p e r t i e s which w i l l enab le us t o prove t h e r e s u l t we a r e t r y i n g t o o b t a i n , namely, t h a t m a - nl c r f o r some cho ice o f m and n.

The f i r s t o f t h e s e is e a s i l y s p o t t e d from o u r examples* Note t h a t t h e cross- products o f numerator t imes denominator o f two consecu t ive f r a c t i o n s i n any sequence always d i f f e r by un i ty . Picking a few a t random:

This is no a c c i d e n t . Reca l l i ng t h e r u l e o f i n e q u a l i t y f o r f r a c t i o n s wi th p o s i t i v e numerator and denominator,

;< 5 i f f ad < bc, d

we s e e t h a t , s i n c e we have arranged t h e f r a c t i o n s i n numerical o r d e r f o r each sequence, we must minimize t h e d i f f e r e n c e

b x - a y Y b

g iven 5. Since bx - a y is a p o s i t i v e i n t e g e r , t h e l e a s t it can be is -b

one. Hence bc - ad = 1 i f t h e r e e x i s t s a f r a c t i o n i n o u r list f o r - which t h i s d i f f e r e n c e & u n i t y . I t may be t h a t we cannot f i n d such a f r a c t i o n , and i f we were t o prove t h i s p rope r ty o f Farey f r a c t i o n s , we would have s e t t l e d t h i s i s sue . However, it may be proved, indeed from s o b a s i c a p r i n c i p l e a s E n c l i d ' s a l co r i thm, t h a t we can f i n d a f r a c t i o n wi th t h i s p rope r ty .

A second p r o p e r t y , a consequence o f t h e f i r s t , is t h a t i f we t ake t h r e e consecut ive Farey f r a c t i o n s a / b , c / d , e / f t hen c /d = ( a t e ) / ( b t f ) . For example, cons ide r ( 3 A , 2/3 , 3/41. We have ( 3 t 3 ) / ( 5 t 4) = fi/1 = ?/3 . We can add t o t h e seeming t r i v i a o f t h e hour by observing t h a t , s i n c e bc - ad = 1 and de - c f = 1, then

Farey f r a c t i o n s have a b i t o f unexpected power, a s we s h a l l now s e e . Observe t h a t i f we p l o t t h e k t h sequence o f Farey f r a c t i o n s , s i n c e we have conta ined i n t h i s list 1 /k , 2 /k , 3 /k , ... (k - l ) / k , t h e i n t e r v a l s

between consecut ive f r a c t i o n s is l e s s than l / k . Consequently, any r e a l

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number a between 0 and 1 can be approximated t o w i th in 1/k by a Farey f r a c t i o n i n tl ie k t h sequence. Let r > o be c iven. There is a p o s i t i v e i n t e g e r k such t h a t 1 < r. Let u s form t h e k t h sequence of Farey f r a c t i o n s . ^T? Choose t h e l a r g e s t one which is l e s s 0 ; t han o r e q u a l t o 4 . I-

a , s a y n/m. Now 0 1/k 7/k 3/k n' 1 I look a t t h e m B' next Farey f r a c t i o n - i n t h i s sequence, c a l l it n'/m'. I know t h a t mn' - nm' = 1. Hence

n ' 5 mn' - nm' = 1 - - m ' m m'm m u m

But by ou r above obse rva t ion t h i s d i f f e r e n c e cannot exceed 1 / k , s o

1 < 1 m'm - k .

Hence one o f m,m ' must be a t l e a s t a s l a r g e a s i/K. Let us s a y t h a t m' s. k. Observing t h e f i g u r e , we have, c e r t a i n l y ,

Hence

o r

That i s , we have found, usinp, Farey f r a c t i o n s , a l a t t i c e p o i n t (n.m) t h a t i s c l o s e r t o t h e l i n e y = ox than r.

I n t e r e s t i n g a s a l l t h i s is I have j u s t shown you t h e wrong way t o do t h i s problem. For t h e r e i s a way t o answer both ques t ions we a r e a sk ing i n t h e same b r e a t h . A t t h i s p o i n t Minkowski, i f he were h e r e , would be laughing u s r i g h t o u t o f t h i s room t h a t we could b e s o i n e p t . I f I may be allowed t o paraphrase h i s j e e r i n p comments, t h e y would probably RO something l i k e "You bungl ing i d i o t s ! I can n o t on ly s o l v e t h e f i r s t ques t ion about t h e t r e e s i n t h e f o r e s t , b u t t h e second a s w e l l and inc lude what you s a i d about !ma - nl beinp: a r b i t r a r i l y s m a l l , a s a c o r o l l a r y ! "

lie would f e e l e s p e c i a l l y j i l t e d because it s o happens t h a t one o f h i s famous theorems a p p l i e s t o t h e s i t u a t i o n a t hand. Minkowski made ¥lom innovat ions t o number t h e o r y by h i s use o f convex bod ie s and h i s -.turiy o f a c e r t a i n d i s t ance func t ion .

I 0 I F C is a convex curve

ymmetric about t h e o r i g i n , d e f i n e AB

d(A,B) * o,,

whom All I f t h e euc l idean d i s t a n c e from A t o R and 0U is t h e euc l idean r ad ius o f C p a r a l l e l t o l i n e AB. Tho func t ion rt(A.0) t u r n s o u t t o be a mot r i c f o r t h o p l ane , and one may proceed t o s tudy t h e va r ious p r o p e r t i e s t h i s d i s t a n c e conc rp t has . Such a a tudy belongs t o t h e a r e a o f mathematics known a s Minkowskian geometry. I t I s c l o n r t h a t i f C is any c i r c l e t hen o u r m e t r i c is oucl ldonn. Rut i t I s no t t o o hard t o reason t h a t i f C is any e l l i p s e , then t h e resulting Minkowskian geometry co inc ides wi th Eucl idean geometry.

A d i f f e r e n t s o r t of i dea which Minkowski a p p l i e d t o t h e s u c c e s s f u l s o l u t i o n o f many d i f f i c u l t problems o f number theo ry has come t o be known a s Minkowski's Theorem:

Theorem: Given a convex s e t i n t l e p lane symmetric about one l a t t i c po in t and a r e a g r e a t e r t han o r equa l t o 4 , t h a t convex s e t must con ta in a t l e a s t two f u r t h e r l a t t i c e p o i n t s .

To make t h i s p l a u s i b l e , Minkowski reasoned a s fol lows: Let S be a convex body whose a r e a is K i . 4 and cen te red a t (0,O). C lea r ly we prove t h e theorem i f we prove t h a t S inc ludes a t l e a s t one o t h e r l a t t i c e p o i n t , f o r it w i l l au toma t i ca l ly con ta in its r e f l e c t i o n i n (0.0). Using a mapping l i k e x ' = ax , y ' = ay it is c l e a r we may "enlarge" S by a f a c t o r a u n t i l it j u s t touches one o t h e r l a t t i c e p o i n t , and its r e f l e c t i o n i n ( 0 , 0 ) , bes ides (0 ,0 ) . Then s h r i n k t h i s new s e t back by a f a c t o r 1 /2 . The r e s u l t i n g s e t S' has an a r e a of ( a 2 / 4 ) ~ . Let S' be t r a n s l a t e d t o form a system of concruent symmetric convex s e t s i n t h e plane cen t r ed a t each and every l a t t i c e po in t . I t is c l e a r t h a t t h e s e s e t s look something l i k e t h i s : They a r e non-overlapping bu t c e r t a i n n a i r s w i l l touch a t 1 1 . - - - . . - - - . c e r t a i n p o i n t s . By cons ide r ing u n i t squa res a l s o cen te red a t each l a t t i c e p o i n t , s i n c e t h e convex s e t s do n o t completely f i l l up t h e plane--- there a r e c e r t a i n gaps left---we must have

That is (us ing k > 4 )

Hence, t h e f a c t o r by which we had t o " s t r e t ch" t h e o r i g i n a l s e t S s o it j u s t touched two f u r t h e r l a t t i c e p o i n t s was < 1,bearing wi tnes s t o t h e f a c t t h a t it a l r eady conta ined them i n t h e i r i n t e r i o r .

You can e a s i l y make t h i s r i g o r o u s , b u t I ' m s u r e you g e t t h e i d e a Minkowski had i n mind.

Well, I ' v e k e p t you i n suspense long enough. How should t h e f o r e s t problem he solved? Consider a l i n e 1 i n any d i r e c t i o n pas s ing throurh (0 ,0 ) . Let a r e c t a n g l e be desc r ibed symmetr ical ly about 1 a s a x i s , havinp, width 2 r ( tw ice t h e r a d i u s o f o u r t r e e s ) and l en f lh 7d where d is t h e d i s t a n c e we a r e s e e i n g from (0,0) i n t h e d i r e c t i o n o f 1. The a r e a o f t h i s r e c t a n g l e is 7d 7 r = 4dr . By observing a c i r c l e o f r a d i u s r cen te red a t one o f t h e v e r t i c e s , it is obvious t h a t o u r v i s i o n w i l l be blocked by one o f t h e t r e e s i f and on ly i f t h i s r e c t a n g l e con ta ins a l a t t i c e po in t . Hence, i f my v i s ion is n o t b locked i n t h i s d i r e c t i o n , by Minkowski's theorem, t h e a r e a o f t h e r e c t a n g l e has t o be l e s s t han o r e a u a l t o fou r .

Page 6: PImiMU EPSILON

Thus

Therefore, the distance I can see i n any direct ion does not exceed the r e c i p r i c a l of t h e radius of the t rees . I t ' s Touche' once apain by an elementary theorem of reometry.

UNDERGRADUATE RESEARCH PROJECT

P r o p o s e d b y Kenneth Loewen, U n i v e r s i t y o f Oklahoma

Complex numbers may b e d e s c r i b e d a s a t w o d i m e n s i o n a l v e c t o r s p a c e o v e r t h e ~ r e a l numbers w i t h m u l t i p l i c a t i o n g i v e n by

( a + b i ) (c + d i ) = ( a c - b )

( a + b i ) ( c + d i ) = ( a c - b d ) + ( a d + b e ) i :

w h e r e a , b , c a n d d a r e r e a l numbers . Q u a t e r n i o n s may b e d e f i n e d a s a t w o d i m e n s i o n a l v e c t o r s p a c e o v e r t h e complex numbers w i t h m u l t i n l i c a t i o n g i v e n by

w i t h m, n , p , q complex numbers and i f m = a + h i t h e n ii = a - b i etc. I t is c u s t o m a r y t o w r i t e i j = k; s o t h a t i f n = c + d i , t h e n m + n j = a + b i + c j + d k . I n t u r n a C a y l e y a l g e b r a may b e d e f i n e d a s a two d i m e n s i o n a l v e c t o r s p a c e o v e r t h e q u a t e r n i o n s w i t h m u l t i o l i c a t i o n

( u + v e ) ( x + y e ) = ( u x - v v ) + ( y u + vx-)e.

H e r e u , v , x , y a r e q u a t e r n i o n s so t h a t i f u = a + b i + c j + d k ii = a - b i - c j - d k . O r d e r i s i m p o r t a n t i s t h i s d e f i n i t i o n s i n c e q u a t e r n i o n m u l t i p l i c a t i o n i s n o t c o m m u t a t i v e .

Q u e s t i o n : What s o r t o f s y s t e m s a r i s e i f t h i s p r o c e s s i s c a r r i e d o u t b e q i n n i n q w i t h a s p e c i f i c f i n i t e f i e l d i n s t e a d o f t h e r e a l s ?

The e d i t o r s o l i c i t s c o n t r i b u t i o n s f o r t h i s d e o a r t m e n t . Any p r o b l e m s w h i c h would b e s u i t a b l e u n d e r g r a d u a t e r e s e a r c h a r e welcome.

1 9 6 8 NATIONAL MEETING

A two d a y m e e t i n q o f P i Mu E n s i l o n w i l l b e h e l d i n c o n j u n c t i o n w i t h t h e r e g u l a r summer m e e t i n g s o f t h e M a t h e m a t i c a l A s s o c i a t i o n o f Amer ica and t h e Amer ican M a t h e m a t i c a l S o c i e t y a t Madison , W i s c o n s i n somet ime t h e l a s t week i n A u g u s t .

Your c h a p t e r i s e n c o u r a g e d t o n o m i n a t e y o u r b e s t s p e a k e r who w i l l NOT h a v e a m a s t e r s d e g r e e b y A p r i l , 1 9 6 8 , as a s p e a k e r f o r t h e n a t i o n a l m e e t i n g . N o m i n a t i o n s s h o u l d b e m a i l e d t o DR. RICHARD V. ANDREE, P I MU EPSILON, THE UNIVERSITY OF OKLAHOMA, NORMAN, OKLAHOMA 73069.

SOME USEFIIL PESIILTS IN APPLICATIONS OF VIE POWER

SERIES TRANSFORM TO THE SOLllTInN OF DIFFERENCE EQUATIONS

Ray A. Gaskins, Virginia Polytechnic I n s t i t u t e

1. Introduction. Durinp the past several years difference equations have come i n t o t h e i r own as a techniqae f o r problem solvine in many areas of science and engineering such as s tochas t ic processes, e l e c t r i c a l engineering and engineering mechanics. I t i s desirable, therefore, t o develop a systematic method f o r solving the various types of difference equations which a r i s e . One such method f o r handling a wide range of difference equations i s the power s e r i e s transform.

The power s e r i e s transform as a tool f o r solvinp difference equations i s the d i sc re te counterpart of the Laplace Transform which i s used in solving d i f f e r e n t i a l equations. The transform method of handling d i f fc r - ence equations is superior t o more conventional methods in t h a t i t i s able t o handle non-hoirogeneous equations and equations with variable coefficients . I t i s a l so possible t o obtain d i rec t ly a par t icu la r sol- ution without f i r s t obtaining the general solut ion.

Equipped with a good tab le of t ransfonrs and a knowledge of par t ia l f rac t ions one i s able t o deal swi f t ly with complicated difference equations.

2. P a r t i a l Fractions. In order t o f a c i l i t a t e the use of the power s e r i e s transform in solving difference equations it is essen t ia l t h a t one be able t o resolve proper ra t iona l f rac t ions i n t o p a r t i a l f rac t ions . To t h i s end t h e following shortcut i s introduced.

Consider the proper ra t iona l f rac t ion N(x)/D(x), where the polynomial N(x) i s of lower order than the polynomial Dfx):

where. ,m-i

Ai = - N (XI I (m-i) I (x2+bx+c) ' ,al (x-a x=a

Page 7: PImiMU EPSILON

A and B a r e found hy p lac ing A (i=1,m) and 1

B. ( j = l , n ) i n ( 2 . 2 ) F, s e t t i n g t h e r e s u l t equal 3

t o (2.11.

Severa l examples w i l l be given i n t h e fol lowing sec t ions .

3. The Power S e r i e s Transform. L e t (yk) r e p r e s e n t t h e se-

quence (yo, y,,..., yk, ... 1 . We d e f i n e t h e power s e r i e s

t rans format ion of t h e sequence (yk) t o be! - Yk '{yk' a k h 0 ~ (T. 1)

I

From t h e p r o p e r t i e s of power s e r i e s we have by t h e r a t i o

k + 1 t e s t t h a t (T.1) converges f o r s>so =

e x i s t s .

A s a shorthand n o t a t i o n we s h a l l w r i t e Y ( s ) t o denote

t h e c losed form of t h e t ransform, whi le P(yk) w i l l s t a n d

f o r t h e s e r i e s represen ta t ion .

Consider, f o r example, t h e sequence ( 1 ) = (1,1,...,1,...)

and i t s transform P(1) =kzol/sk which has t h e c losed form

Y ( s ) = s/(s-i) , f o r s>l.

The Dower s e r i e s t ransform is a l i n e a r o p e r a t o r , i .e .

P(Ayk + Bzk) = AP(yk) + BP(zk) (T.2)

which fol lows immediately from t h e s e r i e s d e f i n i t i o n .

Consider t h e t ransform of t h e sequence (yk+,,) - ^k+n ^k n-lyk p ( ~ k + n ' " k h 0 ~ S E X O F - k Z 0 ~ - n

The Laurent s e r i e s P(yk) is uniformly convergent f o r

s>so and is t h e r e f o r e a cont inuous func t ion of s f o r t h a t

range. I t may be d i f f e r e n t i a t e d term by term t o g ive :

n a (-1) "k (k+1) . . . (k+n-1) y, sk-n

Hence,

P(k (k+l) . . . (k+n-1) y k ) = (3). (T.4)

Thus knowing Y ( s ) , t h e c losed form of p{yk), we may

o b t a i n by (T.4) many u s e f u l t ransforms.

EXAMPLE 1. Find P(k) . L e t yk=l and n = l i n (T.4).

k Now l e t us cons ider t h e t ransform of t h e sequence ( r 1

k

which converges and i s a continuous func t ion of r a s w e l l a s k

s f o r r>s>so. D i f f e r e n t i a t i n g P ( r In t imes wi th r e s p e c t

t o r we obtain! - k (k-1) . , . (k-n+l) rkmn

= P(k (k-1) . . . (k-n+1) rk-"),

And s i n c e P ' ~ ) ( r k ) = Y'") (s) we g e t t h e i d e n t i t y :

EXAMPLE 2. Find P(k(k-1) ). Let n=2 and r5l i n (T.5). 3

P(k(k-1) 1 = s / ( s - ~ ) . With t h e preceeding theory and examples behind us , we

a r e i n a p o s i t i o n t o handle t h e t rans format ion of almost any

d i f f e r e n c e equa t ion one might encounter. A t a b l e

of t ransforms is given a t t h e end of s e c t i o n 5.

4. The Inverse Transform. Since t h e Laurent s e r i e s repre-

s e n t a t i o n of P(yk) is unique, t h e r e i s a one-to-one cor-

respondence between (yk) and P(yk) , hence t h e i n v e r s e oper-

a t o r P def ined such tha t !

P - ^ Y ( ~ ) I = (yk)

is a l s o unique. We have by (T.2) t h a t P-I i s a l i n e a r

opera tor .

EXAMPLE 3. P-I [s2/(s-1) 2 ] 5 P-I [s / ( s - l ) 1

+ P-I [s/(s-1) 2 l

by l i n e a r i t y and p a r t i a l f r a c t i o n s . Hence

~ - ~ [ s ~ / ( s - l ) ~ ] 5 ( 1 + It).

Page 8: PImiMU EPSILON

5. Applicat ions of the Transform t o t h e So lu t ions of Dif-

ference Equations. We cons ider d i f f e r e n c e equa t ions of t h e

tYpc 8

where a i < i = l , n ) and f k may o r may n o t be func t ions of k.

S ince t h e d i f f e r e n c e equa t ion i s a c t u a l l y a sequence,

we may apply t h e o p e r a t o r P t o it and o b t a i n an a l g e b r a i c

equa t ion i n Y(s) and s. Solving f o r Y ( s ) and us ing p a r t i a l

f r a c t i o n s t o f a c t o r t h e r i g h t hand s i d e , we apply t h e ope-

r a t o r P t o both s i d e s Ad o b t a i n a s o l u t i o n f o r yk.

EXAMPLE 4. I n s t o c h a s t i c processes the usua l ran-

dom walk model w i t h absorbing b a r r i e r s a t

x=0 and x=a may be represen ted by t h e

d i f f e r e n c e equa t ion qk = pqk+l f o r

l ~ k ~ a - 1 , where p i s the p r o b a b i l i t y of a

move one u n i t t o t h e l e f t , q=l-p, and qk

i s t h e p r o b a b i l i t y t h a t x reaches x=0 be-

f o r e x=a, given t h a t i n i t i a l l y x=k. Sub-

s t i t u t i n g k+l f o r k and applying P:

Using p a r t i a l f r a c t i o n s wi th 6=0,m=O,n=2: a - q/p) + s ( q l - 1) ,

y b ) a 9-1

J s-q/e

L e t t i n a k=a we f i n d t h a t s i n c e a =0

But l i m P{yk) = yo = limy (s), s- s-

Hence, C = yo = 1 and by (T.10)

l y I ( (9 1, 04kSn,

Yk = (El.

TABLE OF POWER SERIES TRANSFORMS ---- SEQUENCE TRANSFORM

(T.ll) (wksin k@) w sin Ã

k (T.12) (W cos k + )

Page 9: PImiMU EPSILON

TABLE OF POWER SERIES TRANSFORMS (con't) ---- SEQUENCE TRANSFORM

k (T.14) (w cosh k+)

s , b = cosh #>1

s2 - 2wbs + w2

sL - wbs , b = cosh +>1 s2 - 2wbs + w 2

REFERENCE :

McFadden, Leonard, Clemens, Paul F. and Campbell, Hugh G., The Powe- and To of Difference E a u a t u , Edwards Brothers Inc., 1963.

The New Mexico Institute of Mining and Technology, Socorro New Mexico (enrollment about 500) has found that a mathematical conference is a good way of stimulating interest in mathematics among undergraduate students. Papers are presented both by students and professional mathematicians. Briefing sessions are held in connection with the conference to supply background on the material presented i n the conference. Up to 2 hours may be allowed for each paper.

Students from other schools attend. The first year there were 12 papers presented and students from 12 institutions in three states attending. Before beginning this program the number of students in math was very small but recently as many as 24 percent of upperclassmen enrolled in the institution were math majors.

MATCHING PRIZE FUND

The Governing Council of P i Mu Ensilon has approved an increase in the maximum amount per chapter allowed as a match- inq prize from $70.00 to $25.00. If your chanter uresents awards for outstanding mathematical papers and students, you may apply to the National Office to match the amount spent by your chapter--i. e., $30.00 of awards, the National Office will reimburse the chapter for $15.00 etc.--up to a maximum of $25.00. Chapters are urged to submit their best student papers to the Editor of the Pi Mu Eosilon Journal for possible publication.

ON HULA HOOPS

Dennis Spellman, Temple Un ive r s i ty

Seve ra l months ago L. Ai leen Hostinsky o f Connect icut Col lege gave a l e c t u r e a t Temple Un ive r s i ty e n t i t t l e d "On Groups, Loops, and Hoops"-- t h a t ' s r i g h t -! The purpose o f t h i s paper is t o i n v e s t i g a t e t h e prop- e r t i e s o f hoops - e s p e c i a l l y a c l a s s o f hoops-which we s h a l l c a l l hula hoops.

D e f i n i t i o n 1: Let G be a non-empty s e t . Then a b ina ry ope ra t ion o def ined on G is a mapping from t h e Car t e s i an product G x G i n t o G. I f ( a , b ) E GxG, i t s image is denoted a 0 b.

D e f i n i t i o n 2. Let G be a non-empty s e t on which a binary o ~ e r a t i o n o is def ined. G is a quasi-group under o i f t h e r e e x i s t unique s o l u t i o n s i n G t o each o f t h e equa t ions a 0 x = b and y 0 a = b where a and b a r e f i x e d e lements o f G. A quasi-group G is a group i f and on ly i f it has t h e a s s o c i a t i v e p rope r ty . i.e.. i f and on ly i f (a o b ) o c = a o ( b c ) f o r a l l a , b , c e G. (See ( 1 ) p. 38 Theorem 1 .4 )

D e f i n i t i o n 3 : I f Gl and G a r e quasi- groups , t hen t h e d i r e c t 2

product of G and G is t h e Car t e s i an product G x G2 on which a b ina ry 1 1

o p e r a t i o n i s de f ined a s fol lows (a l ,a2) o ( b b ) = ( a o bl,a2 0 b2) f o r 1' 2

a l l ( a l , a2 ) , ( b , b ) e G x G . I f C i s t h e unique s o l u t i o n o f a, o x = 1 b1

i n GI, and C i s t h e unique s o l u t i o n o f a 2 o x = b i n 9- t hen (Cl,C2)

is t h e unique s o l u t i o n o f <al, a o x = ( b , b ) i n GlxG2. S i m i l a r l y t h e 2

equa t ion y o (a l ,a2) = ( b 1 2 , b has a unique s o l u t i o n i n G x G . Thus,

t h e d i r e c t product of two quasi- groups is a quasi- group.

D e f i n i t i o n 4: Let G and G be quasi- groups. Let f be a mapping 1 2

o f G i n t o G . Then f is a homomorphism i f f ( a 0 b ) = f ( a ) 0 f ( b ) f o r

a l l a , b e GI. I f t h e homomorphism f is a one-one mapping o f Gl onto G2 ,

f is an isomorphism, and G is ismorphic t o G . I n t h i s c a s e we w r i t e 1

Gl = G 2"

D e f i n i t i o n 5: Let Gl and G2 be quasi-groups. Let g be a one-one

mapping o f Gl on to G . Then g is a n anti- ismorphism i f g ( a 0 b ) = g ( b ) 0

g ( a ) f o r a l l a,b, E G . Ev iden t ly t h e c o n c i p t s o f isomorphism and a n t i -

isomorphism co inc ide i f G and G a r e commutative. 1 2

Page 10: PImiMU EPSILON

Defini t ion 6: Let he a quasi-group. Then a e G i s idenpotent i f

a o a=a.

Defini t ion 7: A non-empty s e t tl on which a binary operat ion a i s defined i s a hoop (o r medial systeir) i f t h e following proper t i e s a r e s a t i s f i e d :

11 1) 11 i s a quasi-group under a

I 1 2) a Oa=a f o r a l l a e 11 (every element of H i s idempotent)

tl 3) ( a a h ) a ( c o d) = ( a 0 c) 0 (bo d) f o r a l l a,h,c,d e I I ( the medial property) .

Theorem 1: I f11 i s a hooq, then fo r a l l a ,h ,c E H

1) a 0 (hat) = ( a 0 h ) O (sac) and

?) ( a o h ) o c = ( a o c ) o ( b o c ) (Kie r i g h t and l e f t s e l f - d i s t r i h u t i v e laws).

Proof of ( I ) :

a fh 0 c) = (a 0 a) o (b a C) hy (H 2)

(2) can he proven s imi la r ly .

( i ) An example o f a hoop is t h e s e t K o f r e a l numhers on which a binary

operation is defined a s t h e a r i thmet ic mean i .e . , i f a,b e K, then a o h =

( i i ) We s t i l l have a hoop i f we change t h e hinary operat ion i n t o a weighed a r i t h n e t i c mean, i . e . , i f w and w a r e p o s i t i v e r e a l numbers, 1

wla+w h 2 = "1 then we can def ine aob= - - + (1- - w1 + w2 w + w2 w1 + w2 . 1

We s h a l l prove these asse r t ions l a t e r ( i n g rea te r general i ty) . Another example of a hoop i s t h e s ing le ton H= > on which t h e binary operat ion is defined hy e oe=e. Such a hoop i s c a x e d t r i v i a l . We note t h a t a t r i v i a l hoop i s a l s o a group and s a t i s f i e s t h e following proper t i e s :

E) t h e r e e x i s t s e E tl such t h a t e o.x=x a e=x f o r a l l x E H and A) ( a o b ) Â ¡ c = a o ( b ~ c f o r a l l a,h.c e H.

Theorem 2: No non- t r iv ia l hoop s a t i s f i e s p roper t i e s (E) o r (A) .

Proof: - Let H be a hoop and a e H. Let e e H be an i d e n t i t y element. a 0 a=a and a a e=a

ffl 2) by hypothesis

Both a and e s a t i s f y t h e equation a ox=a. 2. a=e by (H 1)

Since a was a r b i t r a r y , H={e>. Hence, no non- t r iv ia l hoop s a t i s f i e s (E).

A hoop H s a t i s f y i n g ( A ) is a group s i n c e (H 1 ) and (A) a r e s a t i s f i e d simultaneously. -.H has an i d e n t i t y element. ,\H is t r i v i a l . Hence, no non- t r iv ia l hoop s a t i s f i e s ( A ) .

Defini t ion 8: r = (R : R is a r i n g with un i ty 1 # 0 ) .

Defini t ion 9: Let R E .r. a E R is medial i n R provided: 1. a is i n v e r t i b l e and 2. '5'= 1-a is i n v e r t i b l e .

He note a is medial i n R i f and only i f a is medial i n R. I f Rl, R , E .r,

then t h e i r d i r e c t sum R C R i s an clement of I". Moreover, i f a 1 and

a2 a r c medial in R and R respec t ive ly , then (al , a ) i s medial in Pl 0 2 2 -1

R s ince ( a l , a ) " ' = (al , and = %-I). (Dir- 2

e c t sums a r e discussed in (1) pp. 61-62, 77 ahel ian groups, (2) pp. 17-18

r i n g s ) .

Defini t ion 10: Let R E r. Let a he medial i n R and l e t M he a

module over R. M" i s the system .(", ? ) where t h e binary operation i s

defined by a o h = a a + ah fo r a l l a , h e V. (b'odules a r e discussed i n (1) pp. 145-147). We a r e now prepared t o s t a t e our ch ie f r esu l t .

Theorem 3: Let R e T. Let a he medial in R, and l e t M he a module

over R . Then V" i s a hoop.

Proof: Let a, h e - Consider the equation a a x = h. t h i s t r a n s l a t e s i n t o a a + nx = h. We may v e r i f y by d i r e c t s u b s t i t u t i o n t h a t

= ( r l b - r1 ma) is a so lu t ion .

-1 - -1 - -1 aa + a(; h - a aa) = aa + Z h - aa aa

Suppose x, and x a r e any so lu t ions of a 0 x = b. 2

Then a o xl = h and a 0 x2 = h.

.-.XI = x2

Thus t h e so lu t ion of a o x = b i s unique. Similar ly, t h e r e e x i s t s a unique so lu t ion of y 0 a h.

I'.M s a t i s f i e s (I1 1).

Page 11: PImiMU EPSILON

I Let a e

a o a = a a + a a

= a.

:,M s a t i s f i e s (11 2) . Let a , h, c , d E

(a o h) o ( c o d) = a ( a o h) +;(C o d)

= a ( a a + ah) + a ( a c + ad)

= a2a + adh i a n c + a d

Rearranging terms and regrouping, we have

( a 0 p ) 0 ( c o d) = (a2a + anc) + (aah + a d ) a?i = 3 a because

= a (aa + dc) + ;(ah + ad)

F i n a l l y s a t i s f i e s (H 3) and i s a hoop.

d e f i n i t i o n 11: Let R E I'.

!)(I?) - (M-) = M is an R-module and a is medial i n R l . A hoop I1 i s a

h u l a hoop i f 1 1 c IJ Q(R). We have a s imple c o r o l l a r y t o theorem 3. Re r

Coro l l a ry 4: I f R E and M ^ E 0 (R) , then t h e i d e n t i t y mapping - i: M ^ + M ^ is an anti- isomorphism. -

Proof: Let a , h c M. Let M ^ = (M, o ) and M ^ = (M, . ). - i ( a 0 h) = i ( a a + ah)

= i (b) . i ( a ) . - I f M ' ~ ) = is commutative, t hen t h e i d e n t i t y mapping i s j u s t t h e i d e n t i t y isomorphism. ( i i i ) For example t h e a d d i t i v e group K o f r e a l numbers i s a module o v e r t h e r i n g K o f r e a l numbers, and 1/2 i s medial

- i n K. The commutative hoop = K c a y = ^\Â¥1'2 i s j u s t t h e one we met be fo re i n example ( i ) . ( i v ) Consider t h e f o u r element f i e l d F

(See ( 3 ) pp. 158-159 Example 2 ) .

a t b = 1 ; a ' l = b a n d b " = a . :.F ( a ) , F^ e SKF). ~ e t ' s

denote t h e hoop o p e r a t i o n s o f F a and F ' ~ ) by 0 and * r e s p e c t i v e l y .

Then x O y = ax t by and x ft y = bx t ay . F ' ~ ) and F ' ~ ) a r e non-

commutative s i n c e 0 0 1 = 1 * 0 = a S 0 t b . 1 = b a n d

1 o 0 = 0 * l = a . l t b . 0 = a .

Now cons ide r t h e mapping f : F ' ~ ) ¥ F ' ~ ) de f ined by f ( x ) = x2. We

can look a t t h e m u l t i p l i c a t i o n t a b l e o f F t o w r i t e ou t f e x p l i c i t l y .

o + o 1 + 1

f a + b b + a

( b ) We note t h a t f is a one-one mapping o f F ' ~ ) on to F . f(.x 0 y ) = f ( a x t by)

= (ax t by) 2

= a2x2 t 2abxy t bZy2

= a2x2 t 0 t bZy2 s i n c e F is o f c h a r a c t e r i s t i c 2,

2 2 2 2 .'.f(x y ) = a x t b y

2 2 = bx t ay

= b f ( x ) t a f t y )

= f t x ) * f ( y )

Hence, f is an isomorphism, and t h e non-commutative hoops r a n d

F ' ~ ) a r e isomorphic. However, hula hoops come i n ant i- isomorphic

p a i r s which i n g e n e r a l need not be isomorphic. ( v ) For example

Z , z") e " Z ) where Z denotes t h e i n t e g e r s modulo 5.

2 + 4 = 1 ; ~ - ' = 3 a n d 4 ' ~ = 4 .

emm ma 5: Z^ 5 ~ ( 4 ?

Page 12: PImiMU EPSILON

proof".et h= 2 i 2 ) + 2" be an a r b i t r a r y hmo- - morphism. We denote t h e hoop ope ra t ions of

Z(2) and 2' by 0 and * r e s p e c t i v e l y . 5

h(O o 1 ) = h(2 0 t 4 1 ) = h(4)

1. ;.h(O' 0 1 ) = h(4) But h(0 0 1 ) = h ( 0 ) * h ( l )

= 4h(0) t 2 h ( l ) 2. +;h(O a 1 ) = 4h(0) + 2 h ( l ) 3. .%h(4) = 4h(0) t 2 h ( l ) Now h(O 0 4 ) = h(2 ' 0 t 4 4 )

= h(16) = h(1)

4. .',h(O 0 4 ) = h ( l ) But h(O o 4 ) = h(0) * h(4)

= Uh(0) t 2h ( 4 ) 5. ,'.h(O o 4) = Uh(0) t 2h(4) 6. .Â¥.h(l = 4h(0) t 2h(4) Sub t rac t ing ( 3 ) from ( 6 ) we g e t 7. h (1 ) - h(4) = 2h(4) - 2 h ( l ) 8. :.3h(l) = 3h(4)

9. Mult iplying (8 ) by 3'' (=2) we have h(1) = h(4) 9 ¥* h is not one-one.

But h: Z? "* was an a r b i t r a r y hmomorjihism. 5

I f H and K a r e hoops, t hen t h e d i r e c t product H x K is r e a d i l y seen t o be a hoop a l s o . I t is e a s i l y seen t h a t i f Ml is a module ove r Rl and

M2 is a module over R2, then t h e group 'fl 0 M is a module over t h e 2

r i n g Rl R where t h e s c a l a r m u l t i p l i c a t i o n is def ined by 2

and a l l ( a , a 2 ) c M 0 M .

Theorem 6: Let Rl, R2 e l', ~ ( ~ 1 ) 1 e rfRl), and ~ ~ ( ~ 2 ) e (l(R-1. Then

M ^ l ) x ~ ~ ( ~ 2 ) = (Ml 8 M2)("l' '2). Thus, t h e d i r e c t product o f

two hula hoops is a hula hoop.

Proof: Let (a l , a 2 ) , (bl, b ) be elements o f t h e Car t e s i an - 2

product M,x M2. F i r s t l e t ' s cons ide r them a s e lements of - -

M l a 1 x ~ ( ~ 2 ) and denote t h e hoop ope ra t ion by o . Then ( a , a 2 ) 0 (bl, b2) = (al 0 bl, a 2 0 b2)

= ( a a 1 t a 1 1 ' 2 2 b a a + ¡,b2)

Now l e t ' s cons ide r (a l , a,) and ( b , b ) a s e lements o f

(MI 0 ~ ~ ) ( ( ~ 1 ' " 2 and denote t h e hoop ope ra t ion by . - Then (al, a 2 ) (bl, b2) = (al, a 2 ) (a l , a 2 ) t (a l , a2)(bl ,b2)

I f i n theorem (FRl = R2 = R and al = a =a , t hen t h e group M 1 @ M2

is a module over R a s we l l a s over R 8 R, s c a l a r m u l t i p l i c a t i o n being

def ined by ^ (a l , a 2 ) = Oa1, Aa2) f o r a l l A e R and a l l (a l , a ) c

Ml 9 M 2 . I n t h e same s p i r i t a s above, it is e a s i l y seen t h a t

(Ml 9 M , ) ( ~ ) = (M, 8 M2)(("' a ) ) . Hence, we have t h e fol lowing

c o r o l l a r y .

Def in i t ion 12: Let R be a r i n g and M a module over R. A(M) = h e R: Ax = 0 f o r a l l x e MI.

A(M) is non-empty s i n c e 0 is always a member; fur thermore. A(M) is e a s i l y seen t o be an i d e a l i n R which we s h a l l c a l l t h e M-annihilator i n R. ( I d e a l s a r e discussed i n ( 2 ) pp. 21-26, and simple r i n g s a r e d i scussed i n (2 ) pp. 38-39). This d e f i n i t i o n a l lows us t o g i v e a simple cha rac te r- i z a t i o n o f commutative hula hoops.

Theorem 8: I f R E r a n d M") E MR), then M ( ~ ) is commutative i f and

on ly i f a - a (=2a -1) e A(M).

Proof: Necessity: M ( ~ ) is commutative. - a o 0 = 0 o a f o r a l l a e M .

Suff ic iency: a - a E A(M)

( a - ;)a = (a- a l b (=0) f o r a l l a , b e M ^

a a + a b = a b + ;a

a o b = b o a f o r a l l a.b c M ^

M(¡ is commutative.

Coro l l a ry 9: I f R e r , M ^ e B(R), and A(M) = (0 ) then , M )

i s commutative i f and on ly i f 2 is i n v e r t i b l e and a = 2-l.

Page 13: PImiMU EPSILON

Proof: Ma is commutative i f and only i f -

-1 2a - 1 e A(M) = (0 ) . Moreover, a = 2 i f and only

h((2, 1 ) ) = h((2.11- (1,111 = (2 , 1 ) x h ( ( 1 , 1 ) ) = (2, 1 ) x ( h , h )

= ( 2 h , 2 h )

1. :.h((2, 1 ) ) = ( 2 h . 2 h )

Now h( (2 , 2 ) ) = h( (2 , 2 ) (1, 1 ) )

Corol lary 10: I f R e r is simple, 14 # (01, and M") E R(R), then

M ( ~ ) is commutative i f and only is 2 i s i n v e r t i b l e and a = 2 " .

Proof: A(M) is an i d e a l i n R. Since R is simple we - must have e i t h e r A(M) = ( 0 ) o r A(M) = R . Pick

a E M , a # 0. (Recal l M # ( 0 ) ) .

', A(M) = ( 0 ) and i n view of coro l l a ry 9 t h e proof is completed. ( v i ) In t h e r i n g K of r e a l numbers 2 has inverse 1/2. The commutative

' * Â h is not one-one. But h! M + N was an a r b i t r a r y homorphism.

.Â¥. f N. hoop K ( " ~ ) appears i n examples ( i ) and ( i i i ) . ( v i i ) Let t h e module N over t h e r i n g K @ K be t h e abe l i an group K K on which s c a l a r mul t ip l i ca t ion is defined by

( A , p ) ( a , a ) = ( > a , A a ) f o r a l l (1, u ) e K 9 K a n d a l l ( a , a ) E

Although we do no t have a converse t o our r e s u l t , we do have a p a r t i a l converse a s a consequence of t h e next theorem. We s h a l l need t h e following lemma.

Lemma 12: I f S E F', N ( ~ ) e Q(S), H is a hoop, and f : H + N^

i s an isomorphism, then g: H + N") defined by g (x) = f ( x ) - d is an A(N) = ( ( 0 , u) : u e K).

The reader may v e r i f y t h a t N l 2 ) ) is a commutative member of

R(K Ã K) f o r a l l x e K - (0, 1 ) . ( v i i i ) Z is a non-simple r i n g s ince 3Z9 = (0, 3 , 6 ) i s a non- t r iv ia l 9

isomorphism f o r a l l d e N.

Proof: Let yo e N"). Since f is onto, the re - prober idea l . The group Z9 is a module over t h e r i n g Z9, s c a l a r mult i-

p l i c a t i o n coinciding with r i n g mul t ip l i ca t ion i n Z9. A ( Z ) = (0 ) .

The reader may v e r i f y t h a t z ~ ( ~ ) is a commutative member of B(Z9).

(ix) F ' ~ ) of example ( i v ) is non-commutative s ince 2(=0) is not i n v e r t i b l e .

( x ) 2 " ) of example ( v ) is non-commutative s i n c e 2-I = 3 # 2. I f

R E r and M"), N ' ~ ) e n(R) where M and N a r e isomorphic modules over R,

then it is an easy consequence t h a t M ' is isomorphic t o N u .

e x i s t s xe e H such t h a t f ( x o ) = Yo t d.

,: g is onto.

Suppose g(xl) = g(x2)

= x2 s ince f is one - one. The converse of t h i s statement is not t r u e . ( l / 2 , 1 /2 ) i s medial i n

.*g is one-one. 1. g(a o b) = f ( a o b) - d 2. g ( a ) o g(b)= ( f ( a ) - d) o ( f ( b ) -d)

( x i ) Let M be t h e module K 0 K i n which s c a l a r mul t ip l i ca t ion coincides

with r i n g mul t ip l i ca t ion and is denoted (A ,v ) (a,, a 2 ) , ( x i i ) Let

N be t h e module of example ( v i i ) , and l e t s c a l a r mul t ip l i ca t ion be

denoted by (A, u ) x (al, a ) . Clearly M ( (1 /2 , 1 / 2 ) ) ^ .,((1/2, 1 / 2 ) )

= f ( a o b ) -d s ince f is a homomorphism. Lemma 11: M # N. ,' g ( a 0 b ) = g ( a ) o g (b) and g is indeed an ismorphism.

( a ) = ( 6 ) Theorem 13: I f R,S e r , M(') e R(R), N(') e R(s) , and M , , Proof: Let h: M * N be an a r b i t r a r y homomorphism, - and l e t h ( ( 1 , l ) ) = ( h , h ) .

Page 14: PImiMU EPSILON

then M = N where M and N a r e t h e a d d i t i v e groups of t h e modules M and N r espec t ive ly .

Proof: Let Q : M ^ + N(') be an isomorphism.

Then by lemma 12, f : M ^ -+ N") defined by

Y(x) = Q ( x ) - Q ( 0 ) is a l s o an isomorphism, and

Y(0) = Q(0) - Q(0) = 0. Moreover, s ince Y is a hoop

ismorphism, it is a one-one mapping of t h e s e t

M onto t h e s e t N.

1. Y(a^x o 0) = ~ ( a a ' l x t Fo)

= Y(x) .

2. 'i'(a-lx o 0) = ydi'lx) o Y ( O )

= ~ ( a ^ x ) o o

= ~ ~ ( a - ~ x ) + B 0 o

= Bv(a--'-x)

3. BY (a'^x) = Y(X)

4. ~ ( a ' l x ) = 6'S'(x) f o r a l l x e M .

S imi la r ly Y ( F '^XI = 6 "YCX) f o r a l l x e M .

Since Y is a one-one mapping of M onto N , it s u f f i c e s

t o show t h a t Y is a group homorphism.

Let a.b e Mt.

'!(a t b) = ~ ( a a ' l a + EF '^b)

= Y o a -'b)

= ~ ( a ^ a ) o Y ( F b)

= BI 'Ma) o B ~ ( b )

= ~ 6 - l Y(a) t IB ~ ( b )

='+'(a) t Y(b)

Y : M + Nt is an ismorphism.

References:

(1.) Hu, S.T., Elements of Modern Algebra, ~ o l d i n - ~ a ~ Inc. , San ~ r a n c i s c o . 1 9 6 - -

( 2 , ) McCoy, N. H., The Theory o f Rings, The Macmillan CompanyI New York, 1964.

( 3 . ) McCoy, N. H., Introduct ion t o Modern Algegra, Allyn and Bacon, Inc. Boston, 1960.

I thank D r . Hwa Tsang f o r he r aid.

PROBLEM DEPARTMENT

Edited by

U. S. Klamkin, Ford Scientific Laboratory

This department welcomes problems believed to be new and, as a rule, demanding no greater abiISty in problem solving than that of the average member of the Fraternity, but occasionally we shall publish problems that should challenge the advanced undergraduate and/or candidate for the Master's Degree. Solutions of these problems should be submitted on separate signed sheets within four months after publication.

An asterisk (* ) placed beside a problem number indicates that the problem was submitted without a solution.

Address all communications concerning problems to M. S. Klamkin, Ford Scientific Laboratory, P. 0. Box 2 0 5 3 , Dearborn, Michigan 48121.

PROBLEMS FOR SOLUTION

192.* Proposed by Oystein Ore, Yale University. Albrecht Durer's fam- ous etching "Melancholia" includes the magic square

The boxed-in numbers 15-14 indicate the year in which the picture

I was drawn. How many other 4 x 4 magic squares are there which he could have used in the same way?

w 193 . Proposed by William H. Pierce, General Dynamics, Electric Boat Division.

Two ships are steaming along at constant velocities (course and speed). If the motion of one ship is known completely, and if only the speed of the second ship is known, what is the minimum number of bearings necessary to be taken by the first ship in order to determine the course (constant) and range (time-dependent) of the second ship? Given this requisite number of bearings, show how to determine the second ship's course and range.

Editorial Note:

This problem was suggested by problem 186 which was given erron- eously. See the comment on 196 (this issue).

194. Proposed by J. M. Gandhi, University of Alberta. Show that the equation

Page 15: PImiMU EPSILON

has no s o l u t i o n s i n i n t e g e r s except t h e s o l u t i o n s ( i ) x = '1, y = * l , ( i i ) x = 3, y = 9.

195. Proposed by Leon Bankoff, Los Angeles, C a l i f o r n i a . Math. Mag. (Jan. 19631, p. 60 , c o n t a i n s a s h o r t paper by Dov Avishdom, who a s s e r t s wi thout proof t h a t i n t h e a d j o i n i n g diagram, AN = NC t CB. Give a proof .

196. Proposed by R. C. Gebhar t , Pars ippany, N . J . / What is t h e remainder i f xlOO i s d iv ided

by x2 - 3x t 2?

197. Proposed by Joseph Arkin, Nanuet, N. Y.

A box c o n t a i n s (1600 u t 3200)/3 s o l i d s p h e r i c a l metal bear ings . Each bea r ing i n t h e box has a c y l i n d r i c a l h o l e o f l eng th .25 cen t ime te r s d r i l l e d s t r a i g h t through its c e n t e r . The bea r ings a r e t hen melted t o g e t h e r w i th a l o s s o f 4% dur ing t h e me l t ing p rocess and formed i n t o a sphe re whose r a d i u s is an i n t e g r a l number o f cen t ime te r s . How many bea r ings were t h e r e o r i g i n a l l y i n t h e box?

198. Proposed by S tan ley Rabinowitz, Po ly t echn ic I n s t i t u t e o f Brooklyn. A semi- r e g u l a r s o l i d is obta ined by s l i c i n g o f f s e c t i o n s from t h e c o r n e r s of a cube. It is a s o l i d w i th 36 congruent edges , 24 v e r t i c e s and 14 f a c e s , 6 o f which a r e r e g u l a r octagons and 8 a r e e q u i l a t e r a l t r i a n g l e s . I f t h e l e n g t h o f an edge o f t h i s polytope i s e , what is its volume.

199. Proposed by Larry Forman, Brown Unive r s i ty and M . S. Klamkin, Ford S c i e n t i f i c Laboratory. Find a l l i n t e g r a l s o l u t i o n s of t h e equa t ion 'mt 3m = z ,

SOLUTIONS

181. Proposed by D. W. Crowe, Un ive r s i ty o f Wisconsin and M . S. Klamkin, Ford S c i e n t i f i c Laboratory. Determine a convex curve c i r cumscr ib ing a g iven t r i a n g l e such t h a t 1. The a r e a s o f t h e fou r r eg ions ( 3 segments and a t r i a n g l e ) a r e equa l and 2. The curve has minimum pe r ime te r . So lu t ion by t h e proposers . Condi t ion ( 2 ) is i r r e l e v a n t . The on ly convex f i g u r e s a t i s f y i n g ( 1 ) is a c i rcumscr ibed t r i a n g l e whose s i d e s a r e r e s p e c t i v e l y p a r a l l e l t o t h e s i d e s of t h e g iven t r i a n g l e . Our proof depends on t h e known

Lemma* The a r e a o f any in sc r ibed t r i a n g l e i n a g iven t r i a n g l e cannot be s m a l l e r - t han each o f t h e o t h e r t h r e e t r i a n g l e s formed. Consider a convex curve t h a t is no t a t r i a n g l e and draw t h e t h r e e l i n e s o f suppor t a t t h e v e r t i c e s .

For a p r o j e c t i v e proof o f t h i s lemma and o t h e r r e f e r e n c e s , s e e M. S. Klamkin and D. J. Newman, t h e Philosophy and App l i ca t ions o f Transform Theory, SIAM Review, Jan. 1961, p. 14.

Since 1 = I1 = I2 = 1 ( i n a r e a ) , each o f t h e t r i a n g l e s ABR, BCP, and CAQ would have g r e a t e r a r e a than AABC c o n t r a d i c t i n g t h e lemma. The case of two p a r a l l e l suppor t l i n e s is a l s o r u l e d ou t e a s i l y by a r e a cons ide ra t ions .

182. Proposed by G a b r i e l Rosenberg, Temple Un ive r s i ty . Prove t h a t an odd p e r f e c t number cannot be a p e r f e c t squa re .

E d i t o r i a l *: Although Eu le r e s t a b l i s h e d t h e more g e n e r a l r e s u l t t h a t every

odd p e r f e c t number must be of t h e form p'tatl k where p i s a prime o f t h e

fnm 4n+l , t h e above problem was g iven s i n c e it has an e l e g a n t s o l u t i o n i n A--

a d d i t i o n t o t h e f a c t t h a t t h e r e was a sho r t age of p roposa l s .

So lu t ion by H. Kaye, Brooklyn, N . Y. I f N i s an odd p e r f e c t squa re then 1. The sum o f its d i v i s o r s = 2N, 2. The number of its d i v i s o r s , each being odd, must a l s o be odd s i n c e except

f o r / N , t h e d i v i s o r s occur i n p a i r s d and N/d.

Th i s g i v e s a c o n t r a d i c t i o n s i n c e t h e sum o f an odd number o f odd numbers is OL...

Also so lved by Paul J. Campbell (Un ive r s i ty o f Dayton), Paul Myers ( P h i l a d e l p h i a , Pa . ) , Bob P r i e l i p p (Un ive r s i ty o f Wisconsin), Stanley Rabinowitz (Po ly t echn ic I n s t i t u t e of Brooklyn), Dennis Spellman (Temple U n i v e r s i t y ) , M . Wagner (New York, N. Y.), F. Z e t t o (Chicago, I l l i n o i s ) and t h e proposer .

183. Proposed by R. Penney, Ford S c i e n t i f i c Laboratory. I f r ( r r t r r t r r ) = O ,

0 1 2 2 3 3 1

r ( r r + t o r 3 - r r ) = O , 1 0 2 2 3

r ( r r t r r - r r ) = O , 2 0 3 0 1 3 1

r ( r r t r 0 r - r r ) = O , 3 0 1 1 2

Show t h a t a t l e a s t one o f t h e q u a n t i t i e s r0 , r , r2 , r3 vanishes .

So lu t ion by S tan ley Rabinowitz, Polytechnic I n s t i t u t e o f Brooklyn. I f none o f t h e r 's van i sh , t hen t h e expres s ions i n t h e p a r e n t h e s i s must vanish . These c o n d i t i o n s may then be w r i t t e n a s

( 1 ) 1/r t 1 / r t 1 / r = 0 , 1

( 2 ) 1 / r + 1 / r = l/ro ,

( 3 ) 1 / r t l/rl = l/ro ,

( 4 ) 1/r 1 t 1 / r = l/ro . Adding up (2 ) . (31, and ( 4 ) and us ing ( 1 ) g i v e s 3 / r0 = 0.which is impossible . Hence, a t l e a s t one o f t h e r 's must vanish .

E d i t o r i a l E: This problem has a r i s e n i n t h e p r o p o s e r ' s paper , "Geometrization o f a Complex S c a l a r F i e l d . I. Algebra," Jour . o f Math. Phys i c s , March 1966, pp. 479-481. It a l s o fol lows t h a t a t l e a s t 2 r 's must vanish . The problem can be extended t o any number o f v a r i a b l e s , e . g. , f o r 5 v a r i a b l e s we would have:

Page 16: PImiMU EPSILON

I t follows i n a s imi la r fashion a s before t h a t a t l e a s t two of t h e r's must vanish. Also solved by Paul J . Campbell (University of Dayton), Mario E. E t t r ick (Brooklyn, N. Y.), R. C. Gebhart (Parsippany, N. J . ) , Theodore Jungreis (New York University), Bruce W. King (Burnt H i l l s , N. Y.), Edward Lacher (Rutgers un ivers i ty ) , Paul Myers (Philadelphia, Pa. 1, Charles W. Trigg (San Diego, Cal i f . 1, F. Zetto (Chicago, 111.) and the proposer.

184. Proposed by Alan Lambert, University of Miami. Find a parametric representat ion f o r transcendental curve given by the equation

x" = yx . Solution by Bruce W. King (Burnt H i l l s , N . Y.) and Pe te r Lindstrom (Union College). Let y = x t . Then s ince ylogx = xlogy

we have, tlogx = logxt

o r = t l / ( t - l ) and = t t / ( t - l )

Edi tor ia l Note: This parametric representat ion was already known t o Euler. I t can a l s o be used t o determine a l l r a t i o n a l solut ions of the above equation.

Also solved by R. C. Gebhart (Parsippany, N. J . ) H. Kaye (Brooklyn, N. Y.), Stanley Rabinowitz (Polytechnic I n s t i t u t e of Brooklyn, F. Zetto (Chicago, 111.) and t h e proposer.

185. Proposed by A. E. Livingston and L. Moser, University of Alberta.

Given: f (n) = f(n+3) , n = 1,2, . . . ,

Find f (2) .

Solut ion by R. C. Gebhart ( ~ a r s i p p a n ~ , N. J. ).

S t a r t i n g from t h e known summations,

2 cos n x s i n n x IT-x = -1n2sin 2 , E -

1 n 1 =7 ,

we can obtain t h e p a r t i c u l a r sums

Nor mult iply (1) by -a and (2) by -b.

This w i l l give t h e i n i t i a l s e r i e s i f

a = f ( 2 ) , b = f ( 3 ) ,

Whence,

Edi tor ia l Note: A more d i r e c t way of summing these type of s e r i e s would be t o use generat ing functions which lead t o t h e evaluation of d e f i n i t e in tegra l s .

I f

t h e n

and

Similarly,

l i m i t B u n , 5 9 = , , {f(l)G(l) + f(2)H(l) + f(3)1(l)}

I n oilier t o have convergence,

f ( l ) + f ( 2 ) + f ( 3 ) = 0 , and then

Since f ( 1 ) = 1, we have two l i n e a r equations i n f ( 2 ) and f ( 3 ) , leading t o t h e previous resu l t s . Another way of solving the problem is v i a the function

Here, we consider t h e more general problem of summing

where r,s a r e r a t i o n a l and r > 0, 1 < s 5 r.

For convergence, it is necessary and s u f f i c i e n t t h a t

J L + % + . - . +As = 0 .

and

we obtain

= - i j = l A $ ( $ + l > 3 . Since $(x) is known e x p l i c i t y f o r r a t i o n a l x, t h e s e r i e s can

a l s o be e x p l i c i t l y evaluated.

Page 17: PImiMU EPSILON

$(I) = -y - ;.(og3 -2 J $ ( l ) = -Y . 2 /3

Using these resul ts on our or ig inal problem, we f ind tha t

Also solved by the proposers.

186. Proposed by L t . Joned-Bateman, U.S.C.G. and M. S. Klamkin, Ford Scientif ic Laboratory. Two ships are steaming along with constant velocit ies. What i s the minimum number of bearings necessary t o be taken by one ship in order t o determine the velocity of the other ship? Given th i s requisite number of bearings, show hob t o determine the other sh ip ' s velocity. ( I t i s assumed that range-finding equipment is e i ther non-existent o r non-operative).

Solution by William H . Pierce, General Dynamics, Electric Boat Division. Under the assumption tha t both ships are proceeding a t constant velocit ies (courses and speeds), then no number of bearings taken by one ship can determine the velocity of the other ship. To fee t h i s , l e t the parametric coordinates of the two ships be denoted by

Then i t follows easily that the following family of target ships w i l l generate the same bearings as the target ship above with respect t o the f i r s t ship:

Xg = + vg ta sgg , y ' = b ' + v ' t s i n e ' 2 2 2 2

where

Editorial Note: See problem 193, t h i s issue fo r a corrected version of t h i s problem.

MOVING??,

P l e a s e be s u r e t o l e t t h e P i Mu E p s i l o n J o u r n a l know! Send y o u r name, o l d a d d r e s s w i t h z i p code and new a d d r e s s w i t h z i p code t o :

P i Mu E p s i l o n J o u r n a l Depar tment o f Mathemat ics The U n i v e r s i t y o f Oklahoma Norman Oklahoma 73069

BOOK REVIEWS

E d i t e d by Roy B. Deal , Oklahoma U n i v e r s i t y Medical C e n t e r

1. EXements of Real Analysis. By Sze-Ben Ikl, Holden-ky, Inc., San bancisco, California, 1967. A comprehensive, but readable, beginning graduate level introduction t o Real Analysis, following the C U M recommendations with some optional material. It t r ea t s the fundamentals of topological spaces, metric spaces, topological l inear spaces, including Hilbert spaces "studied le isure ly in a single section," measures and integrals, including Daniel1 integrals and. Haar measure, and br ief , but modern, introduc- tions t o differentiation and distributions.

2. Exercises in Mathematics. Ey Jean Bass, Academic Press, Inc., New York, wi+ k57 pp., $14.75.

A valuable collection of carefully worked out problems i n r ea l and complex analysis, ordinary and pa r t i a l d i f ferent ia l equations, and various aspects of applied mathematics.

3 . A Hilbert Space Problem bok . Ey Paul R. Hahas, Van Nostrand Co., LC., Princeton, New Jersey, 1967.

A unique collection of problems and solutions of varying levels of d i f f i cu l ty designed t o i l l u s t r a t e the "central techniques and ideas of the subject. The user of t h i s book should have some knowledge of general topology, r e a l and complex analysis, l inear topological vector spaces, including Hilbert spaces, and r ea l and. complex analysis. The f i r s t one hundred and for ty pages l i s t the problems with br ief h i s to r i ca l and mathematical comments and a few definit ions. The next 22 pages give hints t o the solutions and the l a s t 185 p q e s give f a i r l y complete solutions.

4. Differential Geometry. By Louis Auslander, Harper and Row, New York, 1967.

A f i ne modern treatment of class ica l d i f ferent ia l and global geometry f o r the advanced undergraduate or f i r s t year graduate with two years of calculus and one year of modern algebra. Prerequisite of algebraic topology i s sk i l l fu l ly avoided. An interesting feature i s the t rea t- ment of matrix Lie groups instead of abstract Lie groups.

5. Mundary Value Problem6 of Mathematical Physics. By Ivar Stakgold, MacMillan Co., Riverside, New Jersey, 1967.

For the graduate students i n applied mathematics, engineering, and physical sciences, with advanced calculus and elementary complex variables, t h i s f i r s t of two volumes on l inear boundary value problems i s an excellent in- troduction t o the study of Green's Functions, including the theory Of distribution, and eigenfunction expansions i n l inear in tegra l equations and. the spect ra l theory of second-order d i f ferent ia l operators.

Page 18: PImiMU EPSILON

Addition Theorems. By Henry B. Mann, John Wiley, Inc., New York, 1965, x i + U3 pp.7$8.75.

Ih i s l i t t l e book on additive theory of s e t s i n number theory and more general Abelian groups or semi groups affords one of the best oppor- tuni t ies f o r the interested reader with a l h i t e d bac&round, such as a course i n modern algebra and a minimal introduction t o number theory, t o follow some recent, important research resul ts i n mathematics.

Information A Scient i f ic American Book. By W. H. Freeman and Co., San Francisco, California, 1966,218- i l lus t ra t ions , 15 d a t e s , Cloth- - . . - bound $5.00, ~aperbou& $2.50.

A collection of the faclnating ser ies of a r t i c l e s appearing i n the September, 1966 Scient i f ic American on the s t a t e of the computer art and i t s ' impact on society.

Nunbere and Ideals. By Abraham Robinson, Holden-Cay, Inc., San Francisco, -- California,

A very elementary introduction t o algebraic number theory with i t s own introduction t o mdern algebra i n a small book t h a t could be stuiiied, f o r example, by freshman honor students i n mathematics.

Applied Combinatorial Mathematics. DUted by Etlwir~ F. Beckenback, John Wiley, Inc., New York, 1964, %xi + 591 pp., $13.50.

A collection of eighteen chapters by outstanding mathematicians, covering a wide var ie ty of aspects of t h i s much recently emphasized aspect of mathematics. Varying levels, generally from advanced calculus on are required t o seriously study the ar t ic les .

Mathematical L a i c . By Stephen Cole KLeene, John Wiley, Inc., New York, 1967.

A junior (a lso writ ten f o r juniors) addition of the author's famous "Introduction t o Metamathematics" which should, prone t o be one of the most popular books on the subject f o r Mathematics majors.

Evolution of Mathematical Ih -- 6 y h t . By &schkowski, Holden-Day, Inc., San Pranciso, California, 19 5 155 pp., $5.95.

This book, translated from the German, i s an interesting discussion "about" the foundations of mathematics f o r a wide range of readers i n both in teres t and level of background.

Geometric Pr ramm By Duffin, Peterson and Zener, John Wiley, Inc., ~ e w i&&%!78 pp., $12.50.

Although the material was collected f o r use in solving engineering design problems, and many examples are given, many readers, interested i n optimization problems such as generalization of l inear programming problems, would, f ind t h i s a profitable book t o study.

Ihe Theoq of Gambli and S t a t i s t i c a l Lcgic. By Richard A. Epstein, - Academic P r ~ s d N e w York, 1967, x i i i + 485 pp., $10.00.

A scholarly t r ea t i s e on gambling with many thought-provoking ideas on a wide var ie ty of subjects as well as some interesting mathematical resul ts i n probabili ty theory and game theory fo r readers a t about the advanced calculus level of mathematical maturity.

A Second Course In C --- By William A Veech, W. A. Benjamin, Inc . , New York, ~6f??? =$8.75.

A very good book f o r those with an elementary course i n complex varables and general topol-y who would l i k e fundamental grounding i n some im- portant aspects of the subject including the prime numbers theorem.

Plane Geometr and Its Grou s By Guggenheimer, Holden-Day, Inc., San - Francisco, c a k i ~ n ~ & Although the generation of motions by reflections i s classic, t h i s treatment and selection of topics has such a modern flavor t h a t many students of today w i l l f ind it interesting and profitable t o pursue.

Foundations g Mechanics. By Ralph Abraham, W. A. Benjamin, Inc., New York, 1967, ix + 290 pp., $14.75.

Ih i s i s t r u l y a foundations text. It i s based on a ser ies of lectures directed primarily a t physicists a t the graduate level, but the mathe- matical background i s so well presented tha t it might also serve as an excellent introduction f o r mathematics majors t o a modern treatment of d i f f e r en t i a l manifoldsJ vector and tensor bundles, and integration on manifolds. The study of ce l ec t i a l mechanics contains very recent basic results .

NOTE: All correspondence concerning reviews and all books for review should be sent to PROFESSOR ROY B. DEAL, UNIVERSITY OF OKLAHOMA MEDICAL CENTER, 800 NE 13th STREET, OKLAHOMA CITY, OKLAHOMA 73104.

Topological Vector Spaces and Distributions By John Horvath, Addison- Wesley Publishing Co., Inc., Reading, Mass., Vol. I., 1966.

Ccmmter Pr ranmi and Related Mathematics By R. V. Andree, John Wiley, Inc .:New Y Z k ~ g m pp . , $6 -50.

Abstract Theory of Groups By 0. U. Schmidt, Translated from the Russian by Fred Hol lhg and J. B. Roberts, m i t e a by J. B. Roberts, W. E. Freeman and Co., San Francisco, California, 1966, 174 pp., $5.00.

Value Distribution Theory By Leo Sario and Klyoshi Noshiro, N. J. Van - Nostrand Co., Inc., Princeton, N. J., 1967, x i + 235 pp., $11.50.

Linear Geomet By K. W. Gruenberg and A. J. Weir, Van Nostrand and Co., - b c . , Princet:, I?. J., 1967, viii + I B ~ pp., $6.00.

Set Theory and the Number System6 By May Risch Klnsolving, International Textbook Co., Scranton, Penn.

Vector Analytic Ge-t By Paul A. White, Mckenson Fublishing Co., Inc., - Belmont, C a l i f o r n i a , 5 .

Introduction t o Modem Abstract Algebra By David M. Burton, Addison- Wesley ~ublishingCo., m a d i n g , ass ., 1967.

Abstract Algebra By Chih-Hah Sah, Academic Press, Inc., New York, 1966, x i i i + 335 PP., $9.75.

Theory of Functions of a Complex Variable Vol. 11 By A. I. Marhushevlch, Translated from the & h n by Richard A. Silve--, P r e n t i c e - W , Inc., Englewood CU-ffs, N. J., x i i + 329 pp., $12.00.

Page 19: PImiMU EPSILON

INITIATES

ALABAMA ALPHA, University of Alabama Theory of Functions of a C m p l e x Variable Vol. I11 By A. I. Markushevich, Translated from the Russian by Richard A. Silvern, Printice-Hall, Inc.,

Englewood Cliffs, N. J., 1967, x i + 355 pp., $12.95.

Claud James Irby James R. Jackson Daniel J. Jobson George Thomas Jones Haim Kaspi Emogene Knight Saksiri E. Kridakorn Jerry Kenneth Lewis Jimmie R. Lindsay Larry B. McDonald Robert Lee Manasco Linda K. Manduley Janet Louise Mate Lianna D. Mulliqan Michael Terry Newton Charles M. North, Jr.

Robert R. Parker Jonathon Raffield Fred A. Roberson Ginger Sue Roberts Robert L. Robinson Emily Jean Sabo Mary V. Schorn Linda Carole Smith William B. Smith Billy C. Tankersley Herman E. Thomason Donald G. Turner Claudia E. Vookles James L. Whitson Linda Howell Wise Ronald Lee Wwten James Addison Wright, JI

Elaine E. Alexander Sheila Anne Denholm Ralph 0. Doughty Eleanor Ann Fitts Joyce Gwin Fox Mary E. Garrett Billy Gene Gibbs Eugene F. Glass Charles H. Gunselman James P. Hayes, Jr. Nevin Julian Henson Seth J. Hersh John E. Holleman James 0. Howell John W. Howerton Michael J. Huggins Margret E. Ingrain

Ronald A. ~tkinson Charles A. Barnes Bettve D. Beroen

Introduction t o Real Analysis By Casper Goffman, Harper and Row, New York, 1966.

~ o h n ~ . - ~ l a n d ; Jr. Robert C. Bradley Cecil W. Bridges Marion A. Cabaniss

Vectors, Tensors and Groups By 'Hior A. Bale and Jonas Lichtenburg, W. A. - -- Benjamin, Inc., New York, 1967.

Patrick A. Callahan Donald A. Campbell Wavne Heath Causev ~dwird~.childij ~ r . Ronald James Cornutt Robert W. Cowden Functions of One and Several Variables By Thor A. Eak and. Jonas Lichtenburg,

W. A. Benjamin,Inc.,Mewrk, 1967. . - - - - . -. - - Helen D. Darcey Carolyn Darden

Series, Differential Eouations and Complex Functions W. A. Benjamin, Inc., - New York, 1967.

ALABAMA BETA, Auburn University

Introduction t o Probabilit and Statistics By William Menderihall, Wads- worth P u b ~ s h T n g Co., BelmtnrCalifornia, 1967, xiii + 389 pp., $8.50 text, $11.35 trade.

I

Laura Johnston Glenda M. Jones Steven G. Joseph Judy A. Kennedy Lynne Mielke Rex K. Rainer, Jr.

Frank P. Ramsey, Jr. Beverly J. Upchurch Patricia F. Villafann Phillip A. Wallace Stephanie J. Wallace

Carolyn H. Adkins Ruby Dean Robert 5. Barnes Susan A. Donne11 Phyllis S. Boozer Harold E. Dowler Dianne Burgess Janet S. Farris Lee R. Christian Martha L. Graves

Lawrence M. Gorham Guidelines For Teaching Mathematics By Donovan A. Johnson and Gerald R. Hising, Wadsworth Publishing Co., Inc., Belmont, California, 1967, viii + 439 PP., $6.95.

Introductor Geometr : An Informal A roach Brooks/~ole Publishing Co., Behont, California, 'ljff~, -8*. 50.

ALABAMA GAMMA, Samford University

Sharon Blice Larry T. Bolton E. Delilah Carter Barbara P. Chandler Floyd L. Christian Richard H. Collier Janice E. Cox Wayne B. Faulkner Ernest G. Garrick Charles A. Greathouse

Robert B. Hall Carey R. Herring Charlotte Jarrett M. Carolyn Keyees Dorothy C. King Phyllis I. Laatsch Chester R. Killy Dianne Crews McDaniel Wayne K. Meshejian Helen B. Miller

Stanley W. Milwee Karen Anne Monroe Teresa Ann Morgan Sarah C. Morris Helen B. Murphree Abdul A. Nafoosi William D. Powell Jarrett Richardson Wayne Shaddix Amy Sue Sims

Shera Lynne Thackery Barbara E. Thompson Martha C. Thorpe Rex G . Walker

Statio and Related Stochastic Processes By Harold Cramer and M. R. Leadbe=, John Wiley, Inc., New -7, xii + 345 pp., $12.50.

-. . . - - -. Marjorie Watson Judy L. Wells Ed Wheeler John F. Whirley Aubrev H. White

Combinatorial Methods In the Theory of Stochastic Processes By Lajos Tatoc, John w i n e ; York, 1957, x i + 255 -75.

~rend: P i yarnell Letitia G. Yeager

F i r s t Order Mathematical Loqic B y Angelo 'fargaris, B l a i s d e l l Publishing Co., Waltham, ' lass . , 1967.

ARIZONA ALPHA, University of Arizona

H. Bradford Barber Jack M. Kemler Arts E. Dameani James I. Leong Robert E. Darwin

Catherine E. Parry J. Byron McCormick

Michael J. Merchant Richard E. Rothschild

NOTE: A l l correspondence concerninq reviews and a l l books f o r review should be s e n t t o PROFESSOR ROY B DEAL, UNIVERSITY OF OKLAHOMA MEDICAL CENTER, 800 ME 13th STREET, OKLAHOMA CITY, .-by,- ..,-..- --. .. . ARKANSAS AuPHA, University of Arkansas

Dale B. Nixon Gay R. Reid Van 0. Parker Richard R. Schurtz Michael A. Plunkett Mavis A. Upton Mauer D. Schwartz

John R. Bass James P. Luette James Pi Douglas William H. Ma Russell H. Ewing Jerrel D. Martin Jerry A. Jenkins Susan N. McManus

CALIFORNIA ALPHA, University of California

John March Gardner David B. Gauld Bernard A. Glassman Jerrold M. Gold N. Grossman

Nguyen Huu Anh Charles R. Bosworth Chang-Shing Chen David J. Bailey Jeanne I. Brown Richard J. Clasen M. Kenneth Bates Patrick M. Brown Carol Grace Cook John D. Billings Edgar Coleman Arthur A. Duke Sonia Munn Bloom Lois Ann Chaffee Maxine Eltinqe

NEED MONEY?

The Governing Council o f Pi Mu Epsilon announces a contes t f o r the b e s t expository paper by a student (who has not y e t received a masters degree) s u i t a b l e f o r publication i n the Pi Mu Epsilon Journal.

There w i l l be the fol lowing p r i z e s

$200 f i r s t pr ize

$100 second p r i z e

$50 th ird p r i z e

providing a t l e a s t ten papers are received f o r the contes t .

In addit ion there w i l l be a $20 pr ize f o r the b e s t paper from any one chapter, provided that chapter submits a t l e a s t f i v e papers.

Page 20: PImiMU EPSILON

A. Hales , Young Koan-Kwon Thomas L. Henderson Michael King-Xing Lai Barry Connick Howard Richard Lewis Keith Joseph Wickham Low Jeffrey Asch Karas Victor Marfoe Martha R. Katzin Yashaswini Deval Mittal Melissa Knapp Hadi J. Mustafa Don Krakowski Samuel D. Oman

Nicholas Perceval-Price Gabriel 5. Salzman U. Kurt Scholz Donald J. Secor Charles B. Sheckells Hiroshi Shimada Herbert Siege1 Daniel A. Stock

Arthur Tashima James Thompson Edward Turner Leonard 0. Weisberg Norman Weiss Ellen L. Wong Kenneth J. Wong Virginia Zwnkovich Leon M. Traister

FLORIDA ALPHA, University of Miami

Marilyn Bennan Habib H. Jabali Harvey L. Bernstein William F. Kern Jeffry Ben Fuqua Jeanne Eve Melvin

William Frank Page Geoffrey L. Robb Vaughn D. Rockney, Jr.

Juan Luis Sanchez Kenneth M. Simon

FLORIDA BETA, Florida State University

Richard W. Ross Yaichi Shinohara Ilk0 P. Shulhan James Richard Smotryeki Bennett M. Stern L. Lee Williams

Gary L. Alderman Leila E. Donahue William Walton Alfred Martin J. Fischer Wayne Curtis Barksdale Mary Ellen Gaston John M. Bowling Charles W. Jernigan Charles M. Bundrick Douglas Julian Jones Tanya A. Connor Robert E. Lorraine William R. Crowson Mary Louise Miner

John A. Moore Daniel J. Mundrick Sam S. Narkinsky Brian R. O'Callaghan Susan E. Richardson Kennit Rose

CALIFORNIA ZETA, University of California at Riverside

John Dale Harris David George Hazard Jfadith Marie James Kent D. Mason Kenneth Lee Holnar Robert Norman Reber

Greg W. Scragg Pamela Kay Sierer Suzanne Smith Michael J. Stimson Julianne Tracy Robert James Vrtis

Brian W. Bauske Lynnette Rosalie Daroca Donald P. Brithinee James Dunbrack Wallace P. Brithinee Abraham Dykstra ~obert Earle Carr William A. Fanner Alfred James Delay Sheryl I. Hampton

FLORIDA EPSILON, University of South Florida

Thomas G. Parish Harriet C. Seligsohn LaRay E. Geist Peter T. Laseau

Ilgee Moon CALIFORNIA ETA, The University of Santa Clara

Russell Meredith Richard Schweickert Dennis Smolarski

Irving Sussman Charles Swart Kathleen Weland

Gerald Alexanderson Leonard Klosinski Karel De Bouvere Thomas Kropp Sherrill Ford Bruce McLemore 3EORGIA ALPHA, University of Georgia

Susan Gerald John P. Holmes COLORADD GAMMA, United States Air Force Academy

Robert J. Barnum Charles L. Clements Michael N. Gilea Edward A. Greene, I1 Joseph Hasek

Ronald L. Kerchner Peter L. Knepell Edward C. Land, I1 Frederick K. Olafson Howard L. Parris, Jr.

Arthur R. Miller John J. Watkins, Jr. Eugene A. Rose, 111 Gary N. Wiedenmann Jeffrey E. Schofield Roger L. Wiles Robert Rue Michael C. Wirth Craig C. Squier Donald R. Windham

GEORGIA BETA, Georgia Institute of Technology

E. H. Brownlee D. N. Davis

R. E. Einziqer J. T. Gill

D. F. Jackson w. G. Kelley

J. 0. Scardina J. Hi S c h m e r s Y. S. Yen

Ruth Brady Janice E. Hunter Ronald J. Dale Edwin H. Kaufman, Jr. Roberta S. Haqer Marie S. Ludmar

William P. Mech Edward B. Noffsinger Bilram 5. Rajput

Stephen Mi Taylor Constance L. Thompson William L. Young

CONNECTICUT ALPHA, University of Connecticut

John Kuzma Linda Musco Gail Perry

Nola Reinhardt Judith Rosenberq Sharon Sluboski

Joseph Adamski Elizabeth Baer Bruce Barrett

Donald Bleich Jane Chapitis Sandra Duchnok ILLINOIS DELTA, Southern Illinois University

Johnny M. Brown John R. Crede

Melvin D. Hall Robert Kohn

Rodney R. Oldehoeft Jane M. Pinkstaff Lawrence E. Prichard George Richards

Dale G . Roedl

DISTRICT OF COLUMBIA Alpha, Howard University - - - - - . . . - - - Richard R. Setzekorn Michel B. Samokhine Gloria E. Thurston

~ - - - - - . . . .. . ~ a l t 0. Diltinstraat James L. Mazander John R. Graef Abraham Mazliach Hayford A. Alile Laura E. Parley

Cleo L. Bentley John A. Harris Mohammed A. Bilal William A. Hawkins Millicent S. Carter James A. Jackson Marie A. Cloyd Beverly A. Jacques

Cornel J. Rigby William H. Jones Timothy 0. Olateju Fred D. Price Linda M. Raiford

Pericles D. Stabekis Sudershan K. Vadehra Quinton E. Worrell James T. Yeh ILLINOIS GAMMA, DePaul University

Gerhard Perschke

DISTRICT OF COLUMBIA BETA, Georgetown University

Mary Helen Sullivan Michael J. Sullivan Lewis Suskiewicz John Tallman Edmond Villani

INDIANNA BETA, Indiana University

Anne Marie Hocker Robert Baldwin Thorns Pi Haggerty Joseph A. Ball Denise Huet Charles Byrne Eleanor Christine Iorio Maureen Castle James Jenkins Elroy W. Eckhardt Svetoza Kurepa Andrew Grekas John B. Manning

James Monaqan Larry Panasci Marilyn J. Power Donald J. Propst Kennard W. Reed Stephen M. Soldz

INDIANA DELTA, Indiana State University

Phillip W. Boqan David Keusch Bill Fisher James R. Mayrose Mrs. Charles Flory Ronald L. Melbert Julia Hamilton Gloria Kay Norton Norbert Johnson Gerald Pearson

John Robert Rhoadee Judith Ann Sharps Gerald Slack Artemese E. Smith Barbara Spangle Minnie M. Tashima

Larry Tredway Jim Tremper Tommy Tomlinson Enmitt Tyler Lee Vercoe Timothy D. Voorhies

DISTRICT OF COLUMBIA GAMMA, George Washington University

Kent Minichiello Randy Boss David A. Schedler

Sarah D. Barber Timothy Boehn Richard Epstein

Arthur H. Gardner Jerry W. Gaskill Stephen Z. Kahn

Ruth D. Koidan Linda K. Larsen Thierry Livennan

Page 21: PImiMU EPSILON

INDIANA GAMMA, Rose Pol$technic Institute

LOUISIANA GAMMA, Tulane University David H. Badtke Herbert R. Bailey Karl W. Beeson John D. Borst William R. Brown

Earl E. Creekmore Alan F. Hoskin Charles H. Divine Robert E. Janes Randall E. Drew Gary Roy Johnson Michael J. Hanley Terrence M. Joyce G. Felda Hardymon William D. Schindel

Thomas George Schultz Gholam Ali Semsarzade' George R. Scherfick Robert W. Walden Russell A. Zimennan

Brenda Mi Robinson Christine Robinson Michael P. Saizan Ross Serold Gayle Singer George A. Swan Raymond L. Walker Jocelyn Weinberg John C. Woodward

Jack Aiken Evelyn Alband Jessica Aucoin Claudia G. Avner Don Barrvs Bruce Baguley Wayne Bru Prank R. Buchanan Betty Carriere Mrs. Joan copping

Elizabeth Ford Kenneth Goldberg Richard J. Gonzalez Charles E. Gow Patricia Greene Frances Hayes Thomas W. Kimbrell Karen Klingman Carolyn Kutcher James M. Laborde

Mary Ann Lemley Jonathan Levin Paul E. Livaudais william McCue, Jr. Colyin G. Norwood, Jr. Richard C. Penney stots B. Reels Nicola Ricciuti Roslyn B. Robert

IOWA ALPHA, Iowa State University

David W. Porter James A. Schuttinga Albert D. Sherick Elaine B. Vance Harold R. Warner, Jr. Dennis A. Watts James L. Woo

Sudhir Aqgarwal Robert F. Baur Robert E. Bonnewell Michael M. Burns James F. Frahm David M. Gustafson John S. Hartman

Dean E. Harvey David P. Meyer Gregory F. Hutchinson Alan D. Miller Dean 0. Keppy Loren K. Miller Thomas B. Leffler Michael G. Milligan Steven P. LeMett Eugene J. Mock Jerry Malone Judith K. Noordsy Roy E. MeLuen Mary S. Pickett

LOUISIANA EPSILON, McMeese State College

Terry Franks Theodore Laftin Peter Leathard

Glenn J. Rrejean Ronald Shirley

KANSAS ALPHA, University of Kansas

Sue Meredith Jeff S. Nichols Kathie Phillips Richard E. Sweet Michael C. Rasmussen Louis A. Talman Emil G . Trott, Jr. Roger K. Alexander

LOUISIANA ZETA, University of Southwestern Louisiana Allen Leon Ambler Charles J. Banqert Martin R. Bebb David L. Bitters Sara Ann Bly Paul W. Bock John W. Boushka Ronald R. Brown Michael Sterling Cann

Jose J. Chaparro Mary Ruth Church Barbara Ann Crow Jerald P. Oauer James W, Frane Ned Gibb ns Edward C? Gordon Danille L. Harder John 0. Harris

Martin R. Holmer Lee Martin Hubbell Joseph R. Jacobs David A. Kikel Kurt L. Krey Donald E. Kruse Maria Y. C. Liu Judith Magnuson Dave Mcclain

McKenzie W. Allen Claudia A. Barras ~ e w i s E. Batson John C. Bell Grace A. Blanchard

J. Ray Comeau, Jr. Linda A. Looq Remel R. Cooper, Jr. Anne 8. McGampbell Robert W. H a m Cecil H. McGregor Ossie J. Huval Robert E. Mooney Rose M. Lillie John C. Peck

Matthew D. Peffarkorn James J. Phillips Gospard T. RiZZut0 Constance D. Watler

MAINE ALPHA, University of Maine

Madeline C. S O U C ~ ~ william L. soule, Jr. WilliamFranklin Steam Wilfred Paul Turgeon

Karen Kristine Anderson Stephen Eric Godsoe Martha Ann Parry Barry Michael Beganny Gary Leroy Goss Joanne S. Perry Mary Ann Carson Jane Rachelle Huard Stephen M. Perry Dorothy Rose Dobbins Thomas Peter Hussey Carl Henry Rasmussen Clayton W. Dodge Thomas Childs Jane Sandra Joyce Rogers Meriel F. Duckett Carroll F. Merrill Jeffrey Cole Runnels Sally Ann Emery Van Mourdian Stanley J. Sawyer Elizabeth D. Giusani

KANSAS BETA, Kansas State University

Roae K. Shaw Darrol H. Timmona Jerry D. Wheeler Micha Yadin Shelemyahu Zacks

John Hi Brand I1 R. Phillip DuBois David D. Kreuger Anna Bruce Joseph Arimja Elukpo Raja Nassar Patricia A. Buchan Gary D. Gabrielson Marvin C. Papenfuss John (Chang-Ho) Chu John David Hartman Thomas K. Plant Terry J. Dahlquist Y. 0. Koh Ranjit S. Sabharwal Arthur Dale Dayton Louis D. Kottmann

Helen P. Whitten Janet Marie Viger Clover Jane Willett

MARYLAND ALPHA, University of Maryland LOUISIANA ALPHA, Louisiana State University

Butrus G. Basryaji Paul Gilbert Clemens Gail Laura Lmge Ralph P. Pass I11 John Gerald Belt Robert L. Epstein Roselyn Lowenbach Terri Sue Pierce Anne Bernstein Ellen J. Feinglass Alan M. Mathau Rhoda Shaller William Edward Caswell Marianne T. Hill John Wayne Molino Sue Ellen Stiefel Gary Robert Catzva Paul Don Yon9 Hong Bruce Whitmore Parkinson Nancy Wolf Bryant Stewart Centofanti Fred Jacobson

William F. Beyer Judy M. Dyson Thomas C. Bleach Morgan C. Eiland David M. Bock Daniel J. Fourrier Joan J. Capana Patricia A. Fults Robert P. Cifreo Charles R. Guin Matthew B. Collins Andrew J. Hargooa Howard B. Davis Earl W. H o m e

Jimie J. McKim Gail I. Rusoff William 11. McMillian Florence M. Sistrunk Stuart E. Mills Ray C. Shipley Ronald H. Getting Edwin A. Smith Paul I. Pedroso James E. Phillips Carlton W. Pruden Owen D. Roberts Keith N. Roussel

Carol Seasums Harold 7 . Toups Ernestine Watson Ronnie E. Zammit Stanley J. DeMoss loy ye 0. Howard

Charles N. Dyer Thomas V. McCauley MASSACHUSETTS ALPHA, Wocester Polytechnic Institute

George M. Banks Wayne N. Fabricius Andrew Anthony Lesick Richard Arthur SnaY George R. Bazinet Berton Hal Gunter Edward A. Olszewski Leo T. Sprecher Warren Bentley Mark Hubelbank Michael R. Paige John P. van Alstyne John F. Cyranski Ronald E. Jodoin

LOUISIANA BETA, Southern University

James Altemus James Handy

Isiah Moore Kathleen Morris

Dan V. Snedecor Arthur Turnmr

MASSACHUSETTS BETA, College of the Holy Cross

Daniel G. Dewey Vincent 0. McBrian Patrick Shanahan LOUISIANA DELTA, Southeastern Louisiana College

Sandra Kay Morere David Edward Thomas Naomi Barbara Richardson Harvey L. Yarborough Donna L. Schilling

Judith Ann Alston Pearl Jo Gebauer Thomas Lynn Dieterich Sharon Claire Hein James Russell Dupuy Karen Raye Kistrup MINNESOTA ALPHA, Carleton College

Bruce L. Anderson Ralph W. Carr David R. Barstow Janice Clarke

John W. Gage Jeffrey H. Hoe1

Eric S. Janus Frank M. Markel

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James Louis Morley I1 Vernon Lee Pankonin Elden H. Steeves Paul C. Tangeman Terry A. McKee Marvin Rowher James Robert Stork Joseph E. Tyer Dennis A. Novacek Donald Joe Saa l Janice M. Strasburg Keith W i l l i s Donna Novotny Robert John Schmucker J o e l Richard Swanson Reginald Lee Wyatt Dale Osborn

Stephen R. sav i tzky Edwin D. ~ c o t t Mary G. Whitaker Edward F. Schlenk Timothy I. Wegner Barbara L. Whitten

J. Paul Marks Thomas L. Moore

MINNESOTA BETA, College of S t . Catherine

HEW HAMPSHIRE ALPHA, Universi ty of New Hampshire John R. Chuchel Sr. Cecily Debelak Carol Opatrny

Herman J . Gi t sch ie r Robert G. Leavi t t Charles G . MCLure John L. Sanborn Carolyn Gorski Robert Ludwig Michael C. Malcolm Leon R. Tess ie r Joan A. Kearney Michael A. Minor George B. Phalen Joane Wakefield Marie LaFrance

MISSOURI ALPHA, Universi

Rose Marie Ahee Trudie Akers Bonita J . Allgeyer Manouchehr Assadi Paul W. Barker Stephen Baudendist i l Jane Brightwell Chi Keng Chen Javad S i r u s Chitsaz S a l l y Clayton John A. Criscuolo P h i l l i p Embree Max Engelhardt

t y of Missouri

Sandra Meinershaqen Larry Mot2 Elizabeth Ann M i l l e r Edward J. O'Dell Ronald W. Oxenhandler Robert W. Ramsey Dan Rea Jacqueline Roberts D. M. Robinson

r . Walter J . Rychlewski Husaini Sa i fee B i l l Sangster J e r r y Schatzer

Carol Scrivnmr Larry Smarr Robert H. Smith John B. Spalding Roy G. S tuerz l Lynn Walley Gary Walling Thomas F. Walton Samuel R. Weaver Edward West Elaine Wyett Steven W. Yates Greg Ziercher

Ronald 0. Eulinqer Ed W. Gray Robert B. G r i f f i t h s

NEVADA ALPHA, Universi ty of Nevada Leland R. Hensley Robert A. Har r i s James R. HOUX. J r . J e r r y Ivan B l a i r Michael Paul Ekstrom Thomas 11. Lambert Richard Ross Oliver

Douglas M. Colbart William B. Jacky Erwin McPherson, J r . Larry Anthony Rosa Henry B. Ivy, J r . Christopher W. Jones John E. Jones Harvey L. Johnston, J Franklin Kahl Larry Livengood Larry J. Logan

NEW JERSEY GAMMA, Rutqers Collage of South Jersey

Alice C. Pru tzer Frank G. ROSS Robert D. Turkelson

Orhan H. Alisbah Karen T. Cinkay Richard M. Har r i s Jonathan C. Barron Jesse P. Clay C l a i r e C. Jacobs Robert H. Barron Joseph R. DeMeo Mary T. Kondash Leonard N. Bidwell Ellen 11. Fenwick Leon P. Polek

MISSOURI GAMMA, S t . Louis Universi ty

NEW MEXICO ALPHA, New Mexico S t a t e Universi ty Theodore C. Appelbaum J. Frank Armijo Lynn M i Bar te l s Mary G. Barton Yvonne Bauer S i s t e r M. Luke Bauman David N. Becker Sandra Bellon Pe te r C. Bishop James F. Boyer Aurelia Brennan Axel F. Brisken Catherine P. Browman Kathryn A. Cross Dela Jean Doerr Mary Donnelly Thomas J. Douglas Robert J. Dufford Jeanne Edwards Dennis M. Elking

Francine Endicott Eugene A. Enneking Mrs. Eugene A. Enneking Valer ie Erickson J u l i a E. Fusiek Sue El len Goodman John W. Green Veronica Grob Marjorie E. Hicks Eileen M. Kurd F e l i x 0. I s i b o r Thomas Iverson Anthony Karrenbrock, Jr. William D. Keeley Catherine Kestly Diane Kennedy J . Stephen Kennedy S i s t e r M. Ronald Kozicki Douglas J. Kuhlmann P a t r i c i a A. Laffoon

David R. Lammlein Thomas Madigan Margaret A. Lynn Mary Ann McClintock Joe T. McEwen Margie McNamee Elizabeth Ann Meyer Ann K. Mi l le r Linda K. Montani Ernest R. Moses Robert Murdock Katherine V. Murphy Jane t Nelson Lawrence W. Oberhausen Ellen O'Hara David Pisarkiewicz Nancy C. Poe Mary Louise Radersdorf Maureen Raf te r Donna Rizzut i Michael J. Z l a t i c

James J. Ruder Joseph E. Snyder Francis E. Sohm, J r . Jaroslaw Sobieszczanski Warren F. Speh J o e l J. Tannenholz Brian C. Ternoey Nancy Anne Tlapek Pa t r ick L. Trammel1 Lourdes Triana Benjamin H. Ulrich, J r . Joseph C. Vago Gayle Vincent Robert J. Walsh Barbara A. Wanchek John G. Weber Marie A. Werkmeister William C. Wester Steven R. Wood Mary Joan Woods James R. Wyrnch

Danne E. Coil ing Jim Olin Howard David T. Lindsey Charles P. Oowney Michele Kravitz Darrel Lotz Bernard F. Engebos David W. LeJeune Dennis R. Moon Neal Hart

Elmo LaMont Palmer Dave Peercy Tommy V. Ragland

NEW MEXICO BETA, New Mexico I n s t i t u t e of Mining

Russell E. Erbes Lawrence Lake

and Technology

Paul M. Perry Robert B. P i r o Ronald D. Reynolds Linda N . Sharp

Surendra Singh Henry L. Stuck James W. Tyree, J r . Dennis E. Williams

Lawrence E. F ie lds Frederick C. Frank Charles D. Glazer

Richard W. Lenninq Nicholas J. Nagy Thomas A. Nartker

NEW YORK ALPHA, Syrocus ie Universi ty

Marilyn L. Freedman Halfon N. Hamaoui Donna L. H i r s t Wallace F. Jewell , J r . Frederic H. Katz Michael Kreiqer Robert A. Kre l l J u d i t h Ellen Ladwig Alexander J . Lindsey

Robert W. L i s s i t z Meryll Lizan Joseph E. Paris!

, Michael R. Pike Carolyn A. Quine Charles D. Reinsuer Henry E. Riqer John A. Roulier John D. Schoenfeld

Steven Schulman James Short Pe te r M. Si lvaggio Richard C. S ins Robert L. Smith J o e l M. Tendler Richard Tuacione Emily Wilier David G. West

Walter W. Carey Thomas B. Clarke Anthony Culmone Pe te r M. Danilchick Hai Hong Doan Alexander J . Donadoni Anne L e s l i e Drazen Joann Faconti Rosalind F. Friedman Shar i Frankel

MONTANA BETA, Montana S t a t e Universi ty

Fred J. Balkovetz William J. Haley Kathleen M. Brownell John W. Hemmer Roger L. Brewer David P. Jacobson Eugene A. Enebo Karen McNeal

Spencer E. Minear Edward A. Teppo John B. Mueller William J. Weber Judson L. Temple Larry R. Wedel

NEW YORK BETA, Hunter College of The Cuny NEBRASKA ALPHA, Universi ty of Nebraska

Dennis Cal las Rose Ann Gatto Dianne Mi ls te in

Norbert Moelke Linda Schiffman Diane Peshler Carol Jane Shevitz Shlomo Piontkowaki Leopoldo Tora lba l la

Alice Whitternore William Xanthos Martin Zuckerman

Gary W. Ahlquist Claude Bolton, Jr. John E. Boyer, Jr. Thomas Louis Burger Richard L. Chiburis

Thomas Wesley Copenhaver Larry Dean Eldridqe Jean Marie K h e s Daniel E. Crawford Stephen Ray Gold Alan Lee Larson Thomas B. Cunningham Larry E. Groff John S. Lehigh, Jr. Wendell Dam Linda E. Hammer Ronald Eugene Lux Robert N. Dawaon Allen Lee Harms Norman L. Mejstr ik

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, NEW YORK GAMMA, Brooklyn College

Moshe Augenstein Robert Chalik Joseph Chesir Car l C i r i l l o Barbara R. Comisar Aaron D r i l l i c k Hsrmsnn Edels te in Martin Edels te in Benjamin Fine

Sheldon J. Finke ls te in Allen M. Flusberg Marilyn S. Frankenstein David Freedman Marc M. Friedlander Sylv ia Gelb Sonja M. Gideon Lois Goldstein Seymour N. Goldstein

Keith Narrow Brian Hingerty Mayer M. ~ s c o b o v i t z Margari ta Jimenez Allen Lipmsn Dennis C. Lynch Kenneth Mandelberg Judith R. Markowitz I rane L. May

Michael L. ~ u c h l i s Harry Rutmsn Albert Schon

Richard Freedman Pe te r M. Gsbr ie le Dennis J. Holovach Frank James Jordan John Kiely

Masako Ki ta tan i Drew Koschek John Majewski Fred Mallen David Matos

James Markowsky Norman P e a r l Steven Pincus Michael P o t t e r p h i l i p Rossomando

Robert Schular David Silberman Richard Strako Mark S t e i n e r Robert Vet te r Cameron Wood

Ellen N . schul tz I s raee Sendrovic Jane t S p i t z e r Susan L. Tirschwell George Weinberger Morris Zyskind NORTH CAROLINA ALPHA, Duke Universi ty

Ware BotsfOrd Don W. Brown Marshall Checket Jack Davis Peggy Erlanger Katharine K. Flory Harvey S. Gotts Thomas A. Harr i s Corinna Head

Michael D. Holloway J e f f Kay Freddy Knape Cheryl Kohl Robert Lees Cathy Losey Dennis H. Matthias Linda McKissack

Grady Mi l le r Martha Moore Gwynne Ormaby Pat Paliner

Ron staift-sy ~d Sul l ivan B i l l Taylor Mary E. Thra l l Robert S. Tremlett William Waterfield Charles Whits Ken Young

NEW YORK EPSILON, S t . Lawrence Universi ty

Joyce P. Gurzynski Thomas C. L i t t l e Robert J . Sokol - - - . - - - - Lucy Roberts Jerome Saks Richard Scot t L e s l i e Stanford

NEW YORK LAMBDA, Manhattan College

John A l i n i Diego X. Alvarez Andrew Arenth Richard Basener Alfred Buonaguaro Jerry Carey Gerard Chingas Frank C i o f f i David Concis

Lawrence Cox Frank Ootzler Joseph Femia Alphonse Fennelly Thomas Finn James F. Garside Thomas Heenan Raymond Huntington

Michael A. Infranco Fred Ipavich Christopher Kuretz David Lee Steven Lincks Francis J . Martin John Menick Kevin M. O'Brien

Daniel O'Donnell Robert E. Padian Pe te r Savino Robert Sc ia f f ino Vincent Sgroi John Slowik Vic tor C. S t a s i o Fred Szostek

NORTH CAROLINA GAMMA, North Carolina S t a t e Universi ty

Joseph A. Middleton Tom L. Murdock Wilbert L. Murphy

Frederick Naqle Douglas M. Tennant wil l iam W. Wilfong, J r

~ o b y M. Anthony James A. Blalock Gary M. Brady Dinnis L. C a r r o l l

Ted R. Clem Michael L. Kelly Richard P. Kitson James D. McAdams

NEW YORK NU, New York Universi ty NORTH DAKOTA ALPHA, North Dakota S t a t e Universi ty

Rostom Dagazian Frederick Fein William Gesick

Nancy Makiesky Richard McNulty Melvin Melchner

Marcia L. Parker Westly Parker Lyle R. Paton Harold Paulsen A. rillan Riveland Carol Rudolph Franklyn Schoenbeck J e f f M. Siemers Madeleine Skogen R. Vslborg Smith Larry A. Wurdeman Trudy A. Westrick

Mary Shiffman Karen Sidelmsn

Joseph Tosto I r a j Yaqhoobia

P a t r i c i a M. Beauchan Cur t i s R. Bring Charles L. Comstock Kenneth L. Daniels Kenneth M. Ehley Edward D. Elqsthun Edward D. Eus t ice Wayne W. Fercho Doris M. F i sher Charles R. F r i e s e Betty L. Frigen Vernon L. Gallagher Carol J . Gellner

ip Max 0. Gerling Jeanne M. Glasoe Dale J. Haaland Richard G. Haedt w i l l i a m R. Nauqen Amelia R. Hoffman David L. Hoffman Martin 0. Holoien Lynn H. Jennings Caye M. Johnson Clar Rene Johnson J e r r e l W. Johnson Keith W. Johnson

Paul L. J u e l l Donald J. Lacher Thomas J. Lien Russell E. Long Kenneth A. Losee Quentin D. Lundquist Bruce C. MacDonald Cheryl M. McDougal Dalton D. Meske Terrance E. Nelson E s t e l l e R. Neuharth Donovan M. Olson John F. Ovrebo

NEW YORK XI, Adelphi Universi ty

Gsspar Schweiger Benedict Scot t Robert S i lvers tone John Stevenson Harry S t rauss Jane t Tierney Marian Weinstein Arthur Whelan S t u a r t Wiitamak

Pasquale Arpaia Lynne Bensol Richard Bentz Dorothy Bergmann P h i l i p Bookman Douglas Brsz Alan Chutsky Penelope D'Antonio Joseph Dunn Bernard Eisenberg

Robert E ld i Steven Levinson Doris Fleishman Trueman MacHenry Geoffrey Goldbogen Hubert Msdor Michael Gorelick Boris Mitrohin Whitney Harr i s Vladimir Morosoff Susan Herman Steven Pflanzer James Holwell Barry Reed Alan Hulsaver Martin Rudolph Robert Iosue Judi th Russell John Kruckeberg

OHIO BETA, Ohio Weslayan Universi ty

Sherri-Ann Lancton Lew H. Walters Edward McCleIlan Susan K. Woernsr Ida Eckler Pugh Peggy Wurzburger L e s l i e A. Rodgers

Stephen D. Clements Marjorie J . Ingram Barbara S. Combs James E. Jewett Marjorie E. F e r s t e r Gregg W. Johnson william N. Grunow susan E. Jones

NEW YORK OMICRON, Clarkson College of Technology

William Richard Frsn ics Dale K r e i s l e r Harold King, J r .

Frank J . Lambiase P h i l l i p Washburn

OHIO DELTA, Miami Universi ty

Dale E. McCUnn J e r r y L. P i t tenger Richard D. Se lcer Helen M. Smith

Lawrence A. White Marilyn H i Zanni

NEW YORK RHO, S t . John's Universi ty Michael F. Adolf Robert R. Chandler Candy L. Anthony Michael B. D i l l e r o s e C. Antunes Linda J. F r i t z James E. Brockhoume E. James Gunn

Theresa M. Boyd Gerald S. Korb Donald R. Coscia

David J. Nolan Dennis J. Korchinski Diana M. Pa te rnos te r

S i s t e r Anne Daniel John C. Lumia P a t r i c i a L. S a b a t e l l i

Beatrice J . S e t t e Cr i s t iano S t e i d l e r Marty Tr io la

OHIO GAMMA, Universi ty of Toledo NEW YORK SIGMA, P r a t t I n s t i t u t e

Lawrence I r v i n Hill Wendell C. H U l ?

. w a s G. La Pointm James E. Livingston Msnikant Lodaya Cathay M. McHenry Lawrence E. Mlinac m b e r t Pollex

William Vic tor Backensto Paul Richard C o l l i e r Richard C. Bel l John Gerald Duda Donna Bellmaier James Ear l Geiser t Roberta Boszor P e r s i s Ann Giebel I r i s Elizabeth Coffman James Anthony Grau Alan Roy Cohen Ray A. Grove

Lawrence Adams Gerald A. Aie l lo

William Bel l Ray L. Bieber

Francis Atkins Gordon Brookman

Jef f rey Bromberq Frank Corredine Rudolf Diener

Mary E l l i e n William Erny Stanley Finger

James E. K i t t l e Richard E. Knauer Denise Ann Krauss Martin Pe te r Kummer

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114 1

G e r a l d m a B a b b i t t S t e v e n It. Se lman Thomas Tenney L o u i s e Ann R i d e o u t R i c h a r d N. S n y d a r -

J o h n P. K e a t i n g Gary K o l s t o e B r u c e L i n k Eugene C. McKay S i s t e r Andrea Nenzel w i l l i a m P e t e r s o n James P h i l l i p s G e r a l d S a u n d e r s J o n a t h a n S t a f f o r d Evan Simmons R o b e r t 11. T a r s

J u Wayne C. Teng T r u c k s

G r e g o r y Tye James R. Tye Ronald Wagoner M i l t o n Wannier J u d i t h E. Weinberq Nick Wengreniuk L e s l i e Wilson James Yu B a r b a r a Z e l l e r

Wei Chinq Chang Mark C h r i s t i e J i h J i a n g Chyu E l i z a b e t h Ann C l i f t o n David S. Cohen C l i f f o r d Comisky L a r r y Cowan Winston Dean J e r r y E a g l e Gordon Evans David L. F i s h H a r o l d F i t z p a t r i c k

Jndv P u l l e r Maryann Wojciechowski - --. - ----- R o b e r t S. G a r n e r 0 Ronald G r i s e l l D e n n i s Ileckman Donald G. I l o f f a r d Dean H a y a s h i d a Kuo-Cheng H s i e h Mei-Lee H s i e h

OHIO EPSILON, Kent S t a t e U n i v e r s i t y

Roger B a r n a r d S u s a n Brakua M i c h a e l C l e c k n e r V i t t o r i a C o n s e n t i n o J o e l Edelman Donald F e d e r c h e k Karen F i s h e r

C a r o l Q o r r e r E u n i c e K r i n s k y L i n d a F r y e Nancy Lyons R o b e r t B. Good, Jr. Ann M a l l e t t J e a n Hannah R i c h a r d M a t h i a s S u s a n Hay Myra P a t t e r s o n G a r y H o l u e s T h e r e s a R e c c h i o M a r i l y n K i r s c h n e r

R i c h a r d Shoop Howard S i l v e r . . - - -. . - - -

W a l t e r E. I s l e r Ben J. J o n e s P a u l C. Kang

OREGON BETA, Oregon S t a t e U n i v e r s i t y

M i c h a e l HcDonald Sue Ann McGraw Byran F. HacPherson C a r o l G. M a u r a t t Danny M. N e s s e t t M a r s h a l l E. Noel S t e v e n Q. R a f o t h Alan T. R i c h a r d s

L a r r y W. S a s s e s Harvey W. S c h u b o t h e C a r o l e Anne S e t t l e s M a r b a r e t E. Shay Marion M. S t a l n a k e r Mary J. T a y l o r N e i l 11. Tokerud S u s a n M i Woerndle

OHIO ZETA, U n i v e r s i t y o f Dayton Kenneth H. Anderson Danny Thomas R. M. Avery B a i l e y

S c o t t A. B e u v a i s David G. B e n n e t t David A. B u t l e r R o b e r t D. Erwin J a m e s N. l laek

J a m e s R. Hagan A r t h u r D. Hudson S h i r l e y D. Hudson F r a n c i s C. Hung J o h n R. J a c o b s o n M i c h a e l L. J o h n s o n J o h n L e o n a r d Lewis Lum

N e i l A. B i t z e n h o f e r D e n n i s M. C o n t i C h a r l e s J. DeBrosse D a n i e l Q. Dula R o b e r t C. F o r n e y B e r n a r d J. G a l i l e y

Kenneth M. Herman Mary H. K a l a f u s J o s e p h W. Keim Nancy C. ~ o c h J e f f r e y A. L i n c k

C h a r l e s A. Merk Maureen M o r r i s o n R o b e r t A. Nero Edward N e u s c h l e r P a u l E. P e t e r s

Thomas S a n t n e r James A. Schoen Roger S t e i n l a g e I n e z Young B a r b a r a Zimmerman

OHIO ETA, C l e v e l a n d S t a t a ~ n i v e k s i t ~ OREGON GAMMA, P o r t l a n d S t a t e C o l l e g e

Douglas S. Bouck David C h e y f i t z R a l p h G i g a n t e J o a n n e H a r t

H. ~ i c h a r d l l u t b e r D a r r e l A l l e n Nixon Roger ~ o f o James Reed K e s u l l Thomas M. P e c h n i k Lawrence M. T a u s c h P h i l l i p Kramer V i n c e n t T. P h i l l i p s K e i t h N. Ward David M. Lçwi G e r a l d H. Rosen

L o r e n P. H o t c h k i a s L a r r y G. Meyer

J o y c e S c h l i e t e r R i c h a r d David B a l l James M. Braun

Kent H a l e Byron T r o y J. Downey

PENNSYLVANIA BETA, B u c k n e l l U n i v e r s i t y OHIO IOTA, Denison U n i v e r s i t y

J a m e s R. McSkimin S t e p h e n W. R i d d e l l L o u i s J. M o r a y t i s David A. Robbins B r i a n R. Mund R o b e r t G. Simons Ned W. P o l a n C h r i s t o p h e r S c h e l l J e f f r e y B. P r e s s R o b e r t T a y l o r A l l a n J. R e i c h e n b a c h Mark T o r r e n c e

David L. Ackermann P h i l l i p R. D a n t i n i F r a n k A r e n t o w i c z , Jr. S y l v i a A. G a r d n e r J e f f r e y C. B a r n e t t Thomas L. H o t s l e n w i l l i a m B. B e r t h o l f J u d i t h A. J a c o b s o n George 8. B r i n s e r P e t e r J. J o h a n s s o n Samuel E. C l o p p e r , Jr. B a r b a r a A. M a t t i c k B a r b a r a D. C l o y d V i v i a n K. McDonough

J a n e t A t k i n s o n M a r i l y n Knox P a u l D a v i d N e i d h a r d t Bruce M. S h m a n R i c h a r d B e d i e n t S t e p h e n H. K r i e b e l M a r g a r e t P r i c a M a r g a r e t Enke

A r t h u r S t a d d o n R i c h a r d C. McBurney J a n i c e S h i l c o O k

S u s a n Hardy C l a i r Wvichet

R i c h a r d H a r p e r Meeks

OKLAHOMA ALPHA, U n i v e r s i t y o f Oklahoma

S t e v e Abney C l a r e n c e H. B e s t D a l e T. Bradshaw J o h n W. B y m e Thadeus C a l i n s o n L e l i a Chance J a m e s Chapman J immie W. C h o a t e Mary E. C o a t e s

S t e v e n Doughty J. Rodney Grisham G a r y H e n s l e r David Hughes J a c k J o h n s o n David Kay S i d n e y W. K i t c h e l R u t h A. Loyd C e c i l M. McLaury

PENNSYLVANIA DELTA, P e n n s y l v a n i a S t a t e U n i v e r s i t y George M i t c h e l l George Morrow J a m e s S. Muldownmy David 0 . N i e l s s o n J o h n N o r r i s J o a n n e P a t t e r s o n P a u l 11. P o w e l l M o r r i s E. R i l l Thomas J. S a n d e r s

H. P. S a n k a p p a n a v a r G e r a l d D. S m e t z e r Tom T r e v a t h a n M e l i s s a Uhlenhop L o n n i e Vance James Want land H a r o l d Wiebe E l l i n o r W i e s t L a r r y J. Y a r t z

N e l s o n L. Seaman F r a n k l i n P. S h u l o c k Ted Sims A l l a n K. S w e t t D e n n i s T o r r e t t i S a n d r a M. Z e i l e r

R i c h a r d L. A t k i n s o n B a r b a r a G r o s s C l i f f o r d B a s t u s c h e c k J a m e s Harvey C a r l B e i s w e n g e r Lee H i v e l y J o h n Bobick T e r r y Ann L a n d e r s Lawrence D u n s t Kenneth S. Law, Jr. G e o r g e P r i c k J o h n R u s s e l l Mashey H a r o l d W. G a l b r a i t h A l b e r t Lehman, Jr.

OKLAHOMA BETA, Oklahoma S t a t e U n i v e r s i t y PENNSYLVANIA THETA, D r e x e l I n s t i t e o f T e c h n o l o g y

J u d i t h E. S i s s o n James L. Alexander! W i l l i a m J o s e p h D a v i s Benny Evans C h a r l e s M. B a k e r T e r r y Eas tham P e t e r Chevng

S i s t e r Ann Mark Norman E. A u s p i t z Alan G o l d i n F r a n k Gambl in R i c h a r d E. Myers

David P. P e t e r s R i c h a r d E. R u s s e l Donald R. P o r t e r

Rodney S c h u l t z

OREGON ALPHA, U n i v e r s i t y o f Oregon

David S. Browder Theodore Burrowes

R o b e r t C o d e r g r e e n Chung Hai Chang

M i c h a e l A r c i d i a c o n o C a r l o s B e r k o v i c s Norman Beck R o b e r t Bohac

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c

PENNSYLVANIE IOTA, Villanova Universi ty TEXAS ALPHA, Texas Chr i s t ian Universi ty

Walter 11. Marter Robert J. McDevitt Joseph R. McEnerney Francisco Migliore Michael J . Hruz Lawrence Muth Joseph P e r f e t t e Leon C. Robbins, Jr.

Lucien R. Roy August A. Sardinas Robert L. Smith David Sprows Joseph A. S t rada James Vincenzo John E. Zurad

Nazem M. Marrach Charles W. Mastin Donald F. Reynolds H. Edward Stone

Benjamin M. Adams Ray J o e l Chandler Jeanne M. Ericson Hull Belmore Roy J. Chandler Russell Floyd L i l l a s K. Berzsenyi Char lo t te E. Dickerson James Fugate

Steven D. H i t t

Murray Adelman mi1 Ame10tti Nicholas P. Augell i Robert E. Beck Joseph E. Brennan James 0. Brooks Leo P. Comerford John F. D'Arcy

Richard A. Derrig Robert DeVos William H. DeLone Thomas Dougherty h a n d Innocenzi Richard L. I r v i n e Shelley M. Kam Michael Makynen

TEXAS BETA, Lamar S t a t e College of Technology

Johnny D. Walker John S t Bagby Georgia A. Farr

Muriel D. Huckaby Mary Ann F. Mathews

Wanda L. Savage

RHODE ISLAND ALPHA, Universi ty of Rhode I s land UTAH ALPHA, Universi ty of Utah

Herbert P. Adams Harry Aharonian, Jr. Grace Ann Beinert Sandra Ann Campo J u d i t h Anne Car ro l l Carole Car l s ten

Joyce Davidson Ubiratan D'Ambrosio Leo A. Dextradeur Norman J. F i n i z i o Cheryl Anne Gerry Dennis M. Hagqerty

Constance A. Magner Susan Nadeau Gregory P. Niedzwecki Richard W. Palczynski A l i c i a Parente P h y l l i s A. P e l l a

John Francis Ryan Lynne Helen Stewart S a l l y Taylor M. L. Verrecchia Robert A. Wilkinson Doris E. Wise

Michael Dennis J u l i a n William C. King John P. Lamb Roger G. Larson Marilyn Kay Latham Barbara Jean Laucher N. Wayne Lauritzen David Payne Alexander Peyerimhoff

Walter Reid Robert J. Rich Don K. Richards George W. She l l David A. Simpson Glen Kerr Tolman Allen L. Tuft Sheldon F. Whitaker William A. Williams

Robert Lee Askew George Cleo Barton S h i r l M. B r e i t l i n g Lorin W. Brown A. Ear l Catmull Gary James Clark Eva A. Cranale

Thompson E. Fehr Robert M. F lega l Richard W. F u l l e r Robert Gordon Neils H. Ilansen Joseph H. Har r i s Paul W. Heaton Kenneth E. IIooton Wayne R. Jones

MODE ISLAND BETA, Rhode I s land College -. - ... .-- Gerald L. Despain Sharon Lee Doran

Clancy P. Asselin Bryan L. Barnes J u d i t h A. Bruno Katherine J. Camara Joyce Col l ins Frank B. Correia Michael DiBat t i s ta J u l i e A. Fidrych Edward T. Ford

Francis P. Ford Edmund B. Gamea Valmord Guernon Henry P. G u i l l o t t e Ronald Hamill Nancy L. Humes William E. Ide David Lancaster Marie L. LaPrade

George A. Lozy Raynor Marsland, J r . Ann Maynard Kathleen II. McLee Katherine M. Morgan John Nazarian Pa t r ick J. O'Regan Constance Pasch

Linda L. Read Mariano Rodrigues Arthur F. Smith Elaine Sobodacha Robert Steward Thomas H. Sul l ivan Gertrude M. T e s s i e r P h i l l i p M. Whitman

UTAH BETA, Utah S t a t e Universi ty

Frank Wanq Kai Tak Wong Wen Ming Chen Dale M. Rasmuson Robert Michael Har r i s

VIRGINIA BETA, Vi rg in ia Polytechnic I n s t i t u t e

Jr. Carlton R. Wine John A. Wixson Thomas Woteki Mary Sue Younger

Margie C. Clevinger Steve J . Lahoda Marion R. Reynolds, John A. Cornell Gene Lowrimore John Rogers Daniel Denby William T. McClelland Robert Snider Ralph C u r t i s Epperly J e r r y Moore George T. Vance Samuel V. Givens Glenda Gayle Pace

SOUTH CAROLINA ALPHA, Universi ty of South Carolina

Ada D. Kirkman Gary D. LaBruyere Alonzo Layton 0. William Lever Cynthia S. Lyt le Lorrin F. Martin Gerald H. Morton

Celia Lane Adair David J . Frontz Ronald E. Bowers Robart L. Hart Willys Brotherton Pe te r D. Hoffmann Alpheus Burner Della Hicks William Burns Michael D. Hyman Linda M. Chatham William S. Jacobs Dan Fabrick Eileen B. Ki lpa t r ick Dugenia Felkel

Paul H. Pinson Virg in ia Ann Reeves Betty Ann Shannon John H. S i g l e r Edith Smoak John J. Smoak, J r . P h i l i p Stowell

WASHINGTON BETA, Universi ty of Washington

Alan Aronson Glenn Goodrich Gary R. Houlahan Sandra Burroughs Cl i f ford G. Hale Paul Joppa Pe te r W. Cole Ronald L. Hebron Jane t Lamberg Gael A. Curry Merri lee K. Helmers Susan McDougall Bradley C. Fowler Sidney L. Hendrickson Heidi J . Madden P a t r i c i a J . Gale Ken R. Hinkleman Gary F. Martin

George J. Nett Arthur Pearson Charles J . Schleqel Michael D. Smeltzer Richard Waggoner SOUTH DAKOTA ALPHA, Universi ty of South Dakota

S a l l y Sa i Cam Lau Mary Agnes Lundburq Kathleen E. McMinemee Roy F. Myers Jung J a Park Mona Redmond Elaine Rietz

Shi r ley Jean Schempp Gary Clinton Stamann Alan Thorson Thomas Tucker David Vlasak Gerald Winckler Virginia Mae W i t t

Alyn N . Dull Faryl Fos te r James F. FOX John Hall Cheryl Marie Hone Cynthia Ann Jensen Stephen E. Jones William J . Kabeisman

WASHINGTON GAMMA, S e a t t l e University

William Alan Ayres Gai l Marie Harr i s Mary Ann Hindery Michael G. Severance

WEST VIRGINIA ALPHA, West Vi rg in ia Universi ty

Robert Allen John Atkins S a l l y Beile Donald Beken

Barbara Conaway James C o t t r i l l Alyson Crogan A. B. Cunningham Sara Debolt Paul Deskevich Joy B. Easton Roger F lora

Celes ta J. Gardner Margaret H. Garlow Henry W. Gould William K. Hale Kathryn Hel le r James B. Hickman Franz Hierge is t John W. Hogan

Florence Hunter James M. Handlan Margaret K i r t l e y John Klemm Tze-fen Li Stephen Lipscomb Brenda McFrederick Betty L. Mi l le r

TENNESSEE ALPHA, Memphis S t a t e Universi ty

Gai l D. Clement Ronald L. Fos te r Carolyn A. Cothran Dorothy A. Heidbrink Ada J. Culbreath James D. Holder Betty L. Downs J e r r y A. Matthews

Ronald L. McCalla Henry H . Nabors John L. Pel1 Ben F. Prewi t t

Charles E. Stavely Nancy C. Tice

Larry Brumbaugh ~ o s e p h ~ a r t i s i n o Richard Cerullo Charles Cochran

Page 26: PImiMU EPSILON

M. L. Vest Ronald Virden Dorothy Watt Stanley Wearden

David Moran ' Dawn R. Shelkey Samuel Swallow William C. Nordai ' John S. Skocik Ronald Tomaschko I. D. Peters Joseph K. Stewart Ann Trump John Perrine William Stewart Ronald Vento Deborah Seltz Mary Squire

WISCONSIN ALPHA, Marquette University

Jon Davis Ronald Jodat

Kathleen McElligott Lynn Reppa Sharon Parrot Anthea Smith

Peter Stockllausen

NEW CHAPTERS OF PI MU EPSILON

ALABAMA GAMMA 125-1967

CALIFORNIA ETA 129-1967

MASSACHUSETTS BETA 122-1967

NEW JERSEY GAMMA 131-1967

Prof. William Peeples, Jr., Dept of Math, Samford University, Birmingham 35209

Prof. L. F. Klosinki, Dept of Math., The University of Santa Clara, Santa Clara 95053

Prof. Daniel R. Dewey, Dent of Math, Collage of the Holy Cross. Worcester 01610

Prof. Claire C. Jacobs, Dept of Math, Rutqers Colleqe of South Jersey, Camden 08102

NEW MEXICO BETA 124-1967

Prof. Merry M. Lomanitz, Dept of Math, New Mexico Inst. of Mining 6 Technology, Socorro 97801

NEW YORK SIGMA 132-1967

Prof. A. B. Finkelstein, Dept of Math, Pratt Institute of Brooklyn, Brooklyn 11205

NORTH DAKOTA ALPHA Prof. Martin 0. Holoien, Dept of Math, North Dakota 127-1967 State University, Fargo 58102

PENNSYLVANIA IOTA 130-1967

Prof. James 0. Brooks, Dept of Math, Villanova University, Villanova 19085

RHODE ISLAND BETA 128-1967

Dr. Frank B. Correia, Dept of Math, Rhode Island State College, Providence 02908

TENNESSEE ALPHA 126-1967

Prof. Henry L. Reeves, Dept of Math, Memphis State University, Memphis 38111

WEST VIRGINIA ALPHA Dr. Franz X. Hiergelst, Dept of Math, Weç Virginia 123-1967 University, Morqantown 26506

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